Reviel Netz talks about the aesthetics, or beauty, of mathematics. He finds beauty in the narration of the mathematical texts, especially “this dialectic of freedom and necessity” (261). By this he means that the mathematician has many different routes he can take to reach the solutions to his problems. Since the Pythagorean Theorem was the proposition I had to prepare for class, I found it interesting that the Pythagorean Theorem has these many different routes one can take to reach the solution. You could use Euclid’s simple and reliable proof, or you could use the proofs of Plato and Pythagoras as outlined in Proclus 340. Similarly, as we talked about earlier, one could prove other theorems by different methods such as circles versus superposition. Now, does the fact that these theorems can be proved so many different ways give support to the beauty of mathematics or detract from it? Does a plurality of ways to solve something make it less unique and more common and base, or does it reinforce its truth and beauty? I know that this is really abstract, but I just thought that I would throw this out there since we are reading a paper that discusses the beauty of mathematics.
February 20, 2008 at 3:55 pm
It is an interesting question which I’m not sure there is an definite answer to. Netz recognizes that beauty in mathematics can be appreciated on two different scales: the large and small. On a large scale, one enjoys the mathematical beauty of a text as a whole; here (in response to your question), we would appreciate the many different routes one can use to prove the Pythagorean theorem as a beauty of limitlessness. Mathematics allows for an infinite number of ways in which something can be proved.
On a smaller scale, one can appreciate how the author unfolds a mathematical text to heighten a reader’s anxiety and suprise. (form; 255) We can appreciate the Pythagorean theorem, on this smaller scale, by recognizing the succinct, precise, almost perfect steps Euclid takes in proving proposition 47. We must also recognize how Euclid’s holding out of proposition 47, until the end, adds a dramatic climax to Book I.
I found Hardy’s claims that the utility of a mathematical formula/proof is inversely related to its beauty, to be most engaging and startling. Do you find this true in all mathematics? Can something simple and perfect (like Euclid’s steps) be beautiful as well?
February 22, 2008 at 2:28 pm
Good question, Jared.
Mitch, I really like your observation (vis a vis Hardy) about the utility and beauty of a mathematical formula being inversely related.
One point in history which we might use to think about this is way Copernicus’ theory was received and changed in the sixteenth century. Like Ptolemy (and Aristotle, etc.), Copernicus thought that the planets had to move in perfect circles - after all, the simplicity of a circle, forever turning back on itself in identical motion, is the most beautiful description of eternity!
From our perspective, this beautiful geometry of the celestial bodies failed. Kepler realized that planetary movements had to be described by elliptical motion. Some people thought that ellipses were ugly, because they aren’t nearly as simple (i.e. beautiful) as circles. On the other hand, Kepler found a way to think about how the ellipses captured more elements in their explanation of heavenly movements: the equant describing equal areas in equal times, etc. So he felt that the mathematics of ellipses was even more beautiful, in this instance, than that of circles - it captured more details. So if the mathematics “explains” (whatever that means) more, is it more beautiful? What is utility here? What is beauty here?