In this article we discuss the history and philosophy of the well known Proofs Without Words (PWWs) that have appeared in the MAA's Mathematics Magazine since 1975. We go on to present a new kind of PWW that uses open-source tools like GeoGebra, MathJax, JSXGraph and Processing to create more general, interactive PWWs. We finish with a proposal that Convergence create a regular Proofs Without Words 2.0 column to accommodate these constructions.
In the words of Lynn Arthur Steen, co-editor of the MAA's Mathematics Magazine with J. Arthur Seebach in 1976, when the original "Proofs Without Words" column premiered in that journal [Steen]. (Here, "Steen" links to a "knowl": more about knowls.)
As a teacher, I often urged students to remember geometric diagrams that encapsulate the essence of key mathematical relationships or theorems. For most people, visual memory is more powerful than linear memory of steps in a proof. Moreover, the various relationships embedded on [sic] a good diagram represent real mathematics awaiting recognition and verbalization. So as a device to help students learn and remember mathematics, proofs without words are often more accurate than (mis-remembered) proofs with words.
Here are the first two Proofs Without Words (PWW) as they appeared in Mathematics Magazine in September of 1975:
Visual proofs enjoy a long history, going from some of the earliest recorded mathematical writings through to the Proofs Without Words column that first appeared in Mathematics Magazine (and as of 2012 also in The College Mathematics Journal). Based on the increasing frequency of appearance of these visual proofs in both journals, the popularity of these mathematical artifacts is on the rise.
Proofs, especially proofs in the Euclidean tradition of geometry, often include a full verbalization of the argument as well as a figure that serves to illustrate the key points in the argument and make the logic easier to follow. We use the term "diagram" to refer to a figure that serves as a supplement to a verbally expressed proof. A diagram illustrates a point, a point that is made in language. We use the term "visual proof" to refer to a figure if it expresses mathematical argument or evidence with little or no explanatory text. A visual proof doesn't just illustrate a point, it actually makes the point. We tend to regard the ideal visual proof to be one where very little text occurs. Labels strike us as more in the spirit of visual proof than sentences, but we do not intend to offer a strict classification of degrees of purity for visual proof. For our purposes, a Proof Without Words or a PWW is a figure that has been published in the Proof Without Words feature of either Mathematics Magazine or The College Mathematics Journal. We note that it does not follow that all PWWs are visual proofs. However, we consider all of the PWWs that we discuss at length in this article to be visual proofs.
The top half of Figure 1 is a device for trisecting an angle that was first proposed by Laisant in 1875 [Yates]. The knowledgeable reader may note that the bottom half of Figure 1 that stands as proof of the Pythagorean Theorem in Isaacs' original contribution dates back at least to the ancient Chinese mathematical text, Zhou Bi Suan Jing (c. 200 BCE) [Boyer]. We note that in Zhou Bi Suan Jing this figure was accompanied by explanatory text [Katz]. Hence, as it originally appeared, the figure was a diagram, not a visual proof, and certainly not a proof without words! As such we acknowledge Isaacs' contribution, in this case, as recognizing the diagram's ability to convincingly convey a mathematical result with little or no explanatory text. A further discussion of this kind of contribution as well as an example of the rich history that often accompanies visual proofs follows Siu's Sum of Squares PWW at the end of the Proof Without Words 2.0 section.
The PWWs mentioned in this article are cited by their appearance in Mathematics Magazine. We note that the people named in these attributions may not be the original authors of the figures. Rufus Isaacs, for example, passed away in 1981. We may never know where he first encountered the two PWWs that he contributed, but he believed that these two figures expressed, without words, certain mathematical truths. In other words, he recognized them as visual proofs and offered them as such. The authors of the PWWs cited here deserve credit for creating and/or recognizing figures that can stand on their own as visual proofs. It is in this spirit that we cite the contributors of the PWWs to Mathematics Magazine throughout this article rather than attempting to attribute the PWWs to the originators of the figures employed in the PWWs.
While PWWs have enjoyed broad appreciation as pedagogical devices and recreational puzzles, there remains a natural and straightforward question related to PWWs: Are "Proofs Without Words" really proofs? The answer, on the other hand, is not straightforward. While we do not pretend to provide a definitive answer to this question, we do shed light on the meaning of 'proof' and the ways it might apply to PWWs. We do this in the subsequent sections by examining the received Euclidean conception of proof and the views of Gottlob Frege (1848-1925)—one of the most influential philosophers of mathematics—and applying his ideas in the discussion of specific PWWs.
The rise of ubiquitous computing allows for the extension of traditional PWWs to interactive PWWs. For example, here is an interactive version of Charles Gallant's 1977 PWW "A Truly Geometric Inequality" rendered using the free, open-source, javascript library JSXGraph. To interact with the figure, grab the point on the diameter of the semi-circle with your mouse and move it around. We will return to discuss this and other interactive PWWs at much greater length in the final section.
Figure 2. A Proof Without Words 2.0. This figure is an interactive adaptation of Charles Gallant's original "Proof Without Words: A Truly Geometric Inequality." Move the black point left and right along the base to see different configurations.
In this article, we discuss the history and philosophy of visual proofs—also known as "Proofs Without Words" or PWWs for short. We also propose a new regular feature of the Convergence online journal that would provide a peer-reviewed venue for computer-based PWWs that we dub "Proofs Without Words 2.0" (PWWs 2.0).
As in the Gallant example, we employ freely available tools to create PWWs. The PWWs 2.0 presented here are extensions of PWWs that have been published in Mathematics Magazine. We claim that these computer-based, interactive PWWs are more general and provide more insight to readers than their static counterparts. We also encourage readers to develop their own PWWs 2.0 and to submit them for publication in Convergence.
Throughout the record of intellectual history, people have expressed mathematical ideas with pictures. The proof of the Pythagorean Theorem in the first ever PWWs found in Figure 1 of the introductory section of this article is an example of how ancient mathematicians found evidence of mathematical relationships by drawing pictures. In addition to those found in the ancient Chinese mathematical text, Zhou Bi Suan Jing (c. 200 BCE) [Katz, Boyer], variations of this visual proof have been credited to Pythagoras himself (c. 600 BCE), and to Bhaskara (1114-1185 CE) [Cooke]. This visual style of mathematical proof was so compelling to Oliver Byrne that in 1847 he wrote a version of Euclid's Elements with colored figures subtitled "in which coloured diagrams and symbols are used instead of letters for the greater ease of learners." This wonderful work is discussed in greater detail in the article "Mathematical Treasures - Oliver Byrne's Euclid" by Frank J. Swetz and Victor J. Katz. It is also interesting to note that Byrne's use of color diagrams would certainly have made use of the cutting edge color printing technology of the times (1847). Later in this article, we too advocate the use of new technology to create visual proofs for "the greater ease of learners." In Byrne we find a kindred spirit.
Because mathematics has always enjoyed a close relationship with philosophy, the use of pictures as an effective, accessible way to make mathematical points has not escaped notice by philosophers. Perhaps the most famous use of mathematical diagrams in the history of philosophy occurs in Plato's Meno. This episode has acquired importance because of its centrality in Plato's exposition of his theory of recollection: the idea that what appears to us to be learning should more properly be understood as a kind of recollection. As the dialogue progresses, by Section 80a, Meno has become exasperated trying to follow the strictures of Socrates' method of inquiry. In his frustration he declares that inquiry is impossible by invoking a tricky paradoxical argument. In Socrates' words (Section 80e):
Do you realize what a debater's argument you are bringing up, that a man cannot search either for what he knows or for what he does not know? He cannot search for what he knows—since he knows it there is no need to search—nor for what he does not know, for he does not know what to look for.
In response, Socrates proposes that inquiry is possible because we are capable of achieving vague impressions of the truth that can be sharpened by careful thought. Perhaps he is denying that there is a sharp divide between knowing and not knowing of the sort required for the debater's argument to get off the ground. To illustrate what he is talking about, Socrates takes Meno's slave aside and leads him through some mathematical reasoning to demonstrate how vague impressions of the truth can be sharpened into knowledge by careful thought. The actual demonstration carried out with the slave terminates with the discovery of a method for how to double a square: how to take a square of arbitrary size and generate a square with twice that size. Discerning the structure of the argument is complicated by the fact that it stretches over several pages. The final diagram (Figure 3) Socrates sketches at Sections 84d-e, however, can be read as standing alone as a visual proof of the method he has led the slave to discover:
Figure 3. Socrates' final sketch. Fill colors correspond to the order Socrates adds the squares in Sections 84d-e, darkest first, lightest last. The dashed green lines that form the shaded diamond are added in Section 85a.
The insight provoked is that the inner diamond is made of four halves of identical squares, and so the inner diamond is a square with twice the area of any of the four corner squares. Though this figure is compelling as a visual proof, we do not actually believe that this diagram was intended as a visual proof in the original text because of the manner and care with which Socrates verifies the prior understanding and specific capabilities of the slave in the previous pages. We believe that a specific verbal argument was envisioned and that the figure serves as a diagram in support of that argument in the usual way we see diagrams occurring in, e.g., editions of Euclid's Elements. Nonetheless, the compelling quality of this diagram promotes reading it today as a visual proof that stands alone in justification of the procedure for doubling the square.^{Note}
Despite their ancient roots, visual proofs are still utilized by modern mathematicians. However, they were not often peer reviewed or even widely recognized until the Mathematical Association of America started publishing them regularly in Mathematics Magazine starting in the mid 1970s in the "Proofs Without Words" column.
In September 1975, Rufus Isaacs published an article entitled "Two Mathematical Papers Without Words" in Mathematics Magazine [Isaacs]. This short "paper" appeared at the end of a longer article and included the [two figures] in Figure 1. One was a proof of the Pythagorean Theorem. The other was a drawing of a hypothetical device designed to trisect an angle—a task not possible with standard compass and straightedge construction. While neither was presented as proof of their respective results, they were clearly intended to convey convincingly a mathematical idea in a purely visual manner.
In January 1976, two months after these figures were published, Mathematics Magazine came under the direction of two new co-editors, J. Arthur Seebach, and Lynn Arthur Steen. With the change in editorial leadership came changes in the journal's layout. A new "News and Letters" section, which replaced the old "Notes and Comments" section, was implemented to streamline the process of reader feedback. The new section allowed comments on published articles to be printed within months of the original publication date. As a result, several readers submitted comments regarding articles published in the September 1975 issue. The majority of the comments submitted were regarding "Two Mathematical Papers Without Words." Indeed, of his own PWW, Isaacs in the "News and Letters" feature of the January 1976 issue wrote:
All I intended was to stress the rare and secluded pleasure of of grasping a mathematical truth from visual evidence alone.
As it happens, this pleasure has become far less rare and secluded!
In the same "News and Letters" section, the new co-editors concluded with the following statement [News and Letters]:
Editor's Note: We would like to encourage further contributions of proofs without words for the reasons mentioned by Rufus Isaacs and one other: we are looking for interesting visual material to illustrate the pages of the Magazine and to use as end-of-article fillers. What could be better for this purpose than a pleasing illustration that made an important mathematical point?
Following the publication of this request, figures meeting this description started appearing in Mathematics Magazine under the heading "Proof Without Words" at a rate of approximately one or two per year. By 1987, that rate had increased to five or six per year, averaging to about two per issue. Needless to say, mathematicians took notice of these intriguing mathematical gems. Professor Roger Nelsen, at Lewis and Clark College in Portland, Oregon, was no exception. In June 1987, after several attempted submissions, he published his own PWW entitled "The Harmonic Mean - Geometric Mean - Arithmetic Mean - Root Mean Square Inequality" (Figure 4) [interview, Nelsen].
Figure 4. Roger Nelsen's first Proof Without Words as it appeared in Mathematics Magazine in 1987. [Nelsen]
Later, in the spirit of the peer-reviewed publication process, the editors of Mathematics Magazine asked Nelsen to referee other PWW submissions. Over the years, he saved any PWWs that came to him for feedback. Eventually, he had enough to make a collection, so he published his well-received Proofs Without Words: Exercises in Visual Thinking (1993) [interview] and its sequel, Proofs Without Words II: More Exercises in Visual Thinking (2000). The Proofs Without Words column in Mathematics Magazine (and also in MAA's College Mathematics Journal) continues to be a healthy publication venue for PWWs as of this writing. Nelsen has continued to explore diagrams and visual proof with co-author Claudi Alsina in the books [Math Made Visual] (2006) and [When Less Is More] (2009).
Preliminary philosophical conclusions about PWWs have been somewhat dismissive of the thought that PWWs are or can be genuine proofs. Indeed, Nelsen himself wrote in the introduction of his first anthology of PWWs, "Of course, 'proofs without words' are not really proofs." Nonetheless, PWWs appeared to Nelsen to have real mathematical value. He went on to ask in the introduction to his first volume, "if 'proofs without words' are not proofs, what are they?" His answer was that
PWWs are pictures or diagrams that help the observer see why a particular statement may be true, and also how one might begin to go about proving it true.
Clearly this means that PWWs can be an important vehicle for communicating and stimulating mathematical thought. It seems that for Nelsen, PWWs are something like sketches of proofs rather than complete proofs. No one would deny that a sketch of a proof has substantial value in communicating mathematical evidence. Indeed, a fully explicit proof, one that spares the reader no technical detail, is often a less effective way to communicate with or convince a reader. In the next two sections, we expand on Nelsen's approach and try to fill out more philosophical detail about how PWWs might fit in the spectrum of mathematical argumentation and evidence.
Readers familiar with Nelsen's second anthology may note that his assessment of PWWs as mathematical proofs seems to have become more optimistic. Instead of repeating his earlier line "Of course, 'proofs without words' are not really proofs," Nelsen wrote instead, "Of course, some argue that PWWs are not really "proofs" (emphasis added)." This somewhat ambiguous allusion to the fact that PWWs are not universally accepted as proof is followed by a quote from James Robert Brown that ends:
Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important—but they prove nothing. I want to oppose this view and to make the case for pictures having a legitimate role to play as evidence and justification—a role well beyond the heuristic. In short, pictures can prove theorems.
By proceeding without offering critique, Nelsen seemed to align himself with Brown's much more inclusive position on proof. The next two sections of this essay explore the territory between the common initial reaction that PWWs are not and could not be mathematical proofs and the more inclusive stance adopted by Brown and eventually Nelsen.
Over the next two sections we construct and apply a framework for appraising the mathematical value of visual proofs. We begin in this section by characterizing two somewhat extreme reactions to visual proofs, which we term the Baroque and Romantic reactions to visual proof. We have chosen to use the terms Baroque and Romantic to characterize these positions partly to avoid any specific allusions to schools of thought in the philosophy of mathematics. These terms evoke the notion of a stance, a response pattern, or an approach rather than a specific and developed school of thought. Our goal is to encourage readers to take a fresh look at visual proof and the activity of proof writing in mathematics, and this goal will not be served as effectively if we try to situate our discussion in the range of ongoing discussions in the philosophy of mathematics. We spend much of the rest of this section exploring the doubts raised about the status of visual proofs from the Baroque perspective. We couch our discussion in this conservative framework because ultimately any defense of the mathematical value of visual proofs will be more persuasive if it is grounded in or at least compatible with a relatively conservative conception of proof in mathematics. By the end of the next section, we will conclude that visual proofs often fail to constitute proofs from the Baroque perspective, but that nonetheless, visual proofs are capable of serving many of the same mathematical aims as traditional proof writing—aims that exceed mere pedagogical value. We also prepare the way for continued philosophical discussion in our final section on multicolor, interactive, and animated PWWs 2.0.
We can report anecdotally two common families of reactions that mathematicians have upon encountering PWWs. One reaction is characterized by the thought:
PWWs aren't really proofs, and so a priori they are not convincing.
One might rigidly insist, for example, that a proof must be expressed so as to explicitly draw logical connections between mathematical propositions. Logic operates on the syntax of sentences, so without sentences there can be no proof write-up. One might pick on other logical technicalities in assessing a PWW that depicts a particular case of a general mathematical fact. In mathematics we're usually concerned with general claims, claims about any sum of consecutive cubes or any equilateral triangle, not just the triangle drawn on the board. Some PWWs do a great job of depicting a particular case, but if we wouldn't take the theorem to be proved by examining a single numerical case, why should examining a particular pictorial case change our assessment? In other words, proof by example is no proof at all. Indeed, Fry's PWW (Figure 5) that the sum of the first \(n\) cubes is the square of the sum of the first \(n\) integers depicts only the \(n=3\) case, so the viewer, in the strictest terms, has only been given evidence about one of an infinite range of possible cases:
Figure 5. Fry's Sum of Cubes shows only the case \(n=3.\) [Fry]
Similarly, a given geometric diagram often depicts just one particular choice of an arbitrary point and does not enable inspection of the full range of points in the scope of the theorem, as in Wolf's PWW of Viviani's theorem discussed below and in the [final section]. Something that doesn't address the general phenomenon can't be a proof that the phenomenon holds universally. Because a single example, strictly speaking, leaves a large evidentiary gap, a PWW that depicts a single case of a general phenomenon is not a complete proof and, from this rigid perspective, we should not find it convincing.
For simplicity, we call this first sort of reaction the Baroque response to PWWs. We use the term "Baroque" to indicate the idea that status as a mathematical proof is underwritten by formal correctness, paralleling how a composition's status as a fugue is underwritten by the formal qualities of the music.
The second viewpoint is characterized by another thought:
PWWs can be far more rapidly and deeply convincing than traditional, propositional mathematical argumentation, and are therefore (in such cases) perfectly acceptable, even occasionally preferable proofs.
This second family of reactions emphasizes the capacity of a piece of mathematical reasoning to convince and inspire a mathematician. Formal adequacy, i.e., inclusion in the honorific category of proof, is underwritten by apparent evidentiary force—the extent to which an argument should convince a mathematically astute, reasonable audience. If a PWW presents a mathematical idea with sufficient apparent evidentiary force to convince a mathematician of the truth of a statement, why count it as anything other than a proof? Consider, for example, Figure 6, Rick Mabry's proof that \(\sum\limits_{n=1}^{\infty}(\frac{1}{4})^n=\frac{1}{3}:\)
Figure 6. A convincing proof that \((1/4) + (1/4)^2 + (1/4)^3 + \cdots = 1/3.\) [Mabry]
This PWW puts the reader into a frame of mind that enables her to verify the result in a satisfying, convincing, and interesting way. Indeed, we find this PWW to make an extremely elegant, compelling case for the result. For simplicity, we call this second type of reaction the Romantic reaction to PWWs. We use the term "Romantic" to indicate that a good proof is whatever sparks the mathematical intuition or intellect in the right way, prioritizing the mathematical experience over Baroque genre standards.
The Baroque perspective construes proof as a type of evidence distinguished by meeting particular formal constraints, whereas the Romantic perspective construes any evidence that is powerfully and completely compelling to a mathematically astute, reasonable reader as a proof. We find something appealing in both the Baroque and the Romantic approaches to mathematical thought, and we do not intend to adjudicate between these broad approaches to mathematics. However, from a squarely Romantic perspective there is no in-principle challenge to the status of PWWs as proofs; the true Romantic will only ask whether a particular PWW (or traditional wordy proof, for that matter) is effective in producing the target mathematical insight. We work from within the Baroque perspective in exploring and answering challenges to the status of PWWs because generic challenges to the status of PWWs are not native to the Romantic perspective. In the remainder of this section we explain the Euclidean notion of proof and use this notion, which is a further specification of the Baroque framework adopted for the rest of the paper, to develop a critique of the status of visual proofs as genuine mathematical proofs.
We take the position, at least for the purposes of our present discussion, that a proof is an argument that meets certain evidentiary and formal or syntactic standards. A proof, as a type of argument, is an abstract object rather than an artifact or other concrete entity. Journal pages, chalk marks, or sequences of sounds are concrete objects or events that can merely express or otherwise encode proofs. A series of words and symbols on a page, or a series of sounds in time, can express a proof, and can express a proof well or poorly. From this perspective, a PWW is an artifact, a collection of marks on a page or screen—at this level of abstraction, at least, no different from a traditional "wordy" proof write-up—and we can ask whether it expresses a proof and, if it does, how well.
By writing up a proof we aim to express or encode a particular deduction. Examination of the notion of proof espoused in textbooks used in Sophomore/Junior-level "introduction to proof" courses yields a perspective that aligns fairly well with the classic Euclidean notion of proof. Consider, for example:
Not all textbooks express such straightforwardly Euclidean/Baroque commitments. Consider, for example:
These characterizations blur the distinction between proving a statement and convincing or moving an audience. Such characterizations, however, are less typical, and seem to indicate a more Romantic understanding of proof on behalf of their authors.
Adopting a canonical Euclidean perspective, we take a proof to be a deduction of a mathematical result from basic principles of mathematics and other results previously proved, so that one could, in principle, trace a result all the way back to basic mathematical principles (e.g., axioms and definitions).^{Aside} Practicing mathematicians, it would seem from casual survey, tend to understand their completed proofs in these terms. Actual practice, however, admits of some flexibility. Even examining only traditional "wordy" proofs, we can see that the particular form of expression doesn't uniquely determine a particular proof. In practice, we would hesitate to say that two write-ups express different proofs if they make use of the same starting points and lemmas, but derive the lemmas in a different order. And we would also not hesitate to say that write-ups that employ different starting points express different proofs of the same theorem. Intuition does not quite as cleanly settle cases in the middle, where, for example, two write-ups highlight different lemmas, but employ the same starting points. However we settle this issue, any write-up will have to express a minimal amount of ordering: it will have to specify what is supposed to follow from what. A write up of a proof is more than a mere statement of entailment or consequence, but it is at least such a statement.
Written and spoken language serve us well in expressing proofs because their syntactical conventions involve us immediately in ordered expression. This webpage reads left to right, top to bottom. That makes it easy for us to get a reader to go through one thought before another. In the context of reading a proof, this ordering of thoughts can be used to discern relations of entailment. For example, consider the following written proof of Viviani's theorem, a theorem we will revisit several times in this and subsequent sections.
Viviani's Theorem The sum of the lengths of the shortest lines that connect an arbitrary interior point of an equilateral triangle to each of its sides is the same as the height of the triangle.
Proof. Let \(P\) be some point on the boundary or within an equilateral triangle \(ABC\). Label the perpendiculars to the sides from \(P\) as \(p_a\), \(p_b\), and \(p_c\). Note that \(\triangle ABC = \triangle PBC + \triangle PCA + \triangle PAB.\) Let \(h\) be the height and let \(s\) be the length of a side of \(\triangle ABC\). Then, given that the area of a triangle is \(\frac{1}{2}bh\), we have \(\frac{1}{2}sh = \frac{1}{2}sp_a + \frac{1}{2}sp_b + \frac{1}{2}sp_c\). It follows that \(h = p_a + p_b + p_c.\)
In this proof we are first invited to consider a triangle and the three perpendicular lines from an arbitrary interior point. When we say \("\)given that the area of a triangle is \(\frac{1}{2}bh"\) we are pointing to a previous result, one that many of us proved in high school geometry. The remainder of that sentence applies this fact across the equality established above, and then some (suppressed) basic algebra yields the result. The key observation is that \(s\) is a common factor in all terms because \(\triangle ABC\) is equilateral. There are a number of ways to carry out the final algebra: we could multiply both sides of the equation by \(\frac{2}{s}\), we could divide by \(s\) then multiply by 2, or we could multiply by 2 then divide by \(s\). These divergences do not make for a substantially different proof, so it is perfectly reasonable, given the intended audience for the particular write-up, not to detail all the algebra. Importantly, though, because of the verbal medium of this proof write-up, we are led to see exactly what prior mathematical result is relevant to the deduction: the area formula for an arbitrary triangle. Any write-up of this proof of Viviani's theorem, standard or visual, will need to point out the dependence of the final result on this prior mathematical result.
A picture, however, especially when it is accompanied by no explanatory text, presents everything at once, which makes it more challenging to express an identifiable ordering of thoughts or to express what depicted mathematical fact is supposed to be taken as the basis for another. Hence, pictures alone, because of their nature as static, unordered representations, tend to fail to meet Euclidean syntactic or formal standards of proof.
Figure 7. Wolf's Proof Without Words of Viviani's Theorem shows one difficulty of reading a visual proof: the lack of an obvious ordering of thoughts. [Wolf]
The Wolf PWW of Viviani's theorem in Figure 7, for example, is a geometrical diagram that requires the reader to apply her knowledge of geometry to understand how the diagram can be the basis of a proof of the theorem. It does not, strictly speaking, tell the reader what is supposed to follow from what or which geometric principles Wolf intends the reader to apply. (The Wolf PWW of Viviani's theorem appears to encode a very different proof of the theorem, indeed one in which the area formula for a triangle plays no major role.) We discuss the Wolf PWW of Viviani's theorem at greater length in the [final section]. The same phenomenon—failure to precisely encode particular starting points—is present vividly in the PWW in Figure 8:
Figure 8a. Similar to Wolf's PWW in Figure 7, this PWW of the Law of Cosines lacks an obvious ordering of thoughts, though the bolded triangle does suggest the starting point. [Kung]
In this PWW by Sidney Kung, the triangle with sides \(a, b,\) and \(c\) is bolded, indicating that it is the foundation from which the whole figure is constructed. This is a useful visual indicator of logical ordering in the intended proof, a sort of implied visual syntax. After constructing a circle of radius \(a\) around that triangle, the other labels in the figure can be deduced. Trigonometric PWWs like this one, however, tend to draw on other substantial results without explicitly citing those results as we would in a traditional proof. Kung's PWW, for instance, seems to us to rely on the additional, unstated claim that \(xy' =x'y\) as in this figure:
Figure 8b. Diagram for Lemma for Kung's PWW (Figure 8)
If this result is established as a lemma, then the law of cosines follows easily (with one small algebraic step). But the original figure does not encode a whole, self-standing proof. It is effectively left as a puzzle for the reader to figure out what mathematical facts underlie the diagram and verify them for herself. The diagram could be verified in many ways, and the diagram itself doesn't indicate one way over another. Each different verification might yield a different proof, either by operating very differently on the same starting points, or by employing different mathematical facts as starting points. Reliance on the reader's assumed geometrical knowledge isn't the problem, but it is problematic from the perspective of proof writing and proof assessment that these diagrams don't indicate which geometrical facts the reader is intended to bring to bear in rehearsing the proof indicated by the diagram—they fail to indicate starting points. Some concession to the indication of a starting point comes from bolding the central triangle, but that does not eliminate the need for significant interpretation. PWWs like Kung's from 1990 seem to fall more in the realm of pedagogical and recreational mathematics, and within that framework, where the puzzle is actually part of the pedagogical or recreational value of the PWW, ambiguity about starting points is perfectly reasonable. But from the perspective we adopt in assessing whether the PWW is effective as a proof, these sort of PWWs tend to fall short.
Kung's 1990 PWW is also an example of a published PWW that contains sentences. In the ideal case, a visual proof will contain only a statement of the theorem and a diagram or picture, possibly with labels. We draw only a very rough line concerning included text in our definition of "visual proof"; our definition allows the occurrences of words or even sentences as long as they are not "explanatory". We leave the notion of explanation vague, but the basic idea is that a visual proof shouldn't seem like a very incomplete traditional proof supplemented by a diagram; the figure has to do the heavy lifting on its own. So what of Kung's PWW? The first sentence is an algebraic rearrangement of the second sentence, which is the standard statement of the law of cosines—a statement of the theorem to be proved. The diagram purports to be a proof of the first sentence. This sentence is significant or interesting because it is directly equivalent to the law of cosines. Viewing it in this way, Kung's PWW doesn't seem to contain excessive verbiage and may qualify as a visual proof even though it contains sentences. The PWW might have conformed better to the standards we articulate for visual proof if only the top sentence appeared, and if the title was allowed to make the connection between the displayed theorem and the law of cosines. While it may be a borderline case, from our perspective it seems that Kung's PWW is a visual proof. The editors and reviewers of the existing PWW features have displayed substantial flexibility in what counts as a proof without words. We note that our definition of "visual proof" does not strictly imply that all PWWs are visual proofs, though in practice we believe nearly all published PWWs would count as visual proofs under our definition.
Returning to the main line of thought, we note that some PWWs do a great job indicating some sort of ordering, even if it is not always the logical ordering that frames traditional proofs. For example, in Figure 9 the wonderfully elegant PWW of Viviani's theorem by Ken-ichiroh Kawasaki—not unlike the Fry PWW above—indicates a visual starting point and a couple of transformations, in order, from that starting point.
Figure 9. Kawasaki's Proof Without Words of Viviani's Theorem. This PWW has a clearly indicated ordering of some sort. [Kawasaki]
But the visual starting point indicated is not the same as an identifiable progression of propositions that the author of the proof uses to demonstrate the conclusion. A visual starting point for a transformation is very different from a propositional starting point, i.e., a premise to be employed in a deduction. Even this powerfully convincing PWW doesn't obviously encode the sort of ordering necessary for a Euclidean proof.
It appears to be more difficult to visually encode the basic syntactic or formal features that are important from the Baroque perspective. And, although we see no in-principle bar against a visual proof meeting these standards, our own survey of over 80 published PWWs suggests that few PWWs express sufficient argumentative or deductive structure to count as expressing a particular proof, i.e. a particular argument or deduction with particular, unambiguously identifiable starting points, inferences, lemmas, and conclusions. It's not that the PWWs we surveyed didn't present compelling evidence, but rather that the Baroque perspective puts such tight formal constraints on the category of proof that any ambiguity about intended inferential structure counts against a PWW as an unambiguous write-up of a particular deduction. The problem, really, is that one has to interpret the picture in order to discover the deductive structure hidden within, and that interpretive step—when it admits of any significant wiggle room—is at odds with the Euclidean notion of proof. (As we will see in the final section, "Proofs Without Words 2.0," there are ways in which PWWs 2.0 sometimes do a better job at indicating a particular deduction than their PWWs 1.0 counterpart.)
These shortcomings do not, however, correlate with our impression of whether a visual proof is convincing or satisfying. Indeed, the Kawasaki PWW of Viviani's theorem is notable for being both powerfully convincing and difficult to immediately translate into a traditional proof-writing framework. It seems to be an open question where to draw the line between failing to express a proof and failing to clearly express a particular proof. Surely high levels of unclarity push an attempt to write up a proof below the bar and result in a failure to express a proof at all. In general, though, published PWWs clearly express a mathematically compelling idea. To suggest that they fail to express a proof in Euclidean terms does not settle the question of whether a visual proof can express substantial insights and present compelling mathematical evidence for conclusions—even from a Baroque perspective; the Baroque perspective is a stance on proof, not evidence. Proof is a particular, refined way of presenting a special kind of evidence. It should not surprise us if the same general sort of evidence can be presented in other ways.
From the perspective we have constructed, any proof is an argument. In order to express a proof, a figure must, therefore, express an argument. It is important to remember, though, that the expression of the argument is distinct from the argument itself. The argument itself, a structured ordering of propositions, does the work of proving the theorem. The expression of the proof, be it marks on the page, sentences, or pictures, serves to express this argument.
Though inferential relationships are easy to express in sentences, depicting the dependence of a fact upon another fact is difficult. Failure, in general, to meet the syntactic standards of proof—that propositional starting points or premises be clearly designated, for example—does not immediately mean that a visual proof cannot express cogent, insightful mathematical thought and convincing evidence of truth that is allied to the activity and aims of proof-writing. A hard-line Baroque thinker might say that the syntactic or formal standards are actually the core of the evidentiary standards because they help to ensure that the proof is gap-free.^{Note} But there are other purposes and values encoded in the evidentiary standards of proof, and we find it more meaningful to ask whether a visual proof aligns with these purposes and values rather than dumping the whole enterprise because visual proofs don't tend to meet the syntactic or formal standards emphasized in the Baroque approach to proof.
In the case of powerful, convincing visual proofs that don't seem to unambiguously express a particular proof, the hard-line Baroque thinker can take steps to protect the realm of real proofs from incursion. For instance, the natural distinction between the justification and the discovery of a mathematical fact would let the Baroque thinker refer all mathematical justification to proof, but still allow that reasoning falling short of the standards of proof can contribute to mathematical discovery. Discovery (and rediscovery in the classroom setting) is critical to the practice and advancement of mathematics, so this limited role would still present substantial value for visual proofs.
Even if ultimately correct, this hard-line perspective has to contend with differences between visual proofs and other sorts of non-proof demonstrations that promote mathematical discovery. Unlike the other sorts of things that promote mathematical discovery but fall short of proof, e.g. testing particular numerical cases, visual proofs can align very strongly with an array of values that we would associate with proof, i.e. with the justificatory project in mathematics rather than merely the phenomenon of mathematical discovery.
Even the Baroque perspective has to allow some wiggle room to align itself with the actual practice of mathematics, where proof write-ups, even those published by renowned mathematicians in highly regarded journals, admit of at least a small degree of interpretation. Leaving a little room for interpretation, especially when writing for a particular, limited audience, does not undercut the role of proof in mathematical research. We have to be careful not to take the Baroque perspective so far that it turns out nobody ever writes proofs! When we relax the standard a little bit it becomes interesting to ask whether less canonical forms of proof write-up, like PWWs, can slip through the gap and have a place within—or perhaps just in the neighborhood of—the category of proofs.
Even if a PWW struggles to express clear statements of consequence and inferential ordering, it can illustrate the key idea or main insight that a more completely written-up proof could exploit. And when we return to consider PWWs 2.0 in the final section, we will revisit the Euclidean syntactic or formal standard and discuss ways that interactive and animated PWWs can recover some of the proof-writing syntax/form lost by eschewing prose.
In the next section we turn away from questions about the nature of proof to consider instead the 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 PWWs since they often seem to present ideas of genuine mathematical interest without meeting Euclidean, formal standards of proof.
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.
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 \(64=65\).
Figure 10. Here we see a commonly encountered "joke-proof" that involves the dissection and rearrangement of a region to create a new region with a different area. On page 271 of his book, Dissections: Plane & Fancy, George Fredrickson referred to such "proofs" as bamboozlements.
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:
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.
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.
Figure 11. This untitled Proof Without Words illustrates a number of common trigonometric identities. [Romaine]
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:
Figure 12. This proof without words is intended to establish the general arithmetic - geometric mean inequality,
\({\sqrt{ab}}\leq{\frac{a+b}{2}}.\) [Gallant]
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.
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.
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.
The advent of ubiquitous networked computing provides a new medium of expression for Proofs Without Words. In this section, we demonstrate several interactive Proofs Without Words that we informally dub PWWs 2.0—the 2.0 indicating a second generation of PWWs that leverages this new medium. We discuss some of the philosophical advantages of PWWs 2.0 over traditional PWWs. At the same time, we introduce several open-source technologies that make the creation of PWWs 2.0 accessible to anyone with a computer. We finish by urging the creation of a new PWWs 2.0 column to appear regularly in this journal and can only hope that it enjoys the success of the original Proofs Without Words features in Mathematics Magazine and The College Mathematics Journal.
PWWs like Gallant's "A Truly Geometric Inequality" are powerful in their explanatory capabilities. While not changing its core approach, we argue that the PWW 2.0 in Figure 13 is even more convincing than Gallant's original PWW in Figure 12.
Figure 13. A Proof Without Words 2.0. This figure is an interactive adaptation of Charles Gallant's original Proof Without Words: A Truly Geometric Inequality. Move the black point left and right along the base to see different configurations.
This figure is rendered in a free, open-source, JavaScript-based language called JSXGraph. The mathematical symbols are generated by a free, open-source, JavaScript-based \(\LaTeX\) renderer called MathJax. Furthermore, the documentation for both of these software packages is quite good and freely available. Sketches written using these tools will run in nearly all modern web browsers on nearly all popular computing platforms with no more effort on the part of the user than simply loading the associated web page. (We note that the colors used in the figures throughout this section have been demonstrated to be friendly to readers with any of the three most common types of color blindness.)
The ability of the user to move the point left and right along the baseline generalizes the original PWW in a way that is not possible in a print medium. The interactive nature of the figure allows the user to explore this result. For example, the user may slide the point back and forth and discover that equality holds when \(a=b.\) Here we use solids and color to indicate givens, and dashes and grays to better indicate where a reader/user should focus her attention. This sort of "visual syntax" is much easier to implement in a computational medium.
The use of color also allows the reader to immediately identify the line segments corresponding to the symbols \(a\) and \(b\)—a task somewhat more difficult in the original. The reader is encouraged to work through this PWW and compare the original and the 2.0 version. For convenience, we provide a reading of this PWW tucked away in this knowl (more about knowls).
Having rehearsed Gallant's PWW in both forms, the virtue, in our view, of the 2.0 version of this PWW is that it does a reasonable job of prompting a particular set of questions, the answers to which are easily verified to yield the building blocks of a proof. By prompting questions in this ordered way, the interactive diagram has a "syntax" closer to that of a traditional wordy proof. It doesn't force the reader to go verify the labels in any particular order, but it does suggest rather specific ways to establish the labels. We think of establishing the labels as something like proving lemmas. The significance of these lemmas/labels for establishing the theorem becomes clear when we inspect the geometrical relationship between the labeled lines in the diagram. It's basically the same proof whichever lemma/label we address first, so in this PWW 2.0 we substantially address the earlier concerns that PWWs tend not to encode sufficient logical or propositional structure to count as expressing proofs in the Euclidean sense.
With the interactive nature of PWWs 2.0, sometimes we discover unexpected results. Compare the PWW 2.0 adaptation of Wolf's PWW for Viviani's Theorem in Figure 14 to the original following it in Figure 15.
Figure 14. This interactive adaptation of Wolf's PWW of Viviani's Theorem allows one to position the point \({\mathcal P}\) anywhere within the triangle \(ABC.\) (Rendered using JSXGraph and MathJax)
Figure 15. Wolf's Proof Without Words of Viviani's Theorem. [Wolf]
Because of the nature of proofs without words, we cannot know how Wolf expected the reader to get to the Viviani result, but the use of bolded, dashed and dotted lines is suggestive of a pathway. In trying to get the three perpendiculars to sum to the height of the triangle, we see that the dotted line is already in place. We are left to convince ourselves that the bolded perpendicular at \(F\) is congruent to the bolded segment \(\overline{QG}\) and that the dashed perpendicular at \(G\) is congruent to the dashed segment \(\overline{GC'}\). The three segments are now stacked parallel to the height of \(\Delta ABC\) and we are done. A closer examination of this diagram through use of the PWW 2.0 shows that this suggested pathway is not general enough.
In the PWW 2.0 version, the user is free to move the point \({\mathcal P}\). In its initial configuration (hit reload to reset), this closely resembles Wolf's original PWW. The reader is encouraged to move the point \({\mathcal P}\) in the PWW 2.0 and to observe that in general \(G\) does not lie on segment \(\overline{QC'}\) and that there is no general way to construct the dashed line segment suggested in Wolf's PWW. Of course, Viviani's theorem is still true, just not in the way suggested by casual examination of Wolf's PWW.
Additionally, this PWW 2.0 offers the user the chance to see the extremes of Viviani's theorem by moving \({\mathcal P}\) to one of the vertices of the base triangle, \(\Delta ABC\). It also lets the user ponder the meaning of Viviani's theorem when \({\mathcal P}\) is moved outside of the base triangle and to speculate on new theorems. For example, moving \({\mathcal P}\) out beyond the segment \(\overline{BC}\) might suggest that if we considered signed lengths (i.e. perpendiculars with negative lengths when \({\mathcal P}\) is on the "wrong" side of the line segment), then Viviani's Theorem still holds.
Two other free, open-source software packages called Processing and its sister project ProcessingJS provide another way of creating PWWs 2.0. Consider the 2.0 version of Kawasaki's PWW for Viviani's theorem (Figure 9) in Figure 16. Put your mouse inside the triangle and click.
Figure 16. This PWW 2.0 version of Kawasaki's PWW of Viviani's Theorem is both more general and clearer than the original. Use the sliders to complete the rotational steps, and drag the point \(P\) to reimagine the proof from a different position in the triangle. (Rendered using GeoGebra)
In this PWW 2.0, the user is able to see the animations suggested in Kawasaki's original PWW. Furthermore, the user is able to place the interior point anywhere inside the base triangle before starting the animation with a click. The sketch allows us to see the animation that Kawasaki was only able to suggest in his original PWW. Also, the freedom to place the interior point anywhere within the triangle and restart the animation makes the representation of the result more general and the "proof" more convincing. 2.0 versions of many existing PWWs could be created to reduce the concerns about the reliance on particular examples and to partially address the accusation that many PWWs attempt to encode an essentially fallacious proof by cases.
As we noted in the introduction and our brief history of PWWs, there are a number of visual proofs of the Pythagorean Theorem dating back thousands of years. There are also some excellent on-line interactive visual proofs of the Pythagorean Theorem that are of a distinctly more recent vintage.
Consider, for example, in Figure 17 the PWW 2.0 rendering of Rufus Isaacs' Proof Without Words of the Pythagorean Theorem that we showed in Figure 1 of the Introduction.
Figure 17. A PWW 2.0 adaptation of Isaacs' PWW of the Pythagorean Theorem (Figure 1). Move the endpoints of the segments \(a\) and \(b\) to change the size of the right triangle. The slider translates the pieces into a new configuration. An accompanying student worksheet is available at GeoGebraTube. (Rendered using GeoGebra)
This example is rendered in GeoGebra. GeoGebra is a well-developed, open-source, free development environment that allows the creation of rich on-line interactives. The ability to move the blue points to resize the right triangle, and then to animate the relocation of the various pieces, makes this a general and convincing proof. The use of color allows the user to keep careful track of where the pieces go. The applet allows the user to explore even the most extreme cases of this important result. It is important to note, however, that we must be skeptical that the pieces do not somehow expand or contract or change shape subtly to fit in their destination locations.
A second example, shown in Figure 18, is this applet by Jed Butler which we found on GeoGebraTube and we reproduce here for convenience:
Figure 18. An alternate PWW 2.0 of the Pythagorean Theorem by Jed Butler taken from GeoGebraTube. Move the blue endpoints of the segment and the X to change the right triangle. The slider translates the pieces into a new configuration. This proof is generally attributed to Henry Perigal and is known as "Perigal's Proof" or "Perigal's Dissection." (Rendered using GeoGebra)
The ability to move the black X, the blue points, and then to animate the relocation of the various pieces, again makes this a general and convincing proof. Though embedded here, both of these examples are actually hosted on GeoGebraTube, the free community portal dedicated to GeoGebra.
There are numerous visual proofs of the Pythagorean Theorem discussed in Roger Nelsen's books. The reader is encouraged to try to create 2.0 versions of these with some of the tools mentioned above or with one of the many other tools available.
Another exciting aspect of this new medium of expression is the possibility of rendering in 3 dimensions. While the reliability of 3D rendering on the web is still lower than we would hope at the time of this writing, it is certain to improve over time. To illustrate the possibilities, we finish with a 3D implementation of Siu's elegant "Sum of Squares" PWW which first appeared in Mathematics Magazine in 1984. For reference, we show Siu's original PWW in Figure 19 followed by the 2.0 version in Figure 20.
Figure 19. Siu's original PWW of the formula \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(n+1/2)}{3}.\) The reader is clearly prompted to animate the sequence of figures. [Siu]
Note: The following interactive component relies on a relatively new graphics engine called OpenGL. It may not work on some web browsers. If it fails to run on your computer, try to enable OpenGL. If you do not wish to do so or are unable to do so, you can still watch a video of the applet in action just below the applet window.
(i - zoom in; o - zoom out; mouse drag to orient) |
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Current Problem Size: $n=$3 [] |
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$\displaystyle\sum_{k=1}^{n} k^2$ | $\displaystyle\sum_{k=1}^{n} k^2$ | $\displaystyle\sum_{k=1}^{n} k^2$ |
$3\displaystyle\sum_{k=1}^{n} k^2 = n(n+1)(n+1/2)$ |
The implementation is done using the 3D capabilities of the Processing programming language. The user can click and drag on the animation to see all sides of the animation in progress. By allowing the user to choose a viewpoint, she can convince herself that the blocks fit together and verify that she is not viewing the diagram from a privileged perspective. In addition, this sketch demonstrates the ability of Processing to interact with other elements on the web page. For example, the user can control the problem size and restart the animation using the buttons below the animation.
It is interesting to note that Siu's PWW illustrates a common theme in PWWs that depict results in number theory—the use of blocks or squares to represent units. At the beginning of the animation, each pyramidal stack of blocks represents a sum of squares. Of course we cannot illustrate the sum of the first \(n\) squares using a pyramid. Instead, like a traditional PWW, we are forced to pick a specific value of \(n\) (in this case \(n=3\)) and let the reader generalize. However, the ability to change the problem size makes the PWW 2.0 more convincing.
This PWW 2.0 doesn't have the full generality of a mathematical induction proof, but it does have increased generality over a static PWW. One could still argue that the inspection of a finite number of figures or animations is still an attempt at proof by cases, and proof by a handful of cases is still no proof at all; it leaves just as vast an infinity of cases unexamined. The point isn't that the 2.0 version represents numerically more cases and therefore has more generality. The interesting advance over the static PWW is rather that the 2.0 version allows the user to formulate and answer questions about the generality of the representation. Which cases a user inspects will reflect her concerns about possible ways that the pattern could break down. For example, by setting the problem size to \(n=4,\) the user can address a potentially lingering concern that the result might require a different approach for odd and even numbers, as many traditional induction proofs do. By allowing the user to interact with the animation, we offer a framework for the user to challenge the claim made by the PWW and observe the results. We believe that this interactive capacity, in this case, moves the PWW 2.0 meaningfully beyond the criticism so easily leveled at print PWWs that they cannot constitute proof because proof by cases is no proof at all.
Through this sequence of examples, we have demonstrated that the barrier to creating interactive PWWs and sharing them is quite low. The multitude of free, open-source JavaScript tools grows daily. All of these tools can and should be brought to bear on the task of sharing mathematical insights. In addition, because of the openness of these tools, people interested in learning to create PWWs 2.0 can start by looking at the source code of the examples in this article—all of which are free to learn from, to use, and to modify.
Also, it is interesting to note that many PWWs enjoy rich histories. In preparation for this article, the authors reached out to some of the authors of the PWWs mentioned herein. In an email exchange with Man-Keung Siu, we learned that the origins of his Sum of Squares PWW extend back as far as the 11th century work of Shen Kuo and the later work of Yang Hui (c. 1238-1298), who, in particular, gave a formula for the number of pieces in a pyramidally shaped stack of fruit. We are grateful to Man-Keung Siu for providing this wonderful synopsis of the genesis of his PWW [Siu's email].
We are excited about this new medium of mathematical communication as both a pedagogical tool and a continuation of the historical arc of many of mathematics' most compelling visual proofs. We call for Convergence to create a regular feature called "Proofs Without Words 2.0" to receive, review and, if suitable, publish PWWs 2.0 that provide insight into and/or proof of important mathematical results. As we have seen with Rufus Isaacs' first two Proofs Without Words, there is usually some historical context associated with each Proof Without Words. Since Convergence is "Where Mathematics, History, and Teaching Interact," we feel it is appropriate for contributors to also include an historical note to accompany the PWW 2.0.
Editor's note: Convergence has a definite preference for interactive applets created using the free software GeoGebra, because these applets can be hosted by the MAA channel on GeoGebraTube. Each applet must fit in a window no wider than 680 pixels. We look forward to your submissions of PWWs 2.0!
The authors wish to thank Moira Gresham, a member of the Department of Physics at Whitman College, and E. Sonny Elizondo, a member of the Department of Philosophy at UC Santa Barbara, for their thoughtful feedback on this paper.
Tim Doyle is originally from western Washington, and completed undergraduate work at Reed College and graduate work at UCLA, primarily in philosophy, but with lots of math thrown in for good measure. His research interests include, but are definitely not limited to ancient philosophy, philosophy of mathematics and logic, and early analytic philosophy. When he's not talking to students or buried in books, he moonlights as a winemaker.
Lauren Kutler is a Math for America Teaching Fellow currently teaching algebra and geometry at Malden High School (near Boston, MA). She double majored in mathematics and philosophy at Whitman College and then completed a Master of Arts in Teaching at Boston University. Lauren also enjoys exploring the outdoors, traveling, and cooking with a lot of vegetables.
Robin Miller grew up in Corvallis, Oregon, and attended Whitman College, where she double majored in Mathematics and Economics and was an active member of the college orchestra and choir. After working as a business analyst for two years, she decided to go back to school and study to be a web developer. She is now living in Portland, Oregon and working as a software engineer for a local tech startup. In her spare time, she enjoys cribbage, strategy board games, cooking, and spending time with her family.
Albert Schueller is originally from Philadelphia, PA. He majored in mathematics at Penn State University ('90) and earned a PhD in mathematics at the University of Kentucky ('96) in inverse eigenvalue problems. He has always enjoyed computing and programming. He installed Linux for the first time in 1994 using 120 1.44Mb floppy disks. He is an avid proponent of open-source software and open-source textbooks.