A Philosophy of Learning Arithmetic
Towards an Evaluation of Teaching Methods
MA Thesis (afstudeerscriptie)
Written by Daphne Udo de Haes (born July 10, 1979 in Loenen, The Netherlands)
Under the supervision of dr. Elsbeth Brouwer, and submitted to the Board of Examiners in
partial fulfillment of the requirements for the degree of
MA in Philosophy
at the Universiteit van Amsterdam
Date of the defense: July 4, 2013
Members of the Thesis Committee:
Prof. Dr. Benedikt Löwe
Dr. Maria Aloni
Dr. Elsbeth Brouwer
Graduate School of Humanities
For
Sarah and Ronja
2
Acknowledgements
I am grateful to many people. First of all I would like to thank my supervisor dr. Elsbeth Brouwer for
all her help. Without her illuminating way of questioning I would not have been able to put my ideas
on paper somewhat comprehensible. Her enthusiasm about my first plan for this thesis and her trust
that eventually something good would come out of it was very encouraging. I have enjoyed our
conversations. I owe many thanks to Prof. dr. Benedikt Löwe. Thanks to his enthusiasm about
research at the intersection of the philosophy of mathematics and the (teaching) practice, and his
cordial way of involving students, I became motivated for writing this thesis. I am grateful to Prof. dr.
Michiel van Lambalgen for encouraging me to start with the MoL and for his positive feedback on my
first philosophical paper on arithmetic education (the bachelor thesis). I have learned a lot at the
MoL that proved to be useful for exploring this topic further. I thank dr. Jo Nelissen for his patient
explanations about RME. I owe the children I have had the privilege to teach arithmetic for their
questions and their tireless willingness to explain to me how they had arrived at their solutions. In
particular Camille for showing me her original ways of reasoning. They have inspired me for writing
this thesis. Sarah and Ronja have been very helpful taking distance from writing time and again with
their cheerful presence. I thank my family and friends for all their support. Tamara de Rijk also for
helping with my clumsy writings, Agnes Schuttelaar for correcting my worst language errors and
Oberon Nauta for showing me some layout tricks. My mother, Lous Croon, for providing me a quiet
place to work in her home; my sister, Myrthe Udo de Haes, for her loving care for our girls when I
was writing; and my love, Mick Hijdendaal, for his encouragement to pick up my studies again and for
the sacrifices he has made to make it possible, for his support and belief that all the toil is good for
something.
3
Contents
Introduction............................................................................................................................................. 7
Chapter 1: Logicism ............................................................................................................................... 13
1.1 Foundations ................................................................................................................................. 13
1.1.1 The derivation of the arithmetical concepts ........................................................................ 13
1.1.2 The derivation of the theorems of arithmetic...................................................................... 14
1.2 Epistemology ............................................................................................................................... 16
1.3 Truth ............................................................................................................................................ 17
1.4 Knowledge Conditions ................................................................................................................. 18
Chapter 2: Intuitionism.......................................................................................................................... 19
2.1 Foundations ................................................................................................................................. 19
2.2 Epistemology ............................................................................................................................... 21
2.3 Truth ............................................................................................................................................ 22
2.4 Knowledge Conditions ................................................................................................................. 23
Chapter 3: Formalism ............................................................................................................................ 25
3.1 Foundations ................................................................................................................................. 25
3.2 Epistemology ............................................................................................................................... 27
3.3 Truth ............................................................................................................................................ 28
3.4 Knowledge Conditions ................................................................................................................. 28
Chapter 4: Platonism ............................................................................................................................. 31
4.1 Foundations ................................................................................................................................. 32
4.2 Epistemology ............................................................................................................................... 33
4.3 Truth ............................................................................................................................................ 33
4.4 Knowledge Conditions ................................................................................................................. 34
Chapter 5: Structuralism ....................................................................................................................... 37
5.1 Set Theoretic Structuralism ......................................................................................................... 37
5.1.1 Foundations .......................................................................................................................... 38
5.1.2 Epistemology ........................................................................................................................ 39
5.1.3 Truth ..................................................................................................................................... 39
5.1.4 Knowledge Conditions .......................................................................................................... 39
5.2 Ante Rem Structuralism............................................................................................................... 39
4
5.2.1 Foundations .......................................................................................................................... 39
5.2.2 Epistemology ........................................................................................................................ 40
5.2.3 Truth ..................................................................................................................................... 41
5.2.4 Knowledge Conditions .......................................................................................................... 41
5.3 Modal Structuralism .................................................................................................................... 42
5.3.1 Foundations .......................................................................................................................... 42
5.3.2 Epistemology ........................................................................................................................ 44
5.3.3 Truth ..................................................................................................................................... 45
5.3.4 Knowledge Conditions .......................................................................................................... 45
Chapter 6: Empiricism ........................................................................................................................... 47
6.1 Foundations ................................................................................................................................. 48
6.2 Epistemology ............................................................................................................................... 50
6.3 Truth ............................................................................................................................................ 51
6.4 Knowledge Conditions ................................................................................................................. 52
Chapter 7: Social Constructivism ........................................................................................................... 53
7.1 Foundations ................................................................................................................................. 53
7.2 Epistemology ............................................................................................................................... 54
7.3 Truth ............................................................................................................................................ 55
7.4 Knowledge Conditions ................................................................................................................. 55
Chapter 8: Embodied View .................................................................................................................... 57
8.1 Epistemology ............................................................................................................................... 57
8.1.1 Innate Arithmetic ................................................................................................................. 58
8.1.2 Conceptual Metaphors ......................................................................................................... 59
8.2 Foundations ................................................................................................................................. 62
8.3 Truth ............................................................................................................................................ 62
8.4 Knowledge Conditions ................................................................................................................. 63
Chapter 9: Towards an Evaluation of Teaching Methods ..................................................................... 65
9.1 The Philosophy of Arithmetic: A Summary ................................................................................. 65
9.1.1 A Landscape of Dichotomies ................................................................................................ 66
9.1.2 Knowing Arithmetic; A Discussion ........................................................................................ 69
9.1.3 The Knowledge Conditions ................................................................................................... 72
9.2 Desired Conditions for Teaching Methods .................................................................................. 73
9.2.1 Positioning in the Landscape ................................................................................................ 74
9.2.2 Accommodation of All Knowledge Conditions ..................................................................... 76
5
Chapter 10: Arithmetical Knowledge and Perspectives on Learning .................................................... 79
10.1 Arithmetical Knowledge: A Definition ....................................................................................... 80
10.2 Approaching Learning Arithmetic ............................................................................................. 84
10.2.1 Perspectives on Learning .................................................................................................... 84
10.2.2 Arithmetical cognition ........................................................................................................ 86
Conclusion ............................................................................................................................................. 90
Appendixes ............................................................................................................................................ 93
Case Study: A Philosophical Evaluation of Realistic Mathematics Education ....................................... 95
1. Introduction ................................................................................................................................... 95
2. RME in Theory ............................................................................................................................... 96
2.1 Main Principles ........................................................................................................................ 96
2.2 Learning Goals ......................................................................................................................... 98
2.3 Teaching-Learning-Process ...................................................................................................... 98
2.4 Conclusion ............................................................................................................................... 99
3. A Philosophical Evaluation ............................................................................................................ 99
3.1 The Conception of the Subject of Teaching: A Positioning ..................................................... 99
3.2 Structural Problems and Possible Answers ........................................................................... 101
3.3 The accommodation of the Knowledge Conditions .............................................................. 103
4. Conclusion ................................................................................................................................... 106
Bibliography......................................................................................................................................... 109
6
Introduction
How to test whether one kind of
instruction is better than another?
Instruction serves goals. But what to
do if the goals are formulated in terms
of a philosophy, thereby being a
1
matter of faith?
Hans Freudenthal is talking here about ‘revising mathematics education’ in 1991. His claim is that:
teaching methods should not consist of the traditional algorithm drill but that they should develop
conceptual understanding. About the implementation of his mathematics reform he concludes:
2
...rather than selling textbooks, one has to sell a faith.
I agree with him that which type of mathematics education you prefer is a philosophical question.
However, since philosophy is a matter of arguments it certainly is not a matter of faith. Freudenthal
advocates tolerance,3 but the discussion has ironically stepped into the pitfall of many issues of faith:
it has become a matter of war. These so-called ‘Math Wars’, the widely held disputes on
mathematics education, started in the late 80s of the last century and are still not settled. Especially
primary mathematics education appears to be a concern. Interestingly enough the press reports on
this subject show that this concern comes not only from mathematicians, educational researchers,
textbook authors, teachers and politicians but also from the general public. Mathematics education
proves to be a worry to us all. This makes it even more important that the affair will leave the war
zone and will return to the realm of philosophy, where arguments are sold rather than faith.
In this thesis I present a Philosophy of learning arithmetic. It consists of an overview of eight
philosophical views on the nature of arithmetic analysed in such a way that practical consequences
for arithmetic education become assignable. It’s goal is to offer a systematics for a philosophical
evaluation of educational methods for teaching arithmetic. I claim that the philosophical views on
arithmetic provide the ground on which one can decide whether one kind of instruction is better
than another. The overview of the views that have been put forward in philosophy provides a
landscape in which the conception of the subject of teaching underlying a teaching method can be
positioned. A clear view on how an educational method conceives its teaching subject shed light on
the goals that are pursued in teaching. The positioning of a teaching method in the landscape of the
philosophy of arithmetic provides the arguments and the structural problems for the defense of its
teaching goals. In this thesis I offer an instrument for the evaluation of teaching methods on the basis
of arguments stemming from the philosophy of arithmetic.
The philosophy of arithmetic is presented in the first eight chapters. It treats three related aspects of
the nature of arithmetic: the ontology and foundation of arithmetic, the epistemology of arithmetic
and the status of its truths. These are the three dimensions along which the different positions of the
philosophy of arithmetic will be discussed. The first one contains the naming of the primitive
concepts of arithmetic and a description of the methods that are used to provide its basis. It also
includes a conception of the ontology of arithmetical objects: the numbers. The second issue
1
(Freudenthal, 1991, p. 136)
(Freudenthal, 1991, p. 137)
3
‘In matters of faith tolerance is becoming’ in (Freudenthal, 1991, p. 136)
2
7
concerns the question how we can get epistemic access to arithmetic. Here, the following questions
are central: How is knowledge of arithmetic possible? And: Is knowledge of the truth of arithmetical
statements a priori? The last issue concerns the certainty of arithmetical truth: How absolute and
certain are the truths of arithmetical statements? The three issues are so closely related that a
picture of the status of arithmetical truth will result from the position that is taken on the first two
dimensions. If, for example, the arithmetical entities are viewed to be abstract entities existing
outside the knowing subject, known by a priori intuition, the nature of the truth must be universal,
certain and objective. If the arithmetical entities exist exclusively within a language, the truths are
analytic and certain, but conventional; they exist by virtue of the language system that instantiates
them and they are thus relative to this language system.
The positions that have been defended in the philosophy of arithmetic, concerning the named issues,
are here presented in eight views: Logicism, Intuitionism, Formalism, Platonism, Structuralism,
Empiricism, Social Constructivism and Embodied Arithmetic. In the selection of positions within this
broad research area, I have been guided by the objective to lay down a landscape of possible
positions in which I specific views on arithmetic education can be located. This entails that not all key
positions within the philosophy of mathematics might receive the attention they would deserve in an
overview of the philosophy of mathematics. However this aim does require a proper overview in the
sense that it must at least mention the key positions.
Next to the discussion of the three dimensions of the nature of arithmetic a fourth issue is
mentioned with each view. This concerns the knowledge conditions. Each of the eight views on
arithmetic implies conditions for being able to learn to know that specifically conceived subject. As
the three dimensions provide a ground for a philosophical evaluation of the conception of the
subject of teaching, the knowledge conditions provide a second basis on which teaching methods can
be evaluated. They are expressed in terms of certain types of skills that should be addressed by a
teaching method in order to enable children to access arithmetical knowledge.
I propose a non-restrictive view on learning arithmetic as a guideline for the evaluation of teaching
methods that takes no fundamental position in the philosophy of arithmetic but that instead takes
seriously the possibility of all views. This pragmatic approach with respect to education is based on
the idea that all views on arithmetic present something essential about arithmetic. And even though
they are incompatible in many ways in the foundational discussion, with respect to education they
can be viewed as presenting all a real possibility of acquiring arithmetical knowledge. The nonrestrictive principle comes down to the recommendation to accommodate all knowledge conditions
in a teaching method.
This thesis treats thus the interrelationships of two discussions. The first concerns the question:
What is the nature of arithmetic? And the second: What are desired conditions for a method for
teaching it? The interconnections between these discussions indicate a direction for answering a
third question: What is learning? Teaching is making the right environment for learning. Therefore a
perspective on what learning is precedes the question about the right educational method. From
there one will derive what must be optimized through teaching and what implications this might
have for creating the right environment for learning. Philosophical perspectives on learning and the
philosophy of arithmetic meet in a conception of the notion of arithmetical knowledge for a
perspective on what learning is brings also a picture of what the goal of this process is: a notion of
8
arithmetical knowledge. And a view on the nature of arithmetic brings a conception of arithmetical
knowledge as well.
This interdependence of answering the three questions ‘what is the nature of arithmetic?’, ‘what are
desired conditions for a good method for teaching arithmetic?’ and ‘what is learning?’ makes that a
position in one discussion should be able to be supported by plausible answers to the other
questions. A notion of arithmetical knowledge and a perspective on learning that supports my
proposal of what a good method for teaching arithmetic should contain is therefore presented in the
last chapter of this thesis. There Löwe and Müller’s definition of arithmetical knowledge in terms of
skills is presented. Their context-sensitive notion of arithmetical knowledge turns out as nonrestrictive to the possibility of the views on arithmetic for all knowledge conditions are expressible in
terms of required skills. I sketch two perspectives on learning that both can explain this learning as a
process that aims at arithmetical knowledge conceived as being fundamentally context-dependent
and expressible in terms of skills. These are the embodied perspective – and the practice-based
perspective on learning. The first is a non-representational account of learning, based on
phenomenological studies, expressed by Dreyfus. It locates learning within the interactions between
the learner and it’s environment. The second is an account of learning based on the late
Wittgenstein. It locates learning within an existing social practice. Both accounts of learning result in
a learning product that is fundamentally context-dependent. In support of these conceptions of
learning, both relying heavily on externality, studies on arithmetical cognition are sketched showing
that an embodied way of dealing with external representations plays an important role in
arithmetical cognition.
The philosophical relevance of this thesis lies in bringing the philosophy of mathematics and the
practice of teaching together. A compound of these two research areas is not common, but research
on the intersection of these areas is growing.4 It may give new insight in this notion of arithmetical
knowledge since in both areas an understanding of this is sought. Furthermore a philosophical
perspective on what learning is, is suggested by the conditions for being able to access arithmetical
knowledge that results from the views on the nature of arithmetic.
The structure of the thesis is as follows. The first eight chapters together present the philosophy of
arithmetic. Each chapter treats one view on the nature of arithmetic, each on the basis of four issues.
The first three issues concern in succession the ontology, the epistemology and the status of the
truths of arithmetic. Together these will function as a sketch of a landscape of dichotomies where
4
Since Paul Ernest has founded ‘The philosophy of mathematics education network’ in 1990 the philosophy of mathematics
education, -that is a research area at the intersection of the philosophy of mathematics and learning and teaching
practices-, this has been a rapidly developing area of enquiry. (Ernest, 1994, Introduction). Its central problems concern the
nature of mathematics and the implications of a position in this field for learning and teaching mathematics at the one
hand, and the possibly implicit epistemological assumptions underlying learning and teaching practices at the other hand.
This philosophy of mathematics education is based on the assumption that all mathematical pedagogy rests on a
philosophy of mathematics. Thus, controversies about teaching mathematics come to controversies about what
mathematics is really all about, and these cannot be resolved without facing problems about the nature of mathematics.
This thesis is in line with this philosophy of mathematics education in that it rests on the same assumption and concerns the
same central problems. It differs in that it doesn’t take a position in the philosophy of mathematics while Ernest, being a
Social Constructivist, does. A compound of philosophy of mathematics and educational practice has also become more
common through the shift in the philosophy of mathematics from foundational issues to the practice of mathematics that
naturally includes education. This shift is e.g visible in the research network ‘Philosophy of Mathematics: Sociological
Aspects and Mathematical Practice’, coordinated by Benedikt Löwe and Thomas Müller, that includes social aspects and
mathematical practice into the philosophy of mathematics. See (Löwe & Müller, 2010a).
9
any position in the discussion on the right method for arithmetic education could be located. The
fourth issue handles the knowledge conditions for each view. The conditions for knowing arithmetic
state what the specific view on arithmetic requires for the ability to acquire knowledge of it. In
chapter 9 the landscape of dichotomies of the philosophy of arithmetic is sketched, on the basis of
the first three discussed issues, in which the conception of the subject of teaching underlying any
teaching method can be positioned. This provides the basis for a philosophical evaluation of methods
for teaching arithmetic. Two desired conditions for teaching methods are proposed. The first is a
positioning in the landscape which provides a picture of the conception of the subject of teaching,
the teaching goals and the structural problems for this position. The second is the accommodation of
the knowledge conditions which ensures the opportunity for children to gain knowledge of
arithmetic conceived in all possible ways. Chapter 10 finally gives a definition of the notion
arithmetical knowledge and a perspective on learning that supports this non-restrictive view on
arithmetic education. I conclude with a case study in which Realistic Mathematics Education is
assessed by means of the presented philosophical instrument for evaluating teaching methods.
The explanations of how we gain knowledge of arithmetic, which I find to be suggested by the views
on arithmetic, answer the question how we gain warranted true beliefs in arithmetical truths.
Knowing an arithmetical truth requires that we believe it and that it is true. But knowing that an
arithmetical proposition is true requires something more. Otherwise there would be no difference
between knowing and lucky guessing. This additional element is called ‘warrant’. Although the nature
of this element of knowing is subject of discussion, the idea that this is an essential element of
knowing is not controversial.5 Thus the acquisition of knowledge, suggested by the views on the
nature of arithmetic, is an explanation of the acquisition of warranted true beliefs about a reality or
about a specific construction or invention. It is an explanation of how we gain true beliefs about an
abstract – or concrete reality or about a formal – or social construction, and it is an explanation of
how these believes are warranted.
The discussion is limited to arithmetic. Arithmetic, the theory of numbers, is one of the oldest areas
of mathematics and one of the most fundamental theories to it.6 Before proceeding to the eight
views on arithmetic, below a description of the core subject that is the basis for the proceeding
discussion of the views on arithmetic. The views to be discussed all refer to one general definition of
arithmetic: Peano-Arithmetic. These are the basic principles of the theory of numbers as defined by
Dedekind and Peano at the end of the nineteenths century.7 All views ground this Peano-Arithmetic
in different ways. I will take ‘Peano-Arithmetic’ as a starting point for the description of the system
that is the subject concerned.
5
(Markie, 2012) Introduction
(De Cruz, Neth, & Schlimm, The cognitive basis of arithmetic, 2010, p. 59)
7
(Horsten, 2008) Section 2.1
6
10
Peano-Arithmetic8 is here conceived as the theory of natural numbers and is basically given with five
axioms. The primitive concepts are: ‘0’, ‘natural number’, and ‘successor’ ( ).
1.
2.
3.
4.
5.
0 is a natural number
The successor of any number is a number
No two different numbers have the same successor
0 is not the successor of any number
Every class which contains 0 and which contains the successor of
includes the class of natural numbers.
whenever it contains
Addition is recursively defined with the two equations:
and
And multiplication is defined with the following two:
and
For example, for an addition like
Rewriting this into
addition
induction.
8
you see that:
.
shows a particular form of the associative law for
. These and other familiar arithmetical laws can be proved by
This description of Peano-Arithmetic is taken from (Rundle, 2006, pp. 464-466)
11
Chapter 1: Logicism
The logicist view considers the question of the relation between mathematics and logic. Logicism is
the thesis that mathematics is a part of logic.9 Logic is the study of the general forms that valid
inferences within some (formal) language may take. It is supposed to be neutral about ontological
matters. 10 It does not concern the actual existence of the things that are named in that language. So
grounding arithmetic into logic could give an anti-Platonist account of arithmetical concepts11 and, if
logic is conceived to be a basic form of our thinking, it could also justify arithmetical knowledge as
being a priori.12 To substantiate the logicist thesis, the logicist project consists in showing that
mathematics is reducible to logic.13
I paraphrase Horsten14 for a brief description how the logicist project has developed in history. In the
nineteenth century Frege was the first to carry out this project in The foundations of arithmetic
(1884). Dedekind and Peano defined the basic principles of arithmetic, at the end of the nineteenths
century. These axioms are known as Peano-Arithmetic. Frege managed to derive the Peano axioms
from the basic laws of a system of second order logic. Unfortunately one of his basic principles
turned out to be inconsistent. Russell and Whitehead continued the project in their Principia
Mathematica (1910, 1912, 1913), where they produced a systematization of logic from which they
constructed Peano-Arithmetic. But Russell had to lay down as a basic principle that there exists an
infinite collection of (ground) objects. This could hardly be regarded as a logical principle that is
ontology neutral. Carnap has continued to defend the logicist view in the thirties of the twentieth
century. I will here present logicism on the basis of the views of the above-mentioned authors.
1.1 Foundations
The logicist thesis states that arithmetic is basically a construction built up from logical concepts. The
logicist project -to show that arithmetic is reducible to logic- contains two parts:15
1. The concepts of arithmetic can be derived from logical concepts through explicit
definitions.
2. The theorems of arithmetic can be derived from logical axioms through purely logical
deduction.
I follow Carnap16 in the following discussion on these two parts.
1.1.1 The derivation of the arithmetical concepts
To derive the arithmetical concepts from logical concepts, first the logical concepts to be employed in
the derivation must be specified. The concepts are given in the form of relations between elements
of a domain of discourse. These are relations between names of objects, or between unanalyzed
sentences, or the relations are given in the form of functions. The basic logical concepts are:
9
(Carnap, 1983b, p.41)
(Horsten, 2008) Section 2
11
(Horsten, 2008) Section 2
12
(Parsons, 2006, p. 26)
13
(Parsons, 2006, p. 25)
14
(Horsten, 2008) Section 2.1
15
(Carnap, 1983b, p.41)
16
(Carnap, 1983b, pp. 41-52)
10
13
Logical Concept
Given as a relation between names of
objects
Identity of an object named a and b
Given as a relation between sentences
Negation of a sentence p
Conjunction of two sentences p and q
Disjunction of two sentences p and q
Implication of two sentences p and q
Given as a function
Property f belongs to an object a
Existence: ‘the property f belongs at
least to one object x’
Universality: ‘the property f belongs to
every object x’
Saying
Symbol
‘a and b are names of the same object’
‘not-p’
‘p and q’
‘p or q’
‘if p then q’
‘the property f belongs to the object a’
‘there is an x such that f(x)’
‘for all x, f(x)’
The first part of the logicist thesis is that the definition of all arithmetical concepts requires nothing
more than these logical concepts just given. (Actually less, since not all these concepts are primitive).
Since all concepts of Peano-arithmetic are reducible to the natural numbers, the derivation of
arithmetic from the logical concepts comes to the task to derive the natural numbers from the logical
concepts. This can be done if the natural numbers are viewed as ‘logical attributes’ which belong to
concepts. For example, according to this view, the number 2 can be defined in the following way.
First the term ‘at least two’ is defined as:
=Df
. Where stands
for ‘minimum’ and =Df means ‘is defined as’. Then ‘at least three’ is defined in a similar way:
. And finally the conjunction of
the sentence ‘at least two’ and the negation of ‘at least three’, forms the definition of the natural
number 2:
=Df
. Thus, in this way ‘2’ means: at least two, but not at least three
objects fall under f.
All natural numbers, the arithmetical operations and the concept of natural number itself can be
defined in this line of thought. The derivation of the other kind of numbers, such as the fractions, is
accomplished by the construction of a completely new domain. This means that the natural number
2 and the fraction 2/1 are not identical but merely correlated with one another.
The derivation of the real numbers turned out to yield serious problems. They are constructed by
definitions based on the series of fractions. To every rational real number there corresponds a
fraction and to every irrational real number, -Dedekind showed how-, there corresponds a ‘gap’ in
the series of fractions. Russell developed this line of thought to definitions constructing the entire
arithmetic of real numbers. Here, each real number is defined as the ‘lower’ class of the
corresponding fraction (in the case of the rationals) or as a cut in the series of fractions (in the case of
the irrationals). For example, ‘√2’ is defined as the property (or class) of those fractions whose square
is less than two. I will sketch the problems that occurred in the derivation of the theorems of real
number theory below.
1.1.2 The derivation of the theorems of arithmetic
Since mathematical predicates are introduced by explicit definitions that are not creative, this part of
the thesis comes to this: every provable arithmetical sentence is translatable into a sentence which
contains only primitive logical symbols and which is provable in logic. Where ‘provable in logic’
14
means that they are derivable from six, commonly accepted, logical axioms, and two rules of
inference (for substitution and for implication).
The problem for logicism has been the reconstruction of the theory of real numbers, without falling
into conceptual absolutism. The derivation of the theorems of real number theory turned out to
require more axioms, the one more dubious than the other. Two indispensible axioms, that have
been criticized are the axiom of infinity (which states that for every natural number there is a greater
one) and the axiom of choice (which states that for every set of disjoint nonempty sets, there is, at
least, one set that has exactly one member in common with each of the member sets). The problem
with these axioms is that they are existential sentences and logic is not supposed to make assertions
about existence. Another axiom, posited by Whitehead and Russell in their Principia Mathematic,
called ‘the axiom of reducibility’ is widely viewed as inadmissible. Russell himself abandoned the
axiom in 1925. But he needed this axiom to overcome the serious impossibilities resulting from his
‘theory of types’ that he designed in order to overcome the problem of ‘impredicative definition’.
The problem of impredicative definition is connected with the antinomies that Russell first showed to
appear in set theory. The concept ‘impredicable’ exemplifies these antinomies. A definition is
impredicative if it invokes the set being defined. An example of a contradiction resulting from this is
the set of all sets which do not contain themselves. The question: ‘does such a set contain itself?’
shows the paradox. The problem of impredicative definition is that it generates logical antinomies
and that it violates the vicious circle principle (that no whole may contain parts which are definable
only in terms of that whole).
Russell developed his ‘theory of types’ in order to overcome the problem of impredicative definition.
The theory of types consists of the ‘simple theory of types’ and the ‘ramified theory of types’. The
simple theory of types is a classification of expressions into different ‘types’. Regarding the one-place
functions (the properties) this goes as follows: to type 0 belong the names of objects (a, b,...); to type
1 belong the properties of these objects (f(a), g(a),...); to type 2 belong the properties of these
properties (F(f), G(f),...). For example the concept 2 (f) defined above belongs to this type 2. Now, the
properties of the properties of the properties belong to type 3, etc. The rule of type theory is that
every predicate belongs to a type and can be applied only to expressions of the next lower type. If
this rule is violated (as in ‘f(f)’ or ‘┐ f(f)’), the sentence is qualified as meaningless. Thus, also to say of
a property that it belongs to itself or that it does not belong to itself is meaningless. This is important
for the solution of one part of the problem of impredicative definition: the elimination of the
antinomies. And this theory does not cause new problems in the logicist project in contrast to the
ramified theory of types.
Russell believed the ramified theory of types, which subdivides the properties of each type further
into ‘orders’, necessary to overcome the second problem of impredicative definition: the vicious
circle principle. The rule introduced by this theory is that the expression ‘all properties’ without
reference to a determinate order is inadmissible. This makes that, in the definition of a property,
there never occurs a totality to which it itself belongs.
This ramified theory of types turned out to cause insurmountable difficulties in the treatment of real
numbers. It prescribes that the expression ‘for all real numbers’ cannot refer to all real number
without qualification but only to the real numbers of a determinate order. The result is that there can
be no admissible definition or sentence which refers to all real numbers without qualification. And
15
this results in the impossibility to express the most important definitions and theorems of real
number theory, central to mathematics.
The axiom of reducibility was designed to overcome this impossibility but it could not be regarded as
a logical principle in a anti-Platonist sense.17 This axiom states that for any truth function there is an
equivalent function that is predicative. This made it possible to reduce the different orders of a type
in certain respects to the lowest order of that type. But this required the basic principle that there
exist an infinite collection of ground objects. This could hardly be regarded as a logical principle since
logic is here supposed to consider the valid forms of inferences and thus be neutral about the
ontology of objects and especially about the existence of infinitely many abstract objects.
1.2 Epistemology
Above, logicism is construed as a constructive method as well as a formal one. The natural number
‘2’, for example, is a concept produced from logical primitive concepts in finitely many steps, through
explicit definitions, meaning: ‘two objects fall under it’. Likewise the real numbers are not postulated
but are logical constructions that have, by definition, the usual properties of the real numbers.
Impredicative definitions seems to run counter to this constructivist tendency since they implicitly
refer to the class that they are to define. Logicism is constructivist with regard to the constructive
linking of the steps in a proof. But regarding to the content the proof procedure is not necessarily
constructive since non-constructive existence proofs are in this view acceptable. (These are proofs
demonstrating the existence of a mathematical entity having a certain property without containing a
method for generating an example of such an entity). The logicist method is formal, since logic is
here construed as being about the form of the argument. Although the axioms and rules of inference
are chosen with an interpretation of the primitive symbols in mind, there is no reference to the
meaning of these symbols inside the system as in pure calculus.18
How can this foundation of arithmetic into logic account for arithmetical knowledge? What is the
character of this knowledge and how could our access to it be explained? According to logicism
arithmetical statements are objective, a priori and analytic. In The Foundations of Arithmetic19 Frege
explains these notions as follows. ‘Objective’ means that it is not an object of psychology or a mental
product.20 Frege’s non-psychological interpretation of ‘analytic’ means that the arithmetical truths
can be proved to be grounded into general logical laws and definitions.21 The a priori character of the
17
By the ‘anti-Platonist view in logicism’ I mean the rejection of the idea that the arithmetical objects, such as the numbers,
consists of abstract entities. This forces a neutral stance towards the ontology of objects.
18
(Carnap, 1983b, p. 52)
19
(Frege, 1884)
20
Frege explains: ‘For number is no with more an object of psychology or a product of mental processes than, let us say, the
North Sea is. The objectivity of the North Sea is not affected by the fact that it is a matter of our arbitrary choice which part
of all the water on the earth’s surface we mark off and elect to call ‘North Sea’. This is no reason for deciding to investigate
the North Sea by psychological methods. In the same way number, too, is something objective. If we say ‘The North Sea is
10,000 square miles in extent’ then neither by ‘North Sea’ nor by ’10,000’ do we refer to any state of or process in our minds:
on the contrary, we assert something quite objective, which is independent of our ideas and everything of the sort.’ (Frege,
1884, p. 34)
21
(Frege, 1884, p. 4): ‘If,…,we came only on general logical laws and on definitions, then the truth is an analytic one,...’
Herewith Frege argues against Kant’s qualification of arithmetical statements as synthetic and a priori. According to Kant in
‘7+5=12’ the concept ‘12’ is not reached through analysis of the concepts ‘5’, ‘7’ and ‘a sum’. We must add something in our
intuition, and thus this judgement is synthetic (Kant, 1781-1787, p. 144)B15. Frege says that the truth of propositions like
‘135664+37863=173527’ is given with proof. They can thus be shown to be analytic. The fact that the truth of these types
of propositions are not self-evident is not an argument for holding these propositions to be synthetic, as it is for Kant.
(Frege, 1884, p. 6)
16
arithmetical truths means that the proofs on which these truths rest depend exclusively on general
logical laws so that there is no appeal to empirical facts.22 According to the logicist, the generally
supposed absolute validity of arithmetical laws is out of the question.23 The absolute validity of the
laws of arithmetic is given with a belief in the absoluteness of the formal logical concepts and axioms
in which the concepts of arithmetic are grounded. By grounding arithmetic into a formal system, our
knowledge of it is independent of, or prior to, experience. How can there be a priori knowledge of
absolute, objective arithmetical truths according to logicism?
The reduction of arithmetic to logic does not lead directly to an explanation of how there can be a
priori knowledge of arithmetic for it passes this question on to a conception of our knowledge of
logic.24 By the grounding of arithmetic into logic, the questions on the foundations of arithmetic are
transferred to the foundations of logic. Hereby the epistemic question comes to this: how do we
have access to the primitive logical concepts and how are we able to make these formal logical
constructions that are assumed to be universal, a priori, and absolutely valid? Only if logic is more
basic than arithmetic, in the sense that it could be seen as the basic structure of our thinking, then
this does simplify the problem of a priori knowledge. However the anti-psychologist stance,
qualifying arithmetical truths as objective, does not allow for an explanation like this. Logicism gives a
non-Platonist account for the objective absoluteness. 25 This means that in this view the absoluteness
cannot be accounted for with some reference to existing abstract entities.
1.3 Truth
The Logical positivist accounts for a priori knowledge of objective, analytic truths by adding the
qualification ‘relative’ to the arithmetical truths. Carnap26 gives an anti-Platonist explanation for the
existence and meaning of logical concepts. He relativizes the ‘abstract-concrete dichotomy’ and
claims that arithmetical truths can only be known relative to a linguistic framework. For this he
mentions the distinction between questions that are internal to a linguistic framework and questions
that are external to it. In science only internal questions can be answered. External questions are
philosophical questions like ‘do mathematical entities really exist’. These questions are unintelligible
since there is no linguistic framework in which it could be answered, because they are external to it.
The acceptance of a language, such as a system of numbers, in order to being able to evaluate
internal questions on numbers, does not force one to a belief in the existence of these numbers.
Carnap argues that an empiricist position, denying the existence of entities that are not perceived by
us through immediate experience, is not committed to a rejection of linguistic forms of mathematics
wherein words like ‘five’ are meaningful. Existence is ascribed only to the data; the constructs are not
real entities. They exist only as a linguistic form and questions are only sensible if they can be
evaluated relative to such a framework. So the abstract logical concepts are not ontologically there,
outside the system. Its truths are thus relative to the linguistic framework in which they are
22
See (Frege, 1884, p. 4) on the definition of ‘a priori’ and the argument for the a priori character of arithmetical
propositions (Frege, 1884, pp. 9-17) where Frege argues against Mill’s position that arithmetical truths are basically
empirical facts.
23
(Brouwer, 1983, p. 78)
24
(Parsons, 2006, p. 26)
25
Frege does answers this problem with a Platonist account; with the objective existence of arithmetical objects. According
to Frege ‘every individual number is a self-subsistant object’ (Frege, 1884, pp. 67-72). Since Platonism is discussed in chapter
4, I will not discuss this position here and treat logicism as a non-Platonist position as it has been defended by the logical
positivists, in line with most descriptions of logicism e.g. in (Horsten, 2008) Section 2.
26
(Carnap, 1983a, pp. 241-257)
17
expressed. This view does not view arithmetical truths as absolute for they are always relative to a
framework. However within this framework the truths remain absolutely certain.
Carnap expresses a position where the logical concepts and rules of inference into which arithmetic
is grounded, are valid by convention. Arithmetic is reduced to logic, and logic is analytic in the sense
that it is without factual content; it is a system in which the meaning of arithmetical words and
inferences is given. This is an anti-realist position; not only truth, but also existence is internal to
conventions.27 Thus the objectivity (that means anti-psychologist subjectivity), the certainty (that
means that relative to the system of conventions the truths are certain/ provable) and the analyticity
of the truths of arithmetic are ensured, without having to seek refuge in the existence of a Platonic
world. The Logical Positivist view that knowledge is only possible within a framework cannot view
arithmetic truth as absolute for conventions can vary.28 Since there is no standpoint outside the
framework where one could verify the truth of its statements, arithmetic is exact, certain and a
priori known because they belong to this framework of conventions.
Quine has pointed out a problem for conventionalism: 29 the conception that the logical rules of
inference are valid by convention, leads to an infinite regress. In order to represent a rule of
inference as proceeding according to a convention, another application of this rule is needed. And in
order to represent this rule as based on a convention, again it is necessary to make another
application of this same rule. This leads to the insight that there is no use in reducing arithmetic into
logic other than the simplification and integration of the structure of our theories. Viewing the
elementary rules of logical inferences to be true by convention does not provide this reduction with a
firm ground for this turns out to be the same as viewing them as being hypotheses.
1.4 Knowledge Conditions
Arithmetical statements are a priori true in virtue of conventions. To know the a priori truths of
arithmetic, thus conceived, does not require a faculty that gives insight in its truths from a
perspective outside the system e.g. intuition. What is needed is the ability to learn a (formal)
language; to form and recognize well-formed sentences and to follow valid inferences. A presumed
linguistic ability can be understood as the presence of dispositions in us. Exercise of this ability could
generate beliefs and warrants them. Thus knowledge of analytical, conceptual truths can be reached
through exercising an invoked linguistic ability. Since arithmetic is here conceived as a construction,
knowledge of arithmetic is grounded on the activity of constructing procedures. Therefore exercising
these procedures is indispensible for learning to know it.
Learning to know logicist arithmetic requires the development of a linguistic ability by means of
exercises similar to learning a language. The rules that determine what counts as a well-formed
sentence and what counts as a valid inference are conventional. Learning to know these thus
requires more than natural development; these must be handed over by the teacher somehow. The
learner needs explicit examples to follow. The possibility to learn logicist arithmetic thus requires
exercises in following examples and rules given by the teacher (as opposed to developing these rules
naturally or discovering them by oneself). The explicit examples that are to be followed in the
exercises are examples of procedures for according to this view the nature of arithmetic is formal.
27
(Skorupski, 2005, p. 55) and (Skorupski, 2005, p. 68)
(Skorupski, 2005, p. 53)
29
(Quine, 1936)
28
18
Chapter 2: Intuitionism
Like logicism, intuitionism is a constructive method; according to this view, arithmetic is a construct
built up from logical concepts. However the logic in which the Peano axioms are expressed differs.
The logical axioms and rules of intuitionist arithmetic are those of first order intuitionist predicate
logic. Syntactically this is a restriction of classical first-order predicate logic, in which the law of the
excluded middle
┐ is not accepted as an axiom, and cannot be proved. With respect to the
theory of natural numbers classical Peano-Arithmetic and intuitionist arithmetic share the same basic
axioms and primitives and make the same arithmetical laws true. However, the theory of real
numbers, based on intuitionist logic differs from one that is based on classical logic.30
Intuitionist mathematics originates in the work of Luitzen Brouwer at the beginning of the twentieth
century. 31 His student Arend Heyting gave the first formal development of intuitionist logic and
derived the Peano axioms from this. This system, called ‘Heyting Arithmetic’, is proof theoretically
equivalent to the classically grounded Peano-Arithmetic.32 In the first decades of the twentieth
century intuitionist mathematics received much sympathy. But this decreased when the
mathematics that is based on intuitionist logic appeared to differ rather drastically from the classical
theory in higher mathematics. Nevertheless Brouwers followers, Troelstra and Van Dalen have
continued to develop intuitionist mathematics in the late eighties of the twentieth century.
Intuitionism does not ground arithmetic into intuitionist logic as logicism views classical logic to be
the foundation of classical mathematics. Actually intuitionism treats logic more as a part of
mathematics, both grounded in the human intellect. Yet arithmetic is described by intuitionist logic
and in this sense it is viewed to be reducible to intuitionist logic.33 But unlike the logicist conception
of this construct, here the construction is not formal; instead, arithmetic is viewed as a mental
construction of the human intellect and hereby the logical concepts are linked with something
external to the system itself.34 I will clarify both features of intuitionist method; the anthropological –
and the constructivist characteristics.
2.1 Foundations
The intuitionist anthropologist view states that arithmetic is a production of the human mind, in the
form of language; both natural and formal.35 It is essentially an activity of mental construction by ‘the
ideal mathematician’.36 That is a mathematician abstracted from contingent physical limitations, but
human and therefore a finite being. This grounding in the finite human intellect results in a revision
of classical logic and classical arithmetic.
The intuitionist constructivist position differs from the logicist one in that according to intuitionism
the constructions are human and in that sense subjective. There is no existence attributed to
arithmetical objects independent of human thought.37 This anti-Platonist view describes that e.g. the
30
This first paragraph on intuitionism is based on (Moschovakis, 2010)
These view words on the history of intuitionism is based on (Horsten, 2008) Section 2.2.
32
(Moschovakis, 2010) Section 3 and 4.
33
(Moschovakis, 2010)
34
Based on (Heyting, 1983)
35
(Heyting, 1983)
36
(Horsten, 2008) Section 2.2
37
(Heyting, 1983)
31
19
natural numbers have no other properties than those which are discerned in them by thought. Each
individual natural number can be mentally constructed, but the actual infinite; the whole series of
natural numbers in its entirety, cannot be included in a mental construct.38 More technically this
means that non-constructive existence proofs (proofs demonstrating the existence of a mathematical
entity having a certain property without containing a method for generating an example of such an
entity) are, according to this view, unacceptable.39 The characteristic feature of non-constructive
existence proof is the use of the principle of the excluded middle.
While the principle of the excluded middle (
) is a valid principle in classical logic, intuitionist
logic is obtained by removing this principle from classical logic. To clarify the rejection of this
principle, I must mention two features of intuitionist logic.40 The first is on the meaning of
propositions and assertions and the second concerns the meaning of ‘negation’ and ‘disjunction’ in
the principle at issue. About the first feature: Propositions express a certain ‘intention’ and an
assertion is the fulfilment of this intention. For example, the proposition that Euler’s constant C is
rational, expresses a phenomenological intention of the possibility of ‘thinking of something’. In this
case of finding (and thinking of) two integers a and b such that C=a/b. An assertion is the affirmation
of the proposition. This is the determination of an empirical fact and not itself a proposition.
The second feature concerns the meaning of the logical relations used in the principle of the
excluded middle: the negation and the disjunction. The negation, is a process for forming a new
proposition from a given one. For the intuitionist, negation is something positive. ‘ ’ Expresses the
expectation that one can derive a contradiction from the assumption ‘ ’. The logical relation ‘or’ (‘˅’)
in ‘
’ signifies that intention that is fulfilled if the intention -, or the intention
is fulfilled, or
both. So, one can only assert
if either is proved or a contradiction has been derived from .
Thus the validity of this principle as a general law is dependent on the mathematical proof that either
an arbitrary proposition itself -, or its negation is provable.41
Although the construction of arithmetic, based on the intuitionist attitude, is equivalent to classical
arithmetic with regard to the theory of natural numbers, it differs from classical arithmetic with
regard to the view on the real numbers. Intuitionist mathematics is composed on its most important
primitive concept ‘unity’. The integers are units which differ from one another only by their place in
the series. The fractions are introduced as pairs of integers. The continuum is not thought of as a
mere collection of units, rather it is an immediate intuition in which the continuous and the discrete
are united.42
The real numbers, as defined by Dedekind, based on the series of fractions as mentioned in the
chapter on logicism leads to intuitionist objections. 43 The problem is that when Dedekind’s
definition is transferred into intuitionist arithmetic, there would be no guarantee that an irrational
real number, e.g. Euleur’s constant C, is an irrational number. Intuitionistically interpreted, Dedekind
defines the real numbers by assigning to every rational number the predicate ‘Left’ (from the real
38
(Parsons, 2006, pp. 39-41)
(Horsten, 2008) Section 2.2
40
The features to be discussed are taken from (Heyting, 1983)
41
In this sense, logic is here dependent on mathematics. This shows that intuitionism does not ground arithmetic into
intuitionist logic as logicism views classical logic to be the foundation of classical mathematics. It shows that intuitionism
treats logic as a part of mathematics, both grounded in the human intellect (Moschovakis, 2010).
42
(Brouwer, 1983, p. 80)
43
The description of these objections in this paragraph is based on (Heyting, 1983)
39
20
number to be defined) or the predicate ‘Right’ (from this real number) in such a way that the order of
the rationals is preserved and the ‘gap’ (the rational interval) gets smaller and smaller. The problem
is that this infinite procedure provides us no way of deciding for an arbitrary rational number A
whether it lies left or right of C or is perhaps equal to C. The counterargument is that it does not
matter whether or not this question can be decided, since if it is not the case that A=C, then either
A˂C or A˃C, and this last alternative can be decided after a finite (perhaps unknown) number of steps
N in the computation of C. This argument comes to this: either there is such a finite N after which it
can be decided that ‘A˂C or A˃C’, or there is no such N which shows that A=C. But intuitionistically,
the existence of N means that it is possible to actually produce a number with the requisite property,
and the non-existence of N means the possibility of actually deriving a contradiction from this
property. So, the law of the excluded middle may not be used here as long as we do not know if at
least one of these possibilities exist.
Since Dedekind’s definition of a real number is not permissible to be adopted in intuitionist
mathematics, Brouwer has adjusted it by extending the definition of real numbers to allow so called
‘infinitely proceeding choice-sequences’ in addition to Dedekind’s rule determined sequences. The
difference with Dedekind’s defining rule is that here the series of predicates (Left or Right) need not
be determined to infinity by a rule. Instead the series is determined step by step by free choices. This
sequence determines the spread of all real numbers (the continuum) as a totality. A spread is not the
sum of its elements, since then these elements would be regarded as existing in themselves and this
runs against the constructionist view.
The example of the definition of Euler’s constant sketched the definition of real numbers viewed as a
human action based on free choices and not solely on a rule-determined definition. The infinitely
proceeding sequence is needed in order to prevent an impoverishment of classical number theory44
because the intuitionist arithmetic of real numbers appeared to differ rather drastically from the
classical theory.
2.2 Epistemology
The intuitionist view gives a non-Platonist account for arithmetical knowledge by grounding it in the
human intellect. It is a constructivist view with respect to the logical and arithmetical proof
procedure as well as with respect to the content of arithmetic. This is reflected in the fact that nonconstructive existence proofs are not acceptable according to this view.
By viewing arithmetic as a construct of an ideal human subject, intuitionists qualify arithmetical
knowledge as subjective in a Kantian sense.45 According to Kant arithmetical propositions are
exemplary of propositions exclusively known a priori and their truth is always necessary. Necessity
and a priority are closely connected:
It must first be remarked that properly mathematical propositions are always a priori
judgements and are never empirical, because they carry necessity with them, which
cannot be derived from experience.
44
46
The infinitely proceeding sequence turned out to be needed to preserve the set theoretic theorem about the continuum which states that if every real number is assigned a natural number as a correlate, then all the real numbers have the same
correlate- (Heyting, 1983)
45
(Brouwer, 1983, p. 80) and (Parsons, 2006, p. 30).
46
(Kant, 1781-1787, p. 144) B15.
21
Kant explains how pure mathematical judgements such as ‘5+7=12’ that is a priori and necessary, is
at the same time synthetic. This means that these judgements are known by intuition of the concepts
‘5’, ‘7’ and ‘a sum’. But the concept ‘12’ is not reached through analysis of these concepts, we must
add something in our intuition, and thus this judgement is synthetic. The intuitionist can account for
the a priori access to our mental constructions with a Kantian explanation of the notion
‘constructivist intuition’.
According to Kant knowledge requires two fundamental faculties: sensibility and understanding.47
Space and time are the a priori contributions of the fundamental forms of our sensibility. The a priori
forms of thought contributes to understanding. Sensibility contributes to experience in ‘empirical
intuition’. Empirical intuition gives us the sensations through which the objects are given to us. We
relate the data of this intuition by thinking them under concepts and form judgements about them.
This is called understanding. We have certain concepts a priori: ‘the categories’. These are the
conditions for the possibility of any experience whatever. Together with space and time in empirical
intuition this is the basis of synthetic pure a priori cognition. The general forms of judgements are
fundamental to all judgements about the concepts. Pure mathematics (e.g. ‘7+5=12’) contain pure a
priori cognition.48 These judgements are constructed in (or by) intuition. Constructivist intuition is
grounded in this idea that our minds impose a structure on experience and thought.
2.3 Truth
The anthropological character of intuitionist arithmetic, grounding it in the human intellect, provides
arithmetic with meaning. Arithmetic is a meaningful human activity in the sense that the
contributions of the subject in constructivist intuition is fundamental to arithmetical truth. The
meaning of propositions and assertions is thus given with something external to the system itself.
This in contrast with the formal conception of arithmetic expressed by the logical positivist – and the
upcoming formalist view.
Besides it being meaningful to us, arithmetical truth becomes subjective if grounded in the human
subject. Our contributions in intuition are so fundamental to our cognition of arithmetic (and
anything else) that there is no way of speaking about a purely objective arithmetical truth in the
sense of knowledge of a thing-in-itself. By explaining arithmetical cognition with Kant, intuitionism is
also bound to a form of Transcendental idealism.
The anti-Platonist constructivist starting point concerning the ontology of arithmetical objects, -that
we know that an arithmetical object exists if and only if we can provide a method of constructing
that object-, also holds for arithmetical truths. The limitation of arithmetical truths is given with the
constructivist position: no arithmetical proposition is true unless we can, in a nonmiraculous
constructivist way, know it to be true.49
The anthropological and constructivist character provides a picture of arithmetical truth that is
known a priori and thus, with a classical connection between a priority and necessity, is also
47
The Transcendental Aesthetic deals with sensibility and its pure form: space and time. The Transcendental Logic deals
with the operations of understanding and judgement. The a priori forms of thought named ‘general logic’ and
‘transcendental logic’ contributes to understanding (Kant, 1781-1787), (E.g. in the introduction of the Critique of Pure
Reason (Kant, 1781-1787, pp. 4-10)).
48
(Kant, 1781-1787, p. 144) B15.
49
(Parsons, 2006, p. 39)
22
necessary. Furthermore arithmetical truth is universal, certain (in the sense of provable) and
subjective in a Kantian sense, where arithmetic rests on the fundamental forms of our sensibility and
understanding.
2.4 Knowledge Conditions
In order to explain how we have a priori access to our mental constructions, intuitionists take
recourse to a Kantian explanation of ‘constructivist intuition’. Constructivist intuition is grounded on
this idea that our minds impose a structure on experience and thought.
If logic is understood as describing the natural universal laws of reasoning located inside the human
intellect these must somehow be invoked and naturally developed. Constructivist intuition that is
indispensible for learning to know intuitionist arithmetic is something that must be developed
instead of learned in the sense of externally imposed on the learner. Developing a learner’s
constructivist intuition requires that in teaching the emphasis is on proof, justification and arguments
of the learner for her knowledge productions. Reasoning is the material for the construction that a
learner must build for being able to reach arithmetical knowledge. Learning to know arithmetic thus
requires the development of a learner’s reasoning skills. And teaching is guiding the development of
reasoning that is basically given within the intellect of the learning subject itself.
23
Chapter 3: Formalism
The formalist takes arithmetic as a formal game in which symbols are manipulated according to fixed
rules, without having a direct interpretation.50 The natural numbers are seen as symbols, instead of
mental constructions, as the intuitionist views them.
Like intuitionism, this is an anti-Platonist conception of arithmetic, since there is no external content
that would have to be grasped. But contrary to intuitionism, in this view classical arithmetic is not
abandoned in order to be faithful to a constructivist starting point.51
Formalism as a stance on the foundations of arithmetic has been defended by David Hilbert in the
first decades of the twentieth century. 52 Hilbert’s programme consisted in proving the consistency of
a formalism of classical arithmetic by finitist means,53 which means that he aimed to prove that
number theory does not contain a contradiction. He has based his idea of a proof of consistency on a
theory of proof that is the outgrowth of the logicist project of reducing mathematics to logic.
Hilbert’s programme is thus connected to the logicist project. The difference between logicism and
formalism is the conception of the primitive mathematical entities: the natural numbers. For the
logicist it is a concept, while for the formalist it is a symbol.
3.1 Foundations
Hilbert sought to establish Peano-Arithmetic on a firm foundation by proving the consistency of a
logicist formalisation of it by finitist means. When, in 1931, Gödel proved that there exist arithmetical
statements that are undecidable in Peano-Arithmetic -which means that neither the truth of these
statements nor their negation is provable within the logical finitist proof procedure wherein PeanoArithmetic is grounded- (‘the first completeness theorem’) and shortly after that, also proved that
unless Peano-Arithmetic is inconsistent, the consistency of Peano-Arithmetic is independent of
Peano-Arithmetic -which means that the consistency of Peano-Arithemtic cannot be proved within
its grounding proof system- (‘the second completeness theorem’), Hilbert’s programme failed. This
does not mean that the formalist stance, viewing mathematics as the science of formal systems, has
become untenable. Curry defended the view that mathematics is a collection of formal systems
which have no subject matters and that there can be at most pragmatical reasons for preferring one
system over another, in the fifties of the twentieth century. In the late eighties Detlefsen and
Isaacson took up parts of Hilbert’s programme that are not ruled out by Gödel’s incompleteness
theorems.
According to the formalist view, arithmetic is a game, -an internally closed procedure-, which is
played according to fixed rules, which constructs combinations of primitive symbols on paper.54 With
this view on arithmetic, the questions on the foundations of arithmetic, such as questions regarding
the origins of the supposed absolute certainty of arithmetical truths, are dismissed as nonmathematical. For if arithmetic is considered to be ‘just’ a game, the only important thing for it to be
50
(Horsten, 2008) Section 2.3.
(Parsons, 2006, p. 41)
52
(Horsten, 2008) Section 2.3.
53
A finitist procedure is here understood as a procedure that takes a finite number of axioms and symbols and one rule of
inference (modus ponens) to produce a proof of consistency of all theorems of arithmetic. (Neumann, 1983).
54
The following of this section on the foundation of formalist arithmetic is based on (Neumann, 1983).
51
25
is internally consistent as to the procedure, not necessarily as to content. Even if the statements of
classical arithmetic would turn out to be false as to content, still the primitive symbols and the rules
of inference are considered ‘correct’ and the proof procedure is finitary, and this is the only thing
arithmetic is about, so the only thing that counts. What is important for the foundation of PeanoArithmetic is only its methods of proof and not the content of arithmetical statements itself.
If the method of proof is consistent and valid55 then what is proved, is to be considered as true. In
this view there are no reasons to object to non-constructive existence proofs. There is no need to
construct (derive) p in order to prove its existence, if one can also derive a contradiction from ┐p in
a proof system that is consistent and that generates proofs that are valid.
In order to being able to prove the validity of classical arithmetic, it must be reduced to a finitistic
system, the methods of proof of which should be investigated by checking if it is the case that only
those formulas corresponding to statements of Peano-Arithmetic, can be proved that also have been
shown to be true in Peano-Arithmetic. This is what Hilbert’s theory of proof consists in.
The wider project Hilbert’s theory of proof is a part of, consists in showing the validity of using
classical mathematics as a ‘short-cut’ for validating arithmetical statements. The following tasks are
to be distinguished.
1. The enumeration of all primitive symbols used in mathematics and logic.
2. The characterization of all formulas in classical arithmetic (that is the combinations of the
symbols representing statements classified as ‘meaningful’).
3. The supply of a construction procedure, called ‘proving’, that makes the construction of all
formulas that correspond to the provable statements of Peano-Arithmetic, possible.
3.1
Every axiom is considered proved.
3.2
If and are two meaningful formulas, and if and
have both been proved,
then also has been proved.
4. A proof, by finitary combinatorial means, that all and only those formulas corresponding to
true arithmetical statements (checked by arithmetical methods), can be proved by the
procedure described in (3).
The first three tasks coincide with the logicist project. Hilbert’s programme, then, concerns (4). This
comes to proving, in a finitary way, the consistency of the prove procedure described in (3). For, if
the arithmetical check shows a numerical formula to be false, then from that formula the relation
(where and are two different given numbers), can be derived. According to task (3) this
would give us a formal proof of
from which we could get a logical proof of
What must
be shown is the unprovability of
by the method described in (3). For if the system described in
(1)-(3) is viewed to be the foundation of Peano-Arithmetic, it would be highly undesirable that a
formula like
would be provable in it. The motivation for Hilbert’s programme is that a proof of
consistency would guarantee that this cannot happen.
When Hilbert’s programme was blocked by the appearance of Gödel’s second incompleteness
theorem, the formalist attitude towards the ontology of arithmetic, -viewing it to be basically a
formal game-, has persisted. Within this game a natural number is a symbol. This symbol could have
meaning when it is e.g. represented by physical entities, such as apples in basic arithmetic. However
55
Validity here means that the logical step-by-step-proof-procedure generates conclusions that follow from the premises.
26
in higher arithmetic these symbols are not directly interpretable in a concrete manner. 56 This view
generates purely formal arithmetic.
3.2 Epistemology
As logicism and intuitionism, formalism is a constructivist view with respect to the logical and
mathematical proof procedure, the constructive linking of the steps in a proof. However unlike
intuitionism, formalism is not a constructivist position regarding to content. Non-constructive
existence proofs are in this view acceptable. The constructivism in formalism, viewing mathematical
entities as symbols manipulated by the rules of inference, exists only with regard to this symbol
manipulation.
If a number is a symbol, and (higher) arithmetic is a ‘meaningless’ game, then the first thing one
could conclude concerning arithmetical knowledge is that it is not accessible through empirical
observations of the physical world outside the number system itself. Arithmetical knowledge is
according to the formalist a priori but not in the same way as intuitionist knowledge of arithmetic is.
The difference between the formalist – and the intuitionist view on a priori arithmetical knowledge
lies in the contrast between viewing arithmetic as meaningful and grounded in the knowing subject
or viewing it as a formal outward procedure. Since formalist arithmetic is taken to be a purely
syntactical, empty, outward and closed procedure, our knowledge of its statements is a correct
application of the rules that are set up in advance to specify the truth values. Arithmetical knowledge
is thus: ‘playing the game by the rules’. To know that p means to follow the right procedure for
proving that p in a consistent proof system. While for the intuitionist it means that we construct p in
our understanding. According to the intuitionist the subject contributes in constructivist intuition to
what arithmetic basically is and thus it is always meaningful to this knowing subject. However there
is no reference to meaning in formalist arithmetic. It is grounded external to the knowing subject,
hereby being objective (in the sense of external to the knowing or constructing subject).
Next to being objective, a priori knowledge of formalist arithmetical truth is always relative to the
proof system that generates it. This is in line with the logical positivist conception of truth by
convention and one more contrast with the intuitionist view. While for the intuitionist arithmetical
truth is absolute and universal, -for it is known by constructivist intuition, that is our general form of
experience and understanding-, for the formalist arithmetical truth is always relative to the system of
conventions in which the truth is expressed and proved.
With respect to the formalist view on arithmetical knowledge the question arises how this view can
account for the supposed certainty and applicability of elementary arithmetical knowledge. For if
arithmetic is just a game, there is no reason to belief that this game is of a more fundamental sort
than another. The reduction of arithmetic to logic does not answer this question since this shifts the
question to the foundations of logic, as said in the chapter on the logicist view. The difference with
logicism is that here there is no need to give an account of our knowledge of the meaning of
arithmetical statements since there is no reference made to meaning at all.57
56
(Horsten, 2008) Section 2.3
This is at least the case for some formalist positions. Hilbert did make a distinction between elementary arithmetic and
higher arithmetic. His instrumentalist stance applied at least to higher arithmetic, and not necessarily to elementary
arithmetic that has, in his view, an interpretation. (Horsten, 2008) Section 2.3.
57
27
3.3 Truth
Arithmetical truth is analytic. The rules and symbols that constitute the game include the conditions
for a statement to be meaningful (construed as being qualified for participating in the game) and
true. Thus arithmetical statements are true by convention. Although this makes these truths relative
to this particular game, they remain absolutely certain, since they are basically already there, stated
within the logical formalization of Peano-Arithmetic.
Gödel’s first incompleteness theorem, however, showed that any formalism S that is powerful
enough to express certain basic parts of elementary number theory is incomplete. Which means that
there can be found a formula of S, such that if S is consistent, then neither nor
is provable in
58
S. This implies that within a formalization of Peano-Arithmetic, there are ‘meaningful’ statements
without a truth value. So if there is a proof that , then this truth is certain. But not all arithmetical
statements are decidable.
Gödel’s second incompleteness theorem states that if S includes a statement of its own consistency,
then S is inconsistent.59 For the status of the truths of arithmetic this means that within any
formalisation S of Peano-Arithmetic that is consistent, there cannot be proved that S will not contain
a contradiction. Since there is no guarantee that S is consistent within the proof theory S itself, the
certainty of the truths generated by S is depend on something outside S itself. This thus constitutes a
serious crack in the formalist position, viewing the arithmetical truths to be purely analytical, generated by a closed system-, and at the same time absolutely certain.
3.4 Knowledge Conditions
The ability to construct formal proofs is essential to arithmetical knowledge for knowledge of
arithmetic is a priori and procedural. However the truths that are generated by the proof system are
also dependent on something outside the formal game itself. To know a truth of non-contentful
arithmetic is to know two proofs: the proof of consistency and the proof of the theorem itself.60 Since
Gödel’s second incompleteness theorem showed that there is no guarantee that a formal system is
consistent within that proof theory itself, the absoluteness of the truths generated by it, is
dependent on something outside the system.
In order to being able to know arithmetic one must first of all rely on the faculty required for learning
logicist arithmetic: the ability to learn a (formal) language. This includes the ability to form and
recognize well-formed sentences and to follow valid inferences. A linguistic ability must be invoked
as regard to the syntax of language. Learning formal arithmetic that is based on conventions requires
exercising following explicit examples and rules that are externally imposed on the learner. The game
must be learned in the sense of ‘given to the learner’. Generating arithmetical truths by the use of
this system requires procedural skills. Warranting arithmetical beliefs generated by this game
requires consistency-proofs that are also based on formal procedures. Learning to know formalist
arithmetic thus requires the ability to use ways of reasoning exercised in the game, but also the
ability to use ways of reasoning exercised in consistency-proofs. This reasoning is formal and thus
comes to generating procedures based on fixed rules that are externally given to the constructing
58
(Parsons, 2006, p. 42)
(Horsten, 2008) Section 2.3
60
(Steiner, 1975, pp. 139-140). Steiner remarks that in order to being able to know arithmetical propositions that are taken
to be given in the object language, -that is a formal game-, one must first know a metalinguistic proposition that one cannot
deduce a contradiction from the axioms. (Steiner, 1975, p.29).
59
28
subject. Learning formalist arithmetic is learning how to play a game. The essence of which is that
one learns to use the formal rules properly. This game could become meaningful in different ways. It
can be applied in the physical or social world and within other scientific theories. However these
applications are not part of the core of the game itself thus these applications are not necessarily
part of the conditions for learning to know formalist arithmetic.
29
Chapter 4: Platonism
Platonism is a position regarding the existence of mathematical objects that goes along with
philosophical realism.61 Realism is the idea that arithmetical objects exist and that arithmetic gives us
information about the existing numbers and their structural relations. According to Platonism these
existing objects are abstract.62 The existence of these abstract objects are taken to be independent of
cognitive operations and the mind. The facts concerning them, e.g. the truth values of arithmetical
statements, are not dependent on the possibilities of verification.63 Platonism as a view on the
foundation of arithmetic exists in degrees.
The mildest form is the view in which one accepts Platonism as a mathematical method. According to
this view one takes the language of classical arithmetic at face value, herewith implying the existence
of abstract arithmetical entities without necessarily believing in their existence.64 The most
elementary form of this Platonism is the position in which one postulates the existence of the totality
of the natural numbers, so that one can quantify over all natural numbers and use the law of the
excluded middle, by means of the widely accepted mathematical method in use. This Platonist
position is carried out one step further by the acceptance of the totality of points of the continuum
by means of which a general theory of real numbers can be generated.65 Furthermore the use of
methods of set-theory results in a Platonist position regarding the existence of sets.
The constructivist objection to a commitment to the existence of the totality of points of the
continuum is expressed by intuitionists. A constructivist objection to impredicative definitions definitions of sets in terms of totalities to which they themselves belong- mentioned in the chapter
on logicism, is the problem of the circularity. From a Platonist point of view, this circularity does not
seem to exist. For if a set is understood as existing independently of the language defining it, then
the definition only ‘picks out’ an object from a pre-existing totality.66
A higher degree of Platonism can be seen in set theory, for example within the Zermelo-Fraenkel
axiom system. Most of our mathematical knowledge (including Peano-Arithmetic) can be deduced
from these Zermelo-Fraenkel axioms with the axiom of choice (ZFC). This system allows the iteration
of a process of forming the set of all subsets of a given set and the collection into a set of what has
been obtained by this procedure, into the transfinite.67 Such procedures, that generate a
commitment to larger and larger universes, have shown to lead to the paradoxes of set theory as
mentioned in the chapter on logicism. The appearance of the paradoxes have led to various forms of
attenuation of Platonism.68 Russell’s theory of types and his ‘theory of limitation of size’ are
examples thereof.
In this chapter a strong version of Platonism is discussed: Platonism as expressed by Gödel. According
to his Platonism one believes in the objective existence of arithmetical objects as being abstract
61
(Parsons, 2006, p. 35)
(Horsten, 2008) Section 3.
63
(Parsons, 2006, p. 36)
64
A form of this mild Platonism is called ‘Default Platonism’. It has been defended by Quine. His position will be discussed
in the chapter on empiricism.
65
(Parsons, 2006, p. 36)
66
(Parsons, 2006, p. 36)
67
(Parsons, 2006, p. 37)
68
(Parsons, 2006, p. 38)
62
31
entities and in the objective existence of arithmetical truth that is grounded in the axioms of set
theory.
4.1 Foundations
The core of Gödel’s Platonism is a belief in a conception of arithmetical objects and concepts that
shows a great similarity with physical objects and properties.69 Both are not constructed by humans.
Both are not reducible to mental entities. Both are objective. Both are postulated in order to obtain a
theory of our experience. The difference is that physical objects and properties exist in space and
time, and mathematical objects and concepts not. But our perceptual relation with physical objects
and properties is similar to our ‘quasi perceptual’ relation with arithmetical objects and concepts
through ‘mathematical intuition’.
According to Gödel’s Platonism Peano-Arithmetic is grounded in set theory; in the ZFC- axioms. The
truth of these axioms has a special ontological status: it exists objectively. We have intrinsic evidence
for the truth of these axioms which is given to us by mathematical intuition. Our mathematical
intuition is fallible like our perception. It can be corrected. However this does not mean that the
arithmetical truths themselves are unsure. These axioms are according to Gödel’s Platonism the solid
ground for arithmetic.
A problem for Gödel’s view that arithmetical truth exists objectively and that arithmetic is grounded
in ZFC, is the undecidability of arithmetical hypothesis from ZFC such as Cantor’s continuum
hypothesis (CH). CH concerns the arithmetic of cardinal numbers (the numbers used to describe the
size of a set). Two sets have the same cardinal number if there exists a one-to-one correspondence
between them. The CH is a hypothesis about the sizes of infinite sets. It question whether there is a
cardinal between the cardinal of the integers and that of the continuum. This question appears not
to be decidable. Cantor’s continuum hypothesis (CH) states ‘no’; there is no set whose cardinality is
between that of the integers and that of the real numbers. In other words: the continuum does not
contain subsets of cardinal numbers different from that of the continuum and that of the integers.70
The problem is that the CH is not refutable from ZFC but also not provable from ZFC.
According to Gödel also mathematical statements of which the truth is not determined and whose
truth is independent of ZFC -such as the CH- must be either true or false, even though it might never
be determined which of the two.71 This stems from the idea that the set-theoretical concepts and
theorems describe some determined reality and so the mathematical questions must have definite
answers. Since this hypothesis is independent of ZFC there must be some extrinsic evidence for its
truth or falsity. Gödel allows such evidence in the form of its role in the ‘successfulness’ of a broader
theory. This can be expressed by the ‘fruitfulness of it in verifiable consequences’.72
In summary, according to Gödel’s Platonism Peano-Arithmetic is grounded in ZFC. These axioms are
the primitive concepts. Knowledge of their truth is given by mathematical intuition. And the
arithmetical objects and truths exist in a reality that is independent of the knowing mind.
69
(Horsten, 2008) Section 3.1.
(Parsons, 2006, p. 47)
71
(Parsons, 2006, p. 51)
72
(Horsten, 2008) Section 3.1. It is hard to find philosophical arguments that support Gödel’s position (Parsons, 2006, p.
51). An alternative is expressed by Quine -the view of default Platonism to be discussed below- where the role of a
mathematical theory in a broader scientific theory becomes decisive for the acceptance of a language that implies the
existence of abstract entities.
70
32
4.2 Epistemology
It is hard to find a defensible epistemology for Gödel’s position. One problem for the plausibility of
Gödel’s general conception of our direct epistemological access to arithmetical objects and concepts
through mathematical intuition is the fact that the truth of CH is undetermined. A direct intuition of
the general concept ‘set’ or ‘natural number’ would require that we have an idea of the infinite
before that of the finite.73 But then one would not expect the truth of CH to be undecidable. If these
concepts would be indirectly given to us; developed from an area where we do have direct intuitive
evidence, then it is explicable that the nature of ‘set’ is for an important part indeterminate, and
could be further developed in divergent ways.74
Another famous problem for Platonist epistemology is expressed by Benacerraf.75 His question
comes to this: If Platonism is true, how can we, concrete physical people, stand in contact with
abstract non-physical entities? In all possible answers there must be a connection presupposed
through intuition. But it seems hard to account for a notion of mathematical intuition that is not too
fantastic to believe in.
Gödel’s view, that much of mathematics is seen to be evident in a directly intuitive way could rest on
a naturalist methodology where science itself is the only and best method for judging the claims of
science. If we start reasoning from the fact that we do have arithmetical knowledge in everyday life
and in science, the question shifts to this: ‘How can our arithmetical knowledge be knowledge of
abstract entities that exist independently of our minds?’ We saw that, for Gödel, the evident
character of arithmetical statements is also explained by their consequences for the theory they give
rise to. This character of these statements could also be explained by their role in empirical science
and its applications. This is the heart of the ‘indispensability argument’, developed by Quine, to be
discussed below.
4.3 Truth
Arithmetic is grounded in the ZFC axioms. The abstract arithmetical entities exist independent of the
mind. The statements of arithmetic are either true or false whether it is (yet) determined or not. The
truth or falsity of an arithmetical statement is absolute since in the end they rest on one system of
abstract entities that exists external to the knowing mind. Furthermore the arithmetical truths
themselves are certain. They can be given to us through immediate intuition. Nevertheless our
knowledge of them is fallible since our intuition is fallible in the same sense that our senses can also
deceive us.
73
(Parsons, 2006, p. 47)
(Parsons, 2006, p. 47)
75
(Parsons, 2006, pp. 51-52)
74
33
4.4 Knowledge Conditions
According to Platonism the arithmetical reality exist independently of the mental activities of the
minds that know it and it is known a priori. We have a priori access to this mind-independent
arithmetical reality by means of realist intuition. Platonist epistemology relies heavily on this faculty
because it is the only answer to Benacerraf’s epistemology problem. If arithmetical truths exist in a
Platonic realm, we must somehow reach it in order to gain knowledge of it. What could this faculty
be and how could it be addressed by a teaching method?
The intuitionist ‘constructivist intuition’ could be psychologically accounted for with an innate-based
answer to the question where this intuition comes from. Such a psychological perspective however,
could not account for a priori knowledge of External-Meaningful positions like Gödel’s, for according
to his realism, mathematical objects exist independently of our constructions and can only be
perceived and described. His idea of abstract mathematical intuition is essentially different from
Brouwerian intuition.76
Gödel’s ‘intuition’ must give us an immediate insight into the obviousness of truths while Brouwer’s
intuition accounts for mathematical knowledge that is only obtainable by constructive proof. Gödel’s
conception of ‘mathematical intuition’ must also be obviously much stronger than the concrete
intuition Hilbert’s formalism is based on. Formalism requires an intuition of combinatorial properties
of finite, discrete and concretely representable objects and constructive methods, while Gödel’s
abstract intuition must give us an immediate insight in abstract objects and concepts in order to
provide us the meanings of arithmetical concepts and arithmetical truths.77
Gödel searched for a philosophical account for the feasibility of intuiting real Platonic concepts in
Husserl’s phenomenology.78 Husserl’s inquiry of the structural analogies between sense perception
and our grasping of mathematical truths could give a conception of ‘intuition’ that leads to
knowledge of ideal objects.
The phenomenological notion ‘essential insight’ could provide a possible interpretation of realist
intuition.79 According to Husserl’s phenomenology, a cognitive act has always intentionality in the
sense that a mental act is always about something. To be conscious always means to be conscious
of.80 From a phenomenological point of view, both the perception of physical objects and ideal
objects is not a passive registration, it is an active, goal directed search for anticipated facts.81
Through phenomenological reflection on the most mundane forms of perception, abstract entities
turn out to be indispensible. In order to be able to arrive at necessary truths the focus must be on
the essences that are instantiated by the acts of consciousness, analysed in phenomenology.82 These
essences are higher-order objects that can be perceived through the faculty of intuition.83 ‘Essential
insight’ is ‘seeing’ (directly grasping) concept or essence. Thus essence is embodied in the cognition
of objects. Both, concrete and abstract objects are intuited; the former through perception and the
76
(Liu, 2010, p. 38)
(Liu, 2010, pp. 38-39)
78
(Liu, 2010) and (Hauser, 2006)
79
(Hauser, 2006)
80
(Liu, 2010, p. 39) and (Hauser, 2006, pp. 554-562)
81
(Hauser, 2006, p. 569)
82
(Hauser, 2006, p. 550)
83
(Hauser, 2006, p. 550)
77
34
latter through intuition. The evidence that intuition offers is a judgement about what properties and
relations those objects have although both faculties can deceive you.
Three important features of Gödel’s intuition show similarities with Husserl’s ‘essential insight’.84
These are: 1. Mathematical intuition can decide the truth of the mathematical propositions (e.g. the
axioms of arithmetic and set theory); 2. Intuition is the opposite of proof. By intuition we see
something without a proof: it is a state of consciousness/ an act of cognition by which we can grasp
the essence of abstract concepts; 3. Intuition and perception share similarities.
Since realist intuition is indispensible for being able to learn to know arithmetic, this faculty must be
addressed in arithmetic education. How could this realist intuition be addressed by a teaching
method? A teacher must somehow create optimal conditions for calling this realist intuition of the
learner. She should represent her insight in such a way to the learner that this representation of
what she ‘sees’ invoke a learner’s act of cognition that is ‘grasping the essence of abstract concepts’
and that immediately reveal arithmetical truth.
A representation of an insight of the teacher that has this goal will be most effective if it represents
the insight in such a way that the attention of the learner is drawn to the essence of what needs to
be transferred. A presentation in a very rich context is not at the service of this. If the representation
of an insight in an arithmetical truth is presented in a story or picture that is packed with all sorts of
physical coincidences this will distract the learner’s attention from the real abstract truth that she
must reach by intuition. Of course rich contexts can be very useful in the education of Platonist
arithmetic, for example for enabling learning applications of pure arithmetic or for enabling the
development of the ability to see the bare abstract arithmetic in a contingent physical reality.
Nevertheless poor contexts support the initial insight in an arithmetical truth by a learner’s realist
intuition. To enable this insight is the core aim of Platonist arithmetic education.
Realist intuition is indispensible for knowing arithmetic according to this view and invoking this
requires representations of insights that support the ability to grasp the essence. This requires
representations in poor contexts that reveal the essence of the arithmetical truth as clearly as
possible.
84
(Liu, 2010, p. 40)
35
Chapter 5: Structuralism
Structuralism refers to a collection of views on the nature of arithmetic, expressing the intuition that
arithmetic is the study of structures.85 According to the general structuralist view, an arithmetical
object -a number- is not viewed as an object having a particular nature itself, but as an open place in
a structure.86
According to the structuralist approach the Peano axioms are not viewed as basic assumptions,
which can be true or false, but merely as a condition in order to enable a definition of the subject
arithmetic.87 So Peano-Arithmetic is interpreted as a defining condition on a type of structure.
Within the structuralist views one important distinction is to be made regarding the ontological
status of the structures. This is the distinction between the views compatible with realism or
Platonism on the one hand and the views compatible with nominalism on the other hand.
Nominalism is the contrast of realism. It comes to the view that there are no abstract entities;
everything that exists is concrete.88 Since the rejection of references to abstract arithmetical objects
requires a revision of the language of classical arithmetic, these nominalist views propose
reconstructions or translations of the language. They propose the elimination of references to
abstract mathematical objects (also of ‘structures-as-objects’) from the language of mathematics.89
The noneliminative types of structuralism in contrast, take the structures as absolute objects and
take the language of arithmetic at face value.90
Here I will discuss three types of structuralism; two noneliminative - and one nominalist type. The
noneliminative types are ‘Set Theoretic Structuralism’91 and ‘Ante Rem Structuralism’. The nominalist
structuralism to be discussed is named ‘Modal Structuralism’. 92 The discussion of these three types
presents some issues that play a key role in the philosophy of mathematics concerning the
ontological status of numbers, the corresponding conception of arithmetical knowledge and the
explanation of our access to that.
5.1 Set Theoretic Structuralism
According to Set Theoretic Structuralism numbers are viewed as abstract structures. These structures
are abstract objects, namely sets. One problem for this type of structuralism is that the general
structuralist view -that axioms are to be understood as defining conditions on certain structuresdoes not apply to set theory itself. The existence of structures -meaning the existence of sets- implies
that the structures-as-objects view becomes an abstract object view. Herewith Benacerraf’s
epistemological problem appears to be an issue for this Platonist view on arithmetic. In addition it
85
(Horsten, 2008) Section 4.2 and (Parsons, 2006, pp. 50-51) and (Hellman, 2005, pp. 536-538).
(Horsten, 2008) Section 4.2
87
(Hellman, 2005, pp. 537-538)
88
(Chihara, 2005, p. 483)
89
(Parsons, 2006, pp. 50-51)
90
(Parsons, 2006, p. 50)
91
This name is taken from the first type of mathematical structuralism that Hellman describes in (Hellman, 2005): STS.
92
In order to remove abstract entities from the language of arithmetic, several programs have been developed. There are
two dominant nominalist programs. The first allows points or regions of space-time as physical. The second allows modality.
(Parsons, 2006, p. 50). I will only discuss one of these types as an example of nominalism. This is sufficient for the aim of
this thesis to offer a sketch of the most important philosophical issues concerning the views on arithmetic that could be
relevant for the discussion about good methods for arithmetic education that nonetheless does justice to the actual
philosophical positions.
86
37
encounters another problem also expressed by Benacerraf: ‘Benacerraf’s identification problem’.
This problem holds for all views that identify numbers with sets. It is described below.
5.1.1 Foundations
The abstract structures that are studied in arithmetic, according to this view, are understood as settheoretic constructs. These structures are models of the theory of number. A model of arithmetic
satisfies the arithmetical theorems. Model theory is carried out in set theory. The natural numbers of
arithmetic are identified with sets. These sets are the basic objects of arithmetic.
There exist infinitely many ways in which the numbers can be identified with sets and the question
which way is the right way, considering the ontological real status of the natural numbers, is in
principle unanswerable within this approach.93 This problem is expressed in Benacerraf’s
‘identification problem’. Consider for example the following two ways in which the natural numbers
can be identified with sets.94
I:
0=Ø
1 = { Ø}
2 = {{ Ø}}
3= {{{ Ø}}}
...
II:
0=Ø
1 = { Ø}
2 = { Ø, { Ø}}
3 = { Ø, { Ø}, { Ø, { Ø}}}
...
On both the ‘number-candidates’ of I and II, the successor function as well as the operations addition
and multiplication can be defined in such a way that we arrive at isomorphic structures, which means
that both structures make the same arithmetical sentences true. So there seems to be no reason to
prefer one system above the other. However, Benacerraf’s identification problem shows that it is
impossible that both accounts are correct. This follows from a consideration of extra-arithmetical
questions like the question whether it is the case that
.95 Then the two systems can give
divergent answers. In this case the answer is ‘no’ according to system I, and ‘yes’ according to system
II. If both systems would be correct then the transitivity of identity implies that
93
(Hellman, 2005, p. 539)
The following discussion is taken from (Horsten, 2008) Section 4.1.
95
Meaning: 1 is an element of 3 (or 1 is in 3).
94
38
, which is a set-theoretic falsehood. Benacerraf thus concludes that the natural
numbers cannot be sets at all.
5.1.2 Epistemology
This Set Theoretic Structuralist type is not able to block questions on the set-theoretic reality by
taking sets as the basic objects.96 As Gödels’s Platonism this Platonist view is challenged by
Benacerraf’s epistemological problem too. Also in this view it seems unavoidable to presuppose an a
priori insight into this reality by means of realist intuition in order to being able to have access to
arithmetical knowledge, unless the reduction of the natural numbers into a set-theoretic universe is
not seen as an ontological reduction, holding the view that they are in the end not ontologically
reducible to anything else than those properties attributed to them by the defining axioms of PeanoArithmetic. This last view would dismiss all questions about the nature of numbers and the possible
views on their accessibility are in line with versions of nominalism, of which one type is discussed
below under ‘Modal Structuralism’.
5.1.3 Truth
According to this Platonist account of Set-Theoretic Structuralism, arithmetical statements are true in
a model, in which the numbers-as-structures are ontologically reduced to sets. If one is committed to
the existence of these abstract entities, the arithmetical truths that are generated are objective,
located outside the knowing subject. Despite of the relativeness of the truth to the model in which it
is expressed, the status of it is basically not different from the Platonist truths of arithmetic for if the
sets exist objectively, the truths about them exist too and are in this respect absolute although they
are expressed in a certain model that could have been different.
5.1.4 Knowledge Conditions
The conditions for a priori knowledge of this Platonist conception of structuralism are the same as for
Platonism given in the previous section. Knowing arithmetic thus viewed requires realist intuition.
Teaching arithmetic must according to this view address somehow the invoked realist intuition of the
learner. It is obvious that for this purpose the representations of truths about the structures can best
be given in poor contexts that can reveal the essences of the truths as clearly as possible so that it is
up for grabs by the learner’s intuition.
5.2 Ante Rem Structuralism
According to Ante Rem Structuralism arithmetic is a theory that describes structural relations. This
theory is about real abstract structures. Numbers are open places in these structures. In this sense it
is a Platonist position that asks for answers on Benacerraf’s problems; the epistemological – and the
identification problem. Below a picture of how these problems are answered.
5.2.1 Foundations
This view, articulated amongst others by Shapiro and Resnik, holds that mathematical theories
describe structures, meaning places and positions in structures.97 Sensible questions are the
questions asked from within a particular structure, so that it can be answered on the basis of
structural aspects of the structure that instantiates what is being queried. Benacerraf’s identification
problem is solved by viewing arithmetic as a theory describing structural relations instead of specific
96
97
(Hellman, 2005, pp. 540-541)
(Horsten, 2008) Section 4.2.
39
mathematical objects. The possible systems into which arithmetic can be reduced -such as systems
developed out of I and II described above- are viewed to be instantiations, or representations, of the
natural-number structure. The number 3 is an open place in this structure. For example, in system I,
{{{ Ø}}} only plays the role of this number, and does this equally good as { Ø, { Ø}, { Ø, { Ø}}} in system
II. The question whether it is the case that
, confuses two different structures. While ‘1’ and ‘3’
are places in the structure of the natural numbers,
is a set-theoretic notion.
Ante Rem Structuralism stands for the idea that the structures exist prior to the systems that possibly
instantiate them. So the notion of structure is the primitive concept and is only explicable in terms of
the notion ‘structure’ itself. The description of ‘structure’ as places that stand in relation to each
other, is dependent on the structure to which these relations belong. Some difficulties regarding the
explanation of the possibility to obtain knowledge of arithmetic stem from this circularity.
5.2.2 Epistemology
Learning arithmetic is learning to refer to ante rem structures by a familiarizing with the relevant
arithmetical vocabulary.98 But mastery of arithmetical vocabulary is not possible without reference to
a given ante rem structure. For example, in order to being able to get insight in the structure of the
natural numbers, we must learn the meaning of expressions as ‘the ordering of the natural numbers’.
But in order to make it possible that we understand these expressions, we must know them already
as objects of an ante rem structure.
Only structural aspects are relevant to the knowledge of the truth of arithmetical statements. For
being able to evaluate the truth of a statement, an abstract structure is postulated above the
particular concrete system. Although the ante rem structure is expressible in infinitely many concrete
ways, we must know how to evaluate and understand the arithmetical statements relative to the
structure beyond it, given to us.
Learning arithmetic according to this view, requires somehow an intuition of this a priori structure.
Since the ante rem structure is a primitive concept, it is presupposed when we are doing arithmetic.
An a priori intuition of it is thereby required in understanding arithmetic. It seems natural to hold
nature (inside and/or outside us) responsible for this.
The structure of arithmetic is spelled out in axioms. These structures are presupposed, existential,
abstract entities. In this sense Ante Rem Structuralism is a Platonist position with respect to the
ontology of the primitive concepts; the ante rem structures. This gives reason to worry about the
possible explanations for our epistemological access to arithmetic from this point of view, as
expressed in Benacerraf ‘s epistemological problem.
Benacerraf’s epistemology problem for Platonism, that questions how physical mathematicians can
access truths about an abstract realm of abstract objects, comes here to the question how the
knowing subject can access truths about places in ante rem structures. Shapiro answers this question
by first noting that these structures are abstracta and that these are not located anywhere. So there
is no need to give an account of a journey to a Platonic heaven. From a naturalist methodology,
assuming that there is arithmetical knowledge that is grounded in the mathematical practice itself,
98
(Hellman, 2005, p. 543)
40
the challenge can be adjusted to answering the question: What faculties do (or can) humans invoke
which results in justified beliefs about ante rem structures?
The most important postulated faculties that humans can invoke in order to gain knowledge about
ante rem structures that Shapiro appoints are the faculty of pattern recognition and the faculty of
projection.99 Pattern recognition is the faculty to abstract a pattern or structure from an observation
of systems of objects arranged in various ways. Pattern recognition accounts for knowledge of small
finite structures. Then the faculty of projection eventually provides a subject knowledge of the
natural number structure by the subject’s mentally arrangement of the small finite structures and its
realization that they themselves expose a structure.
By invoking these faculties Shapiro states that generalization is possible from particular knowledge
about a structure having a successor distinct from it to general knowledge about all structures having
a distinct successor. However it does not show why this generalization generates a belief that is
warranted. Shapiro bypasses this problem with an anti-foundationalist, holist epistemology for
mathematics where the belief gradually becomes knowledge when it has proven to be successful and
has taken a central role within mathematics, pure and applied. 100 In acquiring knowledge of
arithmetic the general successor principle could be presupposed or based on other presupposed
general principles for the generalization itself cannot only be justified on the basis of knowledge of
particular structures. Knowledge of such general principles could possibly be innate. But for this
account of knowledge of realist arithmetic also an empiricist epistemology is plausible.
Philip Kitcher101 gives an empiricist epistemology for knowledge of mathematics that is conceived as
knowledge about structures that are present in the physical reality. According to this ‘empiricist
structuralism’, perception is viewed as the process in which our interactions with physical objects
lead us to perceive the structures which they exemplify.
5.2.3 Truth
In this view, the structure provides universality and certainty to our knowledge of arithmetical truths
and makes this knowledge possible in the sense that the structure is prior to it. But nonetheless on a
more fundamental level the truths of general principles about ante rem structures are relative to the
practice of mathematics as a whole. These principles are known a posteriori, through the experience
with the practice of mathematics pure as well as applied and is thus in principle falsifiable by
experience. This is put forward in Shapiro’s naturalist and holist stance with respect to
epistemology.102
5.2.4 Knowledge Conditions
According to Shapiro’s Ante Rem Structuralism, learning arithmetic successfully results in knowledge
about ante rem structures for arithmetic is based on an objectively existing structure, or class of
structures. Natural numbers are places in the natural number structure, that is
99
(Shapiro, 2011, p.136)
(Shapiro, 2011)
101
(Kitcher, 1984)
102
(Shapiro, 2011)
100
41
...the form common to any countably infinite system of objects that has a certain
successor relation obeying certain principles.
103
Concerning ontology this is a Platonist position. However concerning epistemology there is a
difference.104 The knowledge conditions for this view depend on its epistemology -that can be
empiricist or rationalist/Platonist-. Since Platonist epistemology is been discussed above, here I will
focus on the knowledge conditions for an empiricist epistemology.
The knowledge conditions for an empiricist epistemology for knowing structures that have a Platonist
ontology are that the invoked faculty of pattern recognition and the faculty of projection must be
addressed somehow. These faculties enable knowledge about structures that go beyond the
contingent concrete reality. Sensory perception is indispensible for being able to gain knowledge of
structuralist arithmetic for the faculties of pattern recognition and projection build on concrete
systems of physical objects. This means that in arithmetic education the sensory perception of the
learner should be addressed.
Addressing the sensory perception of the learner in order to address these two faculties of the
learner can be done in so many different ways of course. It can be done by recalling the learner’s
perceptions of concrete physical systems in the real world (as it is found outside school; ‘the natural
objects’) by representing these in class, or by working with concrete materials in class that are
designed for the purpose of invoking the special faculties of pattern recognition and projection or by
bringing ‘natural objects’ in the class which can exemplify the structural relations of numbers.
5.3 Modal Structuralism
Another way to respond to Benacerraf’s epistemological problem is to reject ontological Platonism
and develop a nominalist theory. Since the rejection of references to abstract arithmetical objects
requires a revision of the language of classical arithmetic, nominalists have developed divergent
reconstructions of the arithmetical language. Below a sketch of such a translation is given with a
description of Chihara Constructability Theory.
5.3.1 Foundations
According to Modal Structuralism, arithmetic is about the possibility of structural relations. In these
terms also the notion ‘number’ is being expressed.105 Numbers as objects are eliminated, although
number-words can still be used instrumentally in arithmetical reasoning.106
The content of an arithmetical statement is given by every concrete system that satisfies its basic
principles –which are the Peano axioms-. In order to remove abstract entities from the language of
arithmetic, Peano-Arithmetic is reconstructed on the basis of logic, in line with the logicist
program.107 The terms ‘zero’, ‘number’ and ‘successor’ are not viewed as terms which have a
meaning we already know intuitively. Instead they stand for any term that verifies Peano’s five
103
(Shapiro, 2011, p.130)
Shapiro argues for this in (Shapiro, 2011).The difference between ontological Platonism and epistemological Platonism is
also described by Mark Steiner in (Steiner, 1975, pp. 109-137). There he also argues for an ontological Platonist picture of
mathematics that is not accessed through intuition in line with Gödel’s epistemological Platonism.
105
(Hellman, 2005, p. 556)
106
(Hellman, 2005, p. 557)
107
(Parsons, 2006, p. 50) and (Hellman, 2005, pp. 552-558)
104
42
axioms.108 In this way, arithmetical truths are reduced to logical truths. This view results in the
following truth condition for every arithmetical statement A: ‘Every concrete system that makes the
Peano axioms true, also makes A true’.
The problem with this truth condition is that, when there is no concrete instantiation of the Peano
axioms, logically, every statement will come out as being true. Therefore, the background
assumption is needed that there exist concrete physical systems that can serve as models for these
axioms. For if these concrete entities would not exist in the actual world, then the core idea of
nominalism, that mathematics should not refer to abstract entities, is not to be fulfilled. Hilbert has
expressed the concern that there cannot be enough concrete entities in the actual world to play the
role of the abstract entities in classical Platonist arithmetic.109 Putnam has shown that when
modalities are allowed, such an existential background assumption could be weakened to: it is
possible that there exists a concrete physical system that can serve as a model for the basic principles
of an arithmetical theory, (such as Peano arithmetic). The result, if we allow possible worlds, is that
arithmetical sentences no longer depend on physical assumptions about the actual world.110
Chihara has applied this nominalist view on arithmetic in his Constructability Theory.111 The
Constructability Theory deals with open sentences. It tells what open sentences are constructible and
how they can satisfy another open sentence. It is formalized in a standard language of modal logic
but has in addition ‘constructibility quantifiers’, by means of which one can construct a sentence like:
‘It is possible to construct an open sentence A such that A satisfies B’. The formal meaning of
‘possible’ in the expressions of this theory is given by the set of axioms of a modal logic: S5.
Informally, ‘it is possible that P’ means:
112
...the world could be such that, would it be this way, P would be the case.
By means of the constructability quantifiers, open-sentence tokens are constructed. These consist of
particular marks on paper, in a particular time and place. To say that an open-sentence of some sort
is constructible does only presuppose or imply that this physical open-sentence token actually exists.
It does not presupposes the existence of its reference.
By means of the Constructability Theory, Chihara argues against the indispensability argument -an
argument that commits us to the existence of abstract arithmetical entities-. The indispensability
argument expressed by Quine and Putnam comes to this: in the mathematical language one
quantifies over mathematical entities. This language is indispensible for science. Science is the only
judge for the justification of the acceptance of ontological assumptions. This commits us to accepting
the existence of mathematical entities.113 Chihara shows that inferences drawn in (natural) science
can be sound (meaning: if the premises are all true, the conclusion must also be true), even if the
arithmetical theorems used in the inferences are not true (both, construed literally and platonically).
108
(Hellman, 2005, p. 551)
(Horsten, 2008) Section 4.4
110
Parsons has mentioned a problem for this stance by noting that it is hard to believe that there could be physical worlds
that contain too many collections of entities to have a cardinal number (Horsten, 2008) Section 4.4. If this is required, also
this background assumption could be questionable.
111
(Chihara, 2005, pp. 499-505)
112
(Chihara, 2005, p. 500)
113
The indispensability argument will be discussed in the next chapter on Empiricism.
109
43
So the usefulness of arithmetic for natural science does not commit us in any way to a belief in the
truths of this arithmetical language. Here follows an example.114
Consider the following conclusion ‘There are twelve coins on table A at time t’ from the following five
premises:
1) There are five dimes on table A at time t.
2) There are seven quarters on table A at time t.
3) A coin is on table A if and only if it is either a dime or a quarter.
4) Nothing is both a dime on table A at time t and a quarter on table A at time t.
5) 5 + 7 = 12.
For each of the premises as well as for the conclusion there is a corresponding sentence of the
Constructability Theory (the c-version of the sentence) such that it holds for all these sentences that
they are true if and only if their c-version is true. Within the Constructability Theory it can be shown
that the c-version of the presented inference is sound (meaning here that the inference is valid; that
the conclusion is logically entailed by the premises). Which shows that the given inference is sound,
without assuming that premise 5 is literally or platonically true.
The c-versions of the sentences are not viewed to express the actual meaning of the sentences. So
the argument is made without reference to meaning at all. This is a purely formal view on arithmetic
in line with formalism.
5.3.2 Epistemology
The motivation for a nominalist program is based on a concern about the possibility of an
explanation of how we obtain mathematical knowledge. The reason for not taking the language of
mathematics at face value is that it is not possible to ascertain the presupposed ontological status of
the abstract objects. Chihara115 describes his nominalist program as the search for an account of
arithmetic that is compatible with what science tells us about how we obtain knowledge and how we
in fact learn and develop arithmetic. If the epistemology of arithmetic is the starting point of viewing
the nature of arithmetic, then the thought expressed by the indispensability argument, that scientific
theories require a belief in the existence of abstract arithmetical objects seems incompatible with
what we know about how we obtain knowledge. In this view, a belief in arithmetical objects is quite
fantastic.
In contrast to Ante Rem structuralism, there is not one abstract presupposed structure above and
beyond the concrete structures of arithmetic.116 Structures exist only relative to the possible
concrete systems that instantiate them. Thus modal structuralism can get around Benacerraf’s
epistemological problem. Still the question remains how we get epistemological access to the
possibility of a structure.117 The background assumption that it is possible that there exists a concrete
system that can serve as a model for number theory is not justified by a consistency proof for a
114
This example is almost literally taken from (Chihara, 2005, p. 508).
(Chihara, 2005, p. 496)
116
(Horsten, 2008) Section 4.4.
117
(Hellman, 2005, p. 556)
115
44
standard model for that theory.118 This aside, arithmetic is accessible through every system that
satisfies the Peano axioms for there is nothing to know behind these axioms. All structures that can
serve as a model for the Peano axioms can, in principle, be used equally well in natural science. And
the physical marks on paper, which are basically the arithmetical instrument, can be known by
ordinary sense perception.
5.3.3 Truth
According to nominalist structuralism, only structural aspects relative to a system, are relevant to the
truth of mathematical statements and no abstract structure is postulated above and beyond
concrete mathematical systems.119 With regard to their meaning, arithmetical truths cannot be
absolute. Furthermore nominalist truth has the same characteristics as formalist truth. The formal
truths could be known a priori and have nothing to do with empirical facts or Platonist existence. The
a priori truths are given with the system and are in this respect necessary and objective.
5.3.4 Knowledge Conditions
According to nominalists versions of structuralism there are no arithmetical objects. In Chihara’s
constructability theory the constructed open-sentence tokens (that are the physical marks on paper)
have no meaning, they are only instruments in a mechanized process. In this respect nominalist
theories require the same knowledge conditions as formalism. The conditions for gaining a priori
knowledge of conventionalist truths about formal systems are given in the sections on formalism and
logical positivism. These are the ability to learn a syntax and follow valid inferences and a normative
theory for warranting consistency-proofs which all requires an intuition of combinatorial properties
of finite, discrete and concretely representable objects and constructive methods. Since here the
possibility of concrete physical systems serves as a model for arithmetic one could turn to empiricist
epistemology instead of having to seek refuge to realist intuition.
118
119
(Hellman, 2005, p. 556)
(Horsten, 2008) Section 4.4.
45
Chapter 6: Empiricism
Roughly, according to empiricism arithmetic is a theory about the empirical world. This point of view
provides a way out of the ontological gap between pure and applied arithmetic that exists if
arithmetic is conceived as a purely formal procedure. Empiricism answers the question how pure
arithmetic is so useful in its empirical applications with the claim that arithmetic is basically an
empirical theory. The problem with this answer is that the widely supposed necessity of arithmetical
statements is not evident.
The empiricist doctrine states that no informative proposition is a priori.120 Either arithmetical
knowledge is a posteriori, synthetic and fallible, or it is a priori, analytic and infallible. This implies
that the truths of arithmetic must be either synthetic and contingent or necessary and without
factual content.121 By claiming that arithmetic is ultimately an empirical theory the empiricist must
account for the widely held conviction that arithmetical truths are necessities.
The forms of empiricism that will be discussed here are Mill’s empiricism and an elaboration thereof
by Kitcher, and Naturalism as expressed by Quine, Putnam and Maddy. According to Mill, arithmetic
is a theory about the laws of nature.122 The truths are known a posteriori. Naturalism is the view that
the reality is to be determined and described within science itself, and not in a philosophy beyond
science.123 Quine’s holist view takes arithmetic as an inseparable, integral part of natural sciences
and thus ultimately as empirical. While for Maddy naturalism implies that the answers on the nature
of arithmetic are to be found in studies on the practice of mathematics and arithmetic as a science in
history. The answers thus found are ultimately based on historical studies and empirical research on
the mathematical practice.
Since in Peano-Arithmetic the variables range over abstract mathematical objects, one who does not
take these abstract objects as existing must somehow relate her position to this formulation of
arithmetic. One way to go is default Platonism; take this formulation at face value and adopt a belief
concerning ontology only in so far this commitment to abstract arithmetical entities is indispensible
to our best scientific theories. This is in line with Quines’s position. Another way a non-Platonist can
relate to this formulation is rewriting the language. Besides nominalist versions there are also
empiricist formulations of arithmetic adopting a physical paradigm. Mill Arithmetic is an empiricist
reformulation of Peano-Arithmetic. It is not a restriction.
Philip Kitcher discusses ‘Mill Arithmetic’, that is a reformulation of a first order version of additive
Peano Arithmetic in a first order language.124 Mill takes arithmetic as the activity of performing
operations. In this language the primitive notions are: one-operation (Ux: ‘x is a one-operation’), one
operation of being a successor to another (Sxy: ‘x is a successor operation of y’), one operation being
an addition on other operations (Axyz: ‘x is an addition on y and z’), and one operation being the
matchability of operations (Mxy: ‘x and y are matchable’). These notions are comprehensible as
physical activities. To perform a one-operation is to perform a segregative operation in which one
120
(Skorupski, 2005, p. 51)
121
Quine argues against this dogma that there would be a cleavage between analytic – and synthetic
statements in (Quine, 1951).
122
(Skorupski, 2005, pp. 53-55)
(Maddy, 2005, p. 437)
124
(Kitcher, 1984, pp. 112-122)
123
47
object is segregated. The performance of a successor operation of another operation is the
performance of this last operation (by segregating all of the objects there segregated) together with
one singe extra object. To perform an addition on two operations is to combine the objects collected
in two segregative operations on distinct objects. Matchability plays the role of identity. Two
segregative operations are matchable if the objects they segregate can be made to correspond with
one another.
6.1 Foundations
Mill gives a nominalist analysis of arithmetic, which means that the arithmetical objects are not being
construed as abstract - but as concrete entities.125 In his conception, arithmetic is about possible
collections in which the act of collecting is viewed to be concrete and not the things collected.126 In
this concrete form of realism, arithmetic describes the general laws of nature, which are known by a
posteriori methods.127
A full reformulation of Peano-Arithemtic with the primitive notions of Mill Arithmetic is only possible
if principles are adopted which ensure the existence of enough operations. Infinitely many objects
are required to be postulated and since there are not enough physical objects for that, abstract
objects must be called into being. Mill Arithmetic turns out to walk into the same difficulties as
nominalist programs. Thus Kitcher shows the situation:128 Or our knowledge of arithmetic must be
only a fraction from what we had thought that it was (if we view it unjustified to adopt the extra
principles that postulates infinitely many objects), or we must give up the physical paradigm for
understanding the primitive notions of Mill Arithmetic (that enabled us to give an account of the
usefulness of arithmetic in the physical world).
Kitcher proposes to adjust the physical paradigm and conceive the principles of Mill Arithmetic as
implicit definitions of an ideal agent. The capacities of this ideal agent are found by abstracting from
incidental limitations. Different from Brouwer’s ideal mathematician this ideal agent is able to
perform operations that satisfy all principles of Mill Arithmetic that is not a restriction of PeanoArithmetic.
Concerning the ontology of arithmetic all positions to be discussed here share a naturalist starting
point. Quine abandons the project of searching for the foundations of mathematics from a
philosophical point of view; from outside science itself.129 He combines empiricism with holism,
expressed by the doctrine that no claim of theoretical science can be evaluated in isolation.130 For
him, arithmetical theories are part of scientific theories. They are supported by observation in an
indirect way; by their being a central part of natural sciences that is confirmed by experience. In this
way we have at the same time empirical confirmation of arithmetical theories.131
125
(Skorupski, 2005, pp. 64-65)
(Skorupski, 2005, pp. 63-64)
127
(Skorupski, 2005, pp. 53-55)
128
(Kitcher, 1984, pp. 112-122)
129
(Maddy, 2005, p.438)
130
(Resnik, 2005, pp. 413-414)
131
(Horsten, 2008) Section 3.2.
126
48
Quine views that our best scientific theories cover our best account of what exists and how we know
it. When this view is applied to arithmetic then, according to Quine, we appear to be committed to
sets as abstract entities. This stance towards Platonist ontology is called ‘Default Platonism’.132
Default Platonism refers to an uncritical stance towards the mathematical practice, taking the
language of mathematics (implying the existence of mathematical objects), at face value.133 In
general this includes the acceptance of the law of the excluded middle, impredicative definitions and
references to abstract objects forming uncountable and larger totalities.
Quine states his pragmatic attitude towards ontological beliefs in On what there is.134 He claims that
our acceptance of ontology is similar to our acceptance of a scientific theory: we adopt the simples
conceptual scheme. The Platonist ontology is a myth but a good and useful one since it simplifies our
account of physics.135 Mathematics is an integral part of this higher myth. For Quine, the task is then
to see
... how, or to what degree, natural science may be rendered independent of Platonist
mathematics.
136
Quine holds that a reduction of number theory to set theory shows that we can and must do without
numbers. But we cannot do without sets. We need sets anyhow for other mathematical purposes.
Numbers are sets to the extent that they can be reduced to them without impoverishment of
number theory.137 Instead of having to postulate sets and numbers, we can just postulate sets.
Although with this reduction Quine also pleads for the adoption of the view that sets exist in the
universe as abstract entities, this does not lead to an absolute Platonist belief in the existence of sets
for the inquiry into its foundations must be ultimately an inquiry into science itself. Concerning
ontology, we will find there the answer that confirms our belief in arithmetical objects. This thought
is known as ‘the indispensability argument’.
The indispensability argument,138 expressed by Quine, for arithmetical realism, comes to this:
Arithmetic is indispensible to our best scientific theories. It is an integral and important part of our
best scientific theorizing about the world. We can express these theories using the language of set
theory and probably we cannot do without the use of this language. Together with a naturalist
position, this takes us to the acceptance of sets in our philosophical ontology.
Also Putnam notes that there is something wrong with philosophical questions about the nature of
mathematical objects and our knowledge of them, from outside science itself. In Mathematics
without foundations139 he notes that the same mathematical facts are described in different
mathematical descriptions, presenting divergent, and sometimes incompatible, pictures of reality. He
notes that the fact that the primitive terms of the one theory is definable in terms of the other, does
132
Maddy objects to Quines idea that the indispensability argument provides a satisfactory approach to the ontology of
mathematics, on the basis of scientific practice in: (Maddy, 1992). So default Platonism and the indispensability argument
described here, are solely for the account of Quine.
133
(Parsons, 2006, p. 49)
134
(Quine, 1948, pp. 187-192)
135
(Quine, 1948, p. 192)
136
(Quine, 1948, p. 192)
137
(Quine, 1964)
138
This summary of the indispensability argument is based on (Horsten, 2008) Section 3.2.
139
(Putnam, 1983)
49
not make one of these theories fundamental. These divergent pictures are to be used to explain
something and not to prove the existence of an absolute ontology.
While Quine searches the answers to questions concerning the existence of arithmetical entities and
how we obtain knowledge of them inside our best scientific theories,140 for Maddy answers to these
questions are to be found in the mathematical practice itself.141 In Maddy’s naturalistic stance the
focus is on the mathematical practice. She studies how mathematics functions in applications as well
as the nature of pure mathematics, from the perspective of the historical development of ideas on
how mathematics relates to the world.142 She concludes that we are not in fact uncovering an
underlying mathematical structure realised in the world. Instead we construct abstract models and
try to make true assertions about the way in which they do and do not correspond to the physical
world. In elementary arithmetic there are cases (such as the case of ‘2+1=3’), where this
correspondence is very strong, but these cases are exceptional. In most cases of advanced arithmetic
and other areas of mathematics, the correspondence is more complex and in other cases of these
fields, we do not yet understand the correspondence at all.143 We do not have to search for ways to
bridge the ontological gap between pure – and applied arithmetic in cases where pure arithmetic
does not seem to correspond to the empirical world at all. In cases where this correspondence is very
strong there is no ontological gap.
6.2 Epistemology
Mill argues that arithmetical knowledge -known by a posteriori methods- is synthetic by making a
distinction between the connotation and the denotation of a term.144 Both are relevant for the
meaning of it. The statement ‘2+1=3’ is synthetic because ‘2+1’ and ‘3’ has divergent connotations,
while their denotation is the same. A second argument is made by his remark that deductive
reasoning that is analytic, cannot produce new knowledge. In that case there would be no reason for
making arithmetical inferences, since the conclusion would be already present in the axioms. So by
the fact that arithmetic is viewed as producing new knowledge, its statements must be synthetic.
Also according to Quine arithmetic is viewed to be known by a posteriori methods since it is an
integral part of the empirical sciences.145 Gaining arithmetical knowledge is ultimately: discovering
the physical world.
At the beginning of the 1990s Maddy sees a new consensus in the way philosophers are responding
to Benacerraf’s epistemological problem.146 They find forms of ontological thinking that preserve the
use of standard arithmetic without a commitment to an inaccessible Platonist realm so that
elementary arithmetical knowledge can be viewed to be gained by perceptual experience. One
example of a contemporary position that complies with this consensus is Social Constructivism, to be
discussed in the following chapter.
140
(Horsten, 2008) Section 3.2.
Maddy is said to try to save Platonism from epistemological difficulties (Kitcher, 1984, p. 148 in note). By grounding it
into the practice it is to be inquired empirically. Hereby she does not view it as an absolute timeless stable givenness as it is
for a Platonist. See also (Horsten, 2008) Section 3.2.
142
(Maddy, 2008)
143
(Maddy, 2008, p.33)
144
(Skorupski, 2005, p. 60)
145
(Skorupski, 2005, p. 55)
146
(Maddy, 1991)
141
50
In ‘the nature of mathematical knowledge’ Philip Kitcher147 develops an empiricist epistemology for
Mill arithmetic. He argues against apriorism and accounts for mathematical knowledge that is
warranted, in origin, by ordinary sense perception. The growth of mathematical knowledge is
proposed to be accounted for with the help of a study on the growth of mathematics -and science in
general- in history.
Kitcher states his realist stance with the words:
‘...we might consider arithmetic to be true in virtue not of what we can do to the world
148
but rather of what the world will let us do to it.’
According to him arithmetic describes those structural features of the world in virtue of which we are
able to segregate and combine objects. He argues for the view that arithmetic is an activity of
collecting instead of a theory about collections.149
Mankind has become to know the meanings of set, number and addition by combining and
segregating physical objects. Our initial arithmetical knowledge springs from our perception of
performances of collecting. The ‘proto-mathematical knowledge’ thus gained is extended by a
generalization of past experiences that projects this knowledge onto possible (ideal) performances of
collecting of an ideal agent.
A perceptual basis for this realist view on arithmetical knowledge could be supported by a theory of
perception according to which sensory information is rich and which underlines that what is primarily
perceived are the affordances of things for the perceiving subject. Our environment could be seen to
afford us to engage in collective operations and this provides us perceptual knowledge of basic
arithmetical truths. Kitcher suggest that Gibson’s ecological realism could account for such a theory
of perception and proposes that mathematics could be an ideal science of universal affordances. 150
6.3 Truth
From an empirical point of view our knowledge of arithmetic comes for an important part from
sensory experience. If mathematics is viewed as an integral part of empirical science, as expressed by
Quine, and if its truths are judged relative to its role in the empirical sciences, then even the
necessity of the truths of elementary arithmetic seems less obvious.151
According to Mill’s radical empiricism, arithmetical statements are true when they are made to
correspond with the observable facts (meaning: what we are or could be aware of through our
senses).152 The difficulty with this position is stressed by Kant’s point that experience can tell us what
is, but not that it is necessarily so.153 So the arithmetical rules, construed as rules that govern the
behaviour of the physical world, as perceived by our senses, can only be contingent. This stresses an
147
(Kitcher, 1984)
(Kitcher, 1984, p. 108)
149
(Kitcher, 1984, p. 110)
150
(Kitcher, 1984, p. 108) and introduction.
151
(Horsten, 2008) Section 3.2.
152
(Skorupski, 2005, p. 53)
153
(Skorupski, 2005, p. 61)
148
51
inability to discovering laws, in the sense of necessities. Mill’s solution is the rejection of the
necessary-contingent distinction.154 Taking necessity to be natural necessity.
Since for Quine the questions on the ontology and epistemology of arithmetic are questions of
natural science itself,155 it can never claim more certainty than that of natural science. Also according
to Putnam arithmetic might be fundamentally wrong in the sense that the self-evident axioms might
not be evident or even false. So we should accept Peano-Arithmetic not on its ‘absolutely certain
foundations’ but instead on its role as being the basis of a successful scientific system, including
important empirical applications. If someone would propose an alternative that could do the same or
more, there would be no reason not to adopt it.
6.4 Knowledge Conditions
Since arithmetic is, according to this view, basically given with the contingent reality, the actual
appearance of it in the physical world is what it is about. Arithmetic is inextricably bound up with full
reality as it is perceived by us. To know this arithmetic thus requires reference to this reality. An
educational method should somehow accommodate reality, as rich as possible, in some form. This
could be done by enabling children to gain experiences (especially with activities of collecting and
segregating) in their physical environment or by representing reality as rich as possible by
photographs, pictures, stories or materials such as models or maquettes. In short: realistic, rich
contexts are indispensible for learning empiricist arithmetic. Actually these are not contexts wherein
the core, bare arithmetical problem is hidden. The appearance of arithmetic for us in this rich context
is itself arithmetic.
Kitcher notes that this explanation of empiricist knowledge of arithmetic could also accounts for the
empirical genesis of arithmetic in history solely. It does not have to account also for a description of
how children acquire arithmetical knowledge. Current arithmetical knowledge could also be
explained by a transmission of that knowledge from society to an individual and from one society to
its successor.156 According to this picture what is required for gaining knowledge of arithmetic is
primarily authority from mathematical practice. This picture of what is required for learning to know
arithmetic results also from Maddy’s view. If arithmetic is basically an historical phenomenon; a
human practice that develops over time, a clear presence of the current practice for the learner is
required for being able to learn to know it. This picture is in line with the Social Constructivist
conception of the conditions required for a transfer of arithmetical knowledge to be discussed
below.
154
(Skorupski, 2005, p. 61)
(Maddy, 2005, p. 439)
156
Actually this is how Kitcher pictures the acquisition of arithmetic in children (Kitcher, 1984).
155
52
Chapter 7: Social Constructivism
All mathematical knowledge is
embodied in texts, personal
knowledge and contexts of use, and
objective mathematical knowledge is
to be found socially in the
interrelations and interactions of these
texts and persons within the
157
institution of mathematics’.
Social Constructivism as expressed by Paul Ernest, views the nature of arithmetic as linguistic; it is
embedded in the world of human social interaction.158 Arithmetical knowledge rests on socially
situated linguistic practices, rules and conventions and the objectivity of it is inter-subjective; it is
publicly shared and accepted. This means that all mathematical development takes place within a
social and cultural context.
As Maddy noticed at the beginning of the 1990s, the foundational studies in the philosophy of
mathematics has shifted its focus .159 The prescriptive focus of the philosophy of mathematics on ‘the
foundations’ of it has been replaced by a descriptive focus on its ‘nature’ as an existing phenomenon:
‘the actual mathematical practice’. This includes a shift in themes, from ontology and foundations of
mathematical knowledge and truth to themes such as methodology, history and fallibilist
epistemology. This new field of the philosophy of mathematics requires an interdisciplinary approach
including studies in history, education, sociology, cognition and psychology.160 Social Constructivism,
as expressed by Ernest, is situated within this new field.
Ernest proposes a radical reconceptualization of the philosophy of mathematics that accounts for
mathematics as an epistemological phenomenon that is fallible161 and adds to this an account for
mathematics as an historical phenomenon, including social and cultural aspects.162 According to
Ernest a philosophy of mathematics should not only account for the traditional themes concerning
arithmetical objects and knowledge but also for mathematical theories (pertaining to the philosophy
of science), applications of arithmetic and the mathematical practice (including social aspects), and
the learning of arithmetic (including aspects of cognition).163 His ‘social constructivism’ is an
elaboration of such a philosophy of mathematics.
7.1 Foundations
Social Constructivism is a nominalist position with respect to ontology but not so strict that it
restricts itself to nominalist languages. It is supposed to support and recognise existing mathematical
practices.164
157
(Ernest, 1997, p. 132)
(Ernest, 1997, p.126)
159
(Maddy, 1991)
160
See e.g. the preface of (Löwe & Müller (Eds.), 2010a) where the purpose of the research network ‘Philosophy of
Mathematics: Sociological Aspects and Mathematical Practice’ (PhiMSAMP) is expressed.
161
On the basis of Lakatos’ view on the ‘quasi-empirical’ methodology of mathematics (Ernest, 1997).
162
Hereby leaving Lakatos (Ernest, 1997).
163
(Ernest, 1997, pp. 124-125)
164
(Ernest, 2004, pp. 9-22)
158
53
The objects of arithmetic are signs. But not signs in the formalist sense of being empty symbols. The
signs have always meaning, albeit complex and difficult to specify this meaning exactly. The meaning
of the signs of arithmetic are to be found within the arithmetical practice since only there they are
meaningful. The meanings of signs are further signs. But in addition what constitute the meaning of a
sign are sign related activities. These are idealized human actions on signs. The signs are part of an
intersubjective cultural realm. This realm is wider than the understanding and the perception of one
individual, but it does not transcends the knowledge and practice of humankind as a whole. So the
arithmetical objects do not exist in an extra-human reality. For example in Peano-Arithmetic, the
number ‘3’ is defined explicitly by ‘S2’ (the successor of ‘2’). This asserts both a static relationship and
an operation (namely applying the successor operation to 2). Furthermore ‘3’ connotes the act of
establishing a one-to-one correspondence with a prototypical set with cardinality 3, and also it
connotes the act of enumerating (ordinal) a triple set. Both connotations presupposes some
elements of ‘threeness’. Arithmetical objects are arithmetical processes as well as their end products.
7.2 Epistemology
With respect to the foundations of arithmetical knowledge, Social Constructivism is a conventionalist
position.165 based on the Late Wittgenstein.166 In Philosophical Investigations Wittgenstein shows
how the a priori truths of arithmetical statements are implicit within our practice of the use of the
arithmetical vocabulary.167 In his discussion on what it is to follow a rule of arithmetic, it turns out to
be the arithmetical practice that in the end determines which acts are intended in the statement of a
rule and which are not. According to this view, the a priority of arithmetic rests to a significant extent
on an agreement in social customs.
In this view conventionalism -meaning that arithmetical knowledge rests on human conventions,
choices and historical practices- does not make arithmetical knowledge analytically a priori, as it is
the case for logical positivism. Here arithmetical truths are known by means of experiences in the
social world and cannot be known prior to these social practices. Arithmetical knowledge and truth is
thus without absolute validity. It presents a set of choices or possibilities out of a number of
imaginary alternatives. The stability of arithmetical truths stems from the fact that the conventions in
mathematics are not conscious or arbitrary individual decisions but they are reflections of historical
practices that are developed and laid down for very good reasons.
Arithmetic education has a special role in the ongoing process of transmission of arithmetical
knowledge. Here, amongst other places, this knowledge is ‘recontextualized’.168 This is understood as
a realization of the meaning or the content of a text within a different text. Any member of the
mathematical community has had education. Arithmetical knowledge, consisting in their practice, is
recontextualized into the domain of education in the form of physical texts and materials of learning
methods. In the learning conversations using these materials, arithmetical knowledge is
communicated to learners. Through the performance of the personal arithmetical knowledge of the
learner, it can be confronted with corrections in order to continue the learning process. In the form
of a performance in an assessment, which is also a social process, a learner can become certified and
may participate in a new role in a variety of domains in which (new) arithmetical knowledge is
165
(Ernest, 2004, pp. 22-31)
Besides its roots in the work of Lakatos (Ernest, 1997, p. 125).
167
(Parsons, 2006, p. 34)
168
(Ernest, 1997, pp.129-130)
166
54
constituted within an interpersonal negotiation. And in this new role, knowledge of arithmetic is
again communicated in the domain of education. In this process arithmetical knowledge is thus
‘recontextualized’ and is objectified by the publicly accepted texts which embody arithmetical
content.
How is arithmetical knowledge that is socially constructed qualified? All arithmetical knowledge is
the result of different forms of accepted practices, agreements and decisions. All of this knowledge
can thus be questioned and reconsidered. The necessity of arithmetical knowledge that is
experienced widely, is explained by Wittgenstein’s theory of language games. Here, necessity arises
from human agreement in following a rule that is stipulated in a language game. So this necessity
applies only to elements of arithmetical knowledge; practices within a larger whole. As a whole,
arithmetical knowledge is contingently true. This is entailed by the conventionalist starting point. But
since the convention is here not viewed to be a personal decision but instead anchored in historically
determined social practices, this view can explain the great stability of the discipline as well as
historical changes.
7.3 Truth
The arithmetical truths are based on social agreement. This makes knowledge of them at the same
time stable and objective, and fallible and a posteriori. Arithmetical knowledge is dependent on
experiences of the social world, thus a posteriori, but also justified on the basis of convention, albeit
a convention that is sensible to a particular context and situation. Since this convention is not a timeand placeless fixed and static given certainty, the truths are fallible. The epistemological method is so
constituted that the truths can always be refuted. This fallibilist character does not mean that the
truths of arithmetic are equated with consensual belief, since they are subjected to criteria for
acceptance; to proof. It means that the criteria for the acceptance of arithmetical truths do not exist
independently of humankind.
7.4 Knowledge Conditions
If arithmetic is practice-based, how can it be acquired by individuals that are unfamiliar with this
practice? Ernest’s constructivist view applies to arithmetic in general as well as to the individual
acquisition of arithmetical knowledge.169 This constructivist perspective could be supported by a
Piagetian theory of knowledge construction where children are seen to reconstruct knowledge as a
result of their interactions within their physical, social and cultural environment. But a Vygotskian
constructivist perspective is more natural for according to Vygotsky’s social theory of mind learning
precedes development in the sense that the mental functions needed in doing and understanding
arithmetic appear first in social interaction and second psychologically. This Vygotskian constructivist
perspective is adopted by Ernest. 170
According to the Vygotskian perspective the social aspects of classroom interaction are crucial for
learning arithmetic because arithmetic is basically conventional in the sense that it only exists in a
practice of use. Learning happens on the basis of experiences in these interactions. These
experiences improve the fit between the use intended by the community of practitioners (e.g. the
teacher) and the perceived outcomes by the learner. On the basis of knowledge gained in these
169
170
(Ernest, 1994) Chapter 6
(Ernest, 1994) Chapter 6
55
experiences the learner construct new knowledge concepts and hypotheses to be tested again in the
physical and social world. Understanding is thus a process rather than a state.
Arithmetical knowledge of individuals is generated through the educators who pass on knowledge of
the community as it has arisen in history. To become part of the community is enabled by training.
When the practice provides the basis for the meaning and truth of arithmetic, also warranting
arithmetical beliefs is bound to this practice. A prerequisite to be able to learn arithmetic is the
presence of this practice for the pupil. So people that are a visible example by living this practice,
make it possible for the learner to participate. Individual subjects and their social contexts are
interconnected in the practice that constitutes arithmetic. Personal arithmetical knowledge exists
only in this context. Understanding cannot be a state of mind unless one views the mind as extended
into the physical and social world -understands it as conversational-.
In summary, the knowledge conditions for this view are: An actual living practice/ a shared (social)
context. This includes the accepted rhetorical forms of arithmetical language, social relationships and
roles and forms of communication and accepted ways of working with material resources and
symbolic representations. It also asks for a teacher role that is authoritarian is the sense that is must
be responsible for correcting the learner’s knowledge productions and for warranting learner’s
knowledge.
56
Chapter 8: Embodied View
This last view on the nature of arithmetic is an internalist position; it views arithmetic to be ‘bodybased’. The influential book, on which this view is largely based, Where mathematics comes from:
How the embodied mind brings mathematics into being171 by Lakoff and Núñez, presents an
understanding of mathematical concepts as embodied notions. Their approach to the questions
concerning the nature of arithmetic stems from a perspective of cognitive science.172 They claim, on
ground of the state of our scientific knowledge, that arithmetic is purely brain-and-mind-based, and
cannot be something beyond this.173 All realist perspectives on the ontology of arithmetic are hereby
dismissed as being an unscientific religion.174
In this view arithmetic is considered as a cognitive content. Lakoff and Núñez state this to be a
special subsystem of the human conceptual system. Arithmetical cognition is special; it is not to be
equated with ordinary cognition.175 This special status of arithmetical cognition is supported with
reference to brain studies, indicating the parts of the brain that play a crucial role in the mental
representation of numbers as quantities.176 Arithmetic as cognitive content is viewed to be special in
the following ways. It is:
...precise, consistent, stable across time and communities, understandable across
cultures, symbolizable, calculable, generalizable and effective as general tool for
description, explanation, and prediction in a vast number of everyday activities,...
177
The subject of discussion is thus a naturally developed, universal cognitive content, which could be
symbolized in e.g. a description of Peano-Arithmetic.
This cognitive perspective alternates the order of the questions previously discussed. First our
cognition of arithmetic is studied. This will largely be on the basis of the cognitivist perspective of
Lakoff and Núñez and on the linguist perspective of Johnson178. Then, on the ground of the answers
on questions concerning our capabilities to have knowledge of arithmetic, a view on the nature of
arithmetic is provided. So here first a description of the epistemology of arithmetic and then of the
ontological picture.
8.1 Epistemology
As their title states, according to Lakoff and Núñez: ‘the embodied mind brings mathematics into
being’. This mind is viewed to be brain-based and arithmetic is viewed to be based on innate
knowledge of basic arithmetical concepts and a capacity to form metaphors out of everyday
experiences, by means of which the basic concepts are extended to the fully developed number
theory.179 Here, first the innate basis of arithmetic is described and then its development by means of
the so called ‘grounding metaphors’ is discussed.
171
(Lakoff & Núñez, 2000)
(Lakoff & Núñez, 2000, pp. 1-11)
173
(Lakoff & Núñez, 2000, pp. 1-11; 337-379)
174
(Lakoff & Núñez, 2000, pp. 2-3)
175
(Lakoff & Núñez, 2000, pp. 29-30) p.29-30
176
(Lakoff & Núñez, 2000, p. 29)
177
(Lakoff & Núñez, 2000, p. 50)
178
(Johnson, 1987)
179
(Lakoff & Núñez, 2000, pp. 15-103)
172
57
8.1.1 Innate Arithmetic
Brain-based arithmetic is partly motivated by the brain’s innate knowledge of arithmetic.180 An
innate-based answer to questions on the nature of arithmetical knowledge is supported by
developmental psychologist studies. While Piaget in 1952 thought that abstract reasoning skills are
psychologically primitive for understanding number, now, developmental psychologist studies on
numerical competence of non-linguistic creatures (young infants and animals)present the fact that
they have some preconceptual understanding of number, which develops independently of other
abstract reasoning skills.181
Pioneering empirical experiments of Karin Wynn (1992) tested a.o. the ability of five-moths-olds to
perform addition on small quantities.182 In one of the experiments the babies watched a doll on a
display stage. Then a screen temporarily hided it from their view. Thereafter a hand entered the
display stage with another identical looking doll and puts it behind the screen. Then the screen is
lowered to reveal either two dolls or one doll. The babies looked significant longer to the outcome
‘one doll’. Wynn accounted for this result with the idea that infants possess the ability to reason
about number and perform arithmetical operations.183 This is just one out of many experiments
indicating that infants (and nonhuman animals) are gifted with an unlearned capacity, prior to
language, to perform simple arithmetical operations.184
One could comment on the conclusion that this experiment shows that our arithmetical knowledge is
grounded in an innate arithmetic. One could question whether Wynn’s experiment shows that babies
know that ‘1+1=2’ and ‘1+1≠1’. An interesting remark on this result is made by Uller et al. (1999),
who argued that the infants represent the added objects not as integers but as ‘object-files’. An
object-file of ‘two’ is represented as follows:
Meaning: there is an
entity and there is another distinct entity and both are objects, and there is no other object.185 This
formulation shows a resemblance with the logicist idea of ‘number as a concept’, where ‘two’ means:
‘there are at least two distinct entities that fall under the same attribute (in this formulation the
attribute ‘object’) and there are not at least three entities that fall under this attribute’. Also it is
argued that the symbolic notation itself influences arithmetical cognition.186 Furthermore one could
wonder whether there is an overlap between this intuitive arithmetic and the culturally elaborated
formal arithmetic as practiced today in more advanced arithmetic. Despite these comments these
results could make a case for a preconceptual, innate basis of numerical cognition.
180
(Lakoff & Núñez, 2000, pp. 15-26)
(De Cruz, Neth, Schlimm, 2010)
182
In these experiments she relied on the ‘looking time procedure’ (this procedure enables the testing of cognitive abilities
with a minimum of task demands) and the ‘violation of expectation paradigm’ (based on the propensity of humans to look
longer at unexpected predictions on how objects behave).
183
(De Cruz, Neth, Schlimm, 2010)
184
In ‘Where mathematics comes from’ Lakoff and Núñez summarize literature that presents proof of innate arithmetical
abilities of humans in their first chapter (Lakoff & Núñez, 2000, pp. 15-26)
185
(De Cruz, Neth, Schlimm, 2010, p.71)
186
This point is made by (De Cruz, Neth, Schlimm, 2010, p. 73).
181
58
In any case, the innate arithmetic Lakoff and Núñez base their view on, includes the following two
capacities: 1. The capacity to instantly recognizing small numbers of items, and 2. The capacity to add
and subtract small numbers.187
8.1.2 Conceptual Metaphors
The cognitive mechanisms that account for the development of this innate arithmetic into abstract
arithmetic are: ‘conceptual metaphor’ and ‘conceptual blending’.188 A metaphor is a mapping from
one domain to another that preserves inference structure. Conceptual metaphor is a given basic
capacity to understand arithmetical concepts on the ground of repeatable bodily experiences in basic
everyday activities. Johnson gives a linguist perspective on the grounding of abstract arithmetical
reasoning on basic bodily experiences. Lakoff and Núñez appoint four ‘grounding’ metaphors to
conceptualize basic arithmetic. By the concept ‘linking metaphor’, they describe the capacity to mix
these different metaphors in order to form more complex ones. This mixing is called ‘conceptual
blending’.
In The Body in the Mind189 Johnson studies, from a linguist perspective, how reason and the
understanding of abstract meanings arise from a metaphorical extension of physical experiences.
Herewith he argues against the ‘objectivist’ views where rationality comes to a formal logic (purely
abstract logical relations, independent of the reasoner’s mind and body) and where meaning is
viewed as an objective relation between a symbol or a sentence and the facts in the world or in the
possible worlds.
Johnson’s main concern is to point out the crucial role of imagination -that is structured by ‘image
schemata’ and metaphors- in meaning and reasoning, that he views to be neglected by the
disembodied, objectivist views.190 He states that an image schemata is a flexible embodied structure
of an activity that exists preconceptually in our experience. These image schemata allow us to make
sense of our experience and give rise to rational or logical entailments.191 For example: our daily
experience with containment tells us that for everything it holds that it is either inside it or outside it.
This gives the intuitive basis for some laws of classical logic, e.g. the law of the excluded middle.
Furthermore it gives rise to an intuitive understanding of the law of transitivity (if A is in B and B is in
C, then A is in C), and the ‘double negation-law’ (if something is not outside, then it is inside). The
understanding and use of these laws in abstract reasoning is a metaphorical projection upon
container schemata. 192 The experimental basis for abstract reasoning accounts for the possibility to
understand abstract structures. To understand them is explicitly not an operation with a priori
structures of pure reason; it is a connection with meaningful experiences in the material world by
means of a relation to image schemata.193
Metaphor is the second cognitive function, mentioned by Johnson, notable in our understanding of
abstract structures, as present in arithmetic.194 By metaphors, image schemata are elaborated into
arithmetical concepts. We have for example ‘balance’ schemata. The ‘twin-pan balance schema’ is
187
(Lakoff & Núñez, 2000, p. 51)
(Lakoff & Núñez, 2000, pp. 50-76)
189
(Johnson, 1987)
190
(Johnson, 1987) in introduction.
191
(Johnson, 1987, p.22 and pp. 29-30)
192
(Johnson, 1987, p.40)
193
(Johnson, 1987, p.64)
194
(Johnson, 1987, pp.65-100)
188
59
mapped point-by-point onto abstract arithmetical equation as follows. Physical objects are mapped
onto numbers (abstract entities); weight of objects onto numerical values; the physical adding of
weights onto arithmetical addition; and the fulcrum of the physical balance onto the equation sign.195
Thus ‘2+1=3’ is metaphorically constituted by the image schema of balancing a twin-pan balance, by
putting two objects (the one twice as heavy as the other) on one side of the balance-beam and one
object on the other side.
Johnson argues, on the basis of linguist examples, that the given internal structure of image
schemata and metaphorical systems are rich enough to establish the patterns of understanding
abstract structures and to generate our abstract reasoning.196
Lakoff and Núñez appoint four ‘grounding’ metaphors (‘4 G’s’) to conceptualize arithmetic. These
metaphors are: Arithmetic As Object Collection; Arithmetic As Object Construction; The Measuring
Stick Metaphor; and Arithmetic As Motion Along a Path. These are enough to account for our ability
to understand e.g. ‘2+1=3’. But in order to explain our understanding of full number theory, including
our conception of the negative numbers, the fractions and the irrationals, extensions of these 4 G’s,
are given, by means of which our understanding of more advanced arithmetic is explained. Now the
4 G’s will be discussed, one by one, together with an indication of their extensions.
Arithmetic As Object Collection, the first metaphor, maps the domain of the physical objects onto the
domain of numbers. The ‘collection of objects’ of the source domain are mapped onto the ‘numbers’
of the target domain called: arithmetic. ‘Putting collections together’ is mapped onto ‘addition’. Thus
here, ‘2+1=3’ means: ‘putting together a collection of two objects, and a collection of one object’.
And so ‘bigger’ is mapped onto ‘greater’.
By this metaphor the laws of arithmetic are presented as being metaphorical entailments. For
example: commutativity in number theory ((A + B) →(B+A)), is metaphorically entailed by: ‘For object
collections: adding A to B gives the same result as adding B to A’.197 Furthermore, our basic
understanding of adding and subtracting, construed as a combination of innate arithmetic and the
use of this basic metaphor, is extended via conceptual blending, to two versions of multiplication and
division. Multiplication is either the pooling of subcollections of the same size each to form an overall
collection, or it is repeated addition. Division is either splitting up a collection into subcollections of
the same size each, or repeated subtraction. We can also understand ‘zero’ by means of an extension
of this metaphor. Namely, we conceptualize the absence of a collection as a collection with no object
in it. By this we create zero as an actual number.
The second metaphor, Arithmetic As Object Construction, maps the domain of object construction
onto arithmetic. ‘Objects’ (consisting of parts) are mapped onto ‘numbers’ (consisting of other
numbers). The smallest whole object is mapped onto the unit ‘one’. Here ‘2+1=3’ is understood as an
act of constructing one object by putting together an object consisting of two parts of unit size with
an object of one such part. Also this metaphor could be extended in the same two ways as the first
metaphor, via metaphorical blending, resulting in the two metaphors for multiplication and division:
fitting together or splitting up and repeated addition or repeated subtraction. This metaphor gives
195
(Johnson, 1987, p.90)
(Johnson, 1987, p.137)
197
(Lakoff & Núñez, 2000, p. 58)
196
60
rise to our understanding of fractions, since it metaphorical entails that a number can be
decomposed into parts, each of which are to be understood as numbers again.
The Measuring Stick Metaphor is the third grounding metaphor. Here ‘physical segments’ (body parts
or ‘a long, thin thing’), consisting of ultimate parts of unit length, are mapped onto the numbers of
arithmetic. Adding 2 and 1 is the act of putting the basic physical segment end-to-end with a physical
segment with the length of two unit parts. Again multiplication and division are characterized the
same as above, in the same way, via metaphorical blending. This metaphor can also be extended in
order to being able to define fractions (as a part of a physical segment made by splitting a single
physical segment into n equal parts (1/n), or as a physical segment made by fitting together, end-toend, m parts of size 1/n (m/n). ‘Zero’ is defined by the metaphorical extension ‘the lack of any
physical segment’. Moreover there is the ‘Number/Physical Segment blend’ that metaphorically
entails that, given a fixed unit length, it follows that for every physical segment there is a number.
The irrational numbers are defined by means of the Measuring Stick Metaphor and the
Number/Physical Segment blend. For example,
is defined as a construction of a physical segment
of unit length 1 and a right triangle, together with the idea that there must be a number
corresponding to the hypotenuse in this triangle (the Number/Physical Segment blend).
The last grounding metaphor is: ‘Motion Along A Path’. ‘Point locations on a path’ (with an origin
which is mapped onto ‘zero’) are mapped onto ‘numbers’. The sum ‘2+1=3’ means an act of moving
from a point-location ‘2’ away from the origin, a distance that is the same as the distance from the
origin to a point-location ‘1’. This metaphor can only be extended by the iteration extension in order
to provide a metaphorical meaning for multiplication and division. Also the fractions can be entailed
by this metaphor. Furthermore a natural extension gives rise to the negative numbers (as being the
point locations on the opposite direction from the origin of the positive numbers along the same
path).
Where do these metaphors come from? The idea is that the 4G’s arise naturally from correlations
between the innate arithmetic -the two supposed innate capacities: the capacity to instantly
recognizing small numbers of items and the capacity to add and subtract small numbers- and the
source domains of the 4G’s: collection of objects, construction of objects, physical segments and
point locations on a path. These correlations arise in everyday experiences such as the manipulation
of objects, the use of fingers and arms and sticks, taking steps, etc. Not only innate arithmetic but
also the working of these metaphors are viewed to be an innate capacity. The innate arithmetic is
viewed to be a bodily given understanding of abstract inferences (only addition and subtraction up to
the number four). So the babies ‘understanding’ of ‘1+1=2’ in Wynn’s experiment is taken as an
understanding of an abstract inference. In a conflation of this innate arithmetic with a source domain
(e.g. object collection), the abstract innate inference, e.g. ‘2+1=3’, fits those of the source domain (‘if
you put two objects together with one object, you have three objects’). In other words: abstract
innate arithmetic is, within bodily experiences, conceptually blended with the source domain of
object collection. The properties of innate arithmetic ‘pick out’ these four domains of the 4Gs,
because these domains ‘fit’ innate arithmetic in the sense that they correspond structurally.
Arithmetical knowledge, in this way explained, is a correlation between an innate understanding of
abstract basic arithmetic and structures of our experiences with physical objects and acts. According
to this view, the abstract entities and structures referred to, do not exist a priori, since we cannot
61
understand them without our embodied, concrete experiences which are meaningful to us due to
the preconceptual schemata. At the same time we cannot have meaningful experiences without
taking recourse to the preconceptual schemata. So understanding is not something after the event of
reflection on prior experience, it is fundamental to the experience itself. In this sense knowledge of
arithmetic that is knowledge that stems from the basic bodily experiences cannot be falsified by
experience and it thus a priori in that sense that it generates necessary truths.
8.2 Foundations
The meaning of the concepts of arithmetic are provided by innate capacities that give meaning to our
everyday embodied experiences. These innate capacities are: abstract arithmetic, image schemata,
the 4G’s and the function ‘conceptual blending’. The foundation of this embodied arithmetic is thus
given with our bodily capacities in the form of some innate structure in our meaningful bodily
experiences. By viewing metaphor (that is seen as the mapping from one domain to another that
preserves inference structure) as crucial in constituting arithmetic, the basic mathematical notions
‘mapping’ and ‘inference’ cannot be explained.198 So we could view these notions to be the primitive
concepts of this foundation of arithmetic. Note that this foundation is solely about how we
understand numbers and not about what they might be external to us.
This cognitive perspective considers the numbers of innate ‘pure’ arithmetic to be abstract concepts,
grounded inside the knowing subject. In this sense the numbers are human creations. But in each of
the 4Gs, the numbers are things that exist in the external physical world. They thus instantiate a
general metaphor: ‘numbers are things in the external world’. This picture is held to be responsible
for the realist ‘mythology’ Lakoff and Núñez contest. They call this view: ‘The romance of
mathematics’.199 It represents the idea that mathematical objects are real entities external to human
beings. In this myth mathematical truths are absolute and objective, and mathematics characterizes
the very nature of rationality. The main argument against this picture that is brought forward in
Where mathematics comes from is that the only mathematics we can know is the mathematics our
bodies and brains allow us to know, and that this is all there is to say, scientifically, about what
mathematics really is.
8.3 Truth
Arithmetic, thus framed by cognitive science, is an embodied mental creation. The objects, the
numbers, are ideas that are ultimately grounded in human experience and are put together by
normal human conceptual mechanisms, such as image schemata, conceptual metaphors and
conceptual blends.200 This makes arithmetical truth, as any other truth, dependent on embodied
human cognition. However arithmetical cognition is, according to this view, special for it generates
knowledge that is exceptional in its preciseness and stability. Arithmetical truth is naturally grounded
and hereby absolute in the sense of ‘universal’. Knowledge of arithmetic that is knowledge that
stems from the basic bodily experiences cannot be falsified by experience since it is inherent in the
meaning that the basic bodily experiences have to us. Thus it is a priori in the sense that it generates
necessary truths. Since the truths are also grounded in basic experiences in the world, their nature is
not purely subjective. But they obviously cannot be objective in a sense that they could be provided
with a God’s perspective.
198
This is a comment of the reviewer: D.T. Langendoen (Lakoff & Núñez, 2002, p.172).
(Lakoff & Núñez, 2000, pp. 339-340)
200
(Lakoff & Núñez, 2000, p. 366)
199
62
An important consequence of this view on numbers and arithmetical inferences is that there is made
a strict distinction between the numbers (which are always meaningful since they are naturally
understood via metaphors) and the numerals (the cultural-based symbolization of the numbers).201
Performing operations on numerals on the basis of certain procedures cannot have anything to do
with an insight in this natural and meaningful arithmetic since in order to be meaningful it must be
tied to real physical experience.202 In consequence: learning children to carry out operations on
numerals can by no means contribute to an understanding of number theory.
8.4 Knowledge Conditions
According to this internal view, knowledge of an arithmetical truth is created by the knower; it is an
embodied mental creation. Innateness and basic concepts that stems from our basic sensory
experiences with the physical world around us all structured by preconceptual schemata, provides
the basis for arithmetical knowledge that is thus conceived as contentful and a priori. A priori in the
sense that although sensory experience is needed for providing the basic concepts, this experience is
so fundamentally structured by the innate preconceptual schemata that basically any experience
enables us to acquire the concepts involved. As in the case of intuitionism, also here our mind
imposes a structure on experience. For this ‘subjective’ but also universal and necessary arithmetical
knowledge the question is: what is required for learning to know this?
According to the embodied-arithmetic-view, the internal abstract arithmetical concepts are typically
understood via metaphor; that is in terms of more concrete concepts of natural language, implying
that our bodily experiences influence our cognitive life. The idea is that we structure our abstract
thinking by the use of metaphors based on basic life-world experiences.203 For Johnson the theory of
metaphor entails a theory of embodied human understanding. For example: we structure abstract
concepts such as understanding and learning by the basic bodily experience of throwing and
catching things (Did you get it?’; ‘Everything he said flew over my head.’; I couldn’t quite grasp what
he was saying).204 According to this view basic bodily experiences are indispensible for acquiring
arithmetical knowledge. Teaching should follow on these experiences.
More specifically, the four Grounding Metaphors named by Lakoff and Núñez are the basis for an
understanding of embodied arithmetic. These are: Arithmetic As Object Collection, Arithmetic As
Object Construction, The Measuring Stick Metaphor and Arithmetic As Motion Along a Path. By
means of extensions of these metaphors arithmetic is constructed metaphorically. In order to being
able to know this embodied arithmetic the teacher must address the learners basic bodily
experiences with object collection and –construction, with measuring by means of physical segments
and with moving along a path. Arithmetic as a meaningful activity according to this view, can only be
developed by means of reference to experiences in all these four categories. Of course this can be
done in many different ways. By physically experiencing these again in class (e.g. jumping and
201
(Lakoff & Núñez, 2000, pp. 82-86)
(Lakoff & Núñez, 2000, p. 86)
203
(Johansen, 2010, pp. 187-188)
204
Examples of (Johansen, 2010, p. 188). Although the use of the metaphors of arithmetic suggests that we understand
arithmetic using bodily and spatial experience it does not have to determine arithmetical thinking and understanding. The
study of gestures suggest a strong link between gestures and speech, and in addition, brain studies suggest that basic
arithmetic is closely connected to the basic life-world experiences of the body and the physical world, but the thesis that
these metaphors determine what the understanding of arithmetic is, seem too strong to be defended on ground of these
studies (Johansen, 2010, pp. 191-193).
202
63
stepping on a physical number line) and by calling them in the learner’s memory by means of stories
and pictures.
According to this internalist view on learning arithmetic, as is also the case with intuitionism, learning
is primarily development. Teaching is guiding the development of innate arithmetic by means of
conceptual metaphors. For being able to do that, an educational method should at least address the
four G’s for these are constitutive for building this meaningful arithmetic.
64
Chapter 9: Towards an Evaluation of Teaching Methods
When I was seven years old I learned how to count to one hundred and back at school. The whole
class was walking around in one big circle calling at each step a number. At the first step we shouted
‘one’, at the second step ‘two’ and so we walked in a circle around till we were at step ‘hundred’.
Then we stood still while calling again ‘hundred’ and then stepped backwards calling ‘ninety-nine’
and so we walked backwards calling again at each step a number till we were at ‘one’ and then
something strange happened: most children shouted ‘home’ at the last step while some others
secretly said ‘zero’. We were not allowed to say ‘zero’ at the end of this game but instead were
taught to say ‘home’. This happened at a Waldorf school where the ideas about learning and
mathematics underlying the teaching methods are quite specific.
If Freudenthal is mistaken when he says that the goals of teaching are a matter of faith, as I argue,
then there should be reasons by means of which one could evaluate the goals of teaching methods.
With respect to the example of teaching above one could say that if this teaching method would
deprive children from information about how the word - and number ‘zero’ is used in the social
world outside that particular school (which fortunately was not the case in this example for later we
were allowed to say and write the previously forbidden word), then this method excludes at least the
possibility of learning arithmetic viewed according to Social Constructivism. For according to this
view the goal of teaching is to make children familiar with arithmetic as it is practiced in society and
enable them to participate in this practice. A clear presence of the social practice is indispensible for
learning to participate. Excluding this practice from a teaching method would exclude the possibility
of the Social Constructivist view on the teaching subject.
In this chapter I suggest a non-restrictive principle for the evaluation of teaching methods which boils
down to the recommendation to include all knowledge conditions in a teaching method. The
knowledge conditions are suggested by the philosophical views on arithmetic as presented above.
The presentation of the views on arithmetic provides in addition an instrument by means of which
the philosophy of a teaching method -with respect to the conception of its teaching subject and
teaching goals- can be positioned. First, this instrument is presented as an overview; a landscape of
dichotomies stemming from the philosophy of arithmetic. Second, the possible conceptions of what
it is to know arithmetic, based on this overview, are summarized. Third, the knowledge conditions for
all these possible conceptions are listed. This overview of the previous eight chapters provides a basis
for two desired conditions for teaching methods which I propose: a positioning in the landscape which provides reasons to the justification of the teaching goals- and the accommodation of all
knowledge conditions in the teaching method -which provides children the opportunity to access
knowledge of arithmetic conceived in all possible ways.
9.1 The Philosophy of Arithmetic: A Summary
Here I present an overview of the positions on the basis of dichotomies that appears to be an issue in
the above discussion on the three dimensions of the question ‘What is arithmetic?’. The eight
chapters above highlight three dimensions of this question and in addition they state the knowledge
conditions for each view. The first dimension concerns the foundations of arithmetic, including the
conception of the ontology of the numbers. This dimension gives a picture of how the nature of
arithmetic can be conceived. The second concerns the explanation of how we come to know these
numbers and their relations. A position in this dimension pictures the character of arithmetical
65
knowledge. The third is the view on the status of the truths of arithmetic that follows from the
position which is taken in the first two dimensions. Eight positions on these three dimensions have
been discussed. Some were subdivided into several views. In this overview I list only those subviews
which differ from other mentioned views with respect to the dichotomies that are highlighted here,
for the purpose of this overview is to give a sketch of the possible positions on certain dichotomies.
For example, Set Theoretic Structuralism is not different from Platonism, with respect to its position
on the opposites that are mentioned here, and therefore it is not mentioned in the overview.
9.1.1 A Landscape of Dichotomies
With respect to the foundations of arithmetic and the existence of the numbers I recapitulate the
conception of number of each of the views and highlight three dichotomies that came up. The first is
the ‘abstract-concrete-dichotomy’ that concerns the view on numbers. They are either viewed as an
abstract entity or as physical. The second is the ‘internal-external-dichotomy’, meaning that either
the numbers are viewed to exist independent on the subjects mind or viewed as being dependent on
the knowing subject. The third, the ‘meaningful-formal-dichotomy’, concerns the difference in the
views on arithmetic in seeing it as meaningful or as formal. The divergent answers on the first
dimension concerning these dichotomies could be summarized as follows.
The Nature of Arithmetic is:
Logicism
(Logical positivist)
Numbers are
Logical
concepts
Intuitionism
Human constructs
Formalism
Platonism
Ante Rem
Structuralism
Symbols
Real objects
Open places in
Real Structures
Structuralism
(Nominalist)
Open places in
possible
structures
Empiricism
(Mill)
Empiricism
(Quine)
Social
Constructivism
Embodied view
Possible physical
collections
Sets
Abstract
Concrete
Intern
Extern
Meaningful
Formal
*
**
***
Signs; cultural
****
objects
Concepts of the
*****
embodied mind
*An internalist explanation for the existence of an ante rem structure is also possible.
**According to nominalism numbers do not exist but of course number words do. The possibility of a concrete physical
system serves as a model for the basic principles of arithmetic. This makes the nature of arithmetic basically concrete.
***This overview shows Quine’s Default Platonism as a Platonist position with respect to the view on the nature of
arithmetic. However this assumed nature is more a hypothesis then a belief. It is an empiricist position in that it is
exclusively the central role arithmetic plays in natural sciences that commits us to this Platonist picture.
****Arithmetic is grounded in humankind. Although it is human in nature, it is thus grounded outside the individual human
intellect or body.
*****Arithmetic is grounded in human bodily experiences with the physical world. Although these experiences give
meaning to innate abstract arithmetic, they are not themselves arithmetic. Ultimately arithmetic is an abstract human
construct based on innate capacities.
66
On the dimension of the epistemology of arithmetic two main dichotomies appeared. The first is the
‘a priori-a posteriori-dichotomy’. Arithmetical knowledge is mostly viewed as a priori in the sense
that it is independent of -or prior to- experience and in the sense that it is knowledge of necessities205
which are not falsifiable by experience. The second, the ‘invention/construction-discoverydichotomy’ expresses actually two opposites of the realist view that is located on the discovery side.
Arithmetical knowledge is a discovery if it is viewed as something that is somehow given with the
intuition or perception of an external reality (abstract or concrete). This opposes the knowledge of
internalist arithmetic. According to the internalist positions knowledge of arithmetic stems from the
subject and is thus its intellectual – or embodied construction. Knowledge conceived as a discovery
of something real also opposes all views that conceive arithmetic as based on conventions or as
purely formal. According to these formalist/coventionalist views arithmetical knowledge is
characterized as an invention.206
Knowledge of Arithmetic is:
A priori
Logicism (logical positivism)
Intuitionism
Formalism
Platonism
Ante Rem Structuralism
Structuralism (Nominalist)
Empiricism (Mill)
Empiricism (Quine)
Social Constructivism
Embodied view
A posteriori
An Invention/ A
Construction
A Discovery
*
**
*The epistemology for Ante Rem Structuralism could also be Platonist, for example if the ante rem structures are accounted
for by innateness or seen as accessible through realist intuition. In that case knowledge of arithmetic is a priori for it would
be prior to - or independent of experience. Here I picture Kitcher’s empiricist epistemology for Ante Rem Structuralism
based on the perception of concrete systems in the empirical world.
**Arithmetical knowledge is here a priori in the sense of conventional (given with the practice of use). However here a
priori knowledge is conceived as knowing necessities and that is not the case in this view. The truths are explicitly fallible.
Knowledge of arithmetic is always sensitive to the situation, context, place and time. So in this overview it is classified as a
posteriori for the practice is in the end a living empirical perceivable thing.
205
Note that a priori is used here as connected to necessity classically. I have followed the traditional (Kantian) conception
of a priori, in the sense that a priori knowledge is always knowledge of necessities despite of Kripkes examples of necessary
truths known a posteriori and contingent truths known a priori. According to Kant: 'It must first be remarked that properly
mathematical propositions are always a priori judgements and are never empirical, because they carry necessity with them,
which cannot be derived from experience' (Kant,1787, B15). Kripke claims that there are examples of necessary truths
known a posteriori, like some identity statements and statements about natural kinds (Kripke, 1972). About identity
statements pp. 97-105. About natural kinds e.g. pp. 110-134, where he also refers to Putnam. Kripke’s examples of
contingent a priori truths: (Kripke, 1972, pp. 54-56) where he considers the statement ‘Stick S is one meter long at t0’
where the metric system is fixed by reference to stick S. It is not my intention to argue against Kripke’s purely
epistemological notion of a priori, but only want to be clear on how it is used here.
206
These views are characterized as ‘relativist’ on the truth dimension for the truths known according to the formalist views
are always relative to the language system in which they are expressed and according to the conventionalist view of Social
Constructivism the truths are always relative to the social practice.
67
Positions on the first two dimensions are interrelated. If arithmetical truths are known a posteriori, in
the sense that they are known by experiencing the actual world around us (physical or social), then
arithmetic, thus known, must be something that is ontologically external for if it is ontologically
internal, knowledge of it would be prior to experience. But the converse appears not true. If the
numbers are viewed as external concrete things in the world, they do not have to be known by a
posteriori methods. Formalist and nominalist views show that physical number words -the symbolscan express arithmetical truths that could be known a priori. Here the truths are given with the
syntax of the system of which they are part. This makes these views formal: since the a priority is
given with the syntax, it is also limited to the formal rules of the system. Furthermore these formalist
views show that arithmetical knowledge -construed as an invention- is just a construction, the truths
of which are pregiven with this construction itself. Of course also ‘discovered arithmetic’ could be
knowable by a posteriori methods (empiricism), since there it is something that is given by
experience but not prior to experience. The third dimension gives a picture of the interrelated
positions on the first two dimensions. It does not add new information about the possible views but
instead pictures the consequences to the status of the arithmetical truths.
On the dimension of the truths of arithmetic the dichotomies ‘contingent-necessary’, ‘absoluterelative’ and ‘objective-subjective’ appeared to be an issue. All these distinctions are directly
connected to the above-mentioned. The first dichotomy is directly connected to the a priori-a
posteriori-dichotomy. Where arithmetical knowledge is taken to be a priori, the necessity of its truths
can be reassured. While by a posteriori methods necessity, in this strict sense, cannot be reached.
The second distinction concerning the absoluteness of arithmetical truth, points not to the opposite
of ‘fallible’ for even Platonist arithmetic is fallible in the sense that our intuition could mislead us.
Here ‘absolute’ means that the truths also hold outside the system in which they are expressed and
used. Thus a truth can be absolute necessary (Platonist conception of arithmetical truths) and
relative necessary (true in every possible world but always relative to a system that instantiate this
truth, relative to the social practice, to natural sciences, or to the science of mathematics as a
whole). The last dichotomy is connected to the conception of the location of the foundation of
arithmetic. This is either inside the knowing subject or outside it. Thus this dichotomy coincides with
the positions on the ‘inter-extern-dichotomy’.
68
Truths of Arithmetic are:
Contingent
Logicism (logical
positivism)
Intuitionism
Formalism
Platonism
Ante Rem
Structuralism
Structuralism
Nominalist
Empiricism (Mill)
Empiricism (Quine)
Necessary
Absolute
*
Relative
to the
language
Objective
Subjective
to the
language
to
mathematics
to the
language
**
Social
Constructivism
Embodied view
to natural
science
to the social
practice
***
*According to Frege the truths are absolute while for the logical positivists they are relative.
**Mills conception of the truths of arithmetic as ‘natural necessities’ are here taken to be contingencies since although
there is a presupposed law of nature, the necessity of that law is not known as long as the knowledge is taken to be a
posteriori.
***According to the social constructivist view, the truths of arithmetic are intersubjective.
9.1.2 Knowing Arithmetic; A Discussion
What does it mean to know arithmetic according to all these possible positions? What are the
possible pictures of what knowing arithmetic is and what are the structural problems of these
possible positions on the discussed dimensions?
The perspectives on knowing and learning, which I find suggested by the views on arithmetic, take a
position in the rationalist-empiricist-controversy. The acquisition of knowledge of arithmetic is
explained as the acquisition of a priori knowledge or a posteriori knowledge. Each view requires a
particular explanation of how warranted true beliefs in arithmetical propositions are gained. An
explanation of how arithmetical knowledge is acquired therefore depends on the position on the
epistemological dimension.
On the epistemological dimension two dichotomies appeared: a priori versus a posteriori and
invention/construction versus discovery. The first distinction concerns the question to what extent
we gain knowledge of arithmetic independently from sensory experience. The notion a priori is used
in tight connection to necessity and thus an explanation how we know a proposition a priori accounts
at the same time for its necessity. The A-Posteriori-Views must explain how we gain arithmetical
knowledge from sensory perception; they must show why arithmetical propositions do not appear to
be falsifiable by experience and finally they must account somehow for the widely supposed
necessity of its truths. The A-Priori-Views must explain how we gain warranted beliefs in arithmetical
propositions independently from sense experience.
One explanation of a priori knowledge could be given by viewing arithmetic as internal in the sense
that the necessity of the truths are discoverable purely by operation of thought, without any
dependence on what is existent in the world. But if arithmetic is about an external world one could
rely on intuition and deduction. Intuition is an immediate way of ‘seeing’ a truth in such a way that it
constitute a warranted belief in it and deduction is the way of deriving conclusions from intuited
69
premises by means of valid arguments.207 Such a rationalist208 account of gaining knowledge of
arithmetic that is necessary and external on the basis of intuition and deduction still requires an
explanation of how and why this necessity arises in the external arithmetical truth, and also an
explanation of what intuition is and how it is able to provide warranted true beliefs about an external
world.
Knowledge of arithmetic that is a priori could also be accounted for with reliance on innateness.
Innate knowledge is not learned through sensory experience, intuition or deduction. It is just there
with us. Although sensory experiences could trigger a process that brings this knowledge to
consciousness, it does not provide us the knowledge. This rationalist position comes forward in views
where the knowledge is viewed to be gained in earlier existence, or where it is viewed to be given to
us by Devine creation, or where it is viewed to be part of our nature by natural selection. Plato e.g.
presents the doctrine of knowledge by recollection in Meno, where Socrates guides a young slave
from ignorance to mathematical knowledge.209 The doctrine explains how the slave can learn
something new while at the same time he already knows what he learns. When he inquires into the
truth of a mathematical theorem, he recollects the knowledge his soul already possesses prior to its
union with his body.
Like the A-Priori-Views explain our a priori epistemic access to arithmetic that is either a construction
(mental, conceptual or formal) or a mind-independent abstract reality that is there, outside us, to be
discovered (Platonism), also the A-Posteriori-Views explain our epistemic access (on the basis of
sensory perception) either as a construction (a social construction in the case of Social
Constructivism ) or as a discovery of a physical reality outside the perceiving subject (in the case of
Empiricism). The contrast construction-discovery is rooted in the opposition Constructivism versus
Realism.
Constructivism, expressed by Brouwer as the idea that arithmetical existence and truth is
conditioned by the possibility of their construction by an ideal human subject, is positioned opposite
to Realism, where arithmetical existence and truth are to be found in a reality that is independent of
the human mind.210 According to constructivism, arithmetic is invented and according to a realist
view it is discovered. The realist epistemology is either based on sensory perception or on intuition.
Constructivist epistemologies are based on the (im)possibilities of the human mind and body (in the
case of the internalist views); based on the acquisition and use of language (in the case of logical
positivism, the formalist and nominalist views); or based on the ability of adapting to conventions in
the sense of the late Wittgenstein (in the case of Social Constructivism). This last constructivist
position is special for it shows a kinship with the realist positions in that it takes arithmetic as
meaningful and in addition external but besides also explicitly as a construct. 211 All the eight views on
the epistemology of arithmetic are situated in the range between these two opposites and the
conditions for being able to know arithmetic are given with their special position in this range.
207
Arguments are valid iff it is the case that if the premises are true, then the derived conclusions by means of these
arguments, are also true.
208
According to the definition of ‘rationalism’ as opposed to ‘empiricism’ in The Stanford Encyclopedia of Philosophy
(Markie, 2012)
209
Plato’s Meno 80d-81a (Scott, 2006, pp. 75-90)
210
(Parsons, 2006, pp. 35-42)
211
(Ernest, 2004)
70
On the dimension of the ontology the dichotomy ‘extern versus intern’ expresses the difference of
viewing arithmetical objects as either existing independently from the knowing subject or as a
human construct (embodied or intellectual). External views are subdivided into the external-formalviews and the external-meaningful-views.
The external-formal-views are represented by the formalist view and by Chihara’s nominalist
Constructability Theory. According to these views arithmetic is purely syntax. Also the logical
positivist view, expressed by Carnap, rejects an absolute meaning of arithmetical statements. Their
meaning is exclusively relative to the linguistic system that instantiates them. Access to these formal
constructions requires the ability to use these systems properly. Knowing external-formal arithmetic
comes down to the ability to use an external instrument.
According to the external-meaningful-views the external arithmetical objects themselves are seen as
having a real content. Knowing arithmetic thus requires an understanding of an external reality. The
meaning of arithmetic in these views is given by the reference to a world outside us (Platonic,
Physical or Social). Frege’s logicism, Gödel’s Platonism and the Set-Theoretic Structuralist rely on a
conception of a given, real, non-physical realm that we must somehow intuit in order to be able to
gain knowledge of it. The physical world accounts for the content of arithmetic in Mill’s Empiricist
view. The explanation of our access to arithmetical truths could, on this account, rely basically on
sensory perception of the physical world. Finally, Social Constructivism holds that the practice of use
constitutes the meaning of arithmetic. This accounts also for a meaning that is primarily given by
something beyond our individual intellect or bodily being. In summary, knowing arithmetic that is
external and meaningful is having insight in an external abstract, physical or social reality.
Internal views hold that knowing arithmetic is constructing meaning intellectually in constructivist
intuition or bodily in our basic bodily experiences. In the case of embodied arithmetic our basic
bodily experiences in the world are only meaningful to us by virtue of the internal structure of image
schemata. Although the given functions need the input of the external physical world, we only
experience this world by means of these internal structures. Since our knowledge of arithmetic rests
on these experiences, and our arithmetical knowledge is seen as the same as the arithmetical truths,
these truths ultimately rest on the preconceptual structures internal to the agent. In the case of
intuitionalism, the other internalist view, arithmetic is a construction of the human intellect.
According to this view our minds impose a structure on experience and thought in constructivist
intuition in which arithmetic is grounded. Knowing arithmetic that is internal requires the
development of human conceptual mechanisms that gives meaning to our basic bodily experiences
or it requires the development of human reasoning based on constructivist intuition. Knowing
internalist arithmetic basically requires development of human mechanisms instead of learning from
outside.
Knowledge of arithmetic that is conceived as presented by the eight views, in summary, is the ability
to use an external instrument properly, is knowing truths about an external reality, is intellectually
constructing a mental construct or the embodied construction of metaphors. Learning formal
arithmetic is learning to handle a given instrument that must be externally given. Knowing truths
about an abstract reality can be characterized as ‘having insight’ in this reality or as a discovery of
this reality. Knowing arithmetic conceived as discovering natural laws can be viewed as the ability to
perceive the world in a certain way. And knowing social conventions can be understood in terms of
71
participation; as an ability to act in accordance with this social practice. Learning external-meaningful
arithmetic requires that the external reality, which must be discovered by the learner, is presented to
her as clearly as possible. Learning to know internal arithmetic requires guidance of the natural
development of the learner.
9.1.3 The Knowledge Conditions
The possible conceptions of knowing arithmetic summarized above, all require specific faculties or
conditions for being able to access this knowledge. The conditions for knowing arithmetic state what
all possible views on the subject require for the ability to acquire knowledge of it. These knowledge
conditions indicate desired conditions for teaching methods. If the conception of the subject of
teaching underlying a teaching method requires specific conditions for gaining knowledge of this,
then at least those conditions should somehow be addressed by this method.
What is required for being able to acquire knowledge of arithmetic depends on what ontological
status it has and what the nature of the arithmetical knowledge is that must be obtained. So the
starting point is a picture of what must be learned. This can be given with a positioning in the
landscape of dichotomies.
Above three groups of possible conceptions of the teaching subject are indicated: the externalformal-views, the external-meaningful-views and the internal views on arithmetic. Here I list the
conditions for knowing arithmetic, viewed in all these possible ways, according to this classification.
First, learning how to use arithmetic as an external instrument requires exercising the ability to learn
a formal language, which comes down to learning to form and recognize well-formed sentences and
to follow rules for valid inferences. To exercise an invoked linguist ability requires exercises in
following explicit examples of procedures. The rules must be given to the learner within examples of
their application in procedures instead of developed by the learners themselves. Since the systems
are conventional it cannot be expected from the learner that she develops it solely by means of her
own reasoning. These explicit examples and exercises must aim at the development of procedural
skills. Being able to use arithmetic as a formal instrument properly requires what it takes to learn a
language syntactically. It is impossible to learn this without the input of examples of the correct
language use. Learning to use it properly also requires practicing the use of this syntax that is given in
examples. With respect to using arithmetic this comes down to learning to perform algorithms and to
recognize which sentence-constructions belong to this game and which not.
Second, learning arithmetic as an external reality requires representations of insights in poor
contexts and in rich ‘realistic’ contexts and in addition requires the presence of arithmetic as a social
practice for the learner as well as physical systems of patterns perceivable by the learner. Learning
arithmetic as an external abstract reality requires the ability to immediately grasp the essence of an
arithmetical truth by means of realist intuition. To enable this insight, representations in poor
contexts are required that reveal this essence as clearly as possible. Knowledge about structures that
go beyond the contingent concrete reality requires that the invoked faculty of pattern recognition
and the faculty of projection are addressed. Since these faculties build on concrete systems of
physical objects, sensory perception of these are indispensible for learning to know the abstract
structures. In order to learn structuralist arithmetic, the learner must come into contact with sensory
perceivable concrete patterns. Learning to know arithmetic as located in the physical reality requires
the sensory perception of this reality as well as representations in rich contexts, for this enables
72
exercising the ability to do arithmetic as a real activity. Learning to know arithmetic as an actual
social practice requires that this practice (including the accepted forms of arithmetical language,
social relationships and roles, forms of communication and accepted ways of working with material
resources and symbolic representations) is clearly present for the learner. Being able to exercise
arithmetic as it is exercised in practice requires that this practice is present in class. Thus a learner is
able to exercise in participating in this practice. It also asks for a teacher role that takes responsibility
for correcting the learner’s knowledge productions and for warranting learner’s knowledge.
Learning to know arithmetic as a meaningful external reality requires some way to reach knowledge
of this reality. Being able to learn to know Platonist arithmetic requires that a faculty of realist
intuition is addressed. Invoking realist intuition requires a presentation of the truth, that must be
transmitted in education, in an insightful way. This representation of this truth must be such that it
creates the best conditions for bringing this immediate insight into force. Somehow the learner must
see the essence (the truth) behind the representation. Representations of arithmetical truths in poor
contexts are most suitable for these truths concern abstractions. Presenting an abstract truth in an
insightful way requires that contingencies that are not part of it blur the essence as little as possible.
This requires that the representations are highly pre-structured in the service of the clarity that
enables understanding abstract truths by immediate insight. In contrast, a condition for learning
empiricist arithmetic requires rich contexts for knowing arithmetic as an empirical reality cannot
disregard the actual form of existence of the arithmetical truths according to this view. Learning to
know arithmetic as it appears in the physical reality requires that it can be perceived in this reality. If
it is grounded in a social practice then learning to know this requires that this practice is present to
the learner as clearly as possible. Practicing to participate requires experiencing and exercising it as it
is exercised in this practice. In education this means that the teacher-role is crucial for it is the link
between this practice in society and how it is exercised in class.
Third, learning arithmetic that is grounded in the knowing subject requires a particular development
of naturally given abilities of this subject. Learning to know arithmetic as a mental construction
requires the development of constructivist intuition by addressing the learners reasoning skills and
by putting emphasis on justification and argumentation for knowledge productions. Learning to
know arithmetic as an embodied construction, finally, requires that the learners basic bodily
experiences with object collection and object construction, with measuring by means of physical
segments and with moving along a path, are addressed.
Developing arithmetic from a human faculty of reasoning or from an ability to construct conceptual
metaphors of basic bodily experiences requires that these faculties are addressed by exercising it.
Meaningful internal arithmetic can only be developed by the knowing subject itself. Learning to know
it thus requires an active role of the learner in the sense that in the end it is her activity of reasoning
and constructing of metaphorical meaning that brings this arithmetic to life.
9.2 Desired Conditions for Teaching Methods
I propose two directions for a possible philosophical evaluation of teaching methods. The first is to
use the landscape of the philosophy of arithmetic as an instrument that provides a clear view on the
conception of the subject of teaching. A positioning in this landscape will give fundamental
arguments for and against the conception of the teaching subject and it will provide justification for
the contents of the method. The second direction for an evaluation is provided by the knowledge
73
conditions. I propose that a teaching method should accommodate all knowledge conditions for
otherwise it would restrict the possibilities for gaining knowledge of arithmetic which is not what you
may expect from an educational method for learning arithmetic. Both proposed desired conditions
for teaching methods are presented below.
9.2.1 Positioning in the Landscape
Underlying every educational method a position in the previous presented discussion on the nature
of arithmetic is taken. This position can be explicitly formulated in theory or can only be implicitly
present in the contents of the method itself. Anyhow every educational method for learning
arithmetic must conceive its subject in a certain way. The extreme positions in the Math Wars, the
disputes on mathematics education, could easily be identified with one specific view from the views
listed above. For example Mechanistic Mathematics Education, according to which arithmetic is
conceived as the performance of a purely syntactical outward procedure and according to which
teaching should be solely focussed on learning the right performance of this procedure on paper,
must view the subject of teaching as a formal, external construction that generates necessities. This
conception of the subject requires an explanation of the relation between its core subject and its
applications in practical situations in the physical world. Applied arithmetic, that is also part of the
subject taught in school, must somehow be accounted for by this method.
The conviction that the subject of arithmetic education contains both pure – and applied arithmetic
is widely supported. For example, the current core-objectives for primary education in the
Netherlands include:
The pupils learn to understand the general structure and interrelationship of quantities,
whole number, decimal numbers, percentages, and proportions, and to use these to do
arithmetic in practical situations.
212
The differences between the views consist in how these areas (pure and applied) are conceived and
how their relation is explained.
Mostly the conception of the subject of teaching is not simply reflected in the content of the
textbooks for the foundational position does not have to say something about the curriculum but
says more about how the topics are seen to be related. Moreover the position of a specific
educational method in the philosophy of arithmetic, will of course mostly not exactly coincide with
one specific view on arithmetic. The subject of teaching could be seen as a variety of different topics
that all have their own nature. An educational method will for example include ‘games with
numbers’ -that could be an expression of, for example, a formal or a structuralist view on the
subject- and ‘practical problems’ -that could for example be accounted for with an empiricist or
social constructivist view on this specific topic-. Nonetheless all these topics have a specific place in
the method for a certain reason. There will be a certain assumed coherence between the topics. For
example the exercises in following procedures on paper could be justified by the belief that this
supports insight in structural relationships between numbers. Or they could be justified by viewing
them supporting the development of skills for being able to construct proofs. Exercises in applied
arithmetic (e.g. adding apples) could be seen as a service to insight in pure arithmetic or it could be
seen as the core activity of doing arithmetic. It could also be seen as a crucial step in the natural
212
(Stichting Leerplan Ontwikkeling, 2006)
74
development towards a metaphorical understanding of adding. The position an educational method
takes regarding the conception of its teaching subject is thus to be found in the justification of the
topics of teaching and in the relation they are seen to have. By questioning, to the benefit of what is
this included, the contents of teaching methods is questioned and leads to a picture of the
conception of the subject of teaching underlying a teaching method.
I propose that this must be the first way in which an educational method should questioned; by
asking: what are the teaching goals? Therefore, it should make its position in the philosophy of
arithmetic explicit. Uncovering the conception of the subject of teaching comes to questioning the
justification of the curriculum and the relation between the topics . If the position that the
educational method takes in theory is clear then this will provide an overview of the structural
problems that ask for answers as well as the structural capacities of that position. This foundational
evaluation could be held by means of arguments presented in the discussion of the eight views.
Of course education is pragmatic rather than fundamentalist with respect to decisions concerning
the curriculum. And philosophy must do justice to the field. Teachers, authors of educational
textbooks and other education developers can have all kinds of good reasons –other than
fundamental- for teaching certain things. These could for example be didactical reasons about what
can best be taught in a particular age, about what fits the rest of the program or about what
contributes to the necessary variation of tasks that enhance the ability to concentrate and learn with
pleasure. Good reasons for including something in the curriculum could also be pragmatic. If for
example it is known that particular knowledge or skills are tested in an important national test this
could be a reason to include these as extra topics in the teaching method. Other pragmatic reasons
could be about the amount of time a teacher has for giving instruction or guidance. If the class is a
combination of various ages and levels this could be a reason for including more independent work
tasks. There should be no fundamental reason for excluding these from the curriculum even if there
is no relation to ‘the true arithmetic’ at all.
However as said: all teaching methods do view their teaching subject in a certain way and this
conception of the teaching subject gives reasons for decisions about which exercises are present in
the textbooks, which learning materials are offered and what is prescribed in the teachers instruction
manual. Besides the pragmatic reasons for a particular method for teaching arithmetic, fundamental
reasons play a role in the justification of it for ultimately a justification must rest on an idea of what
the goals of teaching are. This conception of the goals of teaching includes a view on the nature of
the teaching subject and an idea of what it means to know this subject.
A philosophical evaluation of teaching methods is possible on the basis of a clear conception of the
position the teaching method takes in the philosophy of arithmetic. This position gives fundamental
reasons for the justification of the contents and coherence of the method. A clear view on this
position in the philosophy of arithmetic indicates the structural weaknesses and strengths of this
philosophical position. Here the fundamental arguments can be found for or against a particular
conception of the teaching subject underlying an educational method, the use of which could alter
the disputes concerning arithmetic education into a proper discussion.
Next to this first direction for a possible philosophical evaluation of teaching methods, a second
direction is provided by the knowledge conditions of the possible views on arithmetic.
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9.2.2 Accommodation of All Knowledge Conditions
A position in the philosophy of arithmetic brings a particular view on what it means to know this
subject. To know arithmetic is conceived as an ability to use an instrument, as having insight in a
reality, and as a developed human activity. The ontology of this instrument, this reality or meaningful
human construct, together with the explanation of our access to it and the assumed nature of this
knowledge indicate conditions for the possibility to acquire this knowledge of this particularly
conceived subject. These knowledge conditions are summarized above.
I propose to adopt a non-restrictive view on arithmetic education. This means taking the following
principle as a guideline for the evaluation of teaching methods: a method for teaching arithmetic
should not exclude the possibility of any serious view on its teaching subject. This comes down to the
recommendation that a teaching method should accommodate somehow all knowledge conditions
for otherwise it would exclude the possibility of gaining knowledge of a serious conception of
arithmetic as expressed and defended in philosophy.
I claim that in order to do justice to the possibility of the views on arithmetic, all knowledge
conditions should somehow be accommodated in the teaching method. The exclusion of conditions
for knowing arithmetic is an exclusion of the possibility of a particular view on arithmetic. All views
on arithmetic present something essential about this teaching subject. Though these views are
incompatible in the foundational discussion, for there they oppose each other on the crucial
dimensions of the question about the nature of arithmetic, as is clear from the above, still, education
does not have to be bound to one fundamental view on arithmetic in the sense that it would exclude
other possibilities. As said a position regarding the conception of the teaching subject is desired, and
this position does exclude the other positions that oppose the view on the subject in a fundamental
way. However this position does not have to exclude the possibility of the other conceptions of
arithmetic in a pragmatic sense. An educational method, positioned in the philosophy of arithmetic,
can include the knowledge conditions of the other views without compromising on its own goals of
teaching. If it would be the case that a knowledge condition of another view on arithmetic is not seen
as contributing in any way to the achievement of the learning goal according to the philosophy of a
teaching method, even then it can do no harm to accommodate this condition anyhow. In fact I think
it would be a missed opportunity when a method would exclude this knowledge condition.
I suggest that education should be pragmatic with respect to the philosophical discussion on the
nature of arithmetic for her aim is to offer children as many opportunities as possible to learn
arithmetic. To address a knowledge condition by a teaching method is giving children an opportunity
to access arithmetical knowledge. If a teaching method would restrict the knowledge conditions it
would not exhaust all possibilities for children to learn arithmetic. So by accommodating all
knowledge conditions in a teaching method, this method does justice to the possibility of all views
for it does not exclude any possibility even though it takes a position itself. In addition this teaching
method would hereby address all opportunities for children to acquire arithmetical knowledge. And I
suppose that is the intention of arithmetic education. Thus I propose a teaching method should take
an explicit fundamental stance in the philosophy of arithmetic and that at the same time it should be
pragmatic by accommodating all knowledge conditions. Let me illustrate this proposal with an
example.
76
If an educational method conceives its teaching subject, for example, in a Platonist way, then the
justification of the topics of teaching and the explanation of their relations are, at least partly, based
on this position. An exercise in adding apples, presented in a rich realist context, could e.g. be
included within this method with the idea that it is an application of previously gained insight into
abstract numerical relations. This exercise can also be justified from a Platonist perspective with the
idea that this exercise somehow supports reaching this insight. Exercising procedures within this
method, for example, could also be justified by viewing it as supporting insight in the real thing. The
Platonist could also justify these exercises in the method by viewing them as exercising the ability to
make deductions from intuited truths, which could also be an aim of Platonist education. The first
desired condition for this method should be, of course, that it must address the realist intuition of
the learner for this is the first requirement for being able to gain knowledge about an abstract reality.
Inclusion of the use of poor contexts would be the most obvious. Next to this, the accommodation of
all knowledge conditions is desired.
How can the Platonist teaching method account for the principle of accommodating all knowledge
conditions? The knowledge conditions of the other views could all be accounted for within the
Platonist position (as in the case of the examples of adding apples and exercising procedures named
above). In case the position cannot account for the accommodation of a specific knowledge condition
it could also be accommodated as an additional single topic glued to the method. If, for example, the
Platonist position would be such that it is solely focussed on reaching immediate insight and if it
could not accommodate exercises in constructing proof, such as exercises in developing - and
evaluating arguments for knowledge productions of learners,213 this would exclude the possibility of
gaining knowledge of constructivist arithmetic and that is not desirable. A pragmatic inclusion of
exercises in constructing and evaluating procedures would be in that case advisable.
In recapitulation, the landscape of the possible positions concerning the conception of the teaching
subject, provides arguments to a teaching method, for the justification of the teaching goals. In
addition this landscape provides an overview of what ‘knowing arithmetic’ may possibly mean. All
conceptions of what it is to know arithmetic indicate conditions for reaching this. These knowledge
conditions can be used as a guideline for an evaluation of teaching methods. The accommodation of
all knowledge conditions in a teaching method ensures that no opportunity to access knowledge of
arithmetic is ruled out.
The proposal to adopt this non-restrictive principle as a guideline for the evaluation of teaching
methods requires the support of a notion of arithmetical knowledge that can in fact include all
conceptions of what it means to know arithmetic. In the next chapter I will present such a notion of
arithmetical knowledge and suggest possible perspectives on learning which are not restrictive to the
possibility of the views on arithmetic either.
213
Which is a position that is improbable in practice
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Chapter 10: Arithmetical Knowledge and Perspectives on
Learning
What appears to be common to all knowledge conditions is that they could all be formulated in
terms of addressing particular faculties which can be expressed in terms of skills. The invoked
faculties for being able to gain knowledge of arithmetic are: a linguist faculty concerning the syntax
of a language, realist intuition, pattern recognition and projection, sensory perception, a faculty to
adjust to a social practice, reasoning, constructivist intuition and metaphorical understanding. The
knowledge conditions could be formulated in terms of skills that are required for knowing arithmetic.
These skills are the ability to construct proof or justification with respect to procedures in a proper
way, wherein the standard is viewed as being understood when one is able to recognize whether a
sentence and a procedure is well formed and in addition when one is able to create new well formed
constructions within this language. These skills -which could be referred to as ‘language proficiency’enables one to use arithmetic as a formal instrument as well as constructing arithmetic intellectually.
Furthermore the skill of perceiving something in ‘the right way’ is named. This immediate way of
understanding could be seen as leading to insight in abstractions or as leading to actions that are in
accordance with what the (social) world as it is perceived is affording. The core of understanding
meaningful external arithmetic is that the knowing subject must stand in a particular relation to
reality. The standard for ‘seeing it in the right way’ is somehow given within this relation.
Phenomenological studies have shed light on proficiency as a particular mutual determination of the
body that acquires skills and the world as it shows up for this body. Platonist – as well as Empiricist
epistemologies rely on these phenomenological studies. The core of the conditions for developing
knowledge of embodied arithmetic could also be expressed in terms of skillful action in the world; as
responding to solicitations of the world as it is experienced by the body. Thus, the knowledge
conditions could be explained as relying on types of skills that are indispensible for learning to know
arithmetic.
The observation that the conditions for being able to reach knowledge of arithmetic, according to all
views, could be seen as relying on skills, indicate that a non-restrictive view on learning arithmetic
could aim at the acquisition of the named types of skills for being able to reach or develop
arithmetical knowledge. Common to all views on knowing arithmetic is that it requires abilities to do
certain things. Teaching arithmetic in a non-restrictive way is thus making the right circumstances for
being able to develop particular types of skills that includes at least procedural skills, argumentation
skills (including the ability to construct valid justifications for arithmetical beliefs) and skills to act in
accordance to reality as it is perceived and experienced in patterns, rules, metaphors (or other
‘essences’).
The aim of this chapter is to describe a particular conception of the notion arithmetical knowledge in
which all discussed views on knowing arithmetic could be expressed and to indicate possible
perspectives on learning that can explain the acquisition of this. This conception of arithmetical
knowledge is a definition in terms of skills by Löwe and Müller.214 It identifies this notion with the
features of modality and context-sensitivity. The modality of the notion arithmetical knowledge is
found in the observation that knowing ‘7x8=56’, for example, is an ability to perform in a certain way.
214
(Löwe and Müller, 2010b) and (Löwe & Müller, 2008)
79
As the objectives for elementary education show, the goals of teaching are formulated in terms of
‘abilities’; it is e.g. the ability to apply procedural rules for the calculation of it and the ability to
recognize and apply this truth in contexts of daily-life-uses and within arithmetic itself. A definition of
arithmetical knowledge in terms of skills brings this feature together with context-dependency in one
description. This means that which specific skills are required for having knowledge of arithmetic
depend on the particular situation. Also the skill level that is adequate for having particular
arithmetical knowledge is found to be context-dependent.
10.1 Arithmetical Knowledge: A Definition
Here I present Löwe and Müller’s definition of mathematical knowledge and argue that this notion is
open to all interpretations of what it means to know arithmetic as presented above. Their definition
shows how the notion mathematical knowledge is context-sensitive.215 Their analysis involves
propositional knowledge and starts from the widely accepted view that mathematical knowledge is a
modal notion.216
An analysis of the standard view, described as ‘S knows that P iff S has available a proof of P’ shows
that ‘having available a proof’ is a modal notion in which the epistemic subject S plays an active role.
More explicit it could mean: ‘S could in principle generate a proof of P’.217 It appears that a traditional
context-independent reading of this cannot give an adequate analysis for all cases of mathematical
knowledge because it cannot account for the context dependency of the notions ‘proof’, ‘could in
principle’ and ‘generate’.
In mathematical practice ‘proof’ (that supplies the justification of mathematical belief) is almost
always informal. What counts as a proof is relative to the situation and the context. There is also
mathematical knowledge without proof.218 Especially knowledge of elementary arithmetic is said to
be known by someone without her being able to generate a (formal) proof of it. ‘Could in principle’
has a temporal component that is dependent on the context. ‘Generate’ is context dependent as to
the kinds of tools or help that should be or should not be available to S when one can justly say that
she generates the proof. These three notions, as they are part of a modal view of mathematical
knowledge, needs to be contextually determined.
Löwe and Müller argue for a definition of mathematical knowledge in which this contextually
determined modality is expressed in terms of ‘skills’ of the epistemic subject S219 and they arrive at
the following definition:
S knows that P iff S´s current mathematical skills are sufficient to produce the form of
220
proof or justification for P required by the actual context.
Note that Löwe and Müller added ‘or justification’ as an optional substitution for ‘proof’ in their
definition. This is important for the case of knowledge of elementary arithmetical propositions
mostly known by children who have no ability to produce any form of proof for it. Herewith this
215
(Löwe and Müller, 2008)
This is based on various analyzes of ‘mathematical knowledge’ as a modal notion of, amongst others, Brouwer, Kitcher
and Steiner, and on Lewis’ general analysis of knowledge (Löwe and Müller, 2008).
217
(Löwe and Müller, 2008, p.92)
218
See also (Steiner, 1975, pp. 93-108).
219
(Löwe and Müller, 2010b) and (Löwe & Müller, 2008)
220
(Löwe and Müller, 2010b, p. 265) and (Löwe and Müller, 2008, p. 104)
216
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analysis can also define knowledge of propositions of elementary arithmetic. The context and
situation determines which skills are required and what level is sufficient for the justification for P.
The advantage of the use of ‘skills’ over formulations in terms of ‘S’s dispositional state of mind’ in
e.g. ‘S knows that P iff S’s dispositional state of mind allows her to produce a proof or justification for
P with the resources allowed by the context’, is that the notion ‘skill’ links the dispositional state of
mind with actual performance, for skills can be tested. It thus includes an empirical side as well as
modality. This definition does not only express the context-dependency of mathematical knowledge
by stating that the context determines the required justification or proof, the notion ‘skill’ itself is
also context-sensitive for the modality inherent to this notion needs also to be contextually
interpreted. What counts as an ability to do something depends on the context and situation.
Dreyfus and Dreyfus’ phenomenological study of skill acquisition that results in their situational fivestep model221 is used to better understand mathematical skills and their role in mathematical
knowledge. In skill acquisition the body that acquires skills and the world as it shows up for this body
determine each other in the following way. Our skills are acquired by dealing with things and
situations in the world and, in turn, the skills determine how things and situations show up for us as
requiring our responses. Our relation to the world transforms when we acquire skills. A
phenomenology of skill acquisition shows that the acquired know-how, as one acquires expertise, is
experienced as finer and finer discriminations of situations paired with the appropriate response to
each. Maximal grip is the tendency of the body to refine its responses to the solicitation of a situation
in the world in such a way as to bring this situation closer to the agent’s sense of an optimal gestalt.
Gestalt222 is a unity of the world and idea, existing for the perceptual consciousness of the perceiving
subject; it is an object of knowledge for it.
The situational five-step model for skill acquisition presents a non-representational account of
learning.223
Stage 1. The novice applies context-free rules, like a computer following a program.
Stage 2. The advanced beginner copes with real situations. She begins to recognize, on the basis of
experience, examples of meaningful additional aspects of the situation. She applies, next to the
context-free rules, rules that are also based on perceived similarities with experienced situations.
Stage 3. The competent performer chooses a perspective that determines which potentially relevant
element of the particular situation is to be treated as important and which ones can be ignored. The
procedure to decide upon a perspective cannot explicitly be learned, like the context-free rules
applied at stage 1 are learned, for there are too many situations differing from each other in so many
ways. They cannot be precisely defined in advance by instructions. The competent performer is thus
involved and must accept risks and responsibility.
221
The model is proposed by Hurbert Dreyfus and Stuart Dreyfus in their book Mind over Machine, 1986. My description
here is based on (Dreyfus, 2002).
222
According to the Stanford Encyclopedia of Philosophy (Flynn, 2011).
223
The Dreyfus-Dreyfus skill model is situational and descriptive. It is not normative; this is not how a learning process
always should run. Although it could be viewed to demonstrate that a learning process always require a period of
application of context-free rules before the learner can move on, this model need not be necessarily taken in this normative
way.
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Stage 4. The proficient performer experiences situations from a perspective. She does not have to
decide to adopt a particular perspective. Her perspective is intuitively invoked and certain aspects
simply stand out as important. After seeing the goal and the important features of the situation, she
must still deliberate what to do to achieve this goal.
Stage 5. The expert performs without calculating or comparing alternatives. With the ability to make
more refined discriminations her situational response is immediate and intuitive. She not only sees
what needs to be done, but also sees how to achieve it. ‘What must be done, simply is done’.224
With this understanding of skill and skill acquisition the question remains which skills the
mathematical skill set consists of. Löwe and Müller link mathematical skills to the historically and
culturally determined requirements of mathematical profession. ‘Mathematical skills’ in the
definition above refers to a situationally determined skill set (those skills that are relevant in the
specific context) that is part of the general mathematical skill set. Whether S’s skills (these are the
situationally determined skill set) are ‘sufficient’, depends on the skill level that is required by the
context. A similar approach to answer this question for arithmetic education could lead to a
formulation of a general skill set that does justice to the historically and culturally determined
requirements of doing arithmetic as a school subject, which includes arithmetical skills for using it in
everyday life situations and as using it in professions including the mathematical profession.225
Can this notion of arithmetical knowledge support the non-restrictive view on arithmetic education?
Or does this definition restrict the possibility of a view on arithmetic? It would restrict the possibility
of a view if it would be incompatible with any plausible interpretation of knowing arithmetic
according to this view. I argue that this is not the case. As mentioned above, the knowledge
conditions of all views can be formulated in terms of certain types of skills. The definition
S knows that P iff S´s current mathematical skills are sufficient to produce the form of
226
proof or justification for P required by the actual context,
leaves all interpretations open with regard to the nature of P. Only, the assumed nature of P
determines which types of skills are required of S in order to know P.
If P is conceived as an external formal truth the skills will have to make recourse to a linguist faculty
and some form of constructivist intuition. Being able to handle this formal external instrument
properly requires the ability to construct procedures in a proper way. This ability, called ‘language
proficiency’, enables one to use arithmetic as a formal instrument as well as constructing arithmetic
intellectually. Knowing an arithmetical proposition that is viewed as a formal external instrument, is
expressible in terms of skills; it is the ability to use this instrument based on linguistic – and
constructive capacities.
224
(Dreyfus, 2002, p.372)
Although Löwe and Müller propose not to analyse mathematical skills as separated skill sets according to the various
subfields of mathematics -and elementary arithmetic as a subject in primary school is part of basic mathematics- it is
nevertheless, in practice, explicitly something else as the first step in a training in becoming a professional mathematician.
Therefore I think it is justified to approach elementary arithmetic as part of elementary education also as an autonomous
field of knowledge. The historically and culturally determined practice of teaching elementary mathematics and the coreobjectives for primary arithmetic education that are part of it, could provide an empirical basis for the general skill set that
is required for a pupil (that is here not the natural kind of human being) to be an expert in (elementary) arithmetic.
226
(Löwe and Müller, 2010b, p. 265) and (Löwe and Müller, 2008, p. 104)
225
82
If P is understood as an external meaningful truth then the skills required by the context will have to
make recourse to realist intuition and/ or sensory perception of a (social) reality. We saw that
Platonist epistemology as well as empiricist epistemology can be explained as a certain relation
between the knowing subject and reality. Knowing arithmetic can be characterized by proficiency,
that is explained by phenomenological studies as a mutual determination of the body that acquires
skills and the world as it shows up for this body. This notion of arithmetical knowledge in terms of
skills could support an Empiricist view on arithmetic as well as a Platonist view on this subject for
both views can rest on this phenomenological account of knowing, as became apparent in the above
description of Gödel’s account of the epistemology of arithmetic and Kitcher’s account of this. Both
take recourse to a phenomenological account of perception. Of course there are other theories
explaining empiricist – or Platonist epistemology of arithmetic, some of which may not be compatible
with a conception of knowing arithmetic in terms of skills. Epistemologies relying on a traditional
cognitivist perspective on knowing in terms of representations and computations, e.g. on the basis of
innate structures, view knowing an arithmetical proposition as a particular state of mind that is
isolated from the environment. This picture may be incompatible with a context sensitive notion of
arithmetical knowledge. However, to find a notion of arithmetical knowledge that supports a nonrestrictive view on learning arithmetic requires that this notion is compatible with at least one
plausible explanation of knowing arithmetic, viewed in all possible ways. A phenomenological
account of proficiency is such a plausible interpretation of a Platonist – as well as an empiricist
conception of knowing arithmetic.
If P is viewed as a ‘subjective’ truth the skills will at least rely on a faculty of metaphorical
understanding and/ or constructivist intuition. In this case P is a construction of the knowing subject.
Only if this construction is interpreted as internal in the sense that it is isolated from the physical and
social environment, locked inside the head of the knowing subject, then a context-sensitive view on
knowledge cannot accommodate it for then the context -that is indispensible according to this
perspective for it basically constitutes what knowledge is- would be meaningless in the account of
this ‘solipsist’ view on knowing P. Both the internal views, intuitionism and the embodied view, are
not solipsist in this sense. Although both rely on structures that are imposed on experience by the
knowing subject, arithmetical knowledge is only meaningful, according to both views, by reference to
the experiences or perceptions in the world outside. This makes a context sensitive interpretation of
knowledge of internal arithmetic obvious.
A context-sensitive definition of knowing arithmetic in terms of skills seems only to exclude a
‘solipsist arithmetic view’ which is not one of the views on arithmetic defended in the philosophy of
arithmetic. Thus it seems that this notion of knowing arithmetic could support the proposal not to
exclude the possibility of any serious view on the nature of arithmetic. It is not restrictive on the
possibility of the views on arithmetic as defended in the philosophy of arithmetic.
This notion of arithmetical knowledge in terms of skills gives indications for possible perspectives on
learning. Since a philosophical evaluation of a teaching method should be able to be supported by a
plausible learning theory, these possible perspectives on learning are sketched below. These
approaches are able to explain learning arithmetic as a process that is non-restrictive with respect to
the conception of the product of learning (that is arithmetical knowledge conceived as described
above).
83
10.2 Approaching Learning Arithmetic
Learning theories explain what the acquisition of arithmetical knowledge basically is. A learning
theory that supports a non-restrictive account of arithmetical knowledge must thus be able to
explain knowledge as the product of a learning process that could be conceived as knowledge of an
external abstract, physical and social reality, an external instrument as well as a mental and bodily
construction. In other words, it should explain learning as a process resulting in knowledge that is
conceivable as a context-sensitive notion, expressible in terms of skills and skill levels that are
situational. I suggest an embodied perspective on learning and a practice-based perspective on
learning that both are capable of explaining learning as a process that results in such a notion of
arithmetical knowledge.
The requested philosophical perspective on learning must of course be in concord with what we
know about how we in fact become to know arithmetic. After the sketch of the possible perspectives
on learning a presentation of findings of studies on arithmetical cognition are presented. These
studies show that external representations play an important role in learning arithmetic.
Furthermore the use of embodied strategies are found to be important in arithmetical cognition. The
proposed perspectives on learning are supported by these features of arithmetical cognition.
10.2.1 Perspectives on Learning
Learning theories explain how we acquire warranted, true beliefs or skills and picture what this
acquisition basically is. Herewith these theories give both an interpretation of the process of learning
and an interpretation of the intended outcome of the process. This could be a traditional cognitivist
picture of ‘understanding’: a mental state as a result of a learning process that is based on internal
representations and computations. True beliefs are according to this perspective warranted by our
universal normative form of rationality. It could also be an explanation of what it is to have insight in
line with Dreyfus’ description of what it is to be an expert in terms of handling a situation as an
immediate response to a situation as it is perceived by the knowing subject. Or in terms of an ability
to act in this skillful way. True beliefs are according to this perspective warranted by our being in the
world and what the world let us do to it. The goal of learning could also be formulated in terms of
‘participation’ based on insight from Wittgenstein’s Philosophical Investigations . Then the intended
outcome of the process is participation in a practice of use. Arithmetical beliefs are according to this
perspective warranted by the current mathematical practice.
I sketch the embodied and practice-based perspectives on learning as possible approaches that
explain how the goals of the learning process are context dependent.
Dreyfus’s five-step model for skill acquisition shows a non-representational account of learning.
While the performance of the novice is characterized by the application of context-free rules, expert
behaviour shows an immediate accordance with a specific situation, in the sense that her perception
of the situation immediately solicits for an action (what she perceives is affording a certain action).
The learning product is skillful behaviour (or the ability to act in a skillful way). The acquired knowhow, as one acquires expertise, is experienced as finer and finer discriminations of situations paired
with the appropriate response to each. The learner acquires the ability to make finer and finer
discrimination of the situation, which affords more and more refined response. The intended product
of a learning process is thus characterized in terms of a way of handling a situation; skillful behaviour.
Warranting arithmetical beliefs must be, according to this perspective, up to the environment for
84
what needs to be done in a specific situation is immediately clear to the experienced expert and her
skillful action simply works in that situation. The experiences in and with the environment comes to
developing this expert behaviour.
This perspective on learning naturally fits the context-sensitive notion of arithmetical knowledge for
this notion of knowledge based on skills is exactly what is aimed according to this embodied
perspective on learning. Next to this perspective on learning also a social orientation to learning
could support a non-restrictive view on learning arithmetic.
Although their analysis of mathematical knowledge in terms of skills is not based on Wittgenstein,
Löwe and Müller’s notion is in line with the practice-based perspective on ‘knowing’ in his
Philosophical Investigations.227 Consider the following three ideas central to Wittgenstein’s PI. The
first is that the role of the (social) context is crucial in learning for the (social) context constitutes the
meaning of the content of the thing that is to be learned and it constitutes what it is to understand
this content. The second idea is that understanding is not a cognitive state. Understanding e.g. the
addition function can only be: acting in accordance with the use of it in an existing practice. And the
third one is the insight that the mind is not a rule-governed machine for a rule itself cannot
determine the right applications of it and the idea that this can be handled by other rules leads to an
infinite regress. All three features of knowing are present in Löwe and Müller’s analysis of
mathematical knowledge.
The context dependency of ‘knowing’ is in many respects present in the definition and the general
skill set is based on the practice of the mathematical profession. This reflects the idea that the
(social) context constitutes the meaning of the content of the thing that is to be learned and that it
also constitutes what it is to understand this content. Furthermore it is stressed that skills have an
empirical side; that the assessment of mathematical skills are dependent on assessment of
performance. This reflects the second idea that understanding something is not a cognitive state but
instead is acting in accordance with the use of it in an existing practice. Lastly Wittgenstein’s idea
that the mind is not a rule-governed machine for the reason that a rule itself cannot determine the
right application of it, is also reflected in the definition of knowledge in terms of skills. Dreyfus and
Dreyfus’ model of skill acquisition show that know-how is in the end not accessible in the form of
facts and rules only. Expert behaviour cannot be taught in the sense that the right application cannot
be given with a rule.
For the same reasons why a rule cannot cause the action that is in accordance with it, teaching
cannot cause learning.228 Instructions from teachers and caretakers rather constitute the justification
for whether a learner’s act is in a certain (community agreed-upon) way in relation to the
instructions. And conversely, a rule does only exist if one acts in a certain way in relation to it.
Although, normally, we can tell when a child has learned how to add, this judgement is always
relative to relevant social criteria and to the particular context in which the learned act is performed.
This judgement thus never stands on absolute certain ground.229
From Wittgenstein’s Philosophical Investigations it can be concluded that when someone
understands the meaning of the usual addition ‘plus’ as in ‘2+1=3’ this consists in her use of this
227
(Wittgenstein, 1953)
(Berducci, 2011, p. 480)
229
(Barsalou, 2008, p. 486)
228
85
function in accordance with the use in mathematical practice. The basic point (and perhaps difficulty)
is that there is no nexus between the past intentions of the learner when she suddenly experiences
that she understands the rule governing the use of ‘plus’ and her present and future performances.
The ‘grasping of the rule’ does not determine that she will produce correct answers in all future
additions for her experience of ‘insight’ can only in a certain context mean that she understands it.
This experience means nothing on its own. And furthermore this experience cannot determine her
future actions since a rule itself does not determine how it is followed or applied in a ‘correct’ way.
Hence understanding must show itself in a perpetual test that is standardized by the mathematical
practice.
This ‘difficulty’ is also present in the notion of skills for the assessability of a person’s skill level shows
a tension between the empirical side of skills and it’s modality. Although skills transcends
performance, assessment of mathematical skills are dependent on assessment of performance. As
Wittgenstein puts it:
‘...When do you know how to play chess? All the time? Or just while you are making a
move? And the whole of chess during each move? – How queer that knowing how to
play chess should take such a short time, and a game so much longer!’
230
Wittgenstein231 explores this tension in his Philosophical Investigations. He describes a situation,
wherein ‘knows’ is used, that can be called an ‘aha-experience’ and concludes that knowing cannot
be something that is hidden behind the visible manifestations in the situation at hand.
‘...it is the circumstances under which he had such an experience that justify him in
232
saying in such a case that he understands,...’
The empirical manifestations in a particular situation are eventually decisive for the existence of
skills. The modality of this notion is thus context sensitive and cannot be explained as a particular
internal state of mind.
The context-dependent notion of arithmetical knowledge in terms of skills is supported by Dreyfus’
embodied perspectives on learning as well as by the practice-based perspective, based on the late
Wittgenstein. So the proposal not to exclude the possibility of any serious view (that is not the
solipsist one) on arithmetic in education, can be supported by a definition of arithmetical knowledge
and philosophical perspective on learning. Finally I present support for these perspectives on learning
in studies on arithmetical cognition.
10.2.2 Arithmetical cognition
Studies on arithmetical cognition provide us insight into the important role of external tools in the
acquisition of arithmetical knowledge. Examples of embodied strategies that are used in arithmetical
cognition evidence that much of arithmetical cognition can only be understood as an interactive
process involving the brain, the body and the environment (social, historical and physical).233 These
230
231
232
233
e
(Wittgenstein, 1953, p. 50 note (b))
e
e
(Wittgenstein, 1953, pp. 50 -52 ; par. 150-154)
e
(Wittgenstein, 1953, p. 52 ; par. 155)
These strategies are discussed in (Johansen, 2010).
86
strategies includes: epistemic actions and cognitive artefacts. I will follow Johansen’s description of
these cognitive tools as presented in his article Embodied strategies in mathematical cognition.234
An epistemic action is a physical action whose purpose is the improvement of cognition by reducing
the memory, the number of steps involved in the mental computation and/or the probability of error
in the computation. An example is the use of the midwife’s wheel, that allows the user to substitute
complicated mental calculation with the physical manipulation of the discs and read off the result.
Figure 1235
This tool, used in an epistemic action, is called a ‘cognitive artefact’.
Artefacts might influence the total outcome of a task. This could e.g. be experienced in the difference
between a talk read from paper or given by the help of Power Point. These artefacts influence both
style and content. Among the artefacts used in arithmetic: fingers, blocks, computers, abacus,..., the
written symbols are the most important ones. In order to answer the question how these artefacts
do influence the content of arithmetic, the written symbols are discussed below.
Our familiar decimal place-value system, originated in India and introduced in Europe in the 10th
century C.E. by way of the Arabic world,236 differ in its computational properties and demands on
cognition for particular tasks, from other systems, e.g. the Roman system. The Hindu-Arabic
numerals are special in that they allow calculations to be performed largely as epistemic actions, so
externally, without reference to the meaning of the symbols.237 This allows for the formalist
interpretation of arithmetical computation as being meaningless manipulations of signs according to
rules. And this characteristic of this tool also causes difficulties in arithmetical calculations. This is
exemplified by studies on error patterns in elementary arithmetic education.238
In Learning and understanding numeral systems, Lengnink and Schlimm review studies that indeed
show that the semantic information is often not being taken into consideration by children and that
this is due to the notational system that does not straightforwardly convey this information. In an
additive numeral system, like the Roman one, each symbol has a fixed meaning. In ‘XX’, both X’s
stand for the value ‘10’ and the referent of this symbol is obtained by adding 10 and 10. While in the
number ‘505’ of our decimal place-value system, the first 5 denotes 500 and the last 5. The ‘base
234
235
(Johansen, 2010)
236
Figure 1: Midwifery Mercantile. (n.d.).
(De Cruz, Neth, Schlimm, 2010, p.87)
237
(Johansen, 2010, p. 186)
238
(Lengnink & Schlimm, 2010)
87
value’ of the cipher ‘5’ is the number ‘5’; this is the explicit meaning of this symbol. The value that ‘5’
represents within a number, its implicit meaning, depends on its position. It appears to be a difficult
task for children to grasp this implicit meaning. In their performance of column-algorithm, the lack of
semantics of the operation causes errors in the proper handling of zeros -(empty) places- especially
when operations like ‘carrying’ and ‘borrowing’ require the movement of digits from one place
(column) to another.
Also the number words in language causes difficulties. For example in German ‘dreiundzwanzig’
mislead into writing ‘32’ instead of ‘23’. The structural mistakes are less likely to be adopted in the
performance of the same operation in the Roman system, or the mistakes would yield correct results
in this system, since this system embodies explicit semantic content, where it is implicit in the HinduArabic system. The semantic tools for the performance of algorithms, used in elementary arithmetic
education, such as graphically representing the grouping of numbers, splitting up numerals, and
providing a meaningful context, convey the implicit meanings of the numerals and must support by
this a more effective use of our system. Lengnink and Schlimm state that an effective use of the
system involve both an understanding of the relation between the symbols and the represented
concepts, and knowing the correct rules for manipulating the notation.
Notation affects arithmetical performance, even when the tasks are performed in the head.239 For
example the summation of two numbers in a tally system (e.g. II + I) is relatively easy compared with
the same task in a decimal place-value system (2 + 1), but determining if the number
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIII is odd or even is harder than the same task in the decimal place-value system
(31). Recent research indeed confirm the prediction that the duration of mental operations is
influenced by properties of numerical notation.240
Thus, epistemic actions and physical cognitive artefacts are constitutive for arithmetic as a cognitive
content. This shows that human cognition of arithmetic is embodied in a concrete sense.241 Namely
in the sense that arithmetical cognition is partly based on the use of cognitive artefacts –which are
concrete things external to the knowing subject- in epistemic action. Arithmetical cognition is an
interactive process involving the brain, the body, and the social and physical surrounding
environment.242 It is an interplay between internal and external representations.243 The role of
notation affects the process of arithmetical computation, but also the representational system itself
is shaped by the need to solve specific problems in particular environments.244 This means that the
cognitive basis of arithmetic can not only be situated internally (inside the head), but must also be
situated externally (e.g. in the notation system).
In summary, firstly, our ability to know arithmetic is dependent on external representations. These
are social and historical products, originated by the need to handle specific problems in real life
situations. This means that the cognitive basis of arithmetic is for an important part situated
externally. Secondly, the use of epistemic actions and artefacts in arithmetical cognition shows that
239
(De Cruz, Neth, Schlimm, 2010)
A review of studies that demonstrate this is given in (De Cruz, Neth, Schlimm, 2010, pp. 91-94)
241
(Johansen, 2010, p. 187)
242
(Johansen, 2010, p. 179)
243
(De Cruz, Neth, Schlimm, 2010, p. 83)
244
(De Cruz, Neth, Schlimm, 2010, p.96). About the relative young origin of pure arithmetic, developed out of the need to
solve concrete problems in the world, see e.g. (Maddy, 2008) and (Steiner, 2005, p. 627).
240
88
arithmetical cognition is embodied in a concrete sense, which means that the mind is extended into
the physical world while learning arithmetic.
The finding that cognition of arithmetic is dependent on the physical use of concrete external tools,
that are social - and historical products, suggests that learning arithmetic can best be explained as a
process that depends for an important part on the (social) environment. The presented perspectives
above, both locate learning within the interaction between the learning subject and its (social)
environment. The goals of these conceptions of learning is arithmetical knowledge that is
understandable as a notion that is context-sensitive and expressible in terms of skills. This
interpretation of arithmetical knowledge leaves all options open as to which types of skills are aimed
to acquire in education. A non-restrictive view on arithmetic education, saying that education should
put all possible entrances to arithmetical knowledge open, is supported by the definition of
arithmetical knowledge in terms of skills, for the knowledge conditions are expressible in terms of
addressing types of skills. Possible philosophical perspectives on learning which can account for
learning goals in terms of skills can be found in the embodied approach to learning as well as in the
practice-based perspective on this phenomenon. The fact that these perspectives locate learning
outside the subject’s mind, within the interaction between its body and its environment, is supported
by studies on arithmetical cognition which show that epistemic actions and artefacts play an
important role in the cognition of arithmetic.
89
Conclusion
This thesis has presented two guidelines for a philosophical evaluation of educational methods for
teaching arithmetic. The first is the recommendation to assess the conception of the teaching subject
underlying teaching methods. This can be done by placing it in the landscape of possible positions in
the philosophy of arithmetic. This positioning provides a view on the structural problems and
capacities. The second is the recommendation to assess whether specified conditions for learning
arithmetic are accommodated in teaching methods. These are specific points that a good method for
teaching arithmetic should contain. I have proposed to apply both guidelines for the evaluation of
teaching methods.
I claim that the first guideline provides reason and clarity in the disputes on arithmetic education and
that the second guideline provides a common basis for all good teaching methods. An explicit
position provides arguments for the justification of the contents of the textbooks and the relation
between the various topics as well as to teaching methods, instructions and guidance. Furthermore a
positioning in this philosophical landscape exposes the structural questions for that position. The
recommendation to put basic ingredients to all teaching methods are based on a non-restrictive
principle regarding education. Meaning that the conditions for knowing arithmetic, according to all
views on this subject, should be accommodated in teaching methods. The knowledge conditions for
each view on arithmetic can be formulated in terms of skills that must be addressed in order to be
able to gain knowledge of it. The non-restrictive principle holds that education should not exclude
the possibility of any view on arithmetic but instead should offer children all opportunities to access
arithmetical knowledge.
The first eight chapters present the philosophy of arithmetic in eight historical views. These views
concern the conception of the ontology- and epistemology of arithmetic and the resulting views on
the status of the arithmetical truths. The positions are discussed on the basis of these three
dimensions. The dichotomies that I found to play a central role in this discussion form a landscape of
possible positions. Concerning the ontology of arithmetic these are the contrasts: internal versus
external, concrete versus abstract and formal versus meaningful. Arithmetic is either grounded in
the knowing subject or it lies outside in an external world. This can be the physical world or a world
of abstractions. Furthermore arithmetic is conceived as formal, generating procedural knowledge, or
as meaningful, generating knowledge about real objects or about concepts that are meaningful to us.
Concerning the epistemology of arithmetic the contrasts are: a priori versus the a posteriori and
construction/ invention versus discovery. Knowledge of realist arithmetic is characterized as a
discovery as opposed to a construction or an invention. The contrast a priori versus a posteriori
expresses the extent of how dependent arithmetical knowledge is of sensory experience. Concerning
the truths of arithmetic the contrasts are: necessary versus contingent, absolute versus relative and
subjective versus objective. Arithmetical truths are mostly seen as necessities if they are know a
priori, and they are seen as contingencies if they are known a posteriori. As opposed to absolute,
arithmetical truths are seen as relative to the language, to mathematics as a whole, to natural
science or to social practices. If they are viewed from an internalist perspective they are subjective, in
the sense of depending on the knowing subject, and absolute, in the sense of universal, for grounded
in humans (the natural kind or an ideal human being). If they are seen as externally grounded they
are objective, in the sense of being about a given externality.
90
This philosophical systematic overview functions as a map on which a position in this philosophical
discussion can be indicated. A positioning in this landscape uncovers the conception of the subject of
teaching that underlies every particular view on how this should be taught. It clarifies the goal of
education, which is a specific conception of knowing arithmetic. Summarizing the views, knowing
arithmetic is roughly conceived in three ways. Knowing arithmetic is an ability to use an external
formal instrument properly, it is having insight in an external reality (which could be abstract,
physical or social), and it is an ability to construct something (an intellectual construction of forms of
proof or an embodied construction of metaphorical meaning).
All views on arithmetic bring conditions for gaining knowledge of it. These knowledge conditions are
specified for all eight views. They all require that specific invoked faculties are addressed. It requires
a linguist faculty, constructivist intuition, sensory perception, realist intuition, a faculty of pattern
recognition and projection, a faculty to adjust to social practice, reasoning and metaphorical
understanding. Aiming at the acquisition of arithmetical knowledge in education requires that
teaching methods should address the invoked faculties of the learner. The knowledge conditions of
the views on arithmetic imply conditions for learning it.
The conditions for learning arithmetic found, are the following. First, explicit examples of following
procedures are indispensible for learning the syntax of the arithmetical language, that is to learn
which sentences are well-formed and which rules are to be applied. Second, a representation of
arithmetical truths in poor contexts are required in order to invoke the realist intuition of the learner
which is indispensible for reaching immediate insights. Third, concrete physical systems of objects
are needed in order to invoke the faculty of pattern recognition and projection that are indispensible
for gaining knowledge of abstract infinite structures. Sensory perception of the physical systems of
objects must be addressed by a teaching method. Fourth, arithmetical truths and activities must be
represented in the context of the physical reality that needs to be present to the learner as rich as
possible. Rich contexts are indispensible for learning to know arithmetic as a real meaningful activity
in the physical world. Fifth, a teacher’s role is required that is authoritarian in the sense that it takes
responsibility for correcting and warranting learner’s knowledge productions and presenting
arithmetic as practiced in society as clearly as possible. This includes the accepted forms of
arithmetical language, social relationships and roles, forms of communication and accepted ways of
working with material resources and symbolic representations. Sixth, constructivist intuition must be
invoked by addressing the learner’s reasoning skills and putting emphasis on justification and
argumentation for arithmetical truths. The final knowledge condition is that the learner’s basic bodily
experiences with object collection and –construction, with measuring by means of physical segments
and moving along a path, are addressed.
In chapter 9, I have argued for an inclusion of all conditions for learning arithmetic in teaching
methods. This proposal is based on a non-restrictive principle regarding education. Meaning that an
educational method should not exclude the possibility of any serious philosophical view on
arithmetic for this would let a real opportunity to access arithmetical knowledge unused. This
chapter presents both guidelines for assessing teaching methods based on the overview of possible
conceptions of arithmetic and the knowledge conditions for these possible positions, presented in
chapter 1-8. The first is the recommendation to assess the conception of the teaching subject
underlying an educational method by means of a positioning in the overview, for this provides a clear
view on the goals of teaching. The second is the proposal to accommodate all knowledge conditions
91
in teaching methods. Thus besides questioning the conception of the subject of teaching, I suggest
educational methods for teaching arithmetic should be evaluated also with respect to the
accommodation of the knowledge conditions.
In the last chapter, a conception of the notion arithmetical knowledge and learning is found that
support this non-restrictive view on learning arithmetic. This has resulted in a presentation of Löwe
and Müller’s analysis of arithmetical knowledge in terms of skills. Conceiving learning arithmetic as
teaching skills does justice to all views because arithmetical knowledge can be understood in terms
of types of skills according to all views. The proposed definition of arithmetical knowledge is context
dependent in a fundamental way. This indicates possible perspectives on learning enabling us to
explain this phenomenon in terms of skill acquisition resulting in knowledge that is context sensitive.
The embodied perspective on learning as well as the practice-based perspective are mentioned as
possible approaches that support the non-restrictive view on learning arithmetic.
As mathematics education is always about philosophy, disputes on mathematics education are
basically about philosophical questions concerning the nature of the subject, the conception of
mathematical knowledge and learning. The answers to these questions are implicitly present in the
positions in the ‘Math Wars’. I think a picture of these underlying philosophical positions of
educational methods provides a ground for this discussion. The philosophical instrument presented,
is designed to provide this picture, which is indispensible for justification, and additionally it points
out a common basis for all positions, which might be a first step towards rapprochement.
In the following case study I will work out this philosophical instrument on the educational method
‘Realistic Mathematics Education’.
92
Appendixes
93
Case Study: A Philosophical Evaluation of Realistic
Mathematics Education
1. Introduction
The ‘Math War’ in the Netherlands focuses primarily on the opposition Realistic Mathematics
Education (RME) versus Mechanistic Mathematics Education (MME)’. ‘Realistic’ stands for arithmetic
as a meaningful human activity within real contexts (which represent the empirical world). The
learning process is here conceived as an interactive process among learners; a reinvention of
arithmetic guided by a teacher. The aim of the process is understanding. MME represents the
opposite of what RME stands for. Arithmetic is here conceived as the performance of a purely
syntactical outward procedure. And the learning process is focussed on the right performance of this
procedure on paper. The RME proponents argue for keeping the discussion on the basis of scientific
research,245 and point out that the results show that RME leads to better learning outcomes than
MME. But obviously this dispute cannot be decided solely on the basis of empirical test results since
always the question remains what one should test (what the aim is of the education) and what can
be tested (what one actually tests when testing). Divergent answers to questions on the view on the
nature of arithmetic, on arithmetical knowledge and on learning with its goals underlie this
opposition. With respect to the opposites RME and MME, the difference between ‘learning to
understand’ and ‘learning to perform’ indicates a difference in the conception of arithmetic as a
meaningful content and arithmetic as an outward procedure.
In this evaluation of RME I focus on the question concerning the conception of the aim of
mathematics education. By this I do not intend to address the question about the curriculum.246 This
related issue I will hardly discuss. I allude to the questions addressed in the philosophy of
mathematics as presented in the above thesis. This evaluation of RME will be held by means of the
two desired conditions for teaching methods that I have proposed on the basis of the overview of the
philosophy of arithmetic.
The first is a positioning in the landscape of possible views on the nature of the subject of teaching.
The second is the accommodation of the knowledge conditions which are required for the ability to
access arithmetical knowledge conceived in all possible ways: as an ability to perform arithmetic
syntactically, according to the accepted procedures; as an ability to see arithmetical truth as a
meaningful abstract or physical reality; as an ability to participate in arithmetical practices; as an
ability to construct justifications for arithmetical truths intellectually and as an ability to construct
arithmetical meaning metaphorically.
The assessment of RME by means of the two directions comes down to answering: How does RME
picture the nature of mathematics and mathematical knowledge? Does it accommodate the
knowledge conditions of all views on arithmetic? Are there improvements in this method to be
made? Before we proceed to the evaluation, first a description of RME; what are the main principles?
How does it conceives the learning goals and the learning process?
245
(Van den Heuvel-Panhuizen, 2010)
Van den Heuvel-Panhuizen advocates a scientific research based answer to the ‘what’ question although she
experienced that instructional principles are not enough to make decisions about subject matter content and that empirical
data from tests are not enough to make decisions about subject matter content (Van den Heuvel-Panhuizen, 2005, p. 38).
246
95
2. RME in Theory
‘The realistic picture of mathematics
fits without brackets into the world
247
picture’.
According to theoretical underpinning of RME, mathematics is rooted in common sense and reality.
Mathematics is a mental activity which creates certainty (inside the system and outside in the
world).248 Mathematical thought is characterized by the interplay of form and content and starts with
‘common sense’. Common sense could be taken as our preconceptions (and not misconceptions!) of
arithmetical truths. It must be systematized and organized in order to become mathematics. This
systematization and organization provides us grip on reality. In this process school and life
experience must not be separated. Thus also performing algorithms is rooted in common sense and
real life experience.
Mathematics is an active process of mathematizing reality. This process is characterized by Treffers’
term: ‘two-way mathematization’: 249 That is horizontal mathematization -that leads from the world
of life to the world of symbols- and vertical mathematization -that is moving within the world of
mathematics which means that symbols are shaped, reshaped and manipulated mechanically,
comprehendingly and reflectingly-. Streefland calls this difference ‘mathematizing reality’ and
‘mathematizing mathematics’.250 Since mathematics is rooted in reality, horizontal mathematization
proceeds vertical mathematization.
2.1 Main Principles
The main principles251 of Realistic Mathematics Education are:
The activity principle: Mathematics is viewed as an activity: ‘mathematizing something’. This refers
to the interpretation of mathematics as a human activity. Since it is an activity mathematics is never
finished. Doing mathematics is asking for reason, leading to improved versions of common sense.
Pupils are treated as active participants in the learning process. Children need to think.
The reality principle: Freudenthal’s interpretation of the nature of arithmetic starts from his
judgement:
252
‘Mathematics starting and staying in reality’.
Pure and applied mathematics are not viewed as something of a different order. Pure mathematics
stems from applied mathematics -that is mathematics in real life context- and pure mathematics is to
be applied in reality again.
Mathematics is the activity of ‘mathematizing reality’. Application is not only at the end of the
learning process but also at the beginning. Mathematics should be taught in meaningful and
247
(Freudenthal, 1991, p. 136)
(Freudenthal, 1991, pp. 1-19)
249
(Treffers, 1987)
250
(Streefland, 1993, p.111)
251
These principles are taken from (Van den Heuvel-Panhuizen, 2010) And (Freudenthal, 1991)
252
(Freudenthal, 1991, p. 18)
248
96
accessible context, linking problems to reality. The realistic contexts are not meant to hide the ‘real’
problem, instead the mathematical problems themselves are viewed as being contextual.
‘Mathematizing reality’ as a didactical principle needs rich contexts to be structured by the learner.
The bonds with reality (the contexts) are provided by a real life location, a story, a project, a theme
or by clippings. This is required because the learning process itself is worth remembering, since this is
part of the understanding (knowing where something comes from).
The level principle: Mathematizing is modelling that starts from a concrete level, that is a level
where children work with concrete physical materials. Mathematizing is progressive organizing,
schematizing and structuring. This means that informal strategies invented by the learner are
important and progressive schematization should be stimulated. This must lead to the acquisition of
insight into how concepts and strategies are related. Also here, mathematizing is straightforwardly
translated into didacticizing. Therefore the didactical principle says: start teaching following what
students themselves come up with and do! From there gradually work towards a standard method.
The intertwinement principle: Mathematical domains are integrated and also the topics within the
domains are taught in close connection to each other. This stems from the importance of that
mathematics need to stay closely related with reality and common sense. A pupil should learn to
understand mathematics as connected to real life and feel free to apply attainments from one
domain to another in order to get grip on the subject and reality. This supports the development of
common sense, so understanding.
The interactivity principle: Learning mathematics is a social activity. Reflection of thought is
characteristic for mathematical thought and is a forceful motor of mathematical invention. In
didactics it has a level-raising function. So pro ‘whole class teaching’ and contra spending too much
time to individual independent work tasks.
The guidance principle: Students should be provided with a ‘guided’ opportunity to ‘re-invent’
mathematics. Didactics is not implementation, but genesis (also social). Children should repeat the
learning process of mankind. According to Freudenthal not factually but rather as it would have been
done if our ancestors had known more.253 Arguments for this approach: 1. Knowledge and ability
stick better and is more available when acquired by one’s own activity; 2. It is enjoyable, so more
motivating; 3. It foster the attitude of experiencing maths as a human activity.
How should a teacher guide? First she should choose learning situations within the learners current
reality (rich contexts), appropriate for horizontal mathematizing. Second she should offer means and
tools for vertical mathematizing. Third the instruction should be interactive. Fourth she should
improvise to stimulate and follow learners own productions. Fifth she should intertwine learning
strands.
253
(Freudenthal, 1991, p. 48)
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2.2 Learning Goals
The aim of mathematics education is to achieve those mathematical skills that are needed in daily life
and professions as well as to achieve understanding of basic mathematics needed for insight in more
abstract mathematics.254
The main goal is getting the children to think mathematically.255 This is to develop a ‘mathematical
attitude’, expressed in mathematical methods and skills. Freudenthal mentions five big strategies as
‘products of the learning process’:256 1. Developing a mathematical language; 2. Change of
perspective; 3. Grasping the degree of precision that is adequate to a given problem; 4. Identifying
the mathematical structure within a context, if any is allowed, and barring mathematics where it
does not apply; 5. Dealing with one’s own activity as a subject matter of reflection in order to reach a
higher level.
How could this main goal of teaching be conceived according to RME? Must ‘thinking
mathematically’ be conceived as constructing mental concepts? Is the ability to use a mathematical
language the same as the ability to speak about real mathematical objects? Freudenthal is very
explicit about this. According to him the final goal of the teaching-learning-process is to access
‘mental objects’ and ‘mental operations’. It is explicitly not the construction of concepts that are
seen to be about ‘how one conceives of an object’.257 Freudenthal believes that the conception of
mathematical knowledge as a hierarchy of concepts (‘New Maths’) is a contempt of common sense.
By organizing and structuring, rather than forming concepts we get grip on reality.
2.3 Teaching-Learning-Process
The emphasis on process is a didactical principle. The learning process is not viewed as a sequence of
states or products. What matters in the learning process are the discontinuities (the ‘jumps’ or ‘ahaexperiences’). These jumps are changes in perspectives rather than new states of mind. 258
‘Realistic learning strands start with the informal context bound working methods of
children, in their personal reality. From there models, schemes and symbolisations are
developed which serve as intermediaries to gradually bridge the gap between these
start situations and the level of formal, more general subject related operations’.
259
The learning process that is characterized as a guided re-invention is based on a coherent long-term
teaching-learning trajectory that is developed in close connection to the didactical practice.
The focus in the process should be on understanding, this means: practising with understanding and
insight. It is not to say that the one leads to the other. Practising and concept formation are cognitive
processes that support each other.260 Practising with understanding in arithmetic education means
that the emphasis is on semantics more than on syntax, it comes to aiming on the retention of insight
during the learning process. Retention of insight is endangered by premature training, too much
training and training as such, and encouraged by the reflection on the learning process.
254
(Verschoor, 2010)
(Van den Heuvel-Panhuizen, 2010)
256
(Freudenthal, 1991, pp. 122-123)
257
(Freudenthal, 1991, pp. 18-19)
258
(Freudenthal, 1991, p. 94)
259
(Treffers, 1993, p.102)
260
(Nelissen, 2010)
255
98
In the teaching-learning process an interplay between form and content is acted out. The change in
viewpoint from content to form and conversely leads to higher levels. The learner can ‘perform
jumps’ guided by, and not lifted by the teacher!
2.4 Conclusion
The characterization of the learning process as a guided reinvention is based on faith.261 It is the faith
of a realistic picture of mathematics. Mathematical knowledge is objective in the sense that it is
based on ‘common sense’ and reality. Mathematics is excellently suited for gradual rediscovery,
according to Freudenthal, since mathematical activity is ‘special’. It distinguishes itself from natural
science in the sense that it is ‘wis’ and ‘zeker’ (sure and certain) since it has been invented many
times in the world in an independent manner.262 The link between didactizing and mathematizing is
justified by the resemblance of the two.
The character of mathematics as understood according to RME is taken over in didactics. This is
visible in the principles: mathematics is an activity so children are active in the learning process.
Mathematics starts and stays in reality so education is about real life situations and also connected to
other subject areas. It is progressive modelling, organizing, schematizing and structuring and
therefore education is focused on these activities. It is reflection of thought and thus children reflect
on their thoughts and strategies in class. It is an ‘invention’ so is (re)invented in education.
How could these theoretical assumptions (or ‘faith’ according to Freudenthal), be philosophically
accounted for? To answer this question I will position RME in the landscape of the philosophy of
arithmetic as sketched in this thesis by means of the framework of dichotomies.
3. A Philosophical Evaluation
First the subject of teaching is pictured by means of the dichotomies on the three dimensions
discussed above: the nature of arithmetic, knowledge of arithmetic and the status of the truths of
arithmetic. In all three dimensions RME appears to take a clear position on at least one dichotomy.
The nature of arithmetic is clearly seen as meaningful and explicitly not as a formal instrument.
Arithmetical knowledge is a discovery of a reality and also a construction of human common sense.
Anyway it is not an invention that could have been essentially different from what it is at this
moment. The truths of arithmetic have a special status: universal, absolutely sure and certain.
Second, structural problems of divergent explanations for this position are discussed and possible
answers to these problems are suggested. Third, an assessment of the accommodation of the
knowledge conditions. This results in more practical recommendations for RME.
3.1 The Conception of the Subject of Teaching: A Positioning
Arithmetic must always be meaningful. It starts and stays in reality and must always be practiced
with insight into its meaning in reality that is given to us by our common sense. This reflects a clear
position opposite to the formal views on arithmetic. This stand opposes the logical positivist view,
the formalist view and the nominalist views. While according to logicism arithmetic is about logical
concepts, for Freudenthal it is not. Instead it is about mental objects and operations. According to
logical positivism its meaning is only given with the linguistic system that instantiate this meaning.
While according to RME, the linguistic system can be developed by means of common sense in
261
262
(Freudenthal, 1991, p. 136)
(Treffers, 1993)
99
different ways, in different levels, in different languages, while expressing the same meanings about
reality. The meaning of arithmetic is thus according to RME not relative to the linguistic system that
expresses it. Chihara’s nominalist Constructability Theory views arithmetic as purely syntax. It is an
external instrument that could be handy but it has no meaning in itself. And also according to
formalism the numbers are in nature physical symbols that are manipulated according to fixed rules.
Knowing external-formal arithmetic is the ability to use an instrument while according to RME it is
understanding reality with a developed common sense.
Access to arithmetic, that is a formal construction according to the formal views, asks for the ability
to use these systems properly while according to RME it is the re-invention of the meaningful activity
of mathematizing reality. Learning to use an external tool requires explicit training in handling this
given instrument (that might as well have been a different instrument if that would be as handy as
arithmetic is). According to RME it is the other way around. Arithmetic should not be taught in the
sense of being imposed on children. Instead they should be enabled to re-invent it, using their own
constructions and their own ways of reasoning and in their own pace. The faith that these
constructions will, under good guidance, lead to the understanding of the sure and certain truths of
arithmetic underlies RME.
With respect to the invention/construction-discovery –dichotomy it is clear from the above that
according to RME arithmetic is not viewed as an invention in the sense of a formal construction or a
convention that could have been essentially different from what it is now. Freudenthal’s statement
that arithmetic is special in the sense that it has been invented many times over in an independent
manner reflects a belief in an universal character of arithmetical truths. This conception of
arithmetical truth allows for the faith in the ability of children to re-invent arithmetic in an quite
independent manner, by means of their common sense and their natural curiosity about the world
around them. This assumed universal character of arithmetic could either be explained by an
internalist conception of the nature of arithmetic or a realist conception. The first perspective
classifies arithmetical knowledge as a construction while the second views it as a discovery. In the
terminology as used in the philosophy of arithmetic, arithmetical knowledge is thus, according to
RME, suited for re-construction (in the internalist perspective) or for re-discovery (in the realist
perspective).
The explicit special status of the truths of arithmetic is according to RME that they are absolute. From
the above it is clear that they do not solely count relative to the language system that expresses
them. Also arithmetical truth is not exclusively relative to a social practice, natural sciences or to
mathematics as a whole for if one of these options would have been the case then it would not be
intelligible that arithmetic is suited for guided re-discovery or re-construction. If for example
arithmetical truths would be relative to a social practice this practice must precede the meaning of
arithmetic and its truths. Re-discovery or re-construction could only be an intelligible expectation of
children when they first have become familiar with the practice of arithmetic in their social
environment. This practice must somehow first be given to children yet unfamiliar with it. A rediscovery by means of reflection of thought individually and amongst peers, who are unfamiliar yet
with this practice too, is not a suitable method for learning to know a social practice. The relativity of
arithmetical truths to mathematics as a whole in the sense of the structuralist view as expressed by
e.g. Shapiro, is according to RME just half the story. In vertical mathematization mathematics is acted
out as if it is about structures. Arithmetical truth is not only relative to an ante rem structure or
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ultimately to mathematics as a whole, for it is also constituted by the activity of horizontal
mathematization. This leads from the physical world to the world of the mathematical hierarchy of
concepts and is acted out by means of common sense.263 Finally the theory of RME opposes the
relativity of arithmetical truth to natural science as viewed e.g. by Quine. Although arithmetic is
rooted in reality and should be taught in connection to other topics, as expressed by the
intertwinement principle, its truths has a special status in contrast to the truths of natural sciences:
they are sure and certain. This seems to contrast an empiricist view according to which arithmetical
truths are not necessities in a strict sense.
Besides its explicit extreme position with respect to the meaningfulness of arithmetic as opposed to a
formal conception of arithmetic and the absolute status of arithmetical truths opposed to a
conception of arithmetical truth that is conceived as being relative, RME thus moves between the
extremes on the dichotomies ‘contingent-necessary’ that is connected to the dichotomy ‘a priori-a
posteriori’ and the conception of arithmetic as being about abstract - or about concrete objects. Also
it moves within the dichotomy ‘intern-extern’ and the related dichotomy ‘construction-discovery’
and ‘subjective-objective’ truth. Tensions in the possible answers on questions concerning its
position on these dichotomies indicate the structural problems and possible answers that the theory
underlying RME encounters.
3.2 Structural Problems and Possible Answers
The truths of arithmetic are seen as sure and certain. This means that if an arithmetical proposition is
true, it is necessarily true, otherwise it could also have been false. If a proposition is contingently true
the truths would not be called sure and certain. If this is the position of RME concerning arithmetical
truth then this is an a prioritist position in the classical sense of these notions. An external empiricist
conception of the subject arithmetic that is said to start and stay in reality, then must explain how
the truths that are in this view about the physical reality can be necessities. Mill’s conception of
arithmetical laws as laws of (external) nature cannot account for the exceptional status arithmetical
truth is given by Freudenthal compared to the truths of other sciences, such as the natural sciences.
If arithmetical necessities would be natural necessities in Mill’s sense they would not be exceptional
sure and certain.
In order to account for the a priority of arithmetical knowledge RME could in theory turn to a
Platonist conception of its subject or an internalist view on it. Freudenthal’s claim that arithmetic is
about mental objects and mental operations indicates an internalist conception of the subject. The
internalist view on arithmetic is supported by the statement that mathematics is the human activity
of mathematizing. The opportunity to mathematize the physical and social reality roots in common
sense. With common sense one is able to reach mathematics that is about the common external
reality. The RME notion ‘re-invention’ is then a human re-construction. Arithmetical truth is then
subjective in the sense of given with the knowing subject.
Arithmetic could be explained not to be falsifiable by experience and thus being a priori and
necessary if the RME notion common sense is taken as a universal givenness that is fundamental to
263
Shapiro’s empiricist epistemology for his Ante Rem Structuralism could be in line with RME, especially when the faculties
of pattern recognition and projection are seen as given with the knowing subject so that it could account for common
sense. However RME explicitly position itself opposed to Structuralism by identifying this as being solely about vertical
mathematization. See e.g. (Van den Heuvel-Panhuizen, 2010, p. 4).
101
the very possibility of having experiences and thought. If common sense would be conceived as
imposing a structure on our experiences and thought that is so fundamental to it that it would be a
condition for being able to have experiences and thoughts then this could explain that arithmetic is
about the physical real environment and at the same time generates truths that are necessities. This
could be an explanation for the idea that arithmetic is rooted in reality and common sense.
Common sense is according to this interpretation ‘common’ in the sense of universal and also it is
fundamental to all experience and thought. As in the case of the embodied view knowledge of
arithmetic is also grounded in experiences in and of reality. It is therfore gained with recourse to
sensible experience (‘common sense’). This could also mean: ‘that what is common in our sensations’
and could thus be interpreted as an internal position in line with Brouwerian (Kantian) constructivist
intuition. Internal arithmetic could be known by making recourse to constructivist intuition or our
bodily-being-in-the-world. The statement that arithmetic stems from reality and stays in reality
indicates the last explanation. Besides the statement that arithmetic is about mental objects instead
of mental concepts points in the direction of an embodied view rather than an intuitionist view on
arithmetic. Common sense in the sense of how we all basically experience and think is the ground for
arithmetical knowledge. The role of our ordinary sensory/bodily experiences in and with the physical
world is crucial for the possibility to understand arithmetic. Structural problems for this view concern
the explanation of how and why arithmetic is applicable to the external world.
The applicability of arithmetic must be explained by the internalist views. For if the truths are
subjective, how and why are they applicable in the external world? The answer to this internalist
problem formulated in the embodied view is that knowledge of arithmetic is grounded in our
everyday bodily experiences in the physical world around us. This gives a very tight connection to the
world as it is for us and could explain by this that for us arithmetic works in the world as we
experience it. It could be supported by evolutionary theory.
This view raises a new problem that also holds for empiricism, namely: How could a law be entailed
by an empirical experience? Then: either the ‘law’ is a posteriori, (and contingent) or we also must
intuit somehow the necessity of our experience. If the applicability-problem for Internal-Arithmetic is
answered by the claim that arithmetic is grounded in our everyday bodily experiences in the physical
world around us, then the problem occurs, how an empirical experience could generate laws that are
necessities. This problem is shared with the empiricist views. Internal-Arithmetic must then
somehow rely on a given intuition of the necessity in our experiences. The internal base leads to
similar questions because here, arithmetical knowledge is basically subjective and this therefore asks
for a way around scepticism. Thus the empiricist ground for the connection of Internal-Arithmetic
with the external world, leads to doubts about the possibility to account for the assumed character
of necessity of the arithmetical truths. This is the problem for RME: or arithmetical truths are not
universal, necessary, sure and certain or there must be an account for an intuition of the necessity
and universality of our experiences. This problem arises for both explanations of the theory of RME;
either if it grounds mathematics in the empirical reality or if it grounds mathematics in the knowing
subject.
The possible answers on the structural problems bring their own position in the philosophy of
arithmetic with respect to the understanding of our access to arithmetical knowledge. An
explanation for the ability to gain knowledge of arithmetic that is seen as grounded in the knowing
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subject and its basic experiences in and with reality must turn to evolutionary arguments in order to
find a way around scepticism. Innateness and a faculty of metaphorical understanding as a
presupposed structure we impose on our experiences are needed in order to account for the
universality and absoluteness of arithmetical truth from an internalist perspective. If one would turn
to a Platonist position for the explanation of the special status of the truths one would have to rely
on a conception of realist intuition. If one would give up the necessity of the truths one could turn to
an empiricist position and takes recourse to a theory of perception. If one would give up the
absoluteness of the arithmetical truths one could also argue for a social constructivist explanation of
RME if the ‘reality’ in the theory of RME is interpreted as the social reality and ‘common sense’ as
‘sense of the community’. In this explanation one would have to rely on the social practice in order to
be able to gain knowledge of arithmetic. The next direction for evaluation, the accommodation of
the knowledge conditions, further examines these possible interpretations. For it may be expected
that the knowledge conditions that are addressed by RME are especially the conditions of that view
on the subject that reflects the conception of the teaching subject according to RME.
3.3 The Accommodation of the Knowledge Conditions
The knowledge conditions specified in the above thesis contain the requirements for the ability of
having knowledge of arithmetic in all eight views presented. The philosophical position RME takes in
theory brings the expectation that this method accommodates at least the knowledge conditions of
the embodied view. Other possible explanations of the theory of RME, as discussed above, turned
out to be an intuitionist position, a Platonist position, an empiricist position and a social
constructivist view on arithmetic. Are the knowledge conditions of these meaningful-arithmeticviews addressed?
The human faculty to construct or re-invent arithmetic, according to the internalsit perspectives,
must rely on constructivist intuition or on the faculty to construct metaphors. Learning arithmetic is a
natural development. The opportunity to develop these human faculties that are essential to the
understanding of arithmetic is ample provided by stressing the importance to rely on children’s
thinking, that is naturally given, and by encouraging them to develop their own ways of reasoning
while handling a problem. The emphasis on the construction of proof underlying arithmetical
judgements that is expressed in constructivist intuition is accommodated in RME by means of the
attention that is given in class to the evaluations of individual ways of reasoning and solving
problems. Children are encouraged to explain and defend their methods for solving a problem and
evaluate it by thinking about possible alternatives. This is expressed in the interactivity principle. The
emphasis of RME on practicing with understanding and insight is accounted for by an ongoing
connection to meanings within the exercises. All four grounding metaphors are addressed in a variety
of ways. (This is the knowledge condition for the embodied view on arithmetic). Arithmetic is
presented as activities of collecting, constructing, moving along a path and measuring with physical
segments, in many different concrete examples. The mapping in metaphorical understanding is
stimulated in both ways: from the world of numbers to the physical domain and vice versa. The
understanding in children is thus also encouraged by asking them to give pure arithmetical sentences
metaphorical meaning by telling a story, or drawing a picture, that fits it. The role of our ordinary
sensory experiences in and with the physical world around us, that is seen as crucial for the
possibility to understand arithmetic according to the embodied view, is accounted for in RME by
giving children the opportunity to start with working with concrete materials in their
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mathematization of these real situations. It is stressed that this concrete level should not be outrun
for learning arithmetic is seen as starting with experiencing the real physical world.
A Platonist explanation of RME should be able to enable children to rely on and develop some form
of realist intuition. Shapiro showed that a realist conception of arithmetic can also rely on a faculty of
pattern recognition (that is the faculty to abstract a pattern from an observation of systems of
objects) and a faculty of projection (that is the faculty of mentally arranging small finite structures
and abstract a structure from these). The development of these faculties could be seen as being
accommodated by the RME term ‘vertical mathematization’. The norms for the ways in which this
‘moving within the world of mathematics’ should be done is, according to Platonism, given with an
external abstract reality that has to be intuited somehow and is not grounded in the human intellect.
As Gödel said, realist intuition is the opposite of proving. It is seeing something in an immediate way.
To develop this faculty by means of an appeal to natural human reasoning about rich contexts that
represents the physical reality is not an obvious method for developing realist intuition. For example,
becoming to know intuitively that all structures have a distinct successor, stems from a
generalization from particular knowledge about a structure having a successor distinct from it.
Teaching this is making the right circumstances for children to have this immediate insight. This
involves examples that makes the essence that has to be grasped as clear as possible. In this case it
would require examples of structures that are not hidden into a real physical context but that are
instead pre-structured (or ‘poor’ according to RME). Offering poor/ pre-structured contexts and
examples in order to enable children to immediately grasp the essence of an arithmetical insight
could be accommodated in RME within the activities characterized as vertical mathematization.
Optimal circumstances for realist intuition are given with ‘poor’, pre-structured materials and
examples where the essence of what is being taught is tried to be as bold and clearly present as
possible. This is not a traditional ingredient of RME and it might deserve more attention in its
teaching methods.
The empiricist knowledge condition is primarily that exercises must represent real life and real world
problems in rich contexts (which means that the situation is not very pre-structured) for the real
activity of doing arithmetic in the empirical world is what arithmetic is about. Kitcher has noted that
if this view accounts for the empirical genesis of arithmetic in history it does not have to say anything
about how children acquire arithmetical knowledge. According to Freudenthal however there is a
tight connection between the genesis of arithmetic and the way children should ‘re-invent’ it. An
empiricist view on arithmetic reflected on teaching suggest that children come to learn the meanings
of set, number and addition by combining and segregate physical objects. Arithmetical knowledge
springs from our perception of performances of collecting and is extended by a generalization of past
experiences. Learning to know arithmetic thus require experiences in the world with activities of
physical combining and segregating objects. The fact that RME views arithmetic as an activity in the
physical world as it is present to us, is reflected in the use of representations of reality in class that
are as rich and real as possible. As Kitcher remarks the acquisition of empiricist arithmetic can also be
explained in a way that is different from its historical roots. Current arithmetical knowledge can also
be explained as a transmission of knowledge from society to an individual and from one society to its
successor. Knowledge conditions for this perspective are the same as those for Social Constructivism.
According to the Social Constructivist picture what is required for gaining knowledge of arithmetic is
primarily authority from the practice of the mathematical community that includes the accepted
104
forms of arithmetical language, social conventions and forms of communication, ways of working
with materials and the accepted use of symbolic representations. The current mathematical practice
is the norm for the correcting of knowledge productions of children and also for warranting learners
knowledge. Secondly classroom interactions are indispensible for making learning possible because
learning happens on the basis of experiences in these interactions (and the confrontation with a
standard of the practice of interactions in the ‘mathematical community’). Understanding is a
process of becoming part of this community. It is enabled by training in social interactions. This
second part of the knowledge conditions of this view are explicitly accommodated in RME with the
activity principle and the interactivity principle. Also here the perspective on learning is that it takes
place in social interaction and that the goal of learning; understanding, is a process rather than a
state. The first knowledge condition however is not accommodated as long as the guiding role of the
teacher is seen as the role of a midwife who helps the understanding of arithmetic to be born.
According to the Social Constructivist view an understanding of arithmetic will not come to life if the
mathematical practice is not clearly presented to the learner since without it there is no norm for the
knowledge productions of children, including their ways of reasoning. And also, knowledge cannot be
warranted without this norm. The role of the teacher must be crucial in making this norm of the
mathematical practice explicit for it cannot be expected to spring from the individual reasoning of
children or from interactions with peers only. Making the norm of the practice explicit boils down to
giving explicit examples of what the learner is expected to do in a specific situation. The role of the
teacher must be authoritarian in the sense of giving this norm rather than guiding.
The knowledge conditions of the possible philosophical positions of RME are accommodated for the
most part. The accommodation of the knowledge conditions suggests an internalist or empiricist
positioning of RME for these knowledge conditions are extensively incorporated in the method.
However, a Platonist interpretation as well as a Social Constructivist interpretation of RME are less
likely. The guiding teacher role prescribed by RME does not account for the knowledge condition of
Social Constructivism according to which learning arithmetic requires a teacher role that presents the
norm. The emphasis on the use of rich contexts and the resistance to the use of pre-structured
materials and poor contexts makes that realist intuition is not intentionally addressed by RME. These
knowledge conditions (poor contexts and a directive teacher role) are traditionally not
accommodated in RME.
How about the accommodation of the knowledge conditions of the formal views? Does the
philosophical position RME take regarding the conception of its teaching subject bring an exclusion of
the possibility of gaining knowledge of arithmetic taken as formal procedures?
Knowledge conditions for learning how to use arithmetic as an external formal instrument, rest on
some form of constructivist intuition for warranting the formal ways of reasoning and on an invoked
linguist ability. Learning formal arithmetic is to develop a linguist faculty in exercising procedures that
generates knowledge. The ability to use an external instrument asks for training; doing it, or handling
it. Since the instrument is given within conventions, the instrument must be given to the learner and
the way in which it is to be used must be demonstrated. Just like learning a language requires the
input of proper language use, learning formal arithmetic requires the examples of a proper use of
procedures. Lack of procedural training is the most frequent criticism of RME.264 Although RME
264
(Van den Heuvel-Panhuizen, 2010, pp. 2-3)
105
textbooks do include exercises in performing algorithmic procedures, these are intended to have
originated from the children’s own reasoning. This is meant by ‘practicing with understanding’.
Procedures are not given to children in the sense of imposed on them. Children are not asked to
follow procedures, instead they are encouraged to re-invent them. Training in following mechanistic
procedures for the calculation of algorithms is not part of RME for it is seen as endangering the
understanding of the meaning of arithmetic. More recently RME proponents see training in
performing procedures on paper and conceptual understanding as mutually supportive processes.265
Nevertheless mechanistic exercises are not a substantial part of the methods of RME and teaching
the procedures does not involve handing over readymade examples by the teacher.
4. Conclusion
RME is positioned in the philosophy of arithmetic on the side of the meaningful-arithmetic-views.
The most obvious interpretation of its theory is an internal - or an empirical perspective on its
teaching subject. The knowledge conditions of these perspectives have a central place in this
teaching method. Other possible interpretations of the theory of RME, a Platonist - or a Social
Constructivist conception of the teaching subject, are not consistent for the method does not
address all knowledge conditions of these views. A Platonist interpretation of RME’s position requires
more emphasis on the use of poor contexts in the representation of arithmetical problems for this is
more likely to activate realist intuition, that grasps an essence in immediate understanding, than rich
contexts will do. A Social Constructivist interpretation asks for a directive teacher’s role instead of (or
next to) a guiding teacher’s role, for learning a social practice requires that this practice preceeds a
possible understanding of it. A teacher should present this practice to the learners for she is
responsible for the link between the practice in class and the arithmetical practice in which they will
participate.
The internalist or empiricist position RME takes in the philosophy of arithmetic regarding the
conception of its teaching subject, bring the structural problem that the absoluteness and necessity
of the arithmetical truths, assumed by RME, is difficult to explain if arithmetical knowledge is seen as
knowledge of the empirical reality. An internalist perspective can account for the universality and
necessity of the truths but asks for a way around scepticism, for how do we know that these
subjective truths reflects truths about the external world? Or arithmetical truths are not so sure and
certain as Freudenthal believes or there must be an account for an intuition of the necessity of our
experiences.
The second direction for the evaluation of teaching methods, proposed in the above thesis, is that all
knowledge conditions should be accommodated in a teaching method. This principle provides
recommendations for RME. As mentioned, the inclusion of more representations in poor contexts
(pre-structured materials) by means of which the learner’s realist intuition is addressed, and a more
directive teacher’s role by means of which the arithmetical practice can be presented to the learner,
is recommended in order to include the possibility of the Platonist – and the Social Constructivist
view on arithmetic in education. Since Shapiro showed that his Platonist conception of structuralist
arithmetic can rely on empiricist epistemology, realist intuition is not indispensible for knowing it.
The knowledge conditions of the formal views require a inclusion of at least one way of performing
265
(Nelissen, 2010)
106
arithmetical procedures given by the teacher which the learner can follow for one can never learn to
perform conventional procedures without the input of explicit examples.
My proposal to adopt a non-restrictive attitude towards arithmetic education, meaning that
educational methods should not be restricted in the possibilities of transmitting arithmetical
knowledge by fundamental reasons, implies that an educational method should accommodate
somehow all knowledge conditions. For RME this generates the recommendation to include more
explicit examples of procedures to follow on the basis of given rules, to include more pre-structured
representations and a more directive teacher’s role. This recommendation is based on the belief that
one should be pragmatic rather than fundamentalist regarding education.
107
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