

Mathematics: Why the brain sees maths as beauty - sytelus
http://www.bbc.com/news/science-environment-26151062

======
chewxy
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](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_...](http://www.ted.com/talks/denis_dutton_a_darwinian_theory_of_beauty)

~~~
hyp0
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.)

~~~
gizmo686
>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.

------
ezequiel-garzon
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...](http://journal.frontiersin.org/Journal/10.3389/fnhum.2014.00068/abstract)

[2]
[http://journal.frontiersin.org/Article/DownloadFile/390079/o...](http://journal.frontiersin.org/Article/DownloadFile/390079/octet-
stream/Data%20Sheet%201.PDF/16)

~~~
ivan_ah
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.

------
tzs
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.

------
kristopolous
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?

~~~
cscurmudgeon
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.

~~~
kristopolous
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.

~~~
cscurmudgeon
> 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.

~~~
kristopolous
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

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

[http://en.wikipedia.org/wiki/Asexuality](http://en.wikipedia.org/wiki/Asexuality)

------
sytelus
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.

------
gballan
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...](http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)

------
ivan_ah
> _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...

~~~
russellsprouts
This essay really drove that point home for me:

[http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...](http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)

------
bane
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.

~~~
nileshtrivedi
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).

------
nly
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.

------
calhoun137
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.

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

~~~
gizmo686
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

~~~
jimmaswell
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.

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

~~~
lmkg
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.

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

[http://www.scottaaronson.com/democritus/lec9.html](http://www.scottaaronson.com/democritus/lec9.html)

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

~~~
jjoonathan
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-...](http://physics.stackexchange.com/questions/32422/qm-without-
complex-numbers)

------
okonomiyaki3000
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.

------
JacksonGariety
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.

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

Thats why i really like to refractor something, i guess

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

