Proofs as Art

Proofs form the basis for all theoretical work in mathematics today. They are the ambrosia of pure mathematicians, the cornerstone upon which all applications stand. Some dedicate their entire lives to pursuing the proof (or refutation) of a single Millenium Prize Problem. Yet, where did this idea of truth - and not simply truth, but a logical, reproducible derivation of that truth - come from? Can there be more to this formal process than a simple conceptual validation or refutation of a theorem?

To that end, I was reading Philosophy of Mathematics by James Rob Brown and became quite intrigued by the idea of visual proofs - those concepts which can be shown, one way or another, via actual images rather than the rigorous notation of conventional mathematical training. Oftentimes, these can be clearer and more easily understood (especially by a layperson) than the latter, as well. Take for example the Pythagorean Theorem. The following is a classical, formal proof of the theorem first advanced in Euclid's Elements:

Statement:	Reason:
1. Draw triangle ABC, a right-angled triangle having angle BAC as the right angle.	1. Construction.
2. Describe on BC the square BDEC, and on BA the square BAGF, and on AC the square ACKH.	2. Construction.
3. Draw a line (AL) through A parallel to either BD or CE, and draw also the lines AD and FC.	3. Construction.
4. CA is in a straight line with AG.	4. Since each of the angles BAC and BAG is a right angle, these adjacent angles are equal to 2 right angles, which is equivalent (by Euclidean Proposition I.14) to saying the lines coincide or are in line with each other.
5. Similarly, BA is in a straight line with AH.	5. As the result of a similar line of reasoning as step #4.
6. Angle DBC equals angle FBA.	6. Two right angles are equal to each other.
7. (Angle ABC + right angle DBC) = (angle ABC + right angle FBA).	7. The same angle, added first to one right angle and then to another right angle produces equal results.
8. Therefore, angle DBA = angle FBC.	8. Since angle DBA = (angle ABC + right angle DBC); and angle FBC = (angle ABC + right angle FBA).
9. Line segment DB = line segment BC, and line segment FB = line segment BA.	9. Since BFGA and DBCE are squares, and on a square all sides are equal.
10. Construction line AD = construction line FC; and triangle ABD is equal to triangle FBC.	10. Since line segment FB = line segment AB (step #9), line segment BC = line segment BD (step #9), and angle DBA = angle FBC (step #8), the triangles FBC and ABD are congruent since triangles are congruent when the corresponding parts of those triangles are equal. (Euclidean Proposition I.4)
11. Parallelogram BDLM is double of the triangle ABD	11. Since triangle ABD and parallelogram BDLM have the same base (BD) and are in the same parallels, BD and AL (by Euclidean Proposition I.41).
12. Square GABF is double the triangle FBC.	12. Since square GABF and triangle FBC have the same base (FB) and are in the same parallels, FB and GC (by Euclidean Proposition I.41).
13. Therefore, parallelogram BDLM is equal to the square GABF.	13. Since the doubles of equals are equal to one another.
14. Draw construction lines AE and BK.	14. Construction.
15. Parallelogram CMLE is equal to the square HACK.	15. By a line of reasoning similar to that which proved step #13.
16. Therefore, the whole square BDEC is equal to the two squares, GABF and HACK. In other words, the square on the side BC is equal to the squares on the sides BA and AC [that is to say, (BC)2 = (AB)2 + (AC)2 ] Q.E.D.	16. Since square BDEC = the sum of the parallelograms BDLM and CMLE.

Source: Khan Amore's Commentary on the Pythagorean Theorem

Now, look at this visual proof and see if you can discern the same resulting equation from the arrangement of shapes provided. Another possibility lies in the dissection of squares, presented here in a series of visual steps. There is literally a multitude of proofs of this famous theorem out there (see Pythagorean Proposition by Elisha Scott Loomis), all drawing the same elegant conclusion, yet all approached in a different way - from a different visual perspective. And that forms the basis for mathematical reasoning. For what is the language of numbers but a more elegant way to represent the world? The heart of all theory lies in understanding, and understanding through manipulation of this language with our logical faculties. Even the most obscure (attempted) solutions to Riemann's Hypothesis essentially began with a window through which to view reality.

Hence, however much visual proofs are abjured by mathematicians as "non-rigorous", I believe they hold a special value in the hearts and minds of men. The whole basis of formal proofs lies in a consistent system to communicate the derivation of an idea to another person. Since mathematicians span a wide range of countries with a wide range of backgrounds, it stands to reason that there would be something to be gained from keeping to these general rules. But. When actually seeking to understand the reasoning for a concept, an image is infinitely helpful - even if by its very nature, it is not as "general" a proof as a written one (the sides of the squares in the visual proof of the Pythagorean theorem are a set length to our eyes, even if they are labeled to be an arbitrary variable length in the picture - because the picture cannot display all different lengths at once). Not only that, but humans are primarily visual creatures, and even when writing the rigorous proof, we oftentimes must manipulate some picture representation on our mental sketchpad to determine the next step. This is most obvious in geometry and trigonometric ventures. The reason for falling back on the language of mathematics is that in much upper level derivations, it is too conceptually difficult to visualize so many different variables, dimensions, etc all at once - so we put them down as pure expressions on paper.

Let me then get to the heart of this post. The idea of picture proofs as a form of art. Now, you may ask, what kind of art could come from this rearrangement of simple geometric shapes? You would have obviously missed the era of modern art. Just look at some famous abstract paintings like Piet Mondrian's compositions and you will see the connection. Islamic mosaics are another example of a geometrically based art form; many of their places of worship were decorated with intricate patterns of squares, circles, and triangles interlinked. Furthermore, anyone who has taken an elementary art course would know that point perspective is very much an exact science. The alignment of vanishing point and horizon line with the rest of the picture is essential to a realistic depiction of buildings, railroads, etc. Tessellations form another area in which mathematics and art meet - the idea of repetition, reversal, and overlap intrinsic in them goes to the fundamentals of both disciplines. In fact, Godel, Escher, Bach by Douglas Hofstadter delves deeply into the relationship between mathematics (Godel's incompleteness theorem), art (Escher's paradoxical illustrations), and music (Bach's Musical Invention) through a very specific pandisciplinary venue - strange loops. Contradictions in Escher's paintings are easy to see. The monks eternally ascending the staircase is a corruption of artistic point of view, which can only exist in the space of a 2-dimensional piece of paper. But it has a mathematical counterpart as well! Kurt Godel discovered that in his incompleteness theorem, essentially using the two statement loop

The statement below is false.
The statement above is true.
in a mathematical sense to show that there cannot be a set number of axioms (statements taken as fact) upon which all future proofs are based. That is, there are always more truths than there are proofs for those truths.

So where am I going with this? Well, essentially to take the relationship between mathematics and art further, to build a larger community of conscientious "diplomats" to mend the gap between the two disciplines (a fault of the false science/humanities divide in today's society). A deep connection between the two areas exist, as I have shown above, and by acknowledging and taking advantage of that fact, both disciplines can grow much fuller in their communication of concepts (mathematics) as well as expand their horizons further through manipulation of new ideas (art). The visual proofs of a past era can provide the inspiration for a number of artistic metaphors, as well as playful corruptions of a theme, such as Escher did with point perspective. Perhaps instead of squares and triangles, someone can arrange buildings and towers in such a way as to illustrate the Pythagorean theorem, while still providing a personalized representation of reality through art. Hidden proofs within a painting - now that is truly a marvelous idea! Similarly, mathematics can derive great benefit from a larger emphasis on visual proofs as a supplement to the rigorous written ones accepted as the formal format today. Besides providing a better understanding of the concepts, it also challenges people to look at a single proof from various different perspectives. Each of the hundreds of proofs of Pythagoras' theorem was a different, correct way of attaining the same mathematical truth! After all, what great discovery in any area wasn't preceded by a new and unique approach to an age-old problem? The language of proofs does not have to be one written in stone. For even our ancestors, scratching on the walls of their caves, realized the beauty of transience...difference...and change.

Disclaimer: I hold neither a degree in mathematics, nor one in art. The above is solely a product of what I have gleaned from books and articles, as well as conversations with other enthusiasts. If you have any corrections to make or points to add, feel free to comment.