Mathematicians often speak of a particular theorem or proof as being “beautiful”. Those of us who have seen beautiful mathematics know exactly what it means, but what if you’ve never seen beauty in mathematics before? Is it one of those “I know it when I see it” kinds of things? Paul Erdös thought so:
Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
Ask your favorite mathematician (we all have one, don’t we?) what they find beautiful, and I’m sure you’ll get an immediate response.
Bertrand Russell said
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
For anyone who asks me, I submit the following excerpt from Leonhard Euler’s Introductio (E101), as translated by George Pólya in Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics, in which Euler presents the thinking that eventually led him to the Pentagonal Number Theorem:
1. Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of the primes, which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order. I am myself certainly far from this goal, but I just happened to discover an extremely strange law governing the sums of the divisors of the integers which, at first glance, appear just as irregular as the sequence of the primes, and which, in a certain sense, comprise even the latter. This law, which I shall explain in a moment, is, in my opinion, so much more remarkable as it is of such a nature that we can be assured of its truth without giving it a perfect demonstration. Nevertheless, I shall present such evidence for it as might be regarded as almost equivalent to a rigorous demonstration.
Euler then proceeds to introduce the divisor function, σ(n), and does some computations that lead to the formulation of a version of the Pentagonal Number Theorem.
8. The examples that I have just developed will undoubtedly dispel any qualms which we might have had about the truth of my formula. Now, this beautiful property of the numbers is so much more surprising as we do not perceive any intelligible connection between the structure of my formula and the nature of the divisors with the sum of which we are here concerned. The sequence of numbers 1, 2, 5, 7, 12, 15, … does not seem to have any relation to the matter in hand. Moreover, as the law of these numbers is “interrupted” and they are in fact a mixture of two sequences with a regular law, of 1, 5, 12, 22, 35, 51, … and 2, 7, 15, 26, 40, 57, …, we would not expect that such an irregularity can turn up in Analysis. The lack of demonstration must increase the surprise still more, since it seems wholly impossible to succeed in discovering such a property without being guided by some reliable method which could take the place of a perfect proof. I confess that I did not hit on this discovery by mere chance, but another proposition opened the path to this beautiful property—another proposition of the same nature which must be accepted as true although I am unable to prove it. And although we consider here the nature of integers to which the Infinitesimal Calculus does not seem to apply, nevertheless I reached my conclusion by differentiations and other devices. I wish that somebody would find a shorter and more natural way, in which the consideration of the path that I followed might be of some help, perhaps.
(For more on the Pentagonal Number Theorem, see this arXiv paper.)
The first time I read through Polya’s book, I didn’t really pay much attention to this passage. (In fact, I didn’t pay attention to anything but the problems.) I read it again two years ago, and I was struck by the elegance with which Euler was conveying his passion for math, his disappointment with the incompleteness of the result, and his determination to keep going. His writing is perspicuous. (Yeah, I just looked that up, but it fits perfectly.) And it’s beautiful. It is the one of the most beautiful pieces of mathematics I have ever read, not because of the quality or utility of the result, but because of the clarity and emotion that permeate it.
Beauty is, of course, subjective, but other often cited examples include the Pythagorean Theorem, Gauss’s Law of Quadratic Reciprocity, and Fermat’s Last Theorem. In 1999, Paul and Jack Abad published a list of The Hundred Greatest Theorems, though I don’t know how many would be considered “beautiful”.
I turn to you now, dear readers. What do you find beautiful in mathematics?