Iconic Proofs more geometrico
Timm Lampert
Abstract
Proofs more geometrico, i.e., proofs in the Euclidean manner, are often
regarded as paradigmatic of the axiomatic proof method. This paper
investigates the suitability of this assessment by comparing the axiomatic
and iconic proof methods. It is argued that Euclidean proofs should be
regarded as a model for an iconic rather than an axiomatic method of
proof. Newton’s experimental proofs and Wittgenstein’s logical proofs are
revealed to be further examples of a non-axiomatic, iconic proof method in
the manner of Euclid’s proofs. By referring to these prominent examples,
the paper aims to demonstrate the significance of an iconic method of
proof and to explain its underlying differences from an axiomatic method
of proof.
1
Introduction
Euclid’s Elements (written circa 300 BC) is acknowledged as a model for scientific proof. This is particularly true in modern times. However, I will not discuss
the influence of Euclid’s Elements on the scientific revolution of modern times.
In particular, I will not reference prominent advocates of rationalism, such as
Spinoza, Descartes, Leibniz and Wulff, all of whom referred to Euclid.1 Instead,
I will discuss Newton’s experimental proof method, which can be seen as an application of Euclid’s method to experimental physics. I do this to illustrate the
significance of a non-axiomatic understanding of Euclid’s proof method with
respect to proofs that extend beyond pure geometric proofs. I am obliged to
ignore many historical and interpretational details that should be discussed if
one wishes to substantiate interpretational issues in the first place. However,
my aim is to substantiate a non-axiomatic proof method with respect to proofs,
presented within different contexts and with prominent advocates from different
ages, that seem, at first glance, to exemplify axiomatic proofs. Thus, I hope
to stimulate (i) further discussion regarding the extent to which an iconic understanding of these proofs is relevant to interpretational questions and (ii) a
discussion of the force and potential of an iconic proof method based on relevant
case studies.
Axiomatic proofs prove propositions (theorems) by deducing them from basic assumptions (axioms) by means of nothing but logical (truth-preserving)
1 Cf.,
for example, Arndt [2] and Schueling [31].
1
rules. According to this understanding, proofs are sequences of sentences expressing logically related propositions that are either true or false. It is the
truth of the content of the sentences (i.e., the truth of propositions) that is
in question. According to a modern understanding of axiomatic proofs, the
sentences that constitute a proof may be formalized to prove their deductive
relations on purely syntactic grounds. However, the proof still proves the truth
of the propositions based on the truth of axioms. Thus, a formalized proof
requires the interpretation of its formalizations. A proof is not identical to a
formal deduction; instead, it consists of such a deduction plus the interpretation
of the related formulas. An axiomatic proof, therefore, consists not of formulas
but of sentences expressing propositions. The representation of sentences in the
form of formulas plus interpretation merely distinguishes the syntactic from the
semantic aspect of sentences. A formalization of axiomatic proofs is part of a
rational reconstruction of those proofs; it is not essential for the proofs themselves. By contrast, referring to the content of sentences is essential to axiomatic
proofs because it is the truth of this content that is in question.
According to the understanding of iconic proofs established in the following,
iconic proofs, by contrast, do not consist of sequences of sentences expressing
propositions.2 Iconic proofs do not prove the truth of the content of sentences
based on the truth of certain foundational assumptions. Regarding the proofs
that are considered as iconic proofs in this paper, the constituents of these
proofs are not sentences but diagrams, in the case of Euclid’s geometric proofs3 ;
experiments, in the case of Newton’s experimental proofs; or logical diagrams,
in the case of Wittgenstein’s logical proofs. These objects are the objects that
are manipulated within a proof, in accordance with certain rules, in order to
2 In positioning iconic proofs in opposition to symbolic proofs, specifically axiomatic proofs,
I refer to Shin [32], who, in turn, refers to Peirce, and to Lampert [13], who argues that
Wittgenstein can be regarded as an advocate to the iconic proof method as well. My emphasis
that the iconic proof method is distinct from the axiomatic method in that it does not depend
on semantics is due to Wittgenstein. For example, I interpret Wittgenstein’s dictum in the
Tractatus that proofs show something without saying something as expressing this distinction
(cf. Wittgenstein [39], 6.1254[1], 6.1265, 6.127f.; cf. also Wittgenstein [41], p. 124, letter 68).
3 Since the “diagrammatic” or “iconic turn”, many studies have addressed the
use of diagrams in proofs; cf.
the survey and bibliography presented at
http://plato.stanford.edu/entries/diagrams/. Discussions of the probative force of diagrams
within Euclid’s proofs play a prominent role in the relevant literature; cf., in particular, Manders [17], Manders [18], Miller [21], Mumma [22], Giaquinto [5], and Mumma [23]. Hinitkka
/ Remes [8], in chapters 4 and 7, distinguish a “propositional interpretation” of the analytic method applied in geometric proofs from an “instantial” or “figural interpretation”; cf.
also section 5. According to the propositional interpretation, geometric proofs are performed
based on sentences, whereas according to the instantial interpretation, geometric proofs are
performed based on geometrical figures or diagrams. The arguments presented by those authors for a figural interpretation also support the view held in this paper. However, they
argue that the difference is merely heuristically significant (Hinitkka / Remes [8], p. 71) and
translate geometric proofs into logical proofs of natural deduction. Thus, among other things,
they ignore the fact that the rules of geometric proofs are not reducible to deductive rules of
logic as soon as one regards diagrams, rather than sentences, as the objects based on which
geometric proofs are performed. Behboud [3], p. 66, argues in the same vein and, in section
4 of his paper, points to further deficiencies of the logical reconstruction of geometric proofs
provided by Hinitkka / Remes [8].
2
serve as means of evidence. According to this understanding, a proof does not
deduce theorems from axioms by means of universal, topic-neutral and theoryindependent rules of logic. Instead, a proof is based on the properties of specific
objects that are used and manipulated within the proof through the application
of theory-dependent rules. It is the properties of the specific constituents of a
proof that serve as the criteria for proof. Thus, a proof is a rule-based construction of means of evidence that possess the necessary properties to prove what
is in question. The identification of the relevant properties, in turn, follows certain rules; these rules, together with the rules for generating the specific means
of evidence, constitute the specific rules of proof. The criteria for proof are
concrete, identifiable properties of perceivable objects (e.g., diagrams or experiments) that serve as means of evidence. It is the properties of the constituents
of a proof that prove the properties that are in question. This reference to
the properties of objects used within a proof as the criteria for proof is what
characterizes iconic proofs.
By contrast, axiomatic proofs are “symbolic” in the sense that the property
in question is the truth of propositions given the truth of certain axioms. The
truth of propositions in general, and of axioms in particular, cannot be identified
based on perceivable properties of the sentences expressing those propositions.
If truth is a property at all, it is a property of the content of the sentences,
not of signs (or other purely syntactic objects). The constituents of axiomatic
proofs refer to the evidence of what is proven. Iconic proofs, by contrast, are
independent of any relation of reference to entities existing outside the proof because the properties of the constituents of a proof provide the relevant evidence.
In this sense, iconic proofs are purely syntactic, whereas axiomatic proofs are
based on semantics.
According to an iconic conception of proof, the specific meaning of a proposition such as “It is possible to construct an equilateral triangle using a ruler
and compass” (cf. section 4), “Light is heterogeneous” (cf. section 5) or “p ↔ p
is a tautology” (cf. section 6) cannot be expressed to its full precision independently of the proof. Instead, the proof is what defines the specific meaning
of a proposition by reducing the properties in question to (formal or syntactic) properties of the constituents of the proof. Independently of the proof, we
possess only insufficient characterizations of those properties and the entities to
which they are applied. We know what a ruler, a compass and an equilateral
triangle are, but as long as we do not know how to construct an equilateral
triangle using a ruler and compass, our understanding of equilateral triangles
is incomplete because it is not connected to an understanding of the relevant
geometrical construction; it lacks the specific “geometrical content” that is in
question. Similarly, we may have a model of light consisting of rays with heterogeneous properties, but without an experimental foundation, such a model
is no more than a hypothesis without “empirical content”. We may define the
property of a tautology as truth with respect to all interpretations, but without
a proof procedure for deciding upon this property, we still do not understand
how to apply it to concrete formulas. The semantic definition of logical truth
does not imply any specific “logical content” based on the properties of formulas
3
Conception Constituents
axiomatic
sentences, i.e.,
formulas + interpretation
diagrams
iconic
Evidence
truth of axioms logical (deductive)
(Euclid)
experiments (Newton)
properties
logical
of constituents
diagrams
Rules
(Wittgenstein)
theory-dependent
Table 1: Comparison of the Axiomatic and Iconic Conceptions of Proof
and not on whatever we relate those formulas to. An advocate of the axiomatic
method emphasizes that it is one thing to specify and understand the content of
a proposition, but it is another to know and prove its truth value. By contrast,
an advocate of the iconic method emphasizes that regardless of what one understands independently of a proof, an iconic proof procedure is the best standard
for specifying the meaning of a proposition in question.
Table 1 summarizes the differences between the two conceptions of proof.
2
Problems with the Axiomatic Method
In this section, to motivate the search for an alternative proof procedure, I will
sketch some of the principle problems with the axiomatic proof method.
From a logical perspective, theorems cannot be less evident than axioms
because they cannot be false while the axioms are true. Therefore, it is questionable why the truth of axioms should serve as evidence, as their truth cannot
be less questionable than the truth of the theorems.
Compared with the iconic proof method, the main problem with the axiomatic method lies in the fact that the evidence on which an axiomatic proof
relies is not a property of a perceivable entity that itself is used within the proof.
Instead, the proof relies on evidence of the truth of propositions stated within
the proof. Such evidence is not suitable as a criterion for proof to decide upon
a property in question. The evidence in an axiomatic proof does not consist
of “internal” properties of the constituents of the proof. Instead, it consists of
external properties that are referred to or expressed by sentences. Thus, the
proof rests on the content of sentences rather than on the properties of signs or
expressions. Abstract content, however, cannot serve as a criterion for truth.
First of all, this would be circular. Second, no property exists that is common
to all true sentences and may be used to decide whether some sentence stating
a proposition is true.
An axiomatic proof cannot do more than prove whether a certain proposition
follows from certain axioms. Thus, from a logical perspective, such proofs prove
the truth not of the theorems but of conditionals: if the axioms are true, then
4
the theorems are also true. Because the truth of the axioms cannot be proven,
the truth of the theorems also is not proven. Furthermore, such a proof is
restricted to a proof of provability; in the case that neither a proposition nor its
negation follows from the axioms, any decision lies beyond the capability of an
axiomatic proof.
Within formal science, an axiomatic conception of proof is based on abstract
and general axioms from realms such as logic, set theory or metaphysics. Within
empirical science, axioms may be falsifiable hypotheses that prove themselves
as long as they are not falsified. Thus, axiomatic proofs seek hypotheticodeductive explanations. Although an axiomatic conception of scientific proof
may be attractive to a rationalist, skepticism may arise from the fact that in
the empirical sciences, the validity of hypotheses is merely indirectly proven,
whereas in the formal sciences, the content of the relevant axioms is general
and thus ambiguous, methodologically difficult to control, or even susceptible
to paradoxes. In section 5, we will come to recognize Newton as a critic of
proofs based on hypotheses in experimental physics. Concerning the formal
sciences, one may think, for example, of Russell’s Paradox questioning Frege’s
axiomatization of naı̈ve set theory or of Wittgenstein’s criticism of the axiom of
identity as a pseudo-proposition that cannot yield a meaningful interpretation
of object language.4
An axiomatic conception of proof relies on the meaning of sentences. Therefore, it is susceptible to semantic problems, such as the problem of how one may
decide whether a grammatically well-formed statement indeed states a proposition that is capable of being true or false. As examples, one may think of proofs
resting on diagonalization, such as the proof of Cantor’s theorem or undecidability proofs in mathematical logic. Such proofs rest on interpretations that
refer to uncountable sets or to formulas that are interpreted as stating their own
unprovability. Again, the indirect, non-constructional manner of proof and the
proximity to paradoxes such as Richard’s or Skolem’s Paradox may invoke skepticism regarding the question of whether the intended content of the sentences
is well defined and, in fact, capable of being either true or false.Philosophical
doubts regarding proofs that rest on semantic concepts such as reference, interpretation, representation, extension and intension, models or truth are transferable to axiomatic proofs. Such doubts concern, for example, set theoretic or
meta-mathematical proofs.
From the perspective of an axiomatic conception of proof, it may be bewildering to relate criteria for proof to perceivable properties of the constituents of
a proof. It is often argued that the concrete properties of perceived objects may
give rise to illusion and lack generality. The case studies in this paper, however,
will illustrate that it is not simply perceivable objects or their properties that
may serve as evidence; instead, only perceivable objects that are constructed
according to certain rules may serve as means of evidence, and only perceivable
properties that are identified according to certain rules may serve as criteria for
proof. It is “intellectual intuition” that is related to proof; neither intuition or
4 Cf.
Wittgenstein [39], 5.53-5.5352.
5
perception on its own, nor abstract concepts, can constitute an iconic proof.
Roughly speaking, an iconic conception of proof relies on nothing but syntax in
terms of the rule-guided construction and manipulation of objects with unambiguously identifiably perceivable properties. Reference to any sort of meaning
of signs referring to entities existing outside the proof is superfluous.
Finally, it should be noted that this paper does not intend to pass judgment
on the axiomatic method. It is merely argued that such a conception of proof
is not without alternatives and that certain reservations or doubts regarding
axiomatic proofs may motivate one to look for such an alternative. In the
following, I will show that such an alternative conception of proof is manifest
in most prominent examples throughout the history of science, although these
examples have often been superficially taken to illustrate or to be related to the
axiomatic method.
3
Problems in Interpreting Euclidean Proofs
This section presents several arguments against analyzing Euclid’s proofs according to the axiomatic method.
Euclid’s Elements is often taken as a model for the axiomatic method because
he begins by presenting certain general presumptions before proving propositions based on these presumptions. These general presumptions, however, are
not axioms. Euclid distinguishes among definitions (‘óρoι), postulates (α’ιτ ήµατ α)
and common notions (κoιναὶ ’´ννoιαι); neither Aristotle’s term αξίωµα (axioms; cf., in particular, Aristotle [1], I. 6, 74b 5) nor the term αρχή (Plato’s and
Aristotle’s term for principles; cf. Menn [19], p. 193) is used.5
Euclid’s presumptions are used as rules within his proofs; they are not used
as axioms that serve as starting points for proofs. Euclid’s definitions establish
the essential properties of relevant geometric figures. The first three postulates
specify the rules for geometric construction (connection of points, extension
of lines, and drawing of circles). The last two postulates allow conclusions that
cannot be identified based on intuition alone to be drawn from certain geometric
properties. Postulate 4 allows one to identify all right angles (according to their
magnitude). Postulate 5 allows one to infer from the angles between two lines
and a third one on which side the two lines will meet. If one regards postulate
5 as a rule of (flat) Euclidean geometry and not as an axiom, then one does
not face the problem of justifying its truth with respect to doubts raised in the
context of non-Euclidean geometries. Like postulates 1-4, postulate 5 prescribes
5 Euclid’s κoιναὶ ’
´ννoιαι are sometimes identified with Aristotle’s αξιώµατ α because Aristotle calls some of Euclid’s κoιναὶ ’´ννoιαι αξίωµα. However, Euclid’s κoιναὶ ’´ννoιαι are not
used as axioms, principles or first sentences within his own proofs. Moreover, Euclid’s rather
uncommon notion of κoιναὶ ’´ννoιαι does not have the meaning of fundamental sentences,
propositions or assumptions (cf. Tannery [33], p. 162-164, and Heath [6], p. 221); instead,
’´ννoιαι is usually translated as notions. Translating ’´ννoιαι as rules may express their opposition to propositions even more clearly. Therefore, I will also call Euclid’s κoιναὶ ’´ννoιαι
general rules. If Euclid had used his κoιναὶ ’´ννoιαι in the sense of Aristotle’s αξιώµατ α, he
would most likely have called them αξιώµατ α.
6
the possible objects and properties of a Euclidean geometry; the postulates do
not describe existing ideal or real objects in a more or less accurate manner.6
In contrast to the specific geometric definitions and postulates, the general
rules (= common notions) concern not geometry specifically but magnitudes in
general. Rules 1-4 are general rules of identity, and rule 5 allows one to infer
the “larger than” relation from the relation of a whole to one of its parts.
Euclidean proofs use these rules (i) to construct certain geometric figures
(postulates 1-3) or (ii) to identify certain geometric properties of the figures
thus generated, either directly based on definitions (such as definitions 1-23 of
Book I) or indirectly based on geometric rules (postulates 4 and 5) or general
rules (common notions 1-5). By no means are the rules used as axioms, in
the sense of starting points for proofs. This is also true for the proofs of theorems (in contrast to the solutions of problems), such as propositions 4 and 5.
Instead, Euclidean proofs start with geometric figures, either through explicit
construction or through reference to previously constructed figures. Based on
these geometric figures, the geometric properties in question are identified in
accordance with fixed rules. Just as logical formulas are the (syntactic) objects
of a formal logical proof and mathematical terms are the (syntactic) objects
of a mathematical proof, so too are geometric figures the objects of Euclidean
proofs. This will be illustrated in the following section in regard to Euclid’s
proof of his proposition 1.
Furthermore, it should be noted that Euclid not only proves theorems but
also solves problems. An axiomatic conception of proof does not do justice to
this difference. In such a conception, problems must be translated into theorems.
Rule-guided constructions must be replaced with statements of propositions to
conform to the categories of logical proof.
In addition, it should be considered that Euclid does not possess a sufficient
theory of deductive reasoning, nor does he refer to deductive reasoning, such as
Aristotle’s syllogisms.
However, the presumably most important objection against a logical reconstruction of Euclid’s proofs is the fact that Euclid’s proofs are incomplete when
analyzed as sequences of sentences. I will demonstrate this in the following
section in regard to Euclid’s first proposition. As soon as one analyzes Euclid’s
proofs in terms of axiomatic proofs, they must be judged as deficient. The
question is whether this deficiency is due to Euclid or to the chosen method of
analysis.
Not without reason, mathematical logicians have proven theorems of Euclidean geometry based on presumptions different from those of Euclid.7 The
question remains, however, whether a non-axiomatic, iconic analysis does better
justice not only to the intended method of Euclidean proofs but also to their
probative force.
6 Giaquinto
[5], p. 285, argues similarly.
addition to Hilbert [7], one may consult Tarski [34] as a typical example of a logical
reconstruction of Euclidean geometry.
7 In
7
Figure 1: Proof of Proposition 1 in Heath [6], p. 241f.
4
Euclid’s Geometric Proofs
This section discusses two alternatives, namely, an axiomatic and an iconic
reconstruction, for analyzing Euclid’s first proof. This proof solves the problem
of how to construct an equilateral triangle from a given finite straight line. I
will focus on the question of the extent to which Euclid’s proof is incomplete.
Figure 1 presents Heath’s translation of Euclid’s first proof based on Heiberg’s
edition of the Greek text.
In the following, I will first reconstruct this proof in a simplified manner
based on an axiomatic conception of proof. Afterward, I will contrast this
reconstruction with an iconic one.
According to the axiomatic understanding, Euclid’s proof consists of a se-
8
quence of sentences. The diagram accompanying the text in Heath [6], p. 241,
merely illustrates the proof; it is neither part of the proof nor does it have any
probative force. I will abstain from a detailed logical analysis of Euclid’s proof.
Such an analysis would be enormously intricate and extend beyond Euclid’s
text. In particular, a full logical formalization of a deductive valid formal proof
would require further axioms that are neither among Euclid’s presumptions nor
mentioned in the proof of proposition 1. Thus, it would be even more evident
that Euclid’s proof is incomplete according to an axiomatic logical analysis.
Instead of a detailed logical reconstruction, I will be satisfied with analyzing
Euclid’s proof in the form of a sequence of sentences as provided in Euclid’s
text and based on his postulates, definitions and common notions. This suffices
to show that judging Euclid’s proof to be incomplete depends on the analysis
of his proof as a sequence of sentences. I will structure the proof according to
Proklos’ (412-485 AD) division. “P” denotes “postulate”, “Def.” denotes “definition”, “R” denotes “general rule” (i.e., common notion), and “A” denotes
“assumption”.
Problem: On a given finite straight line to construct an equilateral triangle.
Division
No. Sentence
Evidence
specification 1.
Let the straight line AB exist.
AE
construction 2.
The circle BCD with center A and distance AB exists.
P3
3.
The circle ACE with center B and distance BA exists.
P3
4.
The two circles BCD and ACE intersect at point C. ?
?
construction 5.
proof
conclusion
The straight lines CA and CB exist.
P1
6.
AC = AB
2 Def. 15
7.
BC = BA
3 Def. 15
8.
CA = AB = BC
6,7 R1
9.
The triangle ABC is equilateral given AB.
8 Def. 20
Table 2: Reconstruction of Euclid’s Proof as a Sequence of Sentences
As is often noted8 , no evidence for the sentence that appears as line 4 of
the proof is available from Euclid’s presumptions. There is no reason for the
assumption that “C” in “the circle BCD” and “C” in “the circle ACE ” refer
to the same point. The existence of a point C that is a point of intersection
between the two circles is not deducible from the postulates, definitions and
common notions that Euclid establishes prior to his proofs. Postulate 5 allows
8 Cf.,
for example, Heath [6], p. 241:
It is a commonplace that Euclid has no right to assume, without premising
some postulate, that the two circles will meet in a point C.
For a discussion, cf., for example, Giaquinto [5].
9
one only to infer the point of intersection between two straight lines; it does
not concern the intersection of two circles. Instead, a further assumption is
required, such as “If the distance between the centers of two circles is smaller
than the sum of their radii, then the two circles intersect” or “If c1 is the center
of a circle A and c2 is the center of a circle B and c1 is on B and c2 is on
A, then A and B intersect”. However, as long as one analyzes the proof as a
sequence of sentences based on nothing but either Euclid’s definitions, postulates
and common notions or previous proofs, then there is a gap in the proof at
line 4. It would be possible to close this gap by referring to the equations
defining the circles and inferring that the two circles intersect from equating
the equations. This, however, requires the application of rules of analytical
geometry and therefore does not refer to Euclid’s calculus. As long as one refers
only to Euclid’s presumptions and analyzes his proof as a sequence of sentences,
his proof of proposition 1 is incomplete.
However, this is merely a consequence of the analysis of his proof and the
underlying proof conception of that analysis. Proklos’ division of Euclid’s proof,
from more than 700 years after Euclid, already suggests an understanding of
Euclid’s proof in terms of a sequence of sentences. If one instead regards the
sentences of Euclid’s text not as the constituents of his proof but only as a
linguistic description of the diagrams that constitute the proof, then one can
refer to the properties of the diagrams as the means of evidence. However,
the text still serves an important function: it identifies the relevant properties
on which the proof is based. For example, it does not matter that the circles
drawn in the presented diagram are not exact circles. The relevant properties
of the diagrams do not simply follow from the diagrams specifically as drawn.
Instead, the diagrams obtain their relevant properties in combination with the
text. Thus, the diagrams described in the proof do contain two circles, even if the
specific drawings of the circles (the “tokens”) are not exact circles. A certain
similarity is sufficient to identify the drawings as circles with the properties
described in the text, namely, with the center A lying on the circle BCD and
with the center B lying on the circle ACE. This, in turn, determines further
perceivable properties that are identifiable via intuition, namely, the fact that
the two circles intersect.
It is especially important to note that the length of the straight line AB
and, therefore, the radii of the circles BCD and ACE are arbitrary. Although
in the concrete diagram, the points A and B are separated by a certain distance,
this distance is not an essential property on which the proof is based. This is
analogous to irrelevant properties of logical formulas: the concrete colors or sizes
of the signs are irrelevant; only the types of the signs are relevant and determine
the relevant logical properties of the formula. The type of a constituent of a
proof allows for variations in the specific tokens used to realize it. As the
constituents of a proof, diagrams are types, not tokens.
In the iconic proof method, the text of a proof is relevant in that it identifies
the specific properties of the diagrams on which the proof is based. In the case
of the proof of proposition 1, the text identifies the relevant property that the
sides of the constructed triangle are the radii of the constructed circles, where
10
Figure 2: Iconic Analysis of Euclid’s Proof; cf. Footnote 9
those radii are identical because the center A of circle BCD is on circle ACE,
while the center B of ACE is on BCD.
However, this property is still a property of the diagrams and, thus, a property of the constituents of the proof. The evidence for the intersection of the
circles BCD and ACE is the diagrammatic fact that two circles constructed
according to the textual specifications necessarily intersect, regardless of the
distance between points A and B. Stating that the point of intersection C is
implied in the rule-guided construction of the two circles is not a separate step
of the proof when those steps are identified with distinct diagrams; cf. figure 2.
According to these criteria for individuating the steps of a proof, the construction or identification of the diagrammatic properties that are inherent within a
step of construction – such as the fact that the two constructed circles intersect
at at least one point (in fact, at exactly two points) – does not constitute a
distinct step of the proof (in the form of a sentence, for instance) that must
be justified by Euclid’s presumptions. There is no gap within the iconic proof
because each step of the proof, when treated in terms of the construction or
manipulation of diagrams, is justified by Euclid’s definitions, postulates or common notions. Obviously, the completeness of the proof depends on how the
constituents of the proof are analyzed. To close the gap within Euclid’s proof,
no further assumption is needed; only a different analysis, based on a different
11
conception of proof, is necessary.
Figure 2 presents an iconic reconstruction of Euclid’s proof that demonstrates its completeness.9 The provision of an iconic proof does not mean that
an understanding of the proof does not require a textual explanation. However,
it does not follow from this requirement that the proof itself consists of sentences. It follows only that the constituents of the proof and its properties are
described in the form of sentences.
According to an iconic reconstruction, Euclid’s presumptions – his postulates, definitions and general rules – are not part of the proof. Instead, they are
rules related to the construction and manipulation of the proof’s constituents
(diagrams) as well as the identification of their relevant properties. They are
not first assumptions from which theorems are deduced. Instead, they are rules
that are applied to diagrams to prove a proposition in question by generating
a diagram with the relevant properties as a means of evidence. According to
this analysis, it is properties of intellectual intuition, namely, properties that are
constructed and identified based on explicit rules (i.e., Euclid’s presumptions),
that serve as criteria for proof. Euclid’s presumptions specify a calculus that is
applied in his proofs. The rules of this calculus must not be confused with the
objects of the proofs that are constructed, manipulated and identified according
to those rules.
5
Newton’s Experimental Proofs
Euclid’s Elements served as a model for Newton’s Opticks. At first glance,
Newton appears to apply the axiomatic method. He presents definitions and
axioms of geometrical optics prior to his proofs of theorems. The appearance of
an axiomatic proof method, however, is misleading. Newton’s proofs are “proofs
by experiment”. Essentially, theorems are inferred directly from experiments
that are described by means of text or diagrams. Newton does not explain how
he infers the theorems from the experiments. He does not refer to his definitions
and axioms to justify his inferences, nor is it possible to do so.
Moreover, the definitions and axioms of Newton’s Opticks are presumptions
of geometrical optics that ground his descriptions of his experiments. This becomes particularly clear from his diagrams of prismatic experiments; cf., for
example, figure 5 below. These diagrams sketch the paths of rays. Obviously,
these sketches presume Newton’s definition of light rays as well as the geometric
laws that apply to ray paths in the cases of refraction, reflection and diffraction. Thus, Newton’s definitions and axioms do not serve as rules that justify
inferences within a proof. Instead, they are rules for geometric descriptions of
experiments. Without the theoretical concept of a ray of light and fundamental
geometric laws concerning those rays, it is not even possible to pose questions
of geometrical optics or to design and describe experiments to answer those
questions.
9 This figure is a modification of a digrammatic proof according to Miller [20], reproduced
as figure 1 at https://plato.stanford.edu/entries/epistemology-visual-thinking/.
12
In his experimental proofs, Newton always emphatically rejects the presupposition of hypotheses and the use of experiments to eliminate alternative hypotheses in order to indirectly derive a hypothesis.10 By contrast, he purports
to derive his theorems “positively & directly” (cf. footnote 10) from experiments. He calls his method of proof “induction”; cf., for example, the quotation
from Newton [25], p. 404, quoted below on p. 13. As I will illustrate by
means of Newton’s experimentum crucis, Newton’s inductive method implies
rules for causal reasoning and, thus, goes beyond the simple generalization of
experimental data.
Newton explicitly compares his method of experimental proof with the geometric proof method from antiquity when he refers to the distinction between
analysis and synthesis; cf. the locus classicus of Newton [25, 404], Query 31:
As in Mathematics, so in Natural Philosophy, the Investigations
[. . .] by the method of Analysis, ought ever to proceed the method
of composition. This analysis consists in making experiments and
observations, and in drawing general conclusions from them by induction [. . .] For Hypotheses are not to be regarded in experimental
Philosophy. And although the arguing from Experiments [. . .] by
Induction be no Demonstration of general Conclusions; yet it is the
best way of arguing which the Nature of Things admits of, [. . .] By
this way of Analysis we may proceed from [. . .] effects to their causes,
[. . .] the Synthesis consists in assuming the causes discover’d, [. . .]
and by them explaining the phaenomena proceeding from them.
The distinction between analysis and synthesis can be traced back to Plato
and Aristotle (cf. Menn [19]) and can also be found in Euclid XIII.11 Similarly
to Plato and Aristotle, this passage describes analysis as proceeding from the
sought to the admitted, whereas synthesis proceeds in the reverse direction, from
the admitted to the sought. Ihmig [11], section 4, argues that Newton refers to
the much discussed locus classicus from Pappus.12 However, the questions of
10 Cf.,
for example, Turnbull [36], p. 96f.:
You know the proper Method for inquiring after the properties of things
is to deduce them from Experiments. And I told you that the Theory wch I
propounded was evinced to me, not by inferring tis thus because not otherwise,
that is not by deducing it only from a confutation of contrary suppositions, but
by deriving it from Experiments concluding positively & directly. [. . .] what I
shall tell [. . .] is not an Hypothesis but most rigid consequence, not conjectured
by barely inferring ´tis thus because not otherwise or because it satisfies all
phaenomena [. . .] but evinced by ye mediation of experiments concluding directly
[. . .].
11 The relevant passage, however, seems to have been inserted in 60 AD by Heron; cf. Heath
[6], p. 138.
12 Cf. Pappus [27], p. 82f., and Heath [6], S. 138f. For discussions of Pappus’ distinction
between analysis and synthesis, cf. Cornford [4], Robinson [29], Hinitkka / Remes [8], Rehder
[28], Behboud [3], and Maenpaa [16]. For the relation between Newton’s method and the
geometric method of analysis, cf. in particular Hinitkka / Remes [8], chapter IX, as well as
Guerlac [15] and Ihmig [11].
13
how to interpret Pappus’ distinction and how to apply it to Euclid’s proofs are
controversial. In the following, I will concern myself only with the application
of this distinction to Newton’s experimental proofs. In “Natural Philosophy”,
Newton related only synthesis to truth-preserving deduction, whereas he related
analysis to induction. This special understanding of analysis as used in experimental proofs is the reason why, in relation to Newton’s use of analysis within
experimental proofs, it is not pertinent to discuss the logical question of whether
analysis proceeds from the sought to the admitted through the drawing of deductive conclusions (Robinson’s downward interpretation) or through a search
for the conditions for a deduction (Cornford’s upward interpretation).
Instead, in the context of Newton’s method of experimental proof, the relevant feature of the distinction between analysis and synthesis is the general and
uncontroversial point that analysis is concerned with finding reasons, whereas
synthesis presupposes them. Newton’s position is that analysis must precede
synthesis. The reasons for a causal explanation must be sought first, before
phenomena may be explained on this basis. According to Newton, analysis is
related to two aspects of experimental proofs: (i) the conducting of experiments
and (ii) inductive reasoning from experiments. The reasons obtained through
analysis are therefore twofold: (i) evidence (ratio cognoscendi) and (ii) causes
(ratio essendi). The first is obtained by performing an experiment, and the
second are derived via induction based on the relevant experimental data. Newton applies the term “analysis” to the method of experimental proof in which
causes are inductively inferred from effects produced in experiments that are
deliberately designed to allow causes to be inferred. In synthesis, by contrast,
causal relations are presumed to explain observed experimental phenomena.
Any causal explanation is therefore based on proving a causal relation.
The crucial point is that Newton’s method of proof does not start from possible hypotheses or axioms that might then be related to experimental phenomena. Instead, an experimental proof consists of establishing both the evidence
and the causes. It does not do justice to Newton’s understanding of proof to
regard analysis merely as an ars inveniendi while identifying synthesis with the
“real proof”.13
Newton’s application of analysis in experimental proofs can be compared to
Euclid’s geometric proofs. Just as Euclid’s geometric proofs are based on the
construction of geometric figures to provide the means of evidence for identifying
the geometric relations in question, Newton’s experimental proofs are based
on experiments that provide the means of evidence for identifying the causal
relations in question. Just as Euclid’s diagrams possess the relevant properties
that serve as proof criteria, Newton’s experiments provide positive criteria for
causal inferences. Moreover, just as an understanding of Euclid’s diagrams
13 Against
Ihmig [11], section 4b, who refers to Newton [26], p. 451:
Through analysis they discovered propositions, and through synthesis they
demonstrated them once found.
It should be noted that Newton reserves the term “demonstration” for truth-preserving deduction, whereas he characterizes experimental proof in terms of induction. The “discovery”
of causal relations based on experimental evidence is what constitutes an experimental proof.
14
Figure 3: Newton’s Experimentum Crucis, from Newton [24], Figure 18
depends on the conception of (i) rules of construction to enable the solution
of a problem and (ii) rules for identifying the relevant properties of diagrams,
an understanding of one of Newton’s experiments depends on understanding its
design against the background of a causal problem and its solution in reference
to the relevant properties of the experiment.
One must learn to “see and read” Newtonian experiments to understand how
they provide positive evidence for the solution of causal problems. As in the
case of Euclid’s geometric proofs, an analysis of Newton’s experimental proofs
as sequences of sentences implies a misconstrual of the evidence on which the
proofs are based. A hypothetico-deductive understanding of Newton’s proofs
does not do justice to Newton’s key assertion that the probative force of his
proofs lies in the experiments. An axiomatic understanding does not make
clear that Newton’s experimental proofs rest on analysis in the form of causal
inference via induction from experimental properties. In the following, I will
illustrate this by means of Newton’s famous experimentum crucis; see Newton
[24], p. 3078f., and figure 5.
With his experimentum crucis, Newton wishes to elucidate the cause of the
oblong form of the spectrum produced when sunlight passes through a prism
in the case that the angle of incidence and the angle of emergence are equal;
cf. Newton [24], S. 3076. The crucial aspect of the experimentum crucis is
that it creates a difference test as a result of its design, which is based on two
prisms. By rotating the first prism around its axis, Newton moves the spectrum
cast on the second board in front of the second prism up and down. To see
that the experimentum crucis makes it possible to compare differences under
homogeneous conditions, one must understand that the angles of incidence of
rays originating from different parts of that spectrum and passing through the
second prism are identical. These angles are identical because of the holes in the
boards behind the first and in front of the second prism. However, although their
incidence angles and all other external conditions are homogeneous, different
rays passing through the second prism refract differently behind the second
15
prism: the red spot M on the wall, which is produced by rays originating from
the red lower part of the spectrum on the board in front of the second prism, is
below the blue spot N on the wall, which is produced by rays originating from
the upper part of the spectrum. It is the realization of different effects (the
locations of M and N on the wall) under homogeneous conditions that proves
that the cause of this difference must be (internal) differences in the properties
of the light. Such an inference from the effects to their cause is based on the
principle of causality (“any variation (or difference) must a have a cause”) and
presumes conditions of homogeneity. Newton is well aware of this kind of causal
reasoning enabled by the design of his experiment, as is clear from a letter he
wrote to Lucas, Turnbull [37], p. 256f. (the brackets are part of the quoted
text):
In ye External causes you name there was no difference. The
incidence of ye rays ye specific nature of ye Glass ye Prisma figure,
&c were the same in both cases, & therefore could not cause ye
difference: [it being absurd to attribute the variation of an effect to
unvaried causes.] All things remained ye same in both cases but ye
rays, & therefore there was nothing but ye difference of their Nature
to cause ye difference of their refraction.
The relevant property of the experimentum crucis that justifies Newton’s
causal inference is the fact that all external factors are identical, yet the positions of the spots on the wall differ. This justifies the identification of internal
differences in the rays of light as the cause of the different refraction behaviors
behind the second prism, under the presumptions of the principle of causation
and the description of the experiment according to Newton’s definitions and
axioms of geometrical optics.
Newton generalizes his inference that proves the differing refrangibility of
the light rays passing through the second prism to infer the heterogeneity of
the sunlight that passes through the first prism and gives rise to the oblong
spectrum on the second board in front of the second prism. This generalization
is based on his second rule of reasoning in experimental philosophy: to identical
effects (= differences in refraction), one must assign, as far as possible (= as
long as no conflict with experimental evidence arises), identical causes (= differences in the properties of light).14 Newton’s experimental proof of a causal
relation therefore relies on principles of causal or, even more generally, inductive
reasoning, such as the principle of causation and other principles of simplicity
such as determinism, continuity, least action and Newton’s rules of reasoning
in experimental philosophy. Newton calls such principles “physical principles
of Science”.15 Causal inferences are based on such principles. This is why
14 Cf.
Newton [26], p. 398.
[24], p. 187:
15 Newton
[. . .] the absolute certainty of a Science cannot exceed the certainty of its
Principles. Now the evidence by wch I asserted the Propositions of colours is in
the next words expressed to be from Experiments & so but Physicall: Whence
the Propositions themselves can be esteemed no more than Physicall Principles
16
Newton regards his experimental proofs as inductive derivations, as opposed to
truth-preserving deductive inferences.16
Newton’s experimental proof of the heterogeneity of sunlight by means of his
experimentum crucis is often misunderstood as an example of deductive reasoning from a hypothesis, in which the experimentum crucis is used to decide upon
an alternative hypothesis. On this basis, Newton’s proof is judged to be incomplete and not forcible.17 Indirect reasoning by eliminating a hypothesis based
on experimental data raises at least two problems: (i) alternative hypotheses
may have been overlooked, and (ii) any hypothesis may be made compatible
with the experimental data by means of auxiliary assumptions. Sabra’s analysis
of Newton’s proof, for example, requires an auxiliary assumption to the effect
that the rays of sunlight are not modified by the first prism. However, a modern
understanding of crucial experiments in terms of experiments for deciding upon
alternative hypotheses does not do justice to Newton.18 Such an understanding
does not consider that Newton does not derive theorems from sentences by applying logical rules but rather derives theorems from experiments by means of
principles of induction (Newton’s physical principles). Just as Euclid’s definitions and common notions allow geometric properties to be identified as proof
criteria, Newton’s physical principles allow experimental properties to be identified as criteria for causal judgments.
Newton never maintained that his experimental proofs were incompatible
with alternative explanations of the experimental phenomena. However, possible alternatives only create gaps in a proof if a deductive axiomatic proof is the
benchmark. Instead, Newton’s benchmark is whether a theorem can be proven
from nothing but experimental evidence according to principles of induction. According to this measure, Newton’s proof by means of the experimentum crucis
is complete because the principle of causality allows one to infer that properties
of the light rays cause the refraction at the second prism and his second rule of
reasoning in experimental philosophy allows one to generalize this inference to
the refraction of the rays of sunlight at the first prism. One might reject such a
non-deductive proof. However, in doing so, one does not identify a gap within
Newton’s inductive experimental proof. According to Newton, a rejection of his
proof method undermines experimental science because such a rejection does
not allow for specific rules that ascribe probative force to experiments alone,
without feigning hypotheses.19
of Science.
16 Cf., for example, Newton’s 4th rule of reasoning in the third edition of Principia, Newton
[26], p. 400.
17 Cf., for example, Sabra [30], p. 249f., and Thompson [35], p. 8f.
18 Note that the term “experimentum crucis” was not established in Newton’s time. Newton
refers to Hooke’s explanation of an experimentum crucis in terms of an experiment “serving
as a Guide or Land-mark, by which to direct our course in the search after the true cause”
(Hooke [10], p. 54). Thus, a crucial experiment enables a causal inference instead of deciding
upon a hypothesis.
19 For a more detailed analysis of the probative force of Newton’s experimental proofs, cf.
Lampert [12].
17
6
Wittgenstein’s Logical Proofs
One might suppose that at least in the case of logical proofs, there is no alternative to a conception of proof in terms of a sequence of sentences. A common
understanding would suggest that this is even true if one identifies a logical
proof with a sequence of formulas, instead of sentences, that are capable of being either true or false. This is because one will still judge the correctness and
completeness of a logical calculus for formal proofs with respect to the extent to
which true sentences, as instances of logical formulas, are derivable from a set
of true sentences. Thus, the conception of formal logical proofs is based on an
axiomatic conception of proof because syntax is commonly measured in terms
of semantics.
However, this understanding is by no means without alternative. For example, Wittgenstein stipulates a logical proof procedure in terms of the translation
of logical formulas into an ideal notation “in which all and only logical equivalents have exactly one and the same expression” (Landini [14], S. 112). “Mere
inspection” (Wittgenstein [39], 6.122) of a suitable notation should enable the
identification of the logical form of the instances of logical formulas: in a proper
notation, the possibilities of the truth and falsehood of propositions, including the limiting cases of tautology and contradiction, must be recognizable from
“the symbol[s]” “alone” (Wittgenstein [39], 6.113). Thus, he argues for an iconic
conception of proof within logic.20 On this basis, he argues against a semantic
foundation for logical proofs. The transformation of logical formulas for the
sake of identifying their logical forms and, thus, the logical possibilities of their
interpretation can be done prior to and independent of any interpretation of
those logical formulas. Of course, this does not mean that there is no need
for an understanding of the symbols of a proper notation. However, it is the
mechanical rules that prescribe how to read symbols that make them understandable; it is not interpretation in the sense of assigning sets, truth values or
intensions of any sort to symbols.
In the Tractatus, Wittgenstein explains his conception of proof only for
propositional logic (cf. Wittgenstein [39], 6.1203). However, Wittgenstein envisaged the realization of his conception of proof within the realm of first-order
logic by means of his ab-notation, which he worked on between 1912 and 1914.
Unfortunately, Wittgenstein never realized this programmatic claim.21 In the
following, to illustrate the analogy to Euclid’s proofs, it is sufficient to sketch
how Wittgenstein’s iconic conception of proof can be realized by means of his
20 This conception can be traced back to Peirce and his existential graphs; cf., in particular,
Shin [32]. Lampert [13] explains Wittgenstein’s understanding of logical proofs in terms of an
iconic proof procedure and also compares it to Peirce’s method of existential graphs. Contrary
to Hinitkka / Remes [8] and [9], who compare Euclidean geometric proofs to natural deduction,
I argue in this section that Euclid’s method of geometric proofs would be better compared
to an iconic logic. This is particularly true in the case of “the figural interpretation” that
Hinitkka / Remes [8] advocate; cf. footnote 3.
21 Lampert [13] explains Wittgenstein’s ab-notation and demonstrates how Wittgenstein’s
iconic conception of proof can be realized for all first-order formulas that do not contain binary
sentential connective within the scope of quantifiers.
18
ab-notation within the realm of propositional logic.
Roughly speaking, a proof within Wittgenstein’s ab-notation consists of two
steps: (i) the construction of an ab-diagram for a given formula and (ii) the
identification of the logical properties of the initial formula based on the relevant
properties of the ab-diagram according to prescribed rules.
To translate a given formula of propositional logic into an ab-diagram, one
must proceed from inside to outside according to the logical hierarchy of the formula. Instead of a propositional variable ξ, one must write aξb (or, in analogy
to Wittgenstein [39], 6.1203, T ξF ). Wittgenstein defines the sentential connectives in terms of ab-operations that assign a- and b-poles to a- and b-poles:
¬ reverses the poles, and binary sentential connectives assign the a-pole and
b-pole to the pairs aa, ab, ba, and bb. Thus, the translation of a formula into an
ab-diagram follows certain prescribed rules.
Wittgenstein illustrates his ab-notation for the example of the ab-diagram of
p ↔ p in a letter to Russell; cf. figure 4.
Figure 4: Wittgenstein’s ab-Diagram of p ↔ p, from Wittgenstein [41], p. 57
ab-diagrams provide identity criteria for logical properties. One can refer to
these criteria by applying mechanical rules that concern only the relations of the
outermost to the innermost poles. For example, the ab-diagram shown in figure 4
identifies the initial formula p ↔ p as a tautology through the application of the
following general identity criterion, which Wittgenstein mentions immediately
afterward:
[. . .] it is tautological because b is connected only with those pairs
of poles that consist of opposite poles of a single proposition (namely
p).
We need not be concerned here with further details of iconic proof procedures
within the realm of logic.It is sufficient to observe that according to an iconic
conception of proof, a logical proof procedure is a translation of formulas into a
suitable notation – in the case of Wittgenstein’s ab-notation, into ab-diagrams
– that allows one to identify the logical properties of the initial formulas in
question based on the properties of the ideal symbols of the notation. As in
the case of the iconic analysis of Euclid’s proof method, the proof procedure is
19
nothing but a rule-guided generation of (purely syntactic) objects that serve as
means of evidence because their (formal) properties can be utilized as identity
criteria for the properties in question. The rules for generating ab-diagrams
are based on nothing but the definitions of the logical constants, and the rules
for identifying logical properties or relations can be used to define the logical
properties in question. Thus, any reference to sentences or to interpretations
that assign extensions (truth values, sets) or intensions (propositions, contents
of sentences, material properties of sets) is superfluous. In this respect, the
logical calculus “look[s] after itself” (Wittgenstein [39], 5.473).
According to Wittgenstein, Euclid’s geometry is an (autonomous) calculus and his geometric proof method is the epitome of a purely syntactic proof
method.22 He seeks a logical conception of proof that follows this Euclidean
model of iconic proof: logical proofs enable the identification of logical forms by
translating logical formulas into ideal symbols whose properties can be used to
identify logical properties.
In contrast to axiomatic calculi of logic, such as Frege’s calculus, the calculus of Russell’s and Whitehead’s Principia or a Hilbert calculus, Wittgenstein’s
iconic conception of logical proof avoids “proof gaps” in three respects: First,
it does not contain any axioms that are stipulated as logical theorems without
proof. Instead, any tautology is identified as a tautology due to one and the
same identity criterion; cf. Wittgenstein [39], 6.1265-6.1271. Second, formal
logical proofs are not confined to the proofs of theorems. Instead, one and the
same criterion is used to decide if a formula is or is not a tautology. There is
no difference between the method of proving logical theorems (tautologies) and
the method of proving that a certain formula is not a logical theorem (tautology).23 Finally, an iconic proof procedure does not require semantics. This is
true not only for the evaluation of formulas but also for the proof procedure. An
iconic proof procedure does not rely on semantics as a measure of correctness,
completeness or even decidability. Instead, it is the iconic proof procedure itself
that defines the logical constants and logical properties and that demonstrates
the extent to which the properties in question are decidable using this procedure. Just as the trisection of an angle does not have a precise meaning within
Euclidean geometry according to Wittgenstein (cf. Wittgenstein [40], p. 36f.),
neither does provability or the decidability of logical validity within logic (cf.
Wittgenstein [38], p. 52, §14-17). Therefore, as a consequence of dispensing
with semantics, Wittgenstein objects to any attempt to represent provability or
a decision function for logical validity within the language of logic. Syntax is
prior to semantics; any semantics may be measured through syntax, but not the
other way around.
Wittgenstein also advocated for an iconic conception of proof in the case
of algorithmic mathematical proofs, such as proofs of arithmetic, algebraic or
22 Cf.,
for example, Wittgenstein [38], p. 50 and 52 in appendix I, §7, §14, and Wittgenstein
[40], p. 38, 61-63, and 162-163.
23 In these two respects, a tree calculus or tableaux procedure of propositional logic is similar
to an iconic proof procedure.
20
analytic equations and even proofs of mathematical induction.24 He considered
logical rules not as universal rules for proof but only as rules within a formal calculus of logic. Arithmetic proofs, by contrast, require arithmetic rules.
The same applies in the case of algebraic, analytic and geometric proofs within
mathematics or for Newton’s proofs in the case of experimental physics. There
is no universal logical calculus that grounds all proofs because not all proofs
can be properly analyzed as proving the truth of sentences based on the truth
of axioms. Instead, proofs in different contexts must be based on specific calculi because the rules for specific proofs depend on the syntax of the formal
properties of their corresponding constituents. Euclidean proofs are examples
of this attitude toward proof, rather than being paradigmatic of an axiomatic
conception of proof.
7
Conclusion
Euclid’s geometric proofs have commonly served as a model for other proofs.
However, they are often regarded as paradigmatic of the axiomatic proof method.
By contrast, I argue that the probative force of Euclid’s proofs can be best understood if one regards his proofs as a model for an iconic proof method. This
also applies to Newton’s experimental proofs and Wittgenstein’s logical proofs.
In contrast to axiomatic proofs, the iconic proof method does not require semantics because the properties of the constituents of an iconic proof serve as
the means of evidence and the iconic proof procedure can be used to specify
the meaning of the propositions in question. In this sense, iconic proofs literally
show their evidence without saying anything. The art of proof is the generation
of means of evidence with properties that can serve as criteria for deciding upon
geometrical, causal, logical or mathematical properties of interest.
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24