I've always been telling people: mathematics is art, just with a smaller audience.
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.
The beauty implicitly assumes the math is correct and consistent, but expresses something much deeper.
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.
Also answering a question. There's a youtube video about topology and solving .. I forgot but say elliptic curves equations. And when encoded in topological terms the problems simply vanishes (some would say the knot unties itself).
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.
Abstract art isn't arbitrary in its judgement. This has been researched, and even people who say "My five year old could have done that" can tell the difference between the work of actual five year olds and very abstract smeary work by professional abstract artists.
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.
I find utility is most beautiful. Various other interpretations of aesthetics in mathematics seem misplaced if the result is that none or a very limited number of verifiably consistent applications exist for some new construct/extension of math that is supposedly art but more or less a useless exercise of mental masturbation. Higher theoretical math is purely a construct of reasoning, and may not always produce verifiable consistency for every domain—like reflecting the real (this) universe being studied. I would prefer mathematicians study what is utilitarian and foundational, instead of yet another questionably useful offshoot of mathematics. There are a million dead ends and very few interest pieces that lead to a goldmine of new applications or improvements.
I found the Apology a delight. You can easily read it in one sitting. But it will stay with you for a long time.
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.)
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)
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.