Research, Learning and Teaching
David A. Reid
with
Christine Knipping
Acadia University, Wolfville, Canada
Research on teaching and learning proof and proving has expanded in recent decades.
This reflects the growth of mathematics education research in general, but also an
increased emphasis on proof in mathematics education. This development is a welcome
one for those interested in the topic, but also poses a challenge, especially to teachers
and new scholars. It has become more and more difficult to get an overview of the field
and to identify the key concepts used in research on proof and proving.
Proof in Mathematics
Education
Research, Learning and Teaching
David A. Reid with Christine Knipping
David A. Reid with Christine Knipping
This book is intended to help teachers, researchers and graduate students to overcome
the difficulty of getting an overview of research on proof and proving. It reviews the
key findings and concepts in research on proof and proving, and embeds them in a
contextual frame that allows the reader to make sense of the sometimes contradictory
statements found in the literature. It also provides examples from current research
that explore how larger patterns in reasoning and argumentation provide insight into
teaching and learning.
Proof in Mathematics Education
Proof in Mathematics Education
SensePublishers
SensePublishers
DIVS
Proof in Mathematics Education
Proof in Mathematics Education
Research, Learning and Teaching
David A. Reid
Christine Knipping
Acadia University, Wolfville, Canada
SENSE PUBLISHERS
ROTTERDAM/BOSTON/TAIPEI
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TABLE OF CONTENTS
Introduction........................................................................................................... xiii
Part I: What is Proof?
1. History of Proof ................................................................................................... 3
The Standard View ............................................................................................ 3
Other Views of the History of Proof................................................................ 10
Summary ......................................................................................................... 24
2. Usages of “Proof ” and “Proving” ..................................................................... 25
Everyday Usages ............................................................................................. 25
Scientific Usages ............................................................................................. 26
Mathematical Usages ...................................................................................... 26
Usages in Mathematics Education Research ................................................... 27
“Demonstration” and “Proof ” in Other Languages......................................... 32
Summary ......................................................................................................... 33
3. Researcher Perspectives..................................................................................... 35
Philosophies of Mathematics........................................................................... 37
Research Based on an a Priorist Philosophy of Mathematics......................... 39
Research Based on an Infallibilist Philosophy of Mathematics....................... 40
Research Based on a Quasi-Empiricist Philosophy of Mathematics ............... 46
Research Based on a Social-Constructivist Philosophy of Mathematics......... 48
Summary ......................................................................................................... 52
Balacheff’s Epistemologies of Proof ............................................................... 53
Diverse or Comprehensive Perspectives?........................................................ 54
Part 2: Important Research Foci, Past and Present
4. Empirical Results.............................................................................................. 59
Key Studies...................................................................................................... 59
Many Students Accept Examples as Verification ............................................ 59
Many Students Do Not Accept Deductive Proofs as Verification ................... 62
Many Students Do Not Accept Counterexamples as Refutation ..................... 63
Students Accept Flawed Deductive Proofs as Verification.............................. 64
Students’ Criteria for the Acceptance of Arguments ....................................... 65
Students Offer Empirical Arguments to Verify................................................ 67
Most Students Cannot Write Correct Proofs.................................................... 68
Ideas for Research ........................................................................................... 70
5. The Role of Proof ............................................................................................. 73
The Roles of Proof in Mathematics ................................................................. 73
v
TABLE OF CONTENTS
What is Proving in School Mathematics? ...................................................... 79
The Roles of Proof for Students..................................................................... 80
Possible Roles of Proof in Teaching .............................................................. 81
Ideas for Research.......................................................................................... 82
6. Types of Reasoning ......................................................................................... 83
Deductive Reasoning ..................................................................................... 84
Inductive Reasoning ...................................................................................... 88
Abductive Reasoning..................................................................................... 99
Reasoning by Analogy ..................................................................................110
Other Kinds of Reasoning ........................................................................... 123
Summary...................................................................................................... 126
Ideas for research ......................................................................................... 127
7. Classifying Proofs and Arguments ................................................................ 129
Proofs and Arguments Described According to the Representations
Involved ....................................................................................................... 130
Other Classifications of Proofs and Arguments ........................................... 142
Ideas for Research........................................................................................ 151
8. Argumentation............................................................................................... 153
Argumentation Versus Proof........................................................................ 155
Argumentation in Accord with Proof........................................................... 158
Argumentation According to Krummheuer ................................................. 161
Summary...................................................................................................... 163
Argumentation in Japan ............................................................................... 164
Ideas for Research........................................................................................ 164
9. Teaching Experiments ................................................................................... 165
Fawcett......................................................................................................... 165
The Debate Approach .................................................................................. 169
Expecting Explanations ............................................................................... 172
Italy.............................................................................................................. 173
Summary...................................................................................................... 175
Ideas for Research........................................................................................ 176
Part 3: Processes of Reasoning and Argumentation
10. Argumentation Structures.............................................................................. 179
Toulmin’s Functional Model and Argumentation Structures ....................... 179
The Source-Structure ................................................................................... 180
The Reservoir-Structure............................................................................... 185
The Spiral-Structure..................................................................................... 187
The Gathering-Structure .............................................................................. 189
Ideas for Research ....................................................................................... 191
vi
TABLE OF CONTENTS
11. Patterns of Reasoning .................................................................................... 193
Deduce-Conjecture-Test cycle ..................................................................... 194
Proof Analysis.............................................................................................. 198
Scientific Verification .................................................................................. 201
Surrender...................................................................................................... 202
Exception and Monster Barring ................................................................... 204
Summary...................................................................................................... 207
Ideas for Research........................................................................................ 208
Part 4: Conclusions
12. Implications for Teaching...............................................................................211
What is Proof and what is it for? ..................................................................211
Formality...................................................................................................... 212
Results from New Math and Two-Column Proof Teaching ......................... 215
Starting where Students and Teachers are.................................................... 217
Teaching Experiments.................................................................................. 218
Summary...................................................................................................... 219
13. Directions for Research ................................................................................. 221
Teaching Proof............................................................................................. 221
Students’ Understandings of Proof............................................................... 223
Conceptional Issues ..................................................................................... 224
Conclusion ................................................................................................... 225
References............................................................................................................ 227
Author Index ........................................................................................................ 241
Subject Index........................................................................................................ 245
vii
LIST OF FIGURES
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:
Figure 9:
Figure 10:
Figure 11:
Figure 12:
Figure 13:
Figure 14:
Figure 15:
Figure 16:
Figure 17:
Figure 18:
Figure 19:
Figure 20:
Figure 21:
Figure 22:
Figure 23:
Figure 24:
Figure 25:
Figure 26:
Figure 27:
Figure 28:
Figure 29:
Figure 30:
Figure 31:
A misleading diagram showing that 8×8 = 5×13 ................................. 14
Word usage in three ESM papers ......................................................... 29
Three dimensions of description for researcher perspectives............... 36
Diagram from Fischbein, 1982, p. 18................................................... 40
Diagram that formed the basis for the triangle angle sum proof
in Fawcett, 1938 ................................................................................... 42
Reid’s PRISM model............................................................................ 58
The Count the Squares problem ........................................................... 86
Drawing used by Bill and John when proving that the sum of two
odd numbers is even............................................................................. 86
Will’s Count the Squares pattern.......................................................... 94
Diagram used in French classroom for Pythagorean Theorem
proof................................................................................................... 104
Handshake diagram for six people .................................................... 105
Sofia’s diagram .................................................................................. 106
The relationship between analogy, generalisation and
specialisation.......................................................................................113
The Arithmagon problem (as posed by Reid, 1995b)..........................115
Triangles showing geometric properties drawn by Wayne
while exploring ...................................................................................116
A tetrahedron with equal masses at its vertices from Hanna and
Jahnke, 2002b..................................................................................... 122
Toulmin model ................................................................................... 180
Overall argumentation structure in the proving process in
Mr. Lüders’ class ............................................................................... 182
Proof diagram from Lüders’ class ...................................................... 182
Argumentation stream AS-4 from Lüders’ class ................................ 183
Proof diagram from Nissen’s class..................................................... 184
The source-structure in Nissen’s class ............................................... 185
The reservoir-structure in Pascal’s class............................................. 186
Diagram used in Pascal’s class for Pythagorean Theorem
proof................................................................................................... 186
Overall argumentation structure in Dupont’s class............................. 187
Argumentation structure in James’ class for the rhombus
proving process .................................................................................. 188
Argumentation structure in James’ class for the side-side-side
proving process .................................................................................. 190
The Deduce-Conjecture-Test cycle .................................................... 194
The first Arithmagon puzzle............................................................... 194
Sandy’s first guess: 5.......................................................................... 195
Sandy’s solution to the first puzzle..................................................... 195
ix
LIST OF FIGURES
Figure 32: Sandy’s symbols for the unknowns and givens in a general
Arithmagon ........................................................................................ 196
Figure 33: Sandy’s puzzle.................................................................................... 199
Figure 34: Sandy’s reasoning, including Proof Analysis ..................................... 201
Figure 35: Scientific Verification......................................................................... 202
Figure 36: Surrender............................................................................................ 202
Figure 37: Exception and Monster Barring ......................................................... 205
x
LIST OF PROOFS
Proof 1:
Proof 2:
Proof 3:
Proof 4:
Proof 5:
Proof 6:
Proof 7:
Proof 8:
Proof 9:
Proof 10:
Proof 11:
Proof 12:
Proof 13:
Proof 14:
Proof 15:
Proof 16:
Proof 17:
Proof 18:
Proof 19:
Proof 20:
Proof 21:
Proof 22:
Proof 23:
Proof 24:
Proof 25:
Proof 26:
Proof 27:
Proof 28:
Elements Book I Proposition 1............................................................... 5
Figure reconstructed on the basis of Liu Hui’s commentary on
Jiuzhang Suanshu ................................................................................ 12
Elements Book I Proposition 4............................................................. 17
Elements Book IX Proposition 20 ........................................................ 19
Proof that the sum of the interior angles of a triangle is
180 degrees .......................................................................................... 44
Textbook proof from Chazan, 1993, p. 365.......................................... 96
Empirical argument from Chazan, 1993, p. 366................................... 97
The sum of the first n integers (by MI) ................................................ 99
The perpendicular bisectors of a triangle meet in a point....................119
The medians meet in a single point, physics proof ........................... 122
The Goldbach conjecture ................................................................... 131
Product of negatives .......................................................................... 132
Try it with 15...................................................................................... 132
The diagonals of a rhombus are congruent ........................................ 133
There are 14 possibilities and all fit ................................................... 134
Not all prime numbers are odd........................................................... 134
Numeric Gauss proof ......................................................................... 134
Divisibility by Nine............................................................................ 135
Action proof of the commutativity of multiplication ......................... 136
Behold!............................................................................................... 137
Schorle proof...................................................................................... 138
Two-column proof for the triangle angle sum.................................... 139
Symbolic Gauss proof ........................................................................ 140
Infinitude of primes............................................................................ 140
The product of two diagonal matrices is diagonal.............................. 140
Proof of an algebraic identity ............................................................. 141
A formal proof.................................................................................... 142
A transformational proof .................................................................... 148
xi
LIST OF TABLES
Table 1:
Table 2:
Table 3:
Table 4:
Table 5:
Table 6:
Table 7:
Table 8:
Table 9:
Table 10:
Table 11:
Table 12:
Table 13:
Table 14:
Table 15:
Table 16:
Table 17:
Table 18:
Table 19:
Table 20:
Table 21:
Table 22:
Table 23:
Table 24:
Table 25:
Table 26:
Table 27:
References in Euclid’s proof of Prop. I.1 ............................................... 6
Chronology of people and publications mentioned in the text ............... 8
Summary of research perspectives ....................................................... 53
Balacheff’s epistemologies of proof..................................................... 54
Key studies reviewed ........................................................................... 60
Results showing that many students accept examples as
verification. .......................................................................................... 61
Summary of results showing that many students and teachers
do not accept deductive proofs as verification ..................................... 63
Summary of results showing students accept flawed deductive
proofs as verification............................................................................ 65
Factors influencing acceptance of arguments....................................... 66
Frequencies of ratings for the familiar and unfamiliar
deductive verifications ......................................................................... 66
Students’ use of empirical arguments................................................... 68
Students able to write correct proofs .................................................... 69
Students who were able to construct a valid proof
for TIMSS item K18 ............................................................................ 69
Results according to the type and truth value of the statements........... 70
Reasoning dichotomies based on certainty .......................................... 90
Types of abductive reasoning used in examples................................. 103
Bill’s analogy ......................................................................................112
Ben’s analogy......................................................................................117
Some links between triangles and tetrahedra ..................................... 120
Kinds of reasoning ............................................................................. 126
Kinds of reasoning and roles.............................................................. 127
Comparison of Balacheff and the preformalists’ classifications of
arguments ........................................................................................... 129
Classification of arguments ................................................................ 131
Overview of categories ...................................................................... 143
Comparison of proof classification systems....................................... 144
Harel and Sowder’s proof schemes ................................................... 149
Notions of argumentation................................................................... 163
xii
INTRODUCTION
Research on teaching and learning proof and proving has expanded in recent
decades. This reflects the growth of mathematics education research in general, but
also an increased emphasis on proof in mathematics education. This development
is a welcome one for those interested in the topic, but also poses a challenge,
especially to teachers and new scholars. It has become more and more difficult to
get an overview of the field and to identify the key concepts used in research on
proof and proving.
When we met, Christine was working on her doctoral dissertation (Knipping,
2003b). She commented on the difficulty she had making sense of the existing
research. David understood this feeling having had the same struggle when
working on his doctoral dissertation (Reid, 1995b). In the interval the amount of
research to be read and understood had increased, but the relationship between the
work of different researchers was no more apparent. Key terms were used differently by different authors, disparate theoretical assumptions were made, phenomena
were classified in incompatible ways, all without comment. We wished that
some synthesis of the literature existed that would explain the discrepancies and
make the links we found missing. And having achieved a better understanding
ourselves of the literature as a result of our efforts, we wondered if we could
attempt such a synthesis, perhaps in a journal article. In our discussions it soon
became clear that a longer piece of writing would be needed, and this book is the
outcome.
This book is intended to help teachers, researchers and students to overcome the
difficulty of getting an overview of research on proof and proving. It reviews the
key findings and concepts in research on proof and proving, and embeds them in a
contextual frame that allows the reader to make sense of the sometimes contradictory
statements found in the literature.
The first part provides this frame. It begins with an outline of the history of
proof in mathematics, both as it is usually presented and as it is interpreted by
scholars who take a wider view. Then the various uses of the words “proof ” and
“proving” in everyday life, science, mathematics and mathematics education are
described and compared. Finally, the various perspectives taken by researchers in
the field are outlined and placed into a structure that allows for comparison.
The second part reviews current research. First, basic findings from empirical
research are summarised. Then important theoretical constructs and classification
systems are discussed in several chapters organised around the themes of the role
of proof, reasoning, types of proof, and argumentation. Finally, several teaching
experiments are described.
The third part focusses on two larger frameworks for examining proving and
argumentation. The first is argumentation processes which are social processes that
occur in classrooms (and elsewhere) through which knowledge changes status.
A method of describing and analysing argumentation processes is outlined which
xiii
INTRODUCTION
reveals differences in the role of argumentation in different contexts. The second is
reasoning processes, which are internal psychological processes which link together
different ways of reasoning about mathematics. Distinctive patterns in reasoning
processes are described and related to the goals of teaching proof.
The final part includes concluding comments, first on the implications research
on proof has for teaching, and then on questions that require further research.
Throughout there is an emphasis on exploring the multiple perspectives different
researchers bring to the study of proof in mathematics education. These perspectives are seen as being not only inevitable in a field where international attention is
brought to problems that often have significant local elements, but also enriching
as a diversity of perspectives offers opportunities to make sense of phenomena that
might be seen in a limited way from a single perspective. Hence, we do not attempt
a combining of perspectives, as Harel and Fuller (2009) have done. While it can be
confusing to encounter multiple perspectives, we believe it is also very valuable.
We hope this book will decrease the confusion and increase the rewards of its
readers in their further explorations of proof in mathematics education.
xiv
PART I
WHAT IS PROOF?
Reading any research literature can be a challenge at first because most authors
make assumptions about what the reader already knows about the field. This is
necessary as there is never space to include all the background underlying a
publication. In the research literature on teaching and learning proof, assumptions
are often made about the historical context of proof in mathematics, the meanings
of words like “proof ” and “proving” and about the theoretical perspective of the
author, which is often assumed to be shared by the reader. In Part 1 we consider
these three sets of assumptions and provide a guide to what assumptions might be
made by authors in the field. Unfortunately, but perhaps necessarily, there is not a
single uniform set of assumptions all researchers on proof share. Hence, we will
describe a range of possibilities, without being able to state definitively what
assumptions a given publication is based on. Given an outline of the possibilities,
however, a reader should be able to pick up on the clues in a publication and
identify the assumptions being made.
Chapter 1 concerns the history of mathematics, and presents an outline of the
“standard” history of proof in mathematics, familiarity with which is often assumed
when proof is discussed. We also present several alternatives to key elements in the
standard history, that some authors refer to in their work.
Chapter 2 discusses the uses of the words “proof ” and “proving” in mathematics,
mathematics education, logic, science and everyday life. Authors sometimes write
from more than one of these perspectives, which means that their terminology can
shift meaning from one paragraph to the next. Being aware of the possible contexts
and the meanings for “proof ” and “proving” associated with them will help readers
find their way through this shifting terrain.
Chapter 3 explores the theoretical perspectives of researchers on proof in mathematics education. From within a given perspective, it seems a natural way of seeing
things, and so authors often do not comment on the perspective they take. However,
for communication with the larger community some awareness of these perspectives
is necessary, and for a reader new to the field understanding that different perspectives exists will aid in making sense of what sometimes seem to be contradictory
statements.
1
CHAPTER 1
HISTORY OF PROOF
Before embarking on a discussion of proof and proving in mathematics education,
a look back at proof and proving in the history of mathematics is in order. This will
provide the necessary background to some of the issues we will discuss in later
chapters, and introduce some important concepts related to the nature of proof, and
the acceptance of proofs.
A student of modern mathematics might be confused at this point, as her or his
experience of mathematical proof in school might have suggested that mathematics
grows by an accumulation of knowledge, so although there are now new theorems
that have been proven since the time of the Greeks, the word “proof ” in the context
of mathematics has meant the same thing since the time of Euclid, at least. However,
as Wilder (1981) points out: “‘proof ’ in mathematics is a culturally determined,
relative matter. What constitutes proof for one generation, fails to meet the standards
of the next or some later generation” (p. 40). By exploring this variation we can
discover other ways of perceiving proof, and other ways of proving.
As you read this chapter you may want to reflect on this question:
– Does the history of proof in mathematics have direct implications for the
teaching and learning of proof? If so, how?
THE STANDARD VIEW
When one reads a history of mathematics (e.g., Anglin, 1994; R. Jones, 1997; Kleiner,
1991; Kline, 1962), one is likely to encounter a version of the history of proof we
call the “standard” view. When the history of mathematics is mentioned in research
on teaching and learning proof, it is usually the standard view that is assumed, and
so it has had significant impacts on proof teaching and research. In this section we
will summarise the standard view. In the next section we will introduce some
critiques and alternatives.
The First Proofs
According to the standard view, proofs originated with the Greeks, specifically
with Thales (c. 600 BCE). Prior to that time mathematics was done without proofs.
A number of theorems are associated with Thales, not because he discovered them,
but because he proved them:
All these theorems were known to the Egyptians and Mesopotamians. The
reason they are associated with Thales is that he was the first person to offer
proofs for them. This was an essential difference between pre-Greek and Greek
3
CHAPTER 1
mathematics: the Greeks established the logical connections among their results,
deducing the theorems from a small set of starting assumptions or axioms.
(Anglin, 1994, p. 14)
A number of authors have speculated on the reason the Greeks began to insist on
proving mathematical statements. Some (e.g., Hannaford, 1998, p. 181; Kleiner,
1991, p. 293) have claimed that the democratic nature of Athenian society created a
context in which logical argument was valued. Others have noted that the existence
of a leisure class meant that there were individuals who had time for philosophical
and mathematical activity without any immediate practical application (e.g., Kline,
1962, p. 45). Kleiner (1991, p. 293) and Arsac (2007, p. 31) also mention the
problem of the incommensurability of the side and diagonal of a square, and
Kleiner adds the need to teach mathematics, as motivations for an emphasis on
proof. Hanna and Barbeau (2002) see the motivation for proving as arising from
the nature of the entities studied in mathematics:
For the early Egyptians, Babylonians, and Chinese, the weight of observational
evidence was enough to justify mathematical statements. But classical Greek
mathematicians found this way of determining mathematical truth or falsehood less than satisfactory. They saw that, unlike other sciences, mathematics
often deals with entities that are infinite in extent or number, such as the set
of all natural numbers, or are abstractions, such as triangles or circles. When
dealing with such entities, mathematics needs to make absolute statements,
that is, statements that apply to every instance without exception. (p. 36)
Whatever the reason, the origin of mathematical proofs is credited to the Greeks,
whose innovation then spread to other cultures.
The Legacy of Logic
The standard view tells us that rather than basing mathematics on observation and
experiment, the Greeks based it on logic. Not only did they make use of logical
arguments, they reflected on their reasoning and their methods. Plato (427–349
BCE) argued for a restriction on the tools that could be used in geometry, and
Aristotle (384–322 BCE) formulated the methods appropriate to mathematics:
In the Posterior Analytics, Aristotle formulates what we call the deductive
method. It was adopted by Euclid and has always been an essential characteristic of mathematics. This method consists of starting with propositions
called axioms and then proving propositions called theorems. Each statement
in the proof has to be justified either by an axiom or by a previously proved
theorem or by a principle of logic. (Anglin, 1994, p. 63)
For example, consider Proof 1, which is the first proof in Heath’s (1956) translation
of Euclid’s Elements (c. 300 BCE). The text consists of several parts: The proposition
to be proven (line 2), a description of what has to be proven given a specific case
(the segment AB in lines 3–4), a construction (lines 5–12), a proof that the object
constructed is what it is meant to be (lines 13–20), and finally a statement asserting
that what has been done is what was required (lines 21–23).
4
HISTORY OF PROOF
Proof 1: Elements Book I Proposition 1
Proposition 1.
On a given finite straight line to construct an equilateral triangle.
Let AB be the given finite straight line. Thus it is required to construct an equilateral
triangle on the straight line AB.
5
With centre A and distance AB let the
circle BCD be described; [Post. 3]
again, with centre B and distance BA
let the circle ACE be described; [Post. 3]
and from the point C, in which the
10 circles cut one another, to the points
A, B let the straight lines CA, CB be
joined. [Post. 1]
Now, since the point A is the centre of
the circle CDB, AC is equal to AB.
15 [Def. 15] Again, since the point B is the centre of the circle CAE, BC is equal to BA.
[Def. 15] But CA was also proved equal to AB; therefore each of the straight lines CA,
CB is equal to AB.
And things which are equal to the same thing are also equal to one another; [C.N. 1]
therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal
20 to one another.
Therefore the triangle ABC is equilateral; and it has been constructed on the given finite
straight line AB.
(Being) what it was required to do.
Heath, 1956, Vol. 1, pp. 241–242, line numbers adjusted
Both the steps of the construction and the proof are justified by references to
common notions, postulates and definitions that are stated earlier in the Elements (see
Table 1 for those referred to in Proof 1). Euclid’s “common notions” and “postulates”
are assumptions that are to be accepted without justification. Nowadays they would
usually both be called “axioms”. Euclid’s distinction between them is that common
notions apply outside of geometry, while his postulates are specific to geometry.
Euclid’s Elements is a structured presentation of the mathematics of that time.
He did not discover any of the theorems he presented, but he did present them as
part of a larger structure. The Elements provided the model for proof in mathematics,
and in other domains, for centuries.
Euclid’s contribution was the logical organisation of the Elements – its
axiomatic structure in which everything is carefully deduced from a small
number of definitions and assumptions. This structure served as a model for
Aquinas’s Summa Contra Gentiles, for Newton’s Principia, and for Spinoza’s
Ethics. The Elements has been the most influential textbook in history. (Anglin,
1994, p. 81)
5
CHAPTER 1
Table 1. References in Euclid’s proof of Prop. I.1
Common Notion 1
Things which are equal to the same thing are also equal to one
another.
Postulate 1
Let the following be postulated: To draw a straight line from
any point to any point.
Postulate 3
To describe a circle with any centre and distance.
Definition 15
A circle is a plane figure contained by one line such that all the
straight lines falling upon it from one point among those lying
within the figure are equal to one another.
Euclid’s axiomatic approach is not only of historical importance. It has become
a central motif in mathematics. Modern mathematical structures are often
systems based on definition and axioms and relying on rules of inference. ...
In such a system, a proposition is considered true if it can be derived from the
axioms in a finite number of logical steps using the permitted rules of inference.
(Hanna & Barbeau, 2002, p. 38)
Descartes (1596–1650) found the inspiration for his philosophical method in the
Elements. What impressed him was the way Euclid based geometry on a few
axioms and proceeded to deduce further statements about which one could have
absolute confidence.
Those long chains of reasoning, each of them simple and easy, that geometricians commonly use to attain their most difficult demonstrations, have given
me an occasion for imagining that all the things that can fall within human
knowledge follow one another in the same way and that, provided only that one
abstain from accepting anything as true that is not true, and that one always
maintains the order to be followed in deducing the one from the other, there is
nothing so far distant that one cannot finally reach nor so hidden that one
cannot discover. (Descartes, 1637/1993, p. 11)
For thinkers from Aristotle and Descartes to the present day, the deductive
method is associated with certainty.
Euclid regarded his starting assumptions not as mere hypotheses, but as
truths. He intended to instantiate the ideal described by Aristotle as the
beginning of the Posterior Analytics: sure basic knowledge is obtained by the
rigorous deduction of the consequences of basic truths. These truths are either
definitions or existence assertions. (Anglin, 1994, p. 82)
Of all those who have already searched for truth in the sciences, only the
mathematicians were able to find demonstrations, that is, certain and evident
reasons. (Descartes, 1637/1993, p. 11)
6
HISTORY OF PROOF
The Restoration of Rigour
In the standard view European mathematics since the Renaissance is a continuation
of the work of the Greeks. True, there were times when the discovery of new
theorems and methods overtook the task of rigourously proving them, for example,
when Newton and Leibniz introduced the calculus in the seventeenth century. Their
justifications for their methods were criticised as not up to the standard of Euclid
by, for example, Berkeley. This difficulty was addressed finally in the nineteenth
century, by Cauchy and others (see Kleiner, 1991, for an outline of this history and
references to other sources).
At the same time criticisms of lack of rigour in analysis were being addressed,
an important development occurred in the history of geometry: the invention of
non-Euclidean geometries. For Descartes and Kant, Euclidean geometry was an
example of knowledge that was undeniably true. Its foundation was the nature of
space itself. When Lobachevsky, Bolyai and Gauss announced that it was possible
to construct a geometry in which one of Euclid’s postulates (the famous parallel
postulate) is false, it became possible to question the truth of Euclidean geometry.
Of course, there was strong temptation to assume there was something wrong with
non-Euclidean geometry, that there was a contradiction somewhere. In 1871 Klein
eliminated this possibility by proving that if there is a contradiction in the new nonEuclidean geometries then there is also a contradiction in Euclidean geometry. This
created the need for a new approach to securing the foundations of geometry and
the rest of mathematics, as it was not longer possible to convincingly claim that
Euclidean geometry was the true geometry of space (Many histories of mathematics
tell this story. Kline, 1972, is thorough.).
This method of establishing the lack of contradiction (the ‘consistency’) of
one mathematical system by showing it is just as consistent as another system,
was applied not only to geometry. Hilbert showed that the various geometries
were as consistent as basic arithmetic, raising the question of the consistency of
arithmetic. The next step was to try to establish arithmetic on the basis of set
theory, and then to establish set theory on the basis of logic (the logicist approach
of Russell, Whitehead, etc.). In the late nineteenth and early twentieth centuries
this led to a new focus on axiomatisation and the axiomatisation of set theory (by
Frege, Russell and others), geometry (by Hilbert) and arithmetic (by Peano and
others).
The next step was to replace traditional mathematical statements with purely
formal statements that could themselves be the objects of calculation. This was
the objective of Hilbert’s formalism. Mathematics, from a formalist perspective,
is the manipulation of symbols, without any reference to any meaning or
interpretation.
Mathematical proof will consist of this process: the assertion of some
formula; the assertion that this formula implies another; the assertion of the
second formula. A sequence of such steps in which the asserted formulas or
the implications are proceeding axioms or conclusions will constitute the
proof of a theorem. Also, substitution of one symbol for another or a group of
7
CHAPTER 1
symbols is a permissible operation. Thus formulas are derived by applying
the rules for manipulating the symbols of previously established formulas.
(Kline, 1972, p. 1205)
Some key authors and publications in the history of mathematics, and the history of
thought more generally are listed in Table 2, to provide an overview of the standard
version of the history of proof as we have presented it here. Note that there is a
considerable gap between the work of Aristotle and Euclid and the European
philosophers and mathematicians who found those works significant. Authors and
works who are often not discussed when relating the standard history of proof are
omitted here, resulting in this gap. We are not claiming that no mathematics
happened in this period, only that it is usually ignored when the standard version is
presented. Some Chinese works we will discuss later are included as a hint that the
standard version may be incomplete.
Table 2. Chronology of people and publications mentioned in the text
600 BCE
427–349 BCE
384–322 BCE
300 BCE
250 BCE–100
CE
0
200
263
300–600
800
1000
1200
1258–1264
1400
1500–1557
1596–1650
1632–77
1642–1727
1646–1716
1685–1753
1724–1804
1789–1857
1830
1845–1918
1848–1925
1849–1925
1858–1932
1862–1943
1872–1970
8
Thales, first historically recorded proofs
Plato
Aristotle (335 BCE Posterior Analytics)
Euclid’s Elements
Jiuzhang Suanshu (exact date of composition uncertain, and parts may
be much older)
Liu Hui’s commentary on the Jiuzhang Suanshu
Theoretical phase in Chinese mathematics
Thomas Aquinas’s Summa Contra Gentiles composed
Tartaglia
Descartes (1637 Method)
Spinoza (1677 Ethics published posthumously)
Newton (1687 Philosophiæ Naturalis Principia Mathematica)
Leibniz
Berkeley (1734 The Analyst)
Kant
Cauchy
Bolyai and Lobachevsky publish first treatises on non-Euclidean geometry
Cantor
Frege
Klein (1871 Proof that a contradiction in non-Euclidean geometry
implies a contradiction in Euclidean geometry)
Peano
Hilbert (1899 Foundations of Geometry)
Russell (1910–1913 Principia Mathematica, with Whitehead)
HISTORY OF PROOF
Degrees of formality. It is unlikely that the reader will have encountered many
proofs that meet the formalist definition of mathematical proof (an example is
Proof 27 in Chapter 7). However many proofs are called “formal” that do not meet
the formalist definition. This can be confusing when trying to interpret a statement
like “students of mathematics should understand formal proof ” (Moore, 1990, p. 57).
Lakatos (1978) distinguishes between three different degrees of formality in
proofs: pre-formal, formal, and post-formal. Lakatos uses “formal” in the same sense
as the formalists: A sequence of symbols that makes it possible to “mechanically
decide of any given alleged proof if it really was a proof or not” (p. 62). By “preformal” he means a proof which is accepted as such by mathematicians, convincing,
but not a formal proof.
[In a pre-formal proof] there are no postulates, no well-defined underlying
logic, there does not seem to be any feasible way to formalize this reasoning.
What we were doing was intuitively showing that the theorem was true. This
is a very common way of establishing mathematical facts, as mathematicians
now say. The Greeks called this process deikmyne and I shall call it thought
experiment. (pp. 64–65)
A similar description has been adopted by a group of mathematics education
researchers we refer to as “preformalists”. We will discuss their work in Chapters 3
and 7. Note, however, that Lakatos refers specifically to informal proofs acceptable
to mathematicians, and the preformalists include proofs that could be made
acceptable. To avoid confusion we will follow the preformalists’ usage. We will use
“semi-formal” to refer to informal proofs acceptable to mathematicians, instead of
Lakatos’s term “pre-formal” and use the word “preformal” (without a hyphen) to
refer to the proofs discussed by the preformalists.
Lakatos’s “post-formal” proofs are proofs about formal proofs. For example, the
proof of the Duality Principle in Projective Geometry “Although projective
geometry is a fully axiomatized system, we cannot specify the axioms and rules
used to prove the Principle of Duality, as the meta-theory involved is informal.”
(p. 68). Other examples include the consistency and completeness proofs of formal
systems such as Gödel’s proof of his Incompleteness Theorem.
Most proofs are either preformal or semi-formal. However, the influence of the
formalists has made proofs in general more formal, and within the sub-disciplines
of mathematical foundations and computer science formal proofs are the norm.
In the early years of the twentieth century the mathematical community began to
have confidence that the formal structures they were developing would, for mathematics at least, achieve what Leibniz had dreamed of in the eighteenth century, “an
exhaustive collection of logical forms of reasoning—a calculus ratiocinator—which
would permit any possible deductions from initial principles” (Kline, 1980, p. 183).
The standard history of proof suggests that this has, in fact, been achieved.
It is believed that every mathematical text can be formalised. Indeed, it is
believed that every mathematical text can be formalised within a single formal
language. This language is the language of formal set theory. (Davis & Hersh,
1981, p. 136)
9
CHAPTER 1
By formalising mathematics, it was possible to revise the proofs of the past to new
standards of rigour inspired by, but improving upon, Euclid’s Elements.
During the period from about 1821 to 1908 ... mathematicians restored and
surpassed the standards of rigour which had been established during the
period of classical Greek mathematics. (R. Jones, 1996)
Formalist work at the foundations of mathematics inspired the Bourbaki group in
France to apply axiomatic approaches to algebra and analysis, which in turn inspired
some of the reforms of the New Math curriculum reforms of the 1960s. This brings
us to the present day. In the standard view today’s proofs are direct descendants of
the proofs of Euclid, although today’s proofs make more use of symbols to make
formalisation easier. Like Euclid’s proofs they start from axioms and lead to results
that are “true” within the structure defined by the axioms.
OTHER VIEWS OF THE HISTORY OF PROOF
Not everyone accepts the standard view of the history of proof, and alternative
viewpoints have emerged that challenge many claims of the standard view. These
challenges are usually based on historical evidence and sociological analyses. In
this section we will discuss the challenges to several claims in standard view of the
history of proof, including these:
– Proof began in Greece and was limited to cultures with an intellectual connection
to Greece, primarily those in Europe. “For the early Egyptians, Babylonians,
and Chinese, the weight of observational evidence was enough to justify mathematical statements” (Hanna & Barbeau, 2002, p. 36). If one accepts that “the
deductive method. ... has always been an essential characteristic of mathematics”
(Anglin, 1994, p. 63) then one must conclude that what the early Egyptians,
Babylonians, and Chinese did was not mathematics.
– In Euclid’s Elements “everything is carefully deduced from a small number of
definitions and assumptions” (Anglin, 1994, p. 81). Euclid’s proofs are models
of mathematical rigour.
– The work of Russell, Frege, etc. re-established mathematics on firm foundations.
In principle, every mathematical proof can be reduced to a sequence of formal
statements, in which each statement follows from previous statements according
to the rules of symbolic logic.
– Proofs transmit truth from established axioms to the theorems they prove. The
purpose of proofs is to make this connection from the axioms to the theorems.
We will consider each of these beliefs in turn, but first, it is important to note a
feature of the history of mathematics that makes any discussion of specific practices
problematic.
The history of mathematics is spread over a wide time period and a wide range of
cultures, and in many cases the data available is far from ideal. In the cases of Greek
and Chinese mathematics, the original sources are lost, and most of what we know
about them comes from sources written a thousand years after the originals. In the
case of the Greek texts the copies we have come from Arab sources that had their
own rich mathematical culture, which may have influenced the transcription and
10
HISTORY OF PROOF
translation of the texts they had available. We can see this process in more recent
cases; for example, in Heath’s translation of Euclid’s Elements from Greek to English
his footnotes indicate places where he chose to translate passages into formulations
more accessible to contemporary readers, but differing from the original Greek text
(this occurs often; see, for example, the footnotes to Book I Proposition 4, Heath,
1956, p. 248). Aside from purposeful changes to the texts, there are also accidental
changes and missing sections that have forced later translators, transcribers, and
historians to interpolate material that might not be the same as in the original. With
Egyptian and Mesopotamian sources, we are a bit better off in that some original
papyri and cuneiform tablets have survived, but the interpretation of them is a
challenge for experts as the original languages fell into disuse and had to be
reconstructed.
In addition, we have no way of knowing if the mathematical texts we have
from these cultures are representative of their mathematical practices. If one plucked
a book about mathematics at random from all those printed in the twenty-first
century, the likelihood is that one would end up with a school book, as many
more school books are printed than university texts or specialists’ monographs.
Such a school book would hardly represent the present level of development of
mathematics.
The lack of original data is one problem facing historians of mathematics, but
the diversity of the data is another. Anyone attempting to look at the whole picture
is forced to work from secondary or even tertiary sources, as the number of
academics with strong backgrounds in both mathematics and history, and able to
read Arabic, traditional Chinese, ancient Egyptian, classical Greek, Sanskrit and
Sumerian (to name only a few of the languages in which important mathematical
texts have been written) is probably very small. As Smoryński (2008) comments:
The further removed from the primary, the less reliable the source: errors are
made and propagated in copying; editing and summarising can omit relevant
details, and replace facts by interpretations; and speculations can become
established fact even though there is no evidence supporting the “fact”.
(p. 11)
These considerations alone should make one cautious of accepting the standard
view of the history of proof (or any view of the history of proof), and there are also
some other reasons to be wary.
Proof in China
The standard view of the history of proof claims that proof originated in Greece,
and that while Chinese mathematics includes many significant discoveries, the
Chinese did not prove. “Mikami [1913] considered the greatest deficiency in old
Chinese mathematical thought was the absence of the idea of rigorous proof ”
(Needham, 1959, p. 151). Since the 1960s, however, Western scholars have been
aware that there is at least one work of ancient Chinese mathematics in which
proofs play an important role:
11
CHAPTER 1
Proof 2: Figure reconstructed on the basis of Liu Hui’s commentary on Jiuzhang Suanshu
A right triangle has sides of 8 steps and 15 steps. What is the diameter of its inscribed circle?
Compute the Xian from the Gou
and the Gu, then add the three
together and divide this sum into
twice the product of the Gou and
the Gu.
[Find c from a and b (using the
Pythagorean theorem), then add
a + b + c and divide by 2ab (the
area of the two figures to the
right)]
Area
= Twice the
product of the
Gou and the Gu
= 2ab
adapted from Siu, 1993, p. 350 and Martzloff, 1989, p. 150,
whose reconstructions are based on that of Li Huang (?–1812)
[While] most Chinese mathematical works contain no justifications ... there is
one major exception, namely a set of Chinese argumentative discourses which
has been handed down to us from the first millennium AD. We are essentially
referring to the commentaries and sub-commentaries of the Jiuzhang
Suanshu, the key work which inaugurated Chinese mathematics and served as
a reference for it over a long period of its history. (Martzloff, 1997, p. 69)
The Jiuzhang Suanshu (Computational Prescriptions in Nine Chapters) is the oldest
Chinese mathematical text known to us, having been compiled beginning in 200
BCE (Martzloff, 1997, p. 124). The proofs in the commentaries by Liu Hui were
made at the end of the third century, at the beginning of a significant period in
Chinese mathematics:
From the third to the sixth century, Chinese mathematics entered its theoretical
phase. For the first time, it seems, importance was attached to proofs in their
own right, to the extent that trouble was taken to record these in writing.
Approximate values for the number π were then derived by computation and
reasoning, rather than simply via an empirical process. (p. 14)
12
HISTORY OF PROOF
It must be said, however, that Liu Hui’s proofs are not like Euclid’s proofs. For one
thing, Euclid expressed his proofs primarily through words, but Chinese mathematicians made extensive use of diagrams. For example, consider Proof 2, which
shows a visual proof based on the text of Liu Hui’s commentary on problem 16 of
Chapter 9 of the Jiuzhang Suanshu. The original diagrams are, unfortunately, lost.
Siu (1993) suggests a reconstruction of the figures, which we have changed slightly
to make the argument clearer. In modern terms, the theorem proven is:
Given a right triangle with sides a, b, c, the diameter α of the inscribed circle
is 2ab / (a + b + c).
Note that the solution is given in general terms (the words “Gou” and “Gu” are
used to refer to the shorter and longer legs of the triangle, instead of using the
numbers given in the problem) and that it is assumed that the reader knows how to
calculate the length of the hypotenuse (the Xian) from the lengths of the legs (i.e.,
that the reader is familiar with the “Pythagorean” theorem).
This practice of basing proofs on visually convincing diagrams continued when
Euclid’s Elements was translated into Chinese after it was introduced by Jesuit
missionaries. For example, Mei Wending (1633–1721) made changes to Euclid’s
diagrams when he incorporated parts of the Elements into his Jihe bubian (Complements of Geometry).
He modified the figures to make them immediately readable, although Euclid
operated in the opposite direction, thus making it necessary to resort to deductive
reasoning. (Martzloff, 1997, p. 113)
This preference for readable figures over verbal descriptions is one reason why
Chinese proofs are still not accepted as proofs by some historians of mathematics
(Siu, 1993, p. 345).
The use of diagrams is sometimes rejected entirely and misleading diagrams are
given to support a claim that basing proofs on diagrams is not reliable.
The prevailing attitude is that pictures are really no more than heuristic
devices; they are psychologically suggestive and pedagogically important —
but they prove nothing. (Brown, 1999, p. 25)
Philosophers and mathematicians have long worried about diagrams in mathematical reasoning — and rightly so. They can indeed be highly misleading.
(p. 43)
One such misleading diagram is shown in Figure 1.
There are also other reasons beyond the use of diagrams for the perception of
the rarity of proofs in Chinese mathematics. Perhaps the most important is the role
of Chinese mathematical texts as textbooks in established schools.
Under the Sui dynasty (518–617), and above all under the Tang dynasty
(618–907), mathematics was officially taught at the guozixue (School for the
Sons of the State), based on a set of contemporary or ancient textbooks as
written support. (Martzloff, 1997, p. 15)
13
CHAPTER 1
Figure 1. A misleading diagram showing that 8×8 = 5×13.
Indeed, to a first approximation, a majority of Chinese mathematical works
may be better represented as pedagogical tools, in other words, didactic aids
used to teach numerical computation, together with prescriptive texts (for
example, user manuals). (p. 47)
As with problem based textbooks today, the intent was that the students should
work out the proofs for themselves. In his commentary on Jiuzhang Suanshu, Liu
Hui “associates in the same sentence two famous passages from the Lunyu
(Confucian Analects) which both suggest an idea of the same order:”
I told him what had gone before and he understood what followed: [I]
showed him a corner [i.e. an aspect of the question] and he replied with the
other three.
Here the author indicates his wish not to disclose all the details of his
reasoning to the student.... Consequently, instead of giving the details of his
own thought processes, he often merely indicates that the solution of a given
problem is analogous to that of some other problem, or that “the remainder
follows in the same way” (Martzloff, 1997, p. 70, bracketed insertions in
quotation from original)
Anyone who has studied from a contemporary university mathematics textbook
will be familiar with hints like “the remainder follows in the same way.” They
often appear in exercises in which new theorems are stated without their proofs,
which are left to be discovered by the students. Herbst (2002b) describes how such
exercises came to be included in high school geometry textbooks in the late
nineteenth century.
A characteristic of Chinese culture that may also have affected the nature of
Chinese proofs was the emphasis in Chinese literary style on conciseness.
Such a knowledge excluded literary ornaments and excessively long passages
such as those found in Euclidean theorems and proofs. In particular, syllogisms
and other logical forms were especially unacceptable for they involved numerous
14
HISTORY OF PROOF
repetitions and redundancies: they were contradictory with the canons of
Chinese literary redaction which, in the case of technical subjects, valued
conciseness above all. (Martzloff, 1997, p. 112)
It is also worth remembering that in Western countries there have been times when
mathematicians, far from publishing proofs of their theorems, kept their results
secret (for example, Tartaglia). The same phenomenon occurred in China:
Last but not least, assuming that an inventor had succeeded in creating novel
procedures, it is not certain that he would have been inclined ipso facto to
reveal the secrets of their creation; in fact, the existence of rivalry between
calendarist astronomers is known. (Martzloff, 1997, p. 49)
In summary, the belief that the practice of proving mathematical results began in
Greece and spread from there to other, primarily European, cultures, is a myth.
Proving was also a part of Chinese mathematics from the third century, and possibly
earlier, but in a different style than Euclid’s. That many historians based a belief
that proof was not a part of Chinese mathematics on their lack of awareness of Liu
Hui’s commentaries is a useful reminder that absence of evidence is not the same
thing as evidence of absence. This might suggest that we approach with caution the
belief that proof was not a part of mathematics in other cultures where significant
mathematical discoveries were made (for example, Egypt, Mesopotamia, and India).
As an illustration of this point we will now briefly consider proof in India before
moving on the to the claim that Euclid’s proofs are models of rigour.
Proof in India
The standard history of mathematics makes claims about proof in India that are
similar to those made about China. For example, in describing Hindu mathematics
in the period 200–1200 CE Kline (1972) writes:
There is much good procedure and technical facility, but no evidence that
they considered proof at all. They had rules, but apparently no logical scruples.
(p. 190)
Joseph (1992, 1994) critiques this claim, pointing out that, as in China, proofs
(“upapatti”) were often included in commentaries on mathematical texts, even if they
were not a part of the texts themselves. There is, however, an important difference
between upapatti and Euclid’s proofs:
The upapattis of Indian mathematics are presented in a precise language,
displaying the steps of the argument and indicating the general principles which
are employed. In this sense they are no different from the “proofs” found in
modern mathematics. But what is peculiar to the upapattis is that while presenting the argument in an “informal” manner (which is common in many mathematical discourses anyway), they make no reference whatsoever to any fixed
set of axioms or link the given argument to “formal deductions” performable
with the aid of such axioms. (1992, p. 194; see also 1994, p. 200)
15
CHAPTER 1
It is this lack of reference to axioms that distinguishes the upapattis of India
from Euclid’s proofs. This is related, as Joseph notes, to a focus on rigour and
certainty that is important to the Euclidean approach as opposed to an emphasis on
understanding and clarity that is present in the Indian tradition. We will encounter
similar differences in the role of proof in subsequent chapters (especially Chapter 5).
Now, however, we will turn to the question of whether Euclid’s proofs are as
rigourous and his theorems are established with certainty as the standard history of
proof claims.
The Euclid Myth
In the standard view Euclid’s proofs are taken to be models of mathematical rigour,
that establish theorems with certainty. While there is no denying that Euclid’s
Elements has had a profound influence on Western mathematics, the claim of
certainty has been questioned:
What is the Euclid myth? It is the belief that the books of Euclid contain
truths about the universe which are clear and indubitable. Starting from selfevident truths, and proceeding by rigorous proof, Euclid arrives at knowledge
which is certain, objective, and eternal. Even now, it seems that most educated
people believe in the Euclid myth. Up to the middle of the nineteenth century,
the myth was unchallenged. Everyone believed it. (Davis & Hersh, 1981,
p. 325)
There are two aspects to this “myth”. One is the assertion that Euclid’s proofs are
rigourous, and the second that the knowledge arrived at using the deductive method
is certain and objective. Here we will discuss the first of these aspects, rigour.
Later we will consider the second aspect in the context of twentieth century
mathematics.
Rigour in Euclid’s proofs. Do the proofs in the Elements live up to the claims
sometimes made about them, that “everything is carefully deduced from a small
number of definitions and assumptions” (Anglin, 1994, p. 81)? In fact, they do not.
Euclid’s proofs make use of assumptions that are never stated, some involve
reference to physical manipulations (as in Liu Hui’s proofs) and some use specific
cases to justify general conclusions.
Since Euclid still has popularity, and even with mathematicians, a reputation
for rigour in virtue of which his circumlocution and longwindedness are
condoned, it may be worth while to point out, to begin with, a few of the
errors in his first twenty-six propositions. (Russell, 1903/1937, p. 404)
Recall, for example, the first proof in Book I, the construction of an equilateral
triangle (see Proof 1, on page 5). Each step in the construction (lines 5–12)
indicates the postulate that states that such a construction is possible, and each step
in the proof (lines 13–22) indicates which definition, postulate or common notion
justifies that step in the argument.
16
HISTORY OF PROOF
Proof 3: Elements Book I Proposition 4
5
10
15
20
25
30
Proposition 4.
If two triangles have the two sides equal to two sides respectively, and have the angles
contained by the equal straight lines equal, they will also have the base equal to the
base, the triangle will be equal to the triangle, and the remaining angles will be equal
to the remaining angles respectively, namely those which the equal sides subtend.
Let ABC, DEF be
two triangles having
the two sides AB, AC
equal to the two
sides DE, DF
respectively, namely
AB to DE and AC to
DF, and the angle
BAC equal to the
angle EDF. I say that the base BC is also equal to the base EF, the triangle ABC will be
equal to the triangle DEF, and the remaining angles will be equal to the remaining
angles respectively, namely those which the equal sides subtend, that is, the angle ABC
to the angle DEF, and the angle ACB to the angle DFE.
For, if the triangle ABC be applied to the triangle DEF, and if the point A be placed on
the point D and the straight line AB on DE, then the point B will also coincide with E,
because AB is equal to DE.
Again, AB coinciding with DE, the straight line AC will also coincide with DF, because
the angle BAC is equal to the angle EDF; hence the point C will also coincide with the
point F, because AC is again equal to DF. But B also coincided with E; hence the base
BC will coincide with the base EF.
[For if, when B coincides with E and C with F, the base BC does not coincide with the
base EF, two straight lines will enclose a space: which is impossible. Therefore the base
BC will coincide with EF] and will be equal to it. [C.N. 4]
Thus the whole triangle ABC will coincide with the whole triangle DEF, and will be
equal to it. And the remaining angles will also coincide with the remaining angles and
will be equal to them, the angle ABC to the angle DEF, and the angle ACB to the angle
DFE.
Therefore etc.
(Being) what it was required to prove.
Heath, 1956, Vol. 1, pp. 247–248, line numbers adjusted
Notice, in line 9, the mention of “the point C, in which the circles cut one another.”
From the diagram it is clear that there are two such points. That Euclid seems to
claim that there is only one is perhaps a minor flaw. He is only trying to prove that
it is possible to construct an equilateral triangle; that the construction might
produce two does not make it invalid.
More significantly, Euclid does not provide a common notion, postulate, or
definition to let us know when we can actually construct a “point C, in which the
circles cut one another.” While these circles intersect, it has not been established
17
CHAPTER 1
under what conditions the intersections will exist, or that these circles satisfy these
unstated conditions. Of course, from the diagram it is clear that C exists, but the
claim made in the Euclid myth is that Euclid reasons “from a small number of
definitions and assumptions” not from diagrams.
We turn now to a proof that, like Liu Hui’s proofs, makes use of physical
manipulations. It is Euclid’s proof of Proposition 4 of Book 1 (see Proof 3). One
thing that is interesting about the proof (lines 19–32) is the lack of references to
common notions, postulates or definitions. In fact, the only such reference (in line 28)
is thought to be a later interpolation (Heath, 1956, p. 249). This is not surprising
when one considers that the whole argument depends on the idea of picking up one
triangle and putting it on top of the other one. The phrase “if the triangle ABC be
applied to the triangle DEF ” (line 19) suggests that ABC be moved so that it
coincides with DEF.
This way of reasoning is not what Euclid is supposed to have done, but it is
quite similar to a way of reasoning used by Liu Hui:
Thus, the argumentation inevitably depends on methods. For example: ...
Recourse to non-linguistic means of communication. This is necessary
because, according to the adage of the Yijing cited by the commentator [Liu
Hui], “not all thoughts can be adequately expressed in words” ... In place of a
discourse, the reader is asked to put together jigsaw pieces, to look at a figure
or to undertake calculations which themselves constitute the sole justification
of the matter at hand. In each of these cases language is purely auxiliary to
such procedures. (Martzloff, 1997, pp. 71–72)
In Euclid’s proof we are asked to make ABC coincide with DEF in our imaginations,
and then to note the correspondences Euclid points out. This is easy to do, and
quite convincing, but it is not the deductive method as described by Aristotle.
To be fair, Euclid did not reason in this way very often, and it is not, in fact,
possible to deduce this proposition from his common notions, postulates and definitions, so he had to depart from the deductive method, or change his postulates.
Hilbert took the latter approach in his Grundlagen der Geometrie (Foundations of
Geometry, 1899/1921) and added this proposition as an axiom.
Reasoning from visual evidence was a mainstay of Chinese and Hindu mathematics, but fell out of fashion in Greece. There is evidence, however, that it was the
basis for Greek mathematics as well for some time. Euclid’s proof of Proposition
I.4 is part of this evidence. And, as Martzloff (1997) points out:
The Greek technical term meaning “to prove” is the verb δείκνυμι. Euclid
uses this at the end of each of his proofs. Originally this verb had the precise
meaning of “to point out,” “to show” or “to make visible.” Thus it appears
that the Chinese proofs of Liu Hui and Li Chunfeng were similar in nature to
the first known historical proofs, an example of which is given by Plato
(well-known dialogue in which Socrates asks a slave how to double the area
of a square); moreover, visual elements remained an essential component of
proofs in China for a long time, while in Greece these were abandoned at an
early stage although figurative references were maintained. (pp. 72–73)
18
HISTORY OF PROOF
Proof 4: Elements Book IX Proposition 20
5
PROPOSITION 20.
Prime numbers are more than any assigned multitude of prime numbers.
Let A, B, C be the assigned prime
numbers; I say that there are more
prime numbers than A, B, C.
For let the least number measured by
A, B, C be taken, and let it be DE; let
the unit DF be added to DE. Then EF is either prime or not.
First, let it be prime; then the prime numbers A, B, C, EF have been found which are
10 more than A, B, C.
Next, let EF not be prime; therefore it is measured by some prime number. [VII. 31]
Let it be measured by the prime number G. I say that G is not the same with any of the
numbers A, B, C.
For, if possible, let it be so. Now A, B, C measure DE; therefore G also will measure
15 DE. But it also measures EF. Therefore G, being a number, will measure the remainder,
the unit DF: which is absurd.
Therefore G is not the same with any one of the numbers A, B, C. And by hypothesis it
is prime.
Therefore the prime numbers A, B, C, G have been found which are more than the
20 assigned multitude of A, B, C. Q. E. D.
Heath, 1956, Vol. 2, p. 413, line numbers adjusted
Martzloff’s comment on the abandonment of visual elements by the Greeks is part
of the standard view of history. Netz (1998, 1999), however, has suggested that the
visual element was not so much abandoned as hidden by the Greeks. He finds
evidence for this in the abundance of points that are undefined except by the diagrams
accompanying the proofs, and by the lettering of the diagrams which suggests that
the proof was created before it was written down.
Netz describes a three stage process:
These three stages are:
(i) drawing a diagram;
(ii) a dress rehearsal in front of the diagram, in which the diagram is dressed,
i.e. letters are inserted
(iii) a full production, writing down the proof. (1998, p. 36)
If Netz is correct, Greek proofs were basically visual and oral, with the written
proof being a record of what came before. We have been equating Greek proofs
with written texts, not because they were, but because all the evidence that survived
was the written text.
We now turn to Euclid’s famous proof of the infinitude of primes (see Proof 4).
In the Elements it has a different form from that usually given in textbooks of
number theory. For an example of a modern version, see Proof 24 in Chapter 7.
19
CHAPTER 1
There are two things of interest about this proof. First, the diagram is not needed,
but it is there anyway. This is true in general of Euclid’s proofs in what we would now
call number theory. Netz (1998) explains the presence of diagrams in these contexts.
He notes that the Greek word “diagramma” refers to more than “a diagram.”
It means something much closer to “a proposition” or “a proof” (see Knorr,
1975, pp. 69–75). This is a notorious fact about Greek practice: it is generally
difficult to tell whether the authors speak about drawing a figure or proving
an assertion, and this is because the same words are used for both. And this
again is because the diagram is the proof, it is the essence of the proof for the
Greek, the metonym of the proof. (Netz, 1998, pp. 37–38)
So while Greek proofs are often taken as the model of modern discursive proofs,
for the Greeks themselves they were fundamentally, essentially, associated with
pictures. This explains why diagrams were included in proofs like Proof 4 which
from a modern perspective do not need diagrams. To the Greeks, if it did not have a
diagram, it was not a proof. The presence of diagrams where they are not needed
may also be related to the origins of proofs as visual arguments, noted above.
The second thing of interest about this proof is that it does not, strictly speaking,
establish that there are an infinite number of prime numbers. What it shows is that
if there are three prime numbers, then there must be four prime numbers. This
specific case is used to stand for all cases, a technique which is also common in
Chinese proofs.
Thus, the argumentation inevitably depends on methods. For example: ...
Passage from the particular to the general, based on a specific, well-chosen
example. (Martzloff, 1997, p. 71)
This method of proving, known as using a generic example (see Chapter 7), is
unavoidable if one does not have a method of representing unspecified numbers
symbolically. Euclid had one such method, representing a number as the length of a
line segment, but he did not have a method for representing an unspecified number
of numbers as he had to do in this proof.
To summarise, Euclid’s proofs do not rigourously use deductive reasoning to
derive propositions from axioms (common notions, postulates), definitions, and
previously established propositions. They use implicit axioms, non-verbal arguments,
and generic examples. This undermines the claim that they establish the propositions they prove with certainty. Nonetheless, they have been the model and measure
of proofs in the Western mathematical tradition for thousands of years. In Chapter 5
we will revisit the role of proving in mathematics and explore some reasons why
the flaws in Euclid’s proofs were not considered serious (or even noticed) until the
beginning of the twentieth century, and why they are still being offered as model
mathematical proofs (e.g., by Hanna & Barbeau, 2002).
The Twentieth Century: Formalism to the Rescue?
The standard view of the history of proof acknowledges that there were some difficulties with less than rigourous proofs at times, especially in the seventeenth century.
20
HISTORY OF PROOF
But the severity of these difficulties is sometimes minimised, and the success of the
efforts to solve them overrated. According to the standard view, even if Euclid’s
proofs are flawed, they were a step in the right direction, and since the work of the
formalists in the early twentieth century, mathematics has once again been placed
on firm foundations. Now, in principle, any mathematical proof can be expressed in
purely formal statements, which can then be checked mechanically, with no chance
of error due to missing assumptions, unclear definitions, use of diagrams, or logical
mistakes. But this is never done, and not only for pragmatic reasons.
An ordinary page of mathematical exposition may occasionally consist entirely
of mathematical symbols. To a casual eye, it may seem that there is little
difference between such a page of ordinary mathematical text and a text in a
formal language. But there is a crucial difference which becomes unmistakable
when one reads the text. Any steps which are purely mechanical may be
omitted from an ordinary mathematical text. It is sufficient to give the starting
point and the final result. The steps that are included in such a text are those
that are not purely mechanical — that involve some constructive idea, the introduction of some new element into the calculation. To read a mathematical
text with understanding, one must supply the new idea which justifies the
steps that are written down. (Davis & Hersh, 1981, p. 139)
The missing steps would first have to be supplied before the proof could be
formalised. This would have to be done by an expert in the field, and even if an
expert could be found with the patience for such a task, there would be no guarantee
that the translation of the proof into a formal language would be free of error. The
problem of checking the correctness of the proof becomes the problem of checking
the correctness of the translation into formal language, and that is not formalisable.
The actual situation is this. On the one side, we have real mathematics, with
proofs which are established by “consensus of the qualified.” A real proof is
not checkable by a machine, or even by any mathematician not privy to the
gestalt, the mode of thought of the particular field of mathematics in which
the proof is located. Even to the “qualified reader,” there are normally
differences of opinion as to whether a real proof (i.e., one that is actually
spoken or written down) is complete and correct. These doubts are resolved
by communication and explanation, never by transcribing the proof into firstorder predicate calculus. Once a proof is “accepted,” the results of the proof
are regarded as true (with very high probability). It may take generations to
detect an error in a proof. If a theorem is widely known and used, its proof
frequently studied, if alternate proofs are invented, if it has known applications
and generalisations and is analogous to known results in related areas, then it
comes to be regarded as “rock bottom.” In this way, of course, all arithmetic
and Euclidean geometry are rock bottom.
On the other side, to be distinguished from real mathematics, we have “metamathematics” or “first-order logic.” As an activity, this is indeed part of real
mathematics. But as to its content, it portrays a structure of proofs which are
21
CHAPTER 1
indeed infallible “in principle.” We are thereby able to study mathematically
the consequences of an imagined ability to construct infallible proofs (Davis &
Hersh, 1981, pp. 354–355)
This leaves us with two sorts of mathematics: first-order logic in which it is possible
to begin with formal axioms and definitions and derive results using formalised
logical rules without fear of error, and what Davis and Hersh call “real mathematics”
in which we begin with axioms and definitions that are expressed in a mixture of
formal and informal language, and derive results using informal logical rules, with
an ever present possibility of error. In the first sort proofs are formal, while in the
second they are semi-formal. One thing these two sorts of mathematics share with
each other, and with the standard view of Euclid’s Elements, is starting from
axioms and definitions that are established beforehand, and on which the truth
(whether absolute or relative, certain or probable) rests. This aspect of the standard
view has also had its critics, as we will see in the next section.
Lakatos and the Retransmission of Falsity
Lakatos (1961, 1976) describes the process by which he claims mathematics is
discovered. In his work he criticises the Euclidean structure of definitions,
postulates, theorems and proofs and the formalist reduction of mathematics to
formal logic.
The history of mathematics and the logic of mathematical discovery, i.e. the
phylogenesis and the ontogenesis of mathematical thought, cannot be developed
without the criticism and ultimate rejection of formalism. (1976, p. 4)
We will concentrate here on Lakatos’s critique of one aspect of the standard view
of proof, the claim that proofs are based on axioms and definitions that are
established beforehand, that it is the (assumed) truth of the axioms that is the basis
for the claimed truth of theorems. As Lakatos notes, this basing of theorems on
axioms is reflected in the presentation of mathematics in textbooks and journals.
Euclidean methodology has developed a certain obligatory style of presentation.
I shall refer to this as ‘deductivist style’. This style starts with a painstakingly
stated list of axioms, lemmas and/or definitions. The axioms and definitions
frequently look artificial and mystifyingly complicated. One is never told how
these complications arose. The list of axioms and definitions is followed by
the carefully worded theorems. These are loaded with heavy-going conditions;
it seems impossible that anyone should ever have guessed them. The theorem
is followed by the proof. (p. 142)
According to Lakatos, the order in this presentation is almost entirely the reverse of
mathematical practice. Rather than beginning with axioms and definitions, he says,
mathematicians begin with conjectures. After a conjecture comes a proof, but a
proof is not a guarantee of the truth of the conjecture. Instead it is “a rough thoughtexperiment or argument, decomposing the primitive conjecture into subconjectures
or lemmas” (p. 127). Proving is a means of analysing the conjecture, part of a
22
HISTORY OF PROOF
process he calls ‘proof-analysis’. The next stage in the process is the emergence of
counterexamples to the conjecture. These counterexamples can reveal problematic
definitions and hidden assumptions. Lakatos divides them into three types: The
first is a counterexample to some step of the proof, but not to the conjecture itself
(It is local but not global.). The second is a counterexample to some step of the
proof, and to the conjecture (It is both local and global.). The third type does not
contradict any step of the proof, and yet it is a counterexample to the conjecture
(It is global but not local.). Each type plays a different role in the proof-analysis
(p. 43). A first-type counterexample signals that there is a problem with the proof;
either a hidden assumption must be revealed, or a definition changed, or a new
proof produced. A second-type counterexample is the most important type for
proof-analysis. When a second type counterexample emerges, the next step is to reexamine the proof to locate the step to which it is a local counterexample, the
“guilty lemma”.
This guilty lemma may have previously remained “hidden” or may have been
misidentified. Now it is made explicit, and built into the primitive conjecture
as a condition. The theorem — the improved conjecture — supersedes the
primitive conjecture with the new proof-generated concept as its paramount
new feature. (p. 127)
The process of proof-analysis is not primarily about proving the conjecture that
was its beginning, but rather improving the definitions and axioms on which it is
meant to be based. “Proof-generated concepts” are important original contributions
to mathematics. They account for the facts that “axioms and definitions frequently
look artificial and mystifyingly complicated” and that theorems “are loaded with
heavy-going conditions” (p. 142).
Counterexamples of the third type exist only if the proof analysis is invalid.
A proof-analysis is ‘rigorous’ or ‘valid’ and the corresponding mathematical
theorem true if, and only if, there is no ‘third-type’ counterexample to it. I call
this criterion the Principle of Retransmission of Falsity because it demands
that global counterexamples be also local: falsehood should be retransmitted
from the naive conjecture to the lemmas, from the consequent of the theorem
to its antecedent. (p. 47)
The Principle of Retransmission of Falsity is very important to Lakatos’s thinking,
and sums up what may be the most important critique in his work of the standard
view of proof. In the standard view, truth is transmitted from axioms to increasingly
complicated theorems. Lakatos claims that this is impossible, but more importantly,
that this does not reflect the way mathematics really works. Mathematics progresses
by the retransmission of falsity from conjectures to axioms and definitions. In this
way counterexamples to conjectures reveal problems with the axioms and definitions.
Many concepts in mathematics have existed since before the time of Euclid, but
these concepts are now much more sophisticated, because conjectures based on
them turned out to give rise to counterexamples which forced (because of the
Principle of Retransmission of Falsity) changes to be made to the concepts.
23
CHAPTER 1
According to Lakatos, Euclid’s presentation of geometry seriously distorted the
nature of mathematical discovery and the role of proof in it by inverting the order
of things and hiding the importance of conjectures and counterexamples.
One should not forget that while proof-analysis concludes with a theorem, the
Euclidean proof starts with it. In the Euclidean methodology there are no
conjectures, only theorems. (p. 107, footnote 3)
This Euclidean version of proof is an integral part of the standard view of the
history of proof, and so it has had significant impacts on the teaching of proof, which
we will discuss in later chapters.
SUMMARY
In this chapter we have summarised what we call the standard view of the history
of proof, and described some important limitations and flaws of this view. Most
notably:
While many sources claim that proof originated in Greece and was not a part
of the intellectual activity of other cultures, there is clear evidence of proving
in ancient China and India, and it is possible that proving was part of
mathematics elsewhere, in spite of the absence of evidence.
Euclid’s proofs are said to be models of rigour, however they make use of
unstated assumptions and evidence from diagrams.
It is believed that mathematical proofs are (or can be made) formal, and that
this means they are absolutely rigourous. In fact, formalisation of most proofs
is not possible, and proofs can only be checked by a “qualified reader”.
Proofs are said to transmit truth from established axioms to the theorems they
prove, but as Lakatos points out, the process can go the other way; proofs
allow us to locate hidden assumptions and flawed axioms by retransmitting
falsity from a conjecture with counterexamples to the underlying definitions
and axioms.
Euclid himself could never have imagined the consequences of his effort to
systematise the mathematics known in his day, and so it is unfair to blame him for
the confusion resulting from the standard view of proof. As his name keeps coming
up, however, it is convenient to use labels like Davis and Hersh’s “Euclid myth”
and Lakatos’s “Euclidean methodology” to describe this point of view. And as long
as we are clear that we are speaking of a particular perspective, held by many
people, even today, and not of a long dead mathematician, we would agree with
Lakatos that:
Euclid has been the evil genius particularly for the history of mathematics
and for the teaching of mathematics, both on the introductory and the creative
levels. (p. 140)
24
CHAPTER 2
USAGES OF “PROOF” AND “PROVING”
The words “proof ” and “proving” are used in everyday life, mathematics, and mathematics education in a number of distinct ways, usually without comment. For researchers in mathematics education this can lead to confusion and may be a serious
obstacle to future research (Balacheff, 2002/2004; Reid, 2005). Without trying to
establish the “right” usages of these words, we will outline here some frequent
ones and describe the differences between them.
As you read this chapter you may want to reflect on these questions:
– What does “proof ” mean to you?
– What should “proof ” mean to students in schools?
– How can you determine what an author means by “proof ”?
EVERYDAY USAGES
In everyday English, “proof ” and “proving” can refer to convincing someone of
something, or to testing something to see if it is correct.
Convincing
When we doubt a statement, we may ask, “Do you have any proof of that? Can you
prove it?” In these questions proof means evidence, and proving means convincing.
When Shakespeare’s Othello says, “Be sure of it; give me the ocular proof ” (Act III,
scene 1) he means that Iago must convince him of the truth of his accusation by
providing visible evidence. What counts as convincing evidence depends on context,
and may include physical force, verbal abuse, social pressure, or anything else that
persuades someone else. In the Sidney Harris cartoon captioned “You want proof ?
I’ll give you proof !” the humour comes from a shift in context, as one mathematician
is shown convincing another mathematician by punching him in the nose, which is
an everyday, but not a mathematical usage of “proof ” as convincing.
Testing
“Prove” is derived from the Latin verb probare, which means to test, to try. The
English verb “probe” still carries this meaning. Taking “prove” as meaning “convince” when it means “test” can lead to odd interpretations of common expressions.
For example, the expression “the exception which proves the rule” is often taken in
the paradoxical sense of asserting that the presence of a counterexample establishes
25
CHAPTER 2
the general truth of a rule, which follows if “prove” is taken to mean providing
convincing evidence. However, the expression is not so paradoxical if “prove” is
being used to mean “test”. Then saying “the exception proves the rules” amounts
to suggesting that examining exceptions closely and reasoning out the way they
occur can lead to a clarification and improvement of the rule. This interpretation
is reminiscent of Lakatos’s (1976) process of proof-analysis in which counterexamples and proving interact to improve theorems in mathematics (see Chapters 1
and 11).
The use of “prove” to mean “test, try” can also occur in the noun form; a
“proof ” can be a test or a trial. In some common phrases, “proof-read,” “proof of
the pudding,” “100 proof,” the word “proof ” is used in this way. Words like
“waterproof” and “fireproof” are also based on this meaning; they describe objects
that have been tested and found to be resistant.
SCIENTIFIC USAGES
When one reads an article about a scientific discovery, one might encounter the
words “proof ” and “proving” used to refer to convincing, but on the basis of
special types of evidence.
Experiments Prove Existence Of Atomic Chain ‘Anchors’
Atoms at the ends of self-assembled atomic chains act like anchors with
lower energy levels than the “links” in the chain, according to new measurements by physicists at the National Institute of Standards and Technology
(NIST).
The first-ever proof of the formation of “end states” in atomic chains may
help scientists design nanostructures, such as electrical wires made “from the
atoms up,” with desired electrical properties. (NIST, 2005, italics added)
When scientists “prove” something they offer convincing evidence, but that evidence
must be of a special type appropriate to science.
MATHEMATICAL USAGES
Godino and Recio (1997, Recio & Godino, 2001) make a distinction between two
usages of the words “proof ” and “proving” in two areas of mathematics: foundations of mathematics and mainstream mathematics. This distinction is similar to
the distinction made by Douek (1998) between “formal proofs” and “mathematical
proofs”, the distinction made by Davis and Hersh (1981) between metamathematics
and “real mathematics”, and our distinction between formal proofs and semi-formal
proofs which we mentioned in Chapter 1.
In foundations of mathematics, proofs give theorems “a universal and intemporal
validity”, “they rest on the validity of the logic rules used,” “the use of formal languages is required,” and proving is a way of coming to grips “with the theoretical
26
USAGES OF “PROOF” AND “PROVING”
problem of organizing and structuring the system of mathematical knowledge”
(Godino & Recio, 1997, pp. 315–316). Hanna (1983) defines formal proofs in this
way:
The term rigorous proof or formal proof ... is understood here to mean a proof
in mathematics or logic which satisfies two conditions of explicitness. First,
every definition, assumption, and rule of inference appealed to in the proof
has been, or could be, explicitly stated; in other words, the proof is carried out
within the frame of reference of a specific known axiomatic system. Second,
every step in the chain of deductions which constitutes the proof is set out
explicitly. (p. 3)
In mainstream professional mathematics, theorems do not have the “character of
absolute and necessary truths”, the validity of proofs is “‘judged by qualified
judges’ (Hersh, 1993, p. 389)”, “proofs are deductive but not formal,” and proving
is a way “to solve new problems, to increase the knowledge body, and, secondarily,
to organize and found the whole system of mathematics” (Godino & Recio, 1997,
pp. 316–317).
Mathematicians working in the foundational domain of proof theory recognise
this distinction, and use “formal proof” to refer to the proofs they study, and
“social proof ” to refer to the proofs of mainstream mathematicians. In Chapter 1
we suggested the adjective “semi-formal” to refer to these mainstream proofs. Davis
and Hersh (1981) note that although an “ideal” mainstream mathematician might
claim his semi-formal proofs meet the same criteria as those of mathematicians
working on foundations, when pressed, he would admit the differences:
What you do is, you write down the axioms of your theory in a formal
language with a given list of symbols or alphabet. Then you write down the
hypothesis in the same symbolism. Then you show that you can transform the
hypothesis step by step, using the rules of logic, till you get the conclusion.
That’s a proof. ... Oh, of course no one ever really does it. It would take
forever! ... [A proof is really] an argument that convinces someone who
knows the subject. (pp. 39–40)
USAGES IN MATHEMATICS EDUCATION RESEARCH
Many researchers in mathematics education use the words “proof ” and “proving”,
in a number of distinct ways. Most use the words in different ways within the same
paper.
For example, consider this sentence:
Even when students seem to understand the function of proof in the mathematics classroom ... and to recognise that proofs must be general, they still
frequently fail to employ an accepted method of proving to convince themselves
of the truth of a new conjecture, preferring instead to rely on pragmatic methods
and more data. (Hoyles & Küchemann, 2002, p. 194, references removed for
clarity, italics added)
27
CHAPTER 2
The use of the singular form “proof ” in the first line instead of the plural “proofs”
suggests that the word is being used to mean a concept or category. In the second
line the plural is used, suggesting that a set of objects is meant. Finally in the third
line, the verb “proving” is used. The fact that “method of proving” seems to include
“accepted methods” as well as “pragmatic methods” suggests that “proving” is used
to mean something different than “constructing a proof” in this case.
This suggests a starting point for an investigation of the usage of the words
“proof ” and “proving” in research in mathematics education. Three categories of
usage can be distinguished on purely grammatical grounds:
1) The use of “proof ” in the singular, without an article to refer to a concept.
2) The use of “proof ” with an article or in the plural to refer to an object.
3) The use of the verb “prove” to refer to an action or process.
Note that “proving” is a difficult case, as it can be a form of the verb “prove” but
also a noun: “Jim is proving the theorem” or “Jim’s proving of the theorem”.
Considering word usage in mathematics education research even at the surface
level of the forms of words reveals some striking differences. For example,
consider the frequency of the use of the words “proof ”, “proofs”, “prove” (including
“proves” “proven” and “proved”), “proving”, words beginning with “argu+”
(“argument”, “arguing”, “argue”, etc.) and “reasoning”. In Figure 2 the frequency
of the usage of these words in three papers published in Educational Studies in
Mathematics is shown. The left hand column shows an example of an author
(Fischbein, 1999) who uses the verb “prove” more often than the nouns “proof ”
and “proofs”. In contrast, the right hand column shows an example of an author
(Uhlig, 2002) who uses the nouns much more than the verb. It is clear from the
centre column that Hanna (2000) uses the word “proof ” much more than “prove”,
but it is not clear whether she means a concept or an object when she writes
“proof ”. A closer look at the article clarifies this. Hanna’s use of “proof ” breaks
down into four categories:
–
–
–
–
“proofs” in the plural form, 32 occurrences, 18%
“proof ” preceded by “a”, 17 occurrences, 10%
“proof ” preceded by “the”, 7 occurrences, 4%
other uses of “proof ”, 122 occurrences, 71%
The final category still contains a few uses of “proof ” to refer to an object (for
example when it is preceded by an adjective, e.g., “an explanatory proof”), but
most uses of the word refer to a concept.
In the following we will go into more detail about the ways mathematics
education researchers use “proof ” and “proving” to refer to a concept, an object, or a
process. Note, however, that we do not claim that any researcher’s usages fall neatly
into a single category, nor that the usages we describe here are themselves disjoint
categories. As the quote from Hoyles and Küchemann at the start of this section
indicates, several usages can occur in a single paragraph. And while the three ESM
articles analysed in Figure 2 show the predominance of some usages over others,
almost all usages appear in all three articles.
28
USAGES OF “PROOF” AND “PROVING”
Figure 2. Word usage in three ESM papers
Proof as a Concept
The use of the word “proof ” to refer to a concept is usually clear from the context
or from syntactical considerations, but once one knows that the word is intended to
refer to a concept, does one know to what concept it is meant to refer? Unfortunately,
no. Researchers in mathematics education have a wide range of perspectives on
proof which make it difficult to know what concept they might mean by the word
“proof ”. In the next chapter we will describe some researchers’ perspectives, but
as many researchers do not provide enough clues in their writing to definitively
identify their perspectives, we can only leave the reader with the advice to be wary.
Proof as an Object
There are a number of different objects “proofs” can refer to in mathematics education
research, and those objects can be distinguished by their forms or by their function.
The two most common usages are to refer to texts, usually written texts, of a certain
form, or to refer to arguments, spoken or written, with the function of convincing.
Proof-Texts.
The majority of the [high attaining 14 and 15 year old] students were unable
to construct valid proofs in [the domain of number and algebra]. (Healy &
Hoyles, 2000, p. 425)
29
CHAPTER 2
In mathematics schoolbooks and journals, one encounters some texts under the
heading “proof ”. Such proof-texts are characterised by a particular form and style.
The proof-texts of schoolbooks are different from the proof-texts of professional
mathematics journals, but there is sufficient unity in the styles to justify the use
of the same term for both. Writing proof-texts is a goal of recent reform documents:
“High school students should be able to present mathematical arguments in written
forms that would be acceptable to professional mathematicians” (National Council
of Teachers of Mathematics [NCTM], 2000, p. 58). That students cannot do this is
what is meant when researchers such as Duval (1990), Senk (1985) and Healy and
Hoyles (2000) conclude that students do not understand proof.
“Proving” can refer to writing a proof-text (e.g., Douek, 1998) but not everyone
who uses “proof ” to refer to proof-texts uses “proving” in this way.
Convincing arguments. The use of “proof ” to refer to a convincing argument by
mathematics education researchers is essentially a return to the everyday usage we
noted above. But the audience to be convinced can vary. For example, for Mason,
Burton, and Stacey (1982) a proof is an argument that convinces an enemy, for
Davis and Hersh (1981) it is an argument that convinces a mathematician who
knows the subject and for Volmink (1990) it is an argument that convinces a
reasonable sceptic. In all cases, it is not the argument itself that makes it a proof,
but rather that fact that it convinces someone. As Manin points out, in this often cited
quotation, “A proof becomes a proof after the social act of ‘accepting it as a proof’.
This is true of mathematics as it is in physics, linguistics, and biology.” (1977,
p. 48). Using proof this way necessarily means whether a given proof-text is a
proof or not can vary. “Proof is that which compels belief. That means that proof is
different in different eras, and indeed, that it is different for different people at any
one time.” (MacKernan, 1996, p. 14).
If “proof ” is used to mean a convincing argument, then “proving” usually refers
to convincing someone of something. This usage is compatible with some ways of
using “proof ” is used to refer to a reasoning process or a social discourse (see
below).
Proof as a Process
“Proof ” can refer to a psychological process of reasoning, or to a social process, a
certain kind of discourse. In both cases, (as with proof as an object) what process is
being referred to can be determined both by the form of the process, and by its
function.
Deductive reasoning. “In fact, ‘proof’ is just ‘reasoning’, but careful, critical
reasoning looking closely for gaps and exceptions” (Hersh, 2009, p. 19). When
“proof ” refers to deductive reasoning it is being used to refer to a psychological
process which takes on a certain form. Because psychological processes are not
directly observable, specifying this form is difficult. It can be loosely described as
a chain or tree of connected statements beginning from some that are taken as true
30
USAGES OF “PROOF” AND “PROVING”
and proceeding to a conclusion according to a few logical rules (modus ponens,
etc.) occasionally supplemented by special rules unique to mathematics (e.g., the
principle of mathematical induction; see Chapter 6 for details). Some authors (e.g.,
Reid 1995b) use the verb form “proving” to refer to reasoning deductively, but use
the noun form “proof ” in another way.
Reasoning for a purpose. For some researchers, “proof ” refers to a reasoning
process, but not necessarily a deductive one. For Harel and Sowder, for example,
“the emphasis is on the student’s thinking rather than on what he or she writes”
(Harel & Sowder, 1998, p. 276) but it is not the nature of the reasoning process that
is important, but instead the function that it serves.
By “proving” we mean the process employed by an individual to remove or
create doubts about the truth of an observation. (p. 241)
In Harel and Sowder’s case, that function is verification of the truth of a statement.
They are careful to distinguish between “proving”, “proof ” and what they call
“proof schemes” but all are related to the purpose of verifying. “Proving” is a
mental act, “the process of removing or instilling doubts about an assertion”
(Harel, 2007, p. 65). “Proof ” is “a particular statement one offers to ascertain for
oneself or convince others” (p. 66), a proof-text. A “proof scheme” is a characteristic
“way of thinking associated with the proving act” (p. 66).
Another function of proof as a process of reasoning is understanding. Raman
seems to use “proof ” in this way, when referring to “the private aspect of proof”:
I distinguish between a private and a public aspect of proof, the private being
that which engenders understanding and provides a sense of why a claim is
true. The public aspect is the formal argument with sufficient rigor for a
particular mathematical setting which gives a sense that the claim is true.
(Raman, 2002, p. 3; see also 2003, p. 320)
Raman’s “public aspect of proof” seems to refer to proof as an object, either prooftexts or convincing arguments.
Discourse defined by function. Knipping (2004, p. 73) discusses “collective proving
processes” or “argumentations” which are “collective processes in which students
and teacher develop the proof together.” The “collective proving process” she
refers to are embedded in a “proving discourse”. This discourse is a social process
whose function is producing reasons for the truth of a statement.
Balacheff (1988b) seems to use “proof ” in the same sense when he writes:
Le passage de l’explication à la preuve fait référence à un processus social
par lequel un discours assurant la validité d’une proposition change de statut
en étant acceptée par une communauté [The transition from explanation to
proof refers to a social process by which a discourse asserting the truth of a
proposition, gives it the status of being accepted by the community.] (p. 29)
Consider, for example, a class attempting to decide the truth of the proposition
“The centre of gravity of a triangle is at the intersection of the medians”. Conjectures
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are voiced and reasons given. Through a process of social negotiation (probably
guided in significant ways by the teacher) an argument is produced that verifies the
truth of the proposition. Note that the criteria for accepting an argument depend on
the class (including the teacher) or more generally on the community. Arguments
might be accepted by some community that would be rejected by others.
Discourse defined by form: Deductive discourse. When deductive reasoning is
expressed in a social context it becomes a method of arguing. Above we noted that
the kind of arguments accepted by a community is community dependent. Some
communities, notably communities of mathematicians, insist on a deductive basis
for acceptable arguments. In such communities if “proving” refers to a collective
process, it will only be used to describe processes with a deductive basis.
To return to the example of the centre of gravity of a triangle, a possible argument
would make use of cut-out triangles of various sizes and angles and empirical
testing of the locations of their centres of gravity. Such an approach might be
acceptable in a physics classroom, but not in a mathematics classroom. In the
mathematics classroom the argument would have to proceed deductively, perhaps
by establishing that each median divides the triangle into two equal areas, that the
three medians meet in a single point, and finally that any line through this point
will divide the triangle into equal areas (However, see Proof 10 in Chapter 6 for a
proof that bridges these two contexts).
“DEMONSTRATION” AND “PROOF ” IN OTHER LANGUAGES
As the research literature on proof in mathematics education includes significant
contributions in languages other than English it is worth being aware of some
issues related to word usage in other languages.
In older English language texts on proof and proving in mathematics education
(e.g., Fawcett, 1938) one encounters the word “demonstration” used to refer to mathematical proofs. This word is now rarely used in this sense, more often being used to
refer to a political protest or a presentation intended to show how something works.
In Romance languages, however, cognate words (e.g., démonstration, dimostrazione,
demostración) continue to be used, and Balacheff, for example, makes a distinction
between “démonstration” and “preuve” (the French cognate of “proof ”).
Nous appelons preuve une explication acceptée par une communauté donnée
à un moment donné. ... Au sien de la communauté mathématique ne peuvent
être acceptées pour preuve que des explications adoptant une forme particulière.
Elles sont une suite d’énoncés organisée suivant des règles déterminées: un
énoncé est connu comme étant vrai ou bien est déduit de ceux qui le précèdent
à l’aide d’une règle de déduction prise dans un ensemble de règles bien
défini. Nous appelons démonstrations ces preuves. (1987, p. 148)
[We call proof an explanation accepted by a given community at a given
moment... Within the mathematical community only explanations adopting a
particular form can be accepted as proofs. They are an organized succession
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USAGES OF “PROOF” AND “PROVING”
of statements following specified rules: a statement is known to be true or is
deduced from those which precede it using a deductive rule taken from a well
defined set of rules. We call such proofs “démonstrations”. ]
When writing in English Balacheff attempts to preserve this distinction by translating
“démonstration” as “mathematical proof”.
By “proof ” we mean a discourse whose aim is to establish the truth of a
conjecture (in French: Preuve), not necessarily a mathematical proof (in
French: Démonstration) (Balacheff, 1991b, p. 109, Note 2)
But most English writers do not use “proof ” and “mathematical proof” in the same
way as Balacheff does, and many authors writing in French, Italian and Spanish
do not make the same distinction between “preuve”, “prova” and “prueba” and
“démonstration”, “dimostrazione”, and “demostración”.
In German the situation is like that of English. “Proof ” can be translated as
“Beweis” and the German word “Demonstration” means approximately the same
thing as the English word “demonstration”.
SUMMARY
The words “proof” and “proving” can be used in a number of ways, even in an
academic discipline like mathematics education where the exact meanings of these
words would seem to be important. As Herbst and Balacheff (2009) note, ignoring
these multiple usages can lead to a deadlock in efforts to communicate. But the
answer is not to insist on one “correct” usage.
If the field is in a deadlock as regards to what we mean by “proof,” we
contend this is so partly because of the insistence on a comprehensive notion
of proof that can serve as referent for every use of the word. ... We have
argued that to make it operational for understanding and appraising the mathematics of classrooms we need at least three meanings for the word. (p. 62)
We have identified a number of usages in this chapter:
– A concept of proof
– Proof-texts
– Convincing arguments
– Deductive reasoning
– Personal verification
– Personal understanding
– A social discourse to verify
– A deductive social discourse
These usages are not disjoint categories, nor does a researcher’s use of “proof ” or
“proving” in one way in one context guarantee that her or his next usage will be the
same. However, being aware that there are different usages is an important step to
being able to decipher mathematics education research.
In our writing we will attempt to use more precise words to say what we
mean, reserving the word “proof ” primarily to refer to a concept. However, to
avoid unnecessary repetitions we may use “proof ” and “proving” in one of their
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other senses when the meaning is clear. Similarly, when quoting others we will
clarify how these words are being used if possible and necessary. If the meaning is
sufficiently clear from the context, or if the meaning is so unclear we cannot
determine what it is, we will not attempt to suggest how the author is using “proof ”
and “proving”.
Word usages can also offer important hints towards larger issues. In the next
chapter we will use three of these usages, proof-texts, reasoning, and discourse, to
distinguish between theoretical perspectives in mathematics education research.
34