Proofs Without Words and Beyond - Why We Write Proofs

In the last section we discussed two families of reactions to visual proofs, the Baroque and the Romantic. Because the Romantic reaction raises no special objections to counting visual proofs as genuine cases of mathematical proof, we primarily explored the shortcomings of visual proofs when assessed from the Baroque perspective, clarified through the lens of the Euclidean notion of proof. At the end we began to take stock of what visual proofs and PWWs seem to accomplish short of constituting fully-fledged mathematical proofs. In this section, we begin to structure those observations and articulate a conception of the potential mathematical value of visual proofs even when they, as a rule, fall short of the Euclidean standards for fully-fledged mathematical proof. We begin by turning away from questions about the nature of proof to consider instead the grounds or reasons that support our proof-writing practices in mathematics, in effect asking: "why do we care about writing proofs as part of the practice of mathematics?" This question is especially relevant to the discussion and appraisal of the mathematical value of visual proofs since they often seem to present ideas of genuine mathematical interest without meeting Euclidean or other formal, Baroque standards of proof. Importantly, we can ask this question and entertain plausible answers without abandoning the Baroque approach to mathematical proof developed in the previous section.

Why Do We Write Proofs?

A proof is a particular, regimented sort of mathematical thought. Fully expressed, completed proofs are perhaps the target products of mathematical knowledge production—the "official" activity of practicing mathematicians—but they are not the substance of most ordinary mathematical reasoning. Mathematical reasoning is about figuring things out, and the proof write-up is just the final, precise, publicly verifiable expression of the gained insights. Why do we ultimately want to force our insights into the form of a proof? What are the purposes of proof-writing?

Our approach to this question already suggests a couple practical answers: we write proofs to share thoughts in a canonical, public way, and we share our ideas partly as a way of seeking further verification through the peer review process. But these are practical answers. Unless there is something special about proof from the outset—unless proof already represents a privileged expression of mathematical knowledge—we could just as well have created or, indeed, could switch over to a mathematical culture that preferred some other form of expression for sharing ideas and peer review. Indeed, outside of Geometry, substantially lower standards of evidence prevailed in mathematics until the mid 19th century. The robust stability of our proof-writing practices suggests that there are additional, not purely practical purposes underlying the proof-writing activity of mathematics.

The simplest characterization of the purpose of proof writing is that we write a proof in order to place the truth of a proposition—the theorem—beyond doubt. The sort of doubt in question is not actual psychological doubt. It is easy to see that no proof can defeat all lingering psychological doubts. For example, serious doubt in one's logical ability or memory will not be allayed by rehearsing a gap-free, absolutely correct proof. The kind of doubt ruled out by a proof is something more abstract than actual psychological doubt. While rehearsing and understanding a correct proof may drive away doubts and produce psychological certainty in many cases, when it doesn't it isn't a failure of the proof, but a failure of the mind surveying it, which may be confronting all sorts of distractions, prejudices, and limitations in the process of rehearsing the proof.

While they present an obstacle in any complex mathematical thought, the possibility of distractions, prejudices and limitations raise special concerns about reliance on diagrams in mathematical reasoning. For example, Figure 10 illustrates a sort of joke-proof that \(64=65\).

Taking the appearance of the diagram at face value, it appears that an \(8\times 8\) square (with area 64) can be divided into four pieces and then rearranged into a \(5\times{13}\) rectangle (with area 65). An algebraic argument, however, reveals that the triangles and the quadrilaterals have different slopes \({0.4}\) versus \({0.375}\) and consequently do not match up perfectly; there is actually a hole of area 1 in the \(5\times{13}\) rectangle and the diagram doesn't present a successful visual proof that \(64=65.\) To believe this "proof" would be to make an error in reasoning, to be led astray by the visually/psychologically convincing aspects of a picture rather than engaging in clear mathematical thought. Insofar as the picture strikes us as convincing, this is because of limits of the visual system and the representational medium, not because it makes a compelling mathematical case. Better geometric PWWs tend to exploit symmetries, (strict geometrical) similarity, and things of that nature, rather than just sensory impressions.

On the other hand, the elimination of (rational, not psychological) doubt isn't the only point of proof. For example, we harbor no doubt that \(1+1=2\) and are absolutely, both psychologically and rationally, convinced of its truth. A proof that \(1+1=2\) from more basic principles can't make us any more certain of the conclusion. Nonetheless, such a proof is still intrinsically mathematically interesting. This interest can't be explained solely on the basis of the elimination of doubt.

We choose to use the term "doubt" to emphasize the ambiguity between psychological and logical/rational interpretations. Readers may be more familiar with the alternate phrasing of the same basic idea: "proof serves to render the theorem certain." We understand "certainty," "convincing," and other such terms to be subject to the same ambiguity between psychological and logical/rational readings, though these latter two lean more clearly toward the psychological. In our view, certainty, or being convinced, is approximately equivalent to belief plus freedom from doubt (with all the ambiguities intact).

The leading 19th century Euclidean, Gottlob Frege (1848-1925), espoused his rationalist, anti-psychologistic doctrine alongside a sophisticated understanding of the purposes of proof. While visual proofs will probably never express an argument that is completely formal and gap free, and will therefore never fully rise to the level of proof in Frege's strict Euclidean terms, visual "proofs" may still satisfy other aims of proof that make sense from a Baroque perspective and be consonant with the Euclidean notion of proof. Indeed, Frege identified at least four aims for proof, none of which have to do with achieving psychological certainty:

1. to put the truth of a proposition (the theorem) beyond rational (rather than psychological) doubt;
2. to afford insight into the pattern of dependence of truths upon one another;
3. to show why a proposition is true; and
4. to reduce the number of basic assumptions underlying mathematics.

The first of these aims is a straightforward justificatory project, but the other three bleed out beyond the basic justificatory aim, the second into the project of generating and extending more synoptic understanding of mathematics (as opposed to piecemeal knowledge of individual mathematical truths), the third more specifically into the project of explanation, and the fourth into the project of converting mathematics into a more deeply unified body of knowledge by sketching connections and carrying out reductions of higher level principles to more basic principles. Taking these principles as the aims of proof, we are now in a position to consider to what extent PWWs or other visual proofs can accomplish some or all of these goals. Frege's Euclidean credentials are unimpeachable, and we also therefore take these aims to be compatible with the Euclidean notion of proof and the broader Baroque perspective on mathematical proof.

In the first half of one particularly pregnant paragraph from §2 of Frege's first major philosophical work, The Foundations of Arithmetic, he stated these four purposes. We add numbers corresponding to the ordering above:

The aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt [1], but also to afford us insight into the dependence of truths upon one another [2]. After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely [3]? The further we pursue these enquiries, the fewer become the primitive truths to which we reduce everything; and [4] this simplification is in itself a goal worth pursuing.

Frege saw a tight connection between all of these aims. Indeed, the third aim is most likely, for Frege, a special case of the second. For Frege the task of articulating why a particular mathematical fact is true, i.e. the task of explanation, is a matter of drawing logical connections between propositions. Mathematical understanding is produced by grasping connections between propositions; we understand why a result is true when we grasp how it is connected to the foundational propositions of mathematics. Indeed, in light of his logicism, his view that the mathematics of number reduces to logic alone, even the fourth aim can be seen as a special case of the third: the case where an explanation, elaborated as a proof, proceeds directly from the most basic foundational facts of mathematics. Frege's logicism and his attendant insistence on complete logical rigor in argumentation binds these four aims very tightly. When we relax the logicist ambitions, though, these four aims come apart and reveal conceptually distinct motivations for carrying out proof in mathematics. It is in this more relaxed mode that we explore these purposes below, fully acknowledging that Frege would take a narrower view of these aims. The broader animating values behind this multi-part understanding of the purpose of proof in mathematics do not require that we adopt strict logicist standards of explanation. Frege cleverly adapted his logicist project to fit these values, but the values are independent of his logicism.

Aligning PWWs with the Aims of Proof

Often one of the appeals of a PWW is the simple and elegant depiction of a mathematical fact. Even if one is familiar with some result because of a traditional proof with words, the PWW of that same result is often intriguing. In general, when given two (or more) proofs of the same result, we can consider whether one is preferable to the other. Although what makes one proof better than another is an open question, Steiner argued that in general the more "explanatory" proof is the preferable one, where "an explanatory proof makes reference to a characterizing property of an entity or structure mentioned in the theorem," and a characterizing property is "a property unique to a given entity or structure within a family or domain of such entities or structures" [Steiner, p. 34] This criterion aligns with the second and third aims Frege sketched. Intuitively, the title 'explanatory' seems to fit a number of PWWs, and an explanatory proof leads to a more satisfying understanding of a result, even though a proof is not any more correct or eliminative of rational doubt because of the use of a characterizing property. Kawasaki's PWW of Viviani's theorem seems to succeed in being explanatory in Steiner's sense because the transformations exploit characterizing features of equilateral triangles and recognizing the height of the initial triangle in the sum of the heights of the component triangles depends on recognizing the midline—the segment formed by the bases of the upper triangles—as parallel to the base of the triangle: a fact that can be easily deduced from the characterizing features of equilateral triangles.

In order to accomplish Frege's first aim, proofs need not be explanatory in Steiner's sense. However, explanatory proofs are more illuminating and thus preferable. Although proofs with words and other reasoned mathematical arguments may also draw on characterizing properties, there are times when a PWW is more explanatory than its wordy counterpart. We claim, for instance, that while both the original "wordy" proof of Viviani's theorem and the [Kawasaki PWW] are explanatory in Steiner's sense, the PWW is more explanatory because it makes deeper use of characterizing features of equilateral triangles. Kawasaki's PWW trades on the rotational symmetry of the equilateral triangle. Each rotation selects and rotates a particular equilateral triangle within the external triangle. Because these rotations correspond to the rotational symmetries of the equilateral triangle, the relevant geometrical structures are preserved throughout the process. Perhaps the envisioned proof would be carried out in analytic geometry, though it is notable that such a formalization would not make the proof any more convincing. In any case, unlike the [wordy counterpart], this proof avoids using the fact that a triangle has area \({\frac{1}{2}}bh,\) which is inessential to the theorem being proved, and emphasizes the properties that characterize equilateral triangles over properties of triangles more generally. Thus, in Steiner's sense, the PWW is more explanatory than the alternative proof with words.

Although many PWWs are elegant, intellectually pleasing, and highly explanatory, that does not necessarily mean they are good proofs. Frege held particularly high standards for proof since his overall goal was to carry aim (4) above to its completion and effect a reduction of large segments of mathematics, especially number theory and analysis, to logical principles alone. In effect, Frege imported logical standards of rigor to the whole of mathematics because he had a goal to reduce mathematics to logic. Even Frege would probably admit that logical standards of rigor go beyond the level of formal rigor required to place the truth of a proposition beyond rational doubt. And even if a PWW or other informal "proof" doesn't present enough evidence to place the truth of a mathematical proposition beyond any rational doubt, it can still make progress in that direction, effectively reducing the space of possible rational doubts even if not entirely eliminating them. It is notable, though, that PWWs tend to bring in more rather than less mathematical machinery, e.g. bringing geometric machinery to bear on simple number theory theorems, and this is generally not conducive to reducing the number of basic assumptions supporting a theorem. So while considering Frege's aim (4) helped us to reach some hopeful conclusions about PWWs, it is fairly clear that PWWs are not especially well-suited to satisfying Frege's third aim for proof.

Kawasaki's PWW of Viviani's theorem proves exceptional in this regard as well. By drawing our attention to the way that the result depends on the fundamental symmetries of the equilateral triangle, we recognize that we can reduce this result to facts about symmetries. Without the full proof elaborated we can't be sure how much other machinery we need to bring to bear, so the strength of this reduction is not clear, but the proof is promising. Indeed any proof, visual or otherwise, that is highly explanatory in Steiner's sense seems to be a good candidate for effecting a meaningful reduction.

On the other hand, precisely because PWWs tend to bring in additional, especially geometrical, mathematical machinery in comparison with traditional proofs, PWWs tend to serve aim (2) fairly well. Indeed, one might reasonably think that mathematical understanding, as contrasted with mathematical knowledge, is a matter of recognizing connections. The capacity of geometry to represent interesting ideas in number theory and analysis is frequently explored in PWWs, and such visual proofs can expand the reader's understanding of the interconnectedness of diverse areas of mathematics. Just as much as any traditional proof, a successful visual proof will show the reader how a result can be linked to other mathematical facts. For example, in the PWW shown in Figure 11, the trigonometric identities listed under the picture can be justified by inspecting the labels on the figure and applying the Pythagorean Theorem. However, in order to fully and convincingly verify the accuracy of the picture, the reader must take a further step and determine whether the labels fit with the geometric definitions of the trigonometric functions.

In verifying the accuracy of the figure, we are led to see the dependence of the result on other truths and more basic facts (e.g. definitions). The first and second aims are satisfied simultaneously. Romaine's PWW does what we expect for a result in trigonometry, but there are other PWWs that reveal sometimes surprising connections. Consider, for example, Gallant's "Truly Geometric Inequality" PWW from 1977 in Figure 12:

The result shown in Figure 12 is the arithmetic - geometric mean inequality: \({\sqrt{ab}}\leq{(a+b)/2}.\) We will begin the next section with a discussion of this PWW, but for now it is worth noting that the PWW encodes a geometrical proof of an inequality that holds in the positive real numbers. This PWW exploits the idea that a line segment on the Euclidean plane can stand in for a positive real number. Because this proof makes use of a geometrical representation of real numbers, it shows or reminds the reader of important structural relationships between geometry and analysis, serving aim (2) in a reasonably substantial way. Indeed, it was commonplace in Ancient Greek mathematics to prove what we would now think of as results in number theory and analysis using a geometrical interpretation of the natural or real numbers. These representational connections seem to have reversed with the Cartesian shift so that we now cultivate procedures for using analysis or algebra, for example, to prove geometrical results. PWWs often bring us back to something more like the Ancient Greek perspective.

Philosophical Conclusions

In the preceding section we introduced a dichotomy between Baroque and Romantic approaches to mathematical thought. The Baroque approach puts a great deal of emphasis on formal correctness and completeness as a necessary dimension of mathematical justification, whereas the Romantic approach—the approach that is intuitively more amenable to terming PWWs "proofs"—emphasizes the degree to which a presentation of mathematical ideas puts the reader into a frame of mind where the target result becomes evident. Baroque approaches guard against the dangers of over-enthusiasm and frenzied thought that sometimes characterize the mathematical experience, whereas Romantic approaches accept the risks and emphasize the value of powerful mathematical experiences in cultivating mathematical understanding. In this section we shifted away from the question of whether PWWs should by termed "proofs" on the basis of their formal features in favor of the question of whether PWWs align with our aims or purposes in writing-up and sharing mathematical evidence in the form of proofs. By shifting to this second question we have provided a way to find mathematical value in PWWs from within a more-or-less Baroque framework. The value we found is not merely pedagogical—though PWWs can occasionally have enormous pedagogical value—but is properly mathematical. By investigating Frege's four aims of proof we demonstrated how PWWs can play a role in the broader mathematical projects that seek to produce explanation, understanding, and comprehension, and appreciation of mathematical interconnectedness rather than just evidence of truth. In these purposes, PWWs can play a role that parallels the role of traditional proof-writing whether or not we elect to extend the term "proof" to cover visual proofs.

Further Reading on Proof

A reader engaged by the ideas surveyed in this and the preceding section may enjoy reading Gottlob Frege's 1884 book The Foundations of Arithmetic [Frege], which is a foundational text for both contemporary philosophy of mathematics and the wider philosophical approach known as "Analytic Philosophy." We found Frege to be a useful inspiration in developing our own defense of the status of PWWs, but his work is of much broader interest and greater importance than our opportunistic use of his ideas might indicate. More substantial analysis of Frege's philosophy of mathematics can be found in the philosophical literature. Part 3 of Tyler Burge's Truth Thought Reason: Essays on Frege [Burge Anthology], the works cited in his Bibliography, and Michael Dummett's Frege: Philosophy of Mathematics [Dummett] are good starting points to appreciate the nuance and power of Frege's thought about mathematics.

The conversations and investigations that spawned this section and the preceding one of this article began in weekly meetings between one of the authors (Kutler) and her undergraduate thesis advisors (Schueller and Doyle). While these sections don't directly reflect his doctrines, the ideas of Imre Lakatos played a large role in our conversations, deeply influenced our understanding of the potential mathematical value of visual proofs and PWWs, and have had a still-unfolding impact on all of our teaching. We encourage interested readers to examine Lakatos's Proofs and Refutations [Lakatos] for themselves.