Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
In the beginning, when there was no language to express general mathematical ideas, proofs without words were the proofs. Martin Gardner wrote, "There is no more effective aid in understanding certain algebraic identities than a good diagram. One should, of course, know how to manipulate algebraic symbols to obtain proofs, but in many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." A classical example concerns triangular numbers:
which is ascribed to the ancient Greeks. I would argue that, for the Ancients, if they ever drew them, the diagrams were more than an effective aid. In the absence of algebraic symbolism, they might have served as a combination of a statement and its proof in the most concise form available at the time. In his Introduction to Arithmetic, Nicomachus of Gerasa (c 100 A.D.) writes, "Every square figure diagonally divided is resolved into two triangles and every square number is resolved into two consecutive triangular numbers, and hence is made up of two successive triangular numbers." Obviously, he refers to the diagram on the right (although there is no indication that he had ever drawn such a diagram), and this is the only proof that is ever given to the identity N2 = N(N+1)/2 + (N-1)N/2, or rather to the fact that the sum of two consecutive triangular numbers is a square number.
In the collection "Proofs Without Words", Roger Nelson credits Nicomachus with the demonstration of another identity: 1 + 3 + ... + (2N-1) = N2. This is probably based on the following passage from Introduction to Arithmetic: "This (square) number also is produced if the natural series extended in a line, increasing by 1, and no longer the successive numbers are added to the numbers in order, as was shown before, but rather all those in alternate places, that is, the odd numbers. For the first, 1, is potentially the first square; the second, 1 plus 3, is the first actuality; the third, 1 plus 3 plus 5, is the second in actuality... In these cases, also it is a fact that the side of each consists of as many units as there are numbers taken into the sum to produce it."W.W.Rouse Ball noted that the book remained a standard authority on the subject of Arithmetic for a thousand years. However, he complained that "geometrical demonstrations are here abandoned, and the work is a mere classification of results then known, with numerical illustrations: the evidence for the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances." Although critical of Nicomachus work, W.W.Rouse Ball appears to accept that (at least at that time) geometrical demonstration was considered a valid proof. As the quote by M.Gardner indicates, it is no longer. The idea of proof has been changing with time.
Up to the time of Thales of Miletus (c 640-546 B.C.), the Ancients had only two ways to communicate mathematical statements: via illustrative examples that served as templates for a general statement, or through a diagram that made the statement obvious. The former told a reader how to obtain a result. The purpose of the latter was to help the viewer internalize the idea by gaining an insight into why the idea was correct.
Thales, one of the Seven Wise Men, conceived of the need to reduce, by the logical argument, mathematical statements to putatively simpler facts, thus making the former more convincing. The understanding of proof as a convincing argument based on intuitive truths was established in human culture by Euclid's Elements. In the nineteenth century, our perception of the role played by intuition in mathematics underwent several crises, and the notion of proof was further formalized. By now, "everybody knows what a mathematical proof is. A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion ([G.-C.Rota])."
The attempt to teach students to create those steps is frustrating for teachers and probably does students more harm than good.
In mathematical instruction, the utility of proof as a convincing argument was questioned already in the seventeenth century (see the very informative article by E. Barbin) when Arnauld and Nicole, the jansenit friends of Pascal, registered dissatisfaction with the Elements on the basis that Euclid was "more concerned with certitude than evidence, more concerned with convincing the mind than enlightening it." In the eighteenth century, Clairaut [Barbin, p 46] made a clear distinction between the role of proof in Mathematics and mathematical instruction. Beginners, he wrote, should be shown a way by which mathematics was being discovered. Barbin further writes that "up to the end of the nineteenth century, authors rejected the idea of proof as deductive reasoning when it was a question of enlightening the beginners."
Reflecting on the words of Clairaut, Barbin asks the following question: "Might it not be that a feeling for proof derives from the very fact of discovering a result?"
Proofs without words (and more generally, a diagrammatic method) answer several concerns that arise at attempts to teach and to learn about proofs. First of all, they help students arrive at the presented mathematical concepts independently - the result may be sensed, or discovered intuitively. Connecting with the desired result is an enlightening experience for the student. Secondly, proofs without words make more discernable relationships between parts or parameters of a mathematical statement. Which usually provides clues to 1 or more steps in a deductive proof. As far as the deductive reasoning is concerned, even a one step proof, even a proof that does not start with fundamental axioms, is a valid proof. In this context, the maxim that making the first step is the most difficult task is true in more than one sense.
Proofs without words indeed very effectively help us not only learn mathematical statements but also develop a feeling for mathematics as a discipline. Those that permit some degree of interaction and experimentation are potentially even more effective. Computers change the way mathematics is done, they change the way mathematics is taught and learned. Computers provide a vehicle for the evolution of an interactive diagrammatic method.
In the introduction to his book, Roger Nelson writes, "I should note that this collection is not intended to be complete. It does not include all PWWs which have appeared in print, but is rather a sample representative of the genre. In addition, as readers of the Association's journals are well aware, new PWWs appear in print rather frequently, and I anticipate that this will continue. Perhaps some day a second volume of PWWs will appear!" I won't be surprised if the second volume of PWWs appears on CD-ROM or even online.
Here is a simulation of an old puzzle. A farmer is to ferry across a river a goat, a cabbage, and a wolf. Besides the farmer himself, the boat allows him to carry only one of them at a time. Without supervision, the goat will gobble the cabbage whereas the wolf will not hesitate to feast on the goat. Clickable are the three shapes by the river and the arrow that indicates the position of the boat (and, of course, the farmer.)
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1997 Alexander Bogomolny