What is the meaning of elegant evidence

Pi and Co .: The pure elegance of mathematics

From current books: excerpt from

As is well known, mathematicians are very different from the majority of their fellow human beings. One only needs to read a few of Enzensberger's poems in which he is amazed to find that they (the mathematicians) mainly stay in curved spaces and easily swap left and right ideals, not to mention lower bodies, which mean something completely different to them than the impartial observer might think.

Here we want to deal with another difference that is right in the middle of life. While the majority agrees on the really important things such as waste separation or dental care, but with relish discusses artistic things (the vernacular says, as is well known: There is no arguing about taste - and of course means the opposite), it is in mathematical terms The other way around the world. Since Plato at the latest, platonists and formalists have been ideologically defended against each other; the dispute as to whether mathematical laws are discovered or invented is a long-running issue; and the question of the primacy of pure versus applied mathematics can excite entire institutes. But when two mathematicians bend over a piece of paper and one of them says: "That is an extremely elegant proof!", He can be sure of the approval of his colleague. There is no dispute about the beauty and elegance of mathematical formulas, sentences and especially proofs, everyone agrees.

»[...] it is by no means a coincidence that most mathematicians are not unfamiliar with aesthetic criteria. It is not enough for them that a proof is stringent; their ambition is aimed at 'elegance'. "
(H. M. Enzensberger)

But what is elegance in mathematics? Strangely enough, nothing can be learned about this from the guild capitalists (but about beauty, as we shall see in a moment). I want to seduce the reader a little into the mathematical world, propose a very personal definition of elegance and illustrate it with some classic examples.

Of the beautiful and the true

When Andrew Wiles proved the most famous open math problem, Fermat’s Conjecture, in 1995, it was a headline on page one in every newspaper. A national newspaper read: "Cambridge mathematician solved 350-year-old mathematical riddle". To describe the virtuoso sequence of logical conclusions and structural statements in Wiles' work as arithmetic could not be further from the truth. But apparently the general opinion is that a mathematician is someone who mixes up 99 formulas and produces a 100th formula.

What some of the greatest mathematicians think sounds different. Aristotle writes in his "Metaphysics": "The mathematical sciences in particular express order, symmetry and limitation - and these are the highest forms of beauty." Johannes Kepler, who already tended to rave about, was entranced by the "golden" proportions of mathematics . Henri Poincaré postulated the astonishing sentence: "The aesthetic, more than the logical, is the dominant component in mathematical creativity." GH Hardy, a master of number theory who produces notoriously complicated formulas, noted with an atypical British overstatement: "There is no permanent place for ugly mathematics! «I hope to have finally shaken your view of the mathematician as a rigorous mathematician when I quote the physicist Paul Dirac:» It is more important that an equation is beautiful than that it agrees with the experiment. «Am Hadamard describes this presence of the beautiful most succinctly in his Psychology of Invention: “Mathematical genius manifests itself in two ways; it chooses the only right one from a multitude of alternatives with unmistakable certainty, and it is guided by the idea of ​​the perfect, a premonition of paradise, of the eternally valid. "

One should not think that the mathematicians only dealt with the beautiful in their meta-writings - on the contrary. The main thing is mathematics as a model of thought, as an image of reality, in short: knowledge and truth. The same Poincaré writes in his "Last Thought" accordingly: Science is the urge for truth on a moral basis. Wittgenstein emphasizes the categorical rigor of logic, and Popper introduced the masochistic principle into science: a theory is only worth something if it can be falsified. One reads sentences like: "Only use ennobles knowledge", "Mathematics is humanism", and in what is perhaps the best recent book "Experience Mathematics" by Davis and Hersh, four out of 400 pages are devoted to questions of aesthetics.