Math Aesthetics: Is Beauty Intrinsic or Subjective?

[fourier jr]

Everybody has heard of a "beautiful" theorem, or proof, or formula, etc but has anyone heard of any work in aesthetics done on things in math? I did a (very) minimal bit or reading on aesthetics, and I read that there are two general points of view. There's the objective point of view, where beauty is an intrinsic property of a sculpture/painting/etc, ie beauty is not in the eye of the beholder and things are beautiful whether or not someone is there to say so. Then there are the people who say it is subjective, and that beauty is in the eye of the beholder. I have NEVER heard of anyone at any time working on this sort of thing in math, only with things like paintings, etc. But what if we did? Is Euclid's proof that there are infinitely many primes a beautiful proof because we say so or because of what it is, regardless of what anyone thinks?

ps - it sounds a bit like the "discover vs invent" argument in math, but that's not what I'm asking. I think the two ideas could be related though.

 

[Janitor]

I know you will get some thoughtful comments on this.

May I throw in a little sub-question of my own on your thread? I wonder if people tend to think that the proof of the four-color map theorem done on a computer by Appel and Haken in 1976 is considered ugly? I have a hard time seeing much beauty in it, though I suspect it is valid.

 

[fourier jr]

I know you will get some thoughtful comments on this.

May I throw in a little sub-question of my own on your thread? I wonder if people tend to think that the proof of the four-color map theorem done on a computer by Appel and Haken in 1976 is considered ugly? I have a hard time seeing much beauty in it, though I suspect it is valid.

I guess the computer went through every possible fundamental case, but it didn't give any insight as to WHY it was true. I don't know the proof but it probably isn't beautiful. Not that computers don't give nice proofs of things though; a computer 'rediscovered' Euxodus' proof that the angles in an isosceles triangle are equal, which is a much nicer one than what's given in textbooks.

 

[chroot]

Personally, I think some mathematical proofs are as beautiful as any sunset, or any starlit night -- that's the subjective bit. On the other hand, I think they definitely have intrinsic beauty: you can almost define mathematical beauty as any proof that combines, amalgamates, or reduces a lot of complex cases into one central, simple, and elegant statement. The more areas of mathematics a theorem brings together while still being simple and easy to follow, the more beautiful it is likely to seem. Thus, little tidbits like e^{i \pi} + 1 = 0 and entire fields like group theory can be said to be very beautiful even in a purely utilitarian sort of way.

- Warren

 

[Loren Booda]

I believe that beauty in mathematics relies foremost on the visualizability of its parts, and comparably, on the simplicity of their relations. It is subjective first in the sense that there is a plurality of proofs to most problems.

How about the beauty of approximate proofs that use probability to demonstrate truth, within a reasonable doubt, of otherwise intractable problems?

 

[Ebolamonk3y]

visualize... I thought your not suppose to visualize after a certaiin point...

 

[matt grime]

beautiful proofs, or perhaps more 'elegant' than beautiful, can be short and direct, and don't involve unnecessarily difficult ideas. sometimes people "confuse" (there is no arbiter, I'm just trying to offer *some* distiction) novel with elegant.

I myself can't decide if the proof that there are infinitely many primes using a topology on Z defined by arithmetic sequences is genuinely elegant or just novel, or if it can't be both. But then the proof of hte fundamental theorem of arithmetic (every polynomial of C has a root in C) using homotopy theory has to be elegantly beautiful doesn't it? Or is it a matter of sophistication?

One thing is for sure, any induction proof has to be inelegant and ugly: it offers nothing insightful to the proof or the result - knowing it true for a trivial case, and inducting on a presumption doesn't seem to offer any illumination.

Perhaps we might take the view that if the argument is one smooth flowing thread that that the proof is elegant, but then that might just come down to the presentation of the author. And are we distinguishing between an elegant result (exp(ipi)+1=0) or an elegant proof of a result?

Here is a case which is an elegant result and (to my mind) is often presented with an ugly proof:

Sylow's Theorems. If G is a finite group of order mq, with q a prime power and m prime to p, then there is a group of order q in G, moreover, all subgroups of order q (the sylow subgroups) are conjugate, every p subgroup is contained in some sylow subgroup the number of sylow subgroups is congruent to 1 mod p (I think, i can never remember that bit correctly).

that result gives you a tremendous amount of information about a group, tells you so much about why groups are so powerful, and is only a couple of lines long, yet its usual proof is horrendously dull. i can think of an elegant proof (of soem parts) using vertices and sources, but you need to know a lot more about groups and their representations before that becomes applicable.

undoubtedly the 4 colout theorem and the classification of finite simple groups are ugly proofs as they are just checking proofs, they offer no ingenuity. ok, that's overly dismissive, since 4-colour required ingenuity to show that all cases could be reduced to one of the computer checked cases, and I'm sure that in CFSG there are bits of ingenuity for different cases, but over all it's a checking proof. But, such are the clouds obscuring all these objects, right now that's the best we can do.

 

[Gokul43201]

Would you consider the 26 variable polynomial that generates primes to be ugly ? Or is it a thing of beauty that you have to use every letter in the English alphabet ?

 

[matt grime]

"the 26 variable polynomial that generates primes"? Which is what?

 

[Gokul43201]

http://functions.wolfram.com/NumberTheoryFunctions/Prime/31/06/

 

[quddusaliquddus]

That sounds nice.

 

[matt grime]

Never seen that before. It can't be beautiful because it is alphabet dependent AND for pity's sake they use "i" in a number theory statement, something you should never do.

 

[quddusaliquddus]

beauty ... in 'i' of beholder

 

[Loren Booda]

Great pun!

 

[matt grime]

the thing that bothers me about the function is that it almost certainly isn't correct. For instance if z were any other number than 1, then the expression is composite and must therefore be negative (or zero) so why not put z=1? although granted the presentation makes it hard to figure out where one factor ends and another starts, similarly k must be odd, so if they were awake they'd replace it with 2k+1. Other such observations apply.

 

[quddusaliquddus]

Wolfram don't usually get things wrong ... u cud b rite

 

[matt grime]

I sincerely doubt it is correct, unless there is something special about that set of primes (ie it is not complete). I just doesn't seem plausible, and assuming I've read the brackets correctly, it contains some very redundant information.


edit: seems my intuition is very wrong indeed. there are lots of such generating functions, must have read the brackets wrong then.

 

[quddusaliquddus]

Thanx ...

 

[Gokul43201]

Matt,
I think I've seen another form of the same polynomial using only 24 variables...so you're probably right there.