Mathematics: Why the brain sees maths as beauty | Hacker News Reader

Mathematics: Why the brain sees maths as beauty

bbc.com

73 pointssytelus45 comments

Here [1] is the study, and here [2] the 60 equations that were rated. I feel the brain scans themselves are more interesting than the scores assigned to the equations by the participants. Still, I would have omitted the equation descriptions, as I feel it could create a bias, and would have placed the great contender, Euler's identity, in a position other than 1.

[1] http://journal.frontiersin.org/Journal/10.3389/fnhum.2014.00...

[2] http://journal.frontiersin.org/Article/DownloadFile/390079/o...

Thx for [2]. It's full of good stuff!

As far as beauty is concerned, I find it difficult compare these equations because they are at very different levels of abstraction. Sometimes vars are numbers, sometimes functions, sometimes groups, etc.

The article explained that brain scans show that when people say particular math is beautiful and particular math is not beautiful, the same parts of the brain are activating in the same way as when people distinguish between beautiful art and not beautiful art.

That just indicates that people who say something in math is beautiful are having the same feeling as someone who describes some art as beautiful. They are not feeling something different and just using the same word for it.

This is very interesting, but the title of the article led me to believe they were going to tell me WHY this is so. I didn't see that in there.

I gave a talk just a couple of days ago on the beauty of Python code[0] - the talk hasn't been transcripted yet. In it I gave an example of how we've been conditioned by evolution to see clever things as beautiful[1][2].

I gave this scenario in the talk: imagine you were a paleolithic craftsman making handaxes. Now, you have a competitor making them handaxes too. We'll assume you make a better hand axe - sharper, cleaner lines and more wedge-shaped than the competitors. This leads to a higher "usefulness" than your competitor's handaxe.

You consistently make better handaxes than your competitor. You get better mates than your competitor. You have a higher probability of spreading your genes. Your "good" handaxes has become a proxy signal for your skill in making handaxes, which itself is a signal of your intelligence.

Over time, if the population uses only the "goodness" of the handaxe as a deciding factor in deciding mating partners, the population needs to be able to discern "good" handaxes from "bad" handaxes. As the lines get smoother, we become more and more tuned towards this standard of "beauty", and we put more and more reverence towards the makers of such items of beauty.

In short, humans see math as beautiful because it is a signa for intelligence, which is a deciding factor of fitness in our evolution. Indeed, our evolution seem to value intelligence over strength anyway.

Now, all the above is personal deduction of course. But I believe if we do go on and figure out the evolution of the medial orbito-frontal cortex of the brain, we'd see it's tied to intelligence.

[0]: https://speakerdeck.com/chewxy/beautiful-python - not very useful slides without the talk. The 4 criteria can also be translated to: usefulness, simplicity, averageness (I used the example: if the man on the street can read your python code and understand it, you did a good job), cleverness

[1]: The Art Instinct by Dennis Dutton

[2]: http://www.ted.com/talks/denis_dutton_a_darwinian_theory_of_...

Related to this is koinophilia, or the "average is beautiful" effect.

http://en.wikipedia.org/wiki/Koinophilia

An alternative view is that, if you make good handaxes, you'll have more money (or patronage), meaning your children have more resources and advantages. You may increase your attractiveness to mates (who value their children's welfare).

Beauty may be to do with simplicity. Simpler ways of seeing things (while retaining accuracy) are more likely to also be correct for situations not yet encountered - have better predictive powe, are better theories. Valuing those has a survival advantage. Simpler theories are also easier to learn, to use, to remember etc.

It's very fundamental: our eyes, when confused by a scene at first, will resolve it to the simplest interpretation.

I think this covers many instances of beauty (e.g. symmetry, neatness, non-arbitrariness, and even the simpler lines that emerge from averages), but it doesn't cover all.

For example, the sworling of ocean waves, the complexity of clouds, of a forest. It is possible that these are in fact "simple", to the theories we have in our minds; just not simple for an intellectual understanding. And of course, fractal patterns are indeed simple, in the sense that they are generated by a very simple process. So, perhaps we somehow intuitively grasp the simplicity of clouds, waves, trees. (Or maybe we just like nature.)

>So, perhaps we somehow intuitively grasp the simplicity of clouds, waves, trees. (Or maybe we just like nature.)

We did evolve in nature, so it makes sense that our brains would have naturally occurring patterns somewhat hardwired.

Imagine that you were a paleolithic craftsman making handaxes. You take your time to make high quality handaxes that are beautiful and durable. Meanwhile another guy only spends 60% as much time making handaxes, only achieving 80% of the beauty and durability. But he has more time to slaughter game, bringing in meat, organs and hide for the tribe - the proxy to status in the tribe. So he is more 'succesful' than you, genetically.

Atlernatively, another tribe manages to make an axe for every man in their tribe before your tribe does, since they are more sloppy but more effective in their craftsmanship. You get your skull cleaved, ending your lineage.

And that, folks, is why people unnderappreciate beauty.

The 'beauty' of paragraph-long evo psych theories is that they can explain pretty much anything.

This is just another one of those speculative stories invented out of thin air that are supposed to be plausible just because they illustrate some reproductive advantage of one trait over another and are hence deemed to be "evolutionary", and hence to be somehow "scientific". It is an easy trap to fall into after learning a little about evolution, but it is a rather bad thinking pattern.

There are hundreds of similar stories one can make up for why the eyes, for example, evolved the way they did and likely all of those just invented at will will be false, while sounding similarly "evolutionary", since almost any complex trait is a product of a whole series of mutations, some of which might not have a single nice intuitive interpretation at all, and those are changes in terms of proteins produced, not necessarily easily translatable into changes in high-level traits. For one, in the case you describe, it could just as well be an accidental byproduct of a series of other small adaptations that were beneficial in other ways. It could have been hundred of ways, we have no clue.

So, please don't bring up evolution when there is absolutely no evidence to base upon. Especially in case of the brain, which we barely understand at all, and which additionally as a soft tissue decays fast and hence even its anatomical evolution is almost impossible to trace back, what to speak about translating this into some sort of cognitive history of human beings.

Was that the same talk you claimed Quicksort was O(n)? ;-)

> In short, humans see math as beautiful because it is a signal for intelligence, which is a deciding factor of fitness in our evolution

I disagree, although I think I see where you are coming from. Perceiving beauty is a form of pleasure, and the brain feels pleasure as the result of a chemical process. The real question is what causes the brain to initiate that process. Certainly cultural influence plays a huge role, as does the need to reproduce. But I don't see any evidence or arguments as to why that has to be the only explanation for why a person perceives beauty.

I think the concept of mathematical beauty is highly subjective, but if I had to put it into words: I think the perception of mathematical beauty is related to the human brains natural capacity to understand patterns and do mathematics.

I'm not trying to be a troll or argumentative, but personally I think it's all ugly and lifeless. Which is interesting because this article says that there's some biological link to people perceiving beauty.

I think all math is some little tool and all code is also some other little tool - some of it hairy, some of it less hairy. But I consider none of it "beautiful".

I think the entire notion is preposterous. I'm totally misanthropic to the whole notion.

Does that make me strange?

Either: you are not a born mathematician or were not trained enough. Math is like music, but unlike music you need to grok quite a bit to get to the beautiful parts.

I'm not a mathematician or anything close and even I can find it quite beautiful... that is, the parts I can grok. :)

I still think some of the Cantor and Gödel work is the most beautiful I've come across that was not from a prolific Swiss polymath.

Relative to most, I've done a lot of math. I have an entire bookshelf dedicated to it, subscribed to journals for a number of years. I worked in cryptography for a while.

The only things I ever think are "why is this presented in such a convoluted manner" or "Why is this so poorly written".

The idea of "beautiful" is so remarkably far removed from the situation --- it would be as if people said they found it sexually arousing.

To me, there is no possible connection.

> I've done a lot of math... I worked in cryptography for a while.

Weirldy, that could be the reason. Doing something beautiful (art, math, etc) for $ usually ends up corrupting it for the person. I did not mean any offense.

Source: Psychology research I am lazy to link and myself.

I'm not living a romantic loss-of-beauty through corrupting money narrative though.

It was not beautiful when I was 5, 15, or 25. I found it a useful tool and that interested me, but aesthetics? no - never.

Sorry to crash the party

I am not doubting your account. I also know there are genuine asexuals.

http://en.wikipedia.org/wiki/Asexuality

And I wonder how much musical training or experience you have. Like mathematics, there is a lifetime (and more) of music theory to grok that will aid in the enjoyment of music.

Not at all; it's just a matter of dwelling on a subject long enough to begin attaching aesthetic notions to it, I think.

For instance many car buffs have well-developed notions of when a car's handling or responsiveness is 'beautiful' (see any car review or episode of 'Top Gear' if you don't believe me) which I also find preposterous! But I bet if I spent nearly as much time as the Top Gear people dwelling on car stuff, I would grow a similar aesthetic appreciation for how the drive handles some curve or whatever they are moaning about in those reviews.

I think that makes you strange, yes. Even in tools we tend to see beauty.

Really? for me it's either competently done with care and precision (call it 'quality') or some degree of not those 3 things.

Maybe it's just a word usage. For me, "beautiful" are things like lush ornamental gardens or rolling green hills on an early spring day.

A, say, 30-in-one screw driver, however, is not - regardless of how suitably manufactured it is. It's just a tool to me.

For me I'd say "what remarkable durability this tool has - at such an affordable price. Look at how compact it is" definitely compliments, but nothing with an emotional attachment.

A lot depends on typography and convention as well. For example, Maxwell's equations in raw form are HUGE. It's only after you compress them by inventing short hand for div and grade, they can be fit on T-Shirt. Even then integral forms does not feel as elegant. Similarly the field equations for General Relativity in "raw" form huge and ugly (involving 4X4 matrix). It's only after you shorten them using tensor notation they start feeling elegant.

I'm not sure we have analog for this in art world. However the technique in this article is interesting because you can extend it to other domains. How about finding out which algorithms are "beautiful"? I think the most beautiful algorithm ever invented is binary search.

Some math is so elegant that many understandably think it beautiful. To assess yourself, perhaps consider the area of the triangle [1, page 4], explained at a stroke -- not a formula in sight. If any math is beautiful, that is.

What is less appreciated is that such elegance is not limited to math. Many (all?) fields in science and engineering and technology are striving for a similar moment of clarity.

[1] A Mathematician’s Lament by Paul Lockhart, http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...

> this beauty of maths was missing from schools and yet amazing things could be shown with even primary school mathematical ability

This.

Most people think that math is about arithmetic and memorization, which couldn't be further from the truth. But can you blame them since the only math teaching they received as kids was about arithmetic and memorization?

It would be interesting how much science will move forward if we had one generation of kids who grow up learning about the more "beautiful" aspects of math and end up interested in it...

This essay really drove that point home for me:

http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...

It's not just the memorization that bothers me, but the belief that there is only one way to do math. I've been in many arguments with parents before that their (and my) kids should learn many algorithms for even the most basic things. Do subtraction by complements sometime, rather than the standard algorithm. Playing with numbers is way more fun than repeating the computations.

Conrad Wolfram is leading a campaign to change that: http://computerbasedmath.org/

I really tried hard to enjoy mathematics...but I never got to the point of ever seeing it as beautiful. I've seen some code or algorithms I thought were beautiful from time to time.

I love art, prose, music...I can frisson, mostly to music, but sometimes to other mediums. I tend to think visually, finding it easier to put a sequence of pictures together to describe what I want to say rather than words. People comment to me that my replies in email are frequently just a link to a relevant picture.

Strangely, I don't enjoy most poetry or music that requires listening to the lyrics to enjoy (songs with lots of humorous lyrics almost always fall flat with me). I did fine with Mathematics in school, studied hard, good grades, but it never really "sung" to me. I never enjoyed it beyond the natural joy one gets when developing a skill. But I guess I never really got to the point where it was speaking to me so I could reach this kind of enjoyment.

I can sit there are appreciate certain cool things. Euler's given in the article provokes some fascination...but beyond being a lucky coincidence it doesn't really stir anything in me.

This kind of makes me sad.

If you're a visual thinker, you may have better luck with geometrical theorems and proofs (such as for pythogoras theorem or fractals or topology).

Geometry and mechanics are beautiful to me, but even many mechanics problems are more elegantly using the methods taught in physics over those taught in maths (mechanics), simply because you if you use the right derivatives or logarithms to start with (e.g. energy instead of speed, decibels instead of absolute power) then you often avoid having to touch calculus, logarithms etc entirely.

The beauty in maths for me has always been in finding ways to use it only to the extent that it makes unintuitive problems more intuitive. Math for maths sake isn't intrinsically beautiful, because for most of us it's more like having all the cheat codes to a game when the levels are too hard than figuring out how to play.

Mathematical beauty is not only for pleasure, according to the great quantum theorist Dirac, who taught us that we should let mathematical beauty guide us in our discovery of new physical laws.

This is how Dirac was able to predict the existence of anti-particles before they were discovered experimentally, and his philosophy about mathematical beauty is a cornerstone of modern theoretical physics.

I never forget the moment when my math teacher revealed e^pi*i+1=0 after dealing with complex numbers. It was beautiful.

I think I am unique in being one of the only people whose first reaction to seeing that equation was to think that it was a mistake (in fact, my first reaction to seeing it is still to think it is a mistake, even knowing that it is not). To me, it feels like the plus sign should be a minus sign, to give the (incorrect) equation `e^pi * i-1=0`. Years after I saw the equation, when I learned how it actually made sense to talk about complex exponents, I realized that the problem is that pi is a stupid constant, but that is another discussion. Returning to the `e^pi * i+1=0` point, I still don't get why people find that beautiful before they have any sense of what taking a complex exponent means (in terms of how to naturally extend the definition of exponent, obviously the geometric result essentially follows from this result).

More to the point, that equation is just a special case of Euler's formula, `e^xi=cos(x) + i * sin(x)`, which is what contains the real beautiful insight.

EDIT: fixed * formatting

The part of it that people find cool is probably e, i, and pi being related, when at first you wouldn't think they had anything to do with each other, before knowing any of that.

I love analytic/holomorphic functions, they're so smooth and beautiful.

The more I studied holomorphic functions, the more I felt that the real numbers were fundamentally incomplete without the imaginary numbers to accompany them. In just the reals, there are tons of ways for functions to misbehave and become irritating to manage, and there's no real rhyme or reason to explain it. In the complex numbers, the 'true' structure of a function is revealed: well-behaved functions are well-behaved everywhere and in all respects, and if a function misbehaves it's clear where and how.

Some people use similar reasoning to "explain" why quantum mechanics uses complex numbers.

http://www.scottaaronson.com/democritus/lec9.html

(scroll down to "Real vs. Complex Numbers", about 2/3rds of the way through the page)

Awesome, I was about to go dig up the link but you saved me the trouble. I found some relevant discussion on stackexchange, but I haven't digested it thoroughly enough to have an opinion yet.

http://physics.stackexchange.com/questions/32422/qm-without-...

This was an amazing read. I love explanations like this which are based on pure concepts that require just a little bit of bending the mind, but then explain so much more.

I got the same feeling learning linear algebra from Axler. He always introduced proofs using complex numbers and only subsequently dealt with the case of real numbers. The real-number proofs invariably devolved into a rat's nest of edge cases that, if one looked closely, accomplished their goals only by inelegantly re-inventing the concept of complex numbers. Admittedly some of the differences in elegance came directly from the fundamental theorem of algebra, but not all of them.

My brain certainly does not see "maths" as beauty. It sees it as a spelling mistake and it hears it as an awkward jumble of sounds that have no business being together.

I feel like the same test could be easily performed with code on programmers. I'd love to see which common bits from a C or JavaScript program will evoke a sense of beauty.

I've got the same thing with code, not with music or anything else.

Thats why i really like to refractor something, i guess

What if your brain sees it but you don't see it?

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