The Aesthetic Beauty of Math
theparisreview.org
theparisreview.org
After all, beauty is the only criterion for a work of pure mathematics. People might use other words, such as elegant and interesting, and also, quite often, surprising to highlight different aspects of that beauty. And quite often, a math paper works in the way a mystery novel does (modern math papers, for silly reasons, are often written backwards and give away the name of the murdered on the first page -- and yet still work in the same way).
In the end, a mathematician would go in a certain direction because it is interesting, and the results will be appreciated if they are beautiful.
The article mentions equations, but quite often, equations are only there out of necessity. The ideas, often better expressed with diagrams, examples, and conversations might still need some algebra to be pinned down, but often enough, they are considered "ugly" and shoved under the rug. The mathematicians' word for this ugliness is technical; during the talks, they'll say somethings likes this:
>And so the limiting distribution of the number of sides in a single cell in a tesselation given by a geodesic curve on a hyperbolic manifold is given by a Poisson process. The proof is quite technical, so I'll skip it...
Which translates to the following:
>Here are two ideas that come from different worlds. It turns out that they are the same in this setting: like two lines in a poem that rhyme. The beauty of the poem is readily seen, but the rhyme is difficult to spell out with the alphabet of proofs and equations - so I will not do it.
I'm no mathematician, but I think correctness and consistency must factor in somewhere.
Analogous to how music adheres to certain rules and structures (these rules and structures are allowed to vary across genres).
Another analogy is how a “beautiful play” in basketball should adhere to the rules of the game. For example, a beautiful dunk during a game implies that the player wasn’t travelling or carrying the ball.
My third analogy would be a “beautiful move” in chess. It’s implied you followed the chess rules to make that move otherwise it wouldn’t be beautiful.
You can invent aesthetically pleasing theories and notations but if it doesn't help solving a problem.. it's a bit moot.
There are __provably correct__ and __provably incorrect__ mathematical works.
That strong binary classification is not found in "other art".
But otherwise, yes.
You might say mathematical proofs have characteristics of quality that once agreed upon, can be somewhat objectively applied: brevity, clarity, innovation, difficulty, relationships to other works.
While abstract art is famously arbitrary in its judgement, most art (dance, music, painting, sculpture) have similarly objective criteria.
The rest is about form and content. There are certainly aesthetic form rules in most of the arts, but the real point of art is the content - emotional distillation of lived experience, psychological reflection, even social commentary and satire. Even terrible kitsch has some element of this, and the best art is very densely packed with it,
A proof cannot be art because it has none of that subjective content. You can maybe - at a reach - apply the results of a proof in a social and political setting, or discover it has some relevance to psychology.
But the proof itself is content-free.
Or the difference between a professional and a monkey?
https://en.wikipedia.org/wiki/Pierre_Brassau
> the real point of art is the content - emotional distillation of lived experience, psychological reflection, even social commentary and satire.
Is Symphony No. 5 really psychological reflection, social commentary, and satire?
Or it is more accurately described as deeply, innately beautiful? A pleasurable combination of patterns, rhythms, and tones?
It addresses a different side of the question. Namely, why, if mathematics is about beauty, most people perceive it as an instrument of torture?
(The TL;DR is: the way we teach math is often a nonsencial torture, to the extent that what we teach is not math).
[1]Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....
If you supplement his rational math with those books from AOPS (art of problem solving curriculum) to practice, it's actually quite easy. Their intro to number theory book is a good example of exploring the 'beauty' (in quotes so I don't sound like a cringeworthy pseudophilosopher)
It is a book you find yourself thinking about long after. It is by one of the great minds of his generation and his thoughts are elegantly expressed.
(Not saying that the views are completely right. Many observers, for instance, have noted that the Number Theory that Hardy cites as without application is used today as the basis of coding. But the ideas are both well-informed and well-stated.)
Basically how primitive tribes did think mathematically and did they start by accounting with pebbles or levers or ..