Do (some) mathematicians think that beauty is evidence of truth in mathematics?

Question

Some physicists have argued that (other things being equal) a beautiful theory is more likely to be true than an ugly one. Dirac famously took this view: "A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data."

It's not uncommon for mathematicians to talk about beauty in mathematics. (I always think of Hardy, "there is no permanent place in the world for ugly mathematics.") But are there mathematicians who think (as some physicists do) that beauty (or elegance, or whatever) is evidence of truth?

I've done some Googling, and I've found many discussions of beauty in maths, but none of the things I've seen so far deal with my question directly.

Answer 1

It's empirically true that reality has so far cleaved rather well to neat models. See The Unreasonable Effectiveness of Mathematics in the Natural Sciences for much more. Since it's always been true so far that "every mystery we've ever solved has turned out to be not magic nicely modellable", we often tend to treat this as evidence that any other given mystery will be nicely modellable.

However, all our models so far have been very much approximations, and we know that they're not Truly Correct because they're not all compatible with each other (c.f. quantum mechanics and general relativity, both excellent but incompatible models). So it's quite possible that reality does not in fact admit any nice compact description.

Note also that there's a huge selection effect going on here: we know how to solve physical problems using mathematics, so of course the problems we have solved are overwhelmingly done using mathematics!

---

I will note, as a software engineer, that coming up with a solution to a software problem that in some sense has mathematical beauty (e.g. it has no edge cases) does seem to be a pretty good indicator that you've found The Right Answer. In my experience, such a solution will be extensible, low in bugs, require fewer rewrites, and so on, and it often turns out to model well entirely new requirements that I hadn't even thought of. But I would argue that this is more likely because an answer with a neat mathematical model is simpler and has fewer moving parts, all of which are more easily described and more easily proved correct, than because there is something inherently "correct" about mathematics as applied to the real world.

Answer 2

It's poetic and all. and beauty $\cong$ simplicity and simple when true will probably have fundimental obvious reasons it is true. But let's not get carried away.

IMO $(a+b)^k = a^k + b^k$ is for more beautiful that then $(a+b)^k = \sum_{k=0}^n \frac {n!}{(n-k)!k!}a^kb^{n-k}$ and the statement "There is harmony in the universe and all values are ratios of whole numbers" is much more beautiful than "The values that can't be express by any finite combination of rational numbers is uncountable more than those that can be".

So here we have the truth is much uglier than the poetic symmetry.

Of course there is beauty in thinking and the reasoning that $(a+b)^k = \sum_{k=0}^n \frac {n!}{(n-k)!k!}a^kb^{n-k}$ contains some clever and "beautiful" thoughts.

BUt... ya know.... this is subjective and romantic but boils down to no real meaning.

I suppose it could be argued that I am being temporally chauvinistic in thinking romance and poetry are outdated and archaic and ...well, just a bit saccharine and twee..... But... I think is wiser to be objective and taking things on the faith that "the universe must have beauty in its method otherwise life is too grim" is just not objective.

......

Of course:

1. All mathematicians are romantics. They find ideas more beautiful than things.
2. A cynic is just a romantic who lived to be forty years old.

Answer 3

My view (PhD Physics, former Visiting Professor, Mathematics), is that given any system of mathematical postulates, there are some resulting theorems (mathematical "truths") that many or even most mathematicians would describe with terms such as "elegant," "parsimonious," even "beautiful," and other theorems that might be called "convoluted," "perplexing," even "ugly." It is the "beautiful" ones that are more easily remembered, more easily disseminated, more easily used in computations and thus more likely to become part of the canon taught and studied. Thus there is an "evolutionary" force that favors these theorems and ideas. The proverbial person-on-the-street knows $a^2 + b^2 = c^2$, in large part because it has properties associated with "beauty."

That being said, mathematicians' notions of "beauty" (and synonyms) certainly changes ("evolves") over time. Some of mathematicians' early responses to Mandelbrot's fractals involved terms like "ugly," "horrendous," and such. But through exploration and familiarity—and especially the representations through computer graphics—are now considered some of the most beautiful math.

Some (misguided) mathematicians even call certain fractal images "art." I disagree... but that's a separate discussion.