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.