February 21, 2014 2:07 PM Subscribe

The emotional experience of mathematical elegance. A new study by the perceptual neurobiologist Semir Zeki and the great mathematician (& Fields Medalist) Sir Michael Atiyah examines fMRI scans of mathematicians viewing formulae which they'd previously rated as beautiful or ugly, and reports that mathematical beauty elicits activity in the same brain regions as great art or music.

posted by Westringia F. (36 comments total) 16 users marked this as a favorite

posted by Westringia F. (36 comments total) 16 users marked this as a favorite

Stokes' Theorem was the first thing that came to mind, yes.

posted by Wolfdog at 2:25 PM on February 21

posted by Wolfdog at 2:25 PM on February 21

Indeed it would, escabeche, just beautiful in a unique way. But to me there's something gratifying about being able to say to those who have not themselves felt breathtaking quality of mathematical elegance that the experience is neurologically the same as the experience of heartbreakingly beautiful poetry or art.

(I'm reminded of the beginning of Brideshead, where Charles Ryder recalls "...Sebastian, idly turning the pages of Clive Bell's*Art*, read, '*Does anyone feel the same kind of emotion for a butterfly or a flower that he feels for a cathedral or a picture?* Yes: I do.' ")

posted by Westringia F. at 2:32 PM on February 21 [2 favorites]

(I'm reminded of the beginning of Brideshead, where Charles Ryder recalls "...Sebastian, idly turning the pages of Clive Bell's

posted by Westringia F. at 2:32 PM on February 21 [2 favorites]

I'm still a little miffed that the section of my Ph.D. thesis where I had to verify that a bunch of technical conditions held for certain non-Noetherian power series rings was what they picked for the "ugly" picture, though.

posted by escabeche at 2:33 PM on February 21 [7 favorites]

posted by escabeche at 2:33 PM on February 21 [7 favorites]

From my point of view, to equate those experiences is to diminish both.

posted by escabeche at 2:35 PM on February 21 [3 favorites]

This PDF from the supplementary material lists all the equations used. I'm going to have to give a shout-out to #54, the Cauchy-Riemann equations, and an anti-shout-out to #28, 3^{2} + 4^{2}= 5^{2}.

posted by Elementary Penguin at 2:38 PM on February 21 [1 favorite]

posted by Elementary Penguin at 2:38 PM on February 21 [1 favorite]

I want to know if analysts and algebraists find different things beautiful/elegant.

posted by weston at 2:41 PM on February 21 [3 favorites]

posted by weston at 2:41 PM on February 21 [3 favorites]

I took a term of vector calculus which consisted mostly of proving Stokes' Theorem and building all this incredibly pretty machinery along the way and it was a ton of fun.

posted by BungaDunga at 2:42 PM on February 21 [2 favorites]

Also: if a person says "I find A and B equally beautiful" does it *matter* whether the fMRI shows the same pattern for A and B? If the fMRI said no, you're perceiving them differently, then what? Who cares?

posted by BungaDunga at 2:45 PM on February 21 [3 favorites]

posted by BungaDunga at 2:45 PM on February 21 [3 favorites]

Also, if you're moved by mathematical beauty, Ted Chiang's "Division By Zero" makes for a lovely little horror story.

posted by Rhaomi at 2:47 PM on February 21 [3 favorites]

The problem with "Division by Zero" is that Presburger arithmetic and Tarski's axiomization of Euclidean geometry are both consistent, so I'm not super worried that there's a proof that 0=1. It is like the mathematical equivalent of sound in space.

posted by Elementary Penguin at 2:54 PM on February 21

posted by Elementary Penguin at 2:54 PM on February 21

EP, from my reading of the Presburger link, isn't it still open that we could prove 0 = 1 if we were allowed to use a language whose signature included multiplication? (Not that I'm staying up night worrying about it either.)

posted by benito.strauss at 3:02 PM on February 21

posted by benito.strauss at 3:02 PM on February 21

This is my surprised fMRI activity pattern.

posted by jfuller at 3:03 PM on February 21 [12 favorites]

posted by jfuller at 3:03 PM on February 21 [12 favorites]

I wish to join Elementary Penguin in the anti-shoutout to 3^{2} + 4^{2}= 5^{2} and would like to add #21, π = circumference / diameter. That's just a definition, not a derivation. Maybe they threw in a few ringers to make sure people weren't responding to the presence of equals signs?

posted by echo target at 3:06 PM on February 21 [1 favorite]

posted by echo target at 3:06 PM on February 21 [1 favorite]

benito.strauss, sure. But if we lose the ability to talk about multiplication without 1!=0, we'll always have addition. Plus, you can do multiplication in euclidean geometry just fine (you just can't define prime numbers in any useful sense) so we'd have multiplication anyway. I'm not enough of a logician to be sure it's enough, but I'm just saying that I'm not losing any sleep about the possibility of all of math being a lie.

posted by Elementary Penguin at 3:12 PM on February 21 [1 favorite]

posted by Elementary Penguin at 3:12 PM on February 21 [1 favorite]

It would have been utterly shocking if this weren't the case— it would suggest that mathematicians have an entirely different pleasure system than other humans, which, some people might suspect, but which seems unlikely to be true.

posted by Maias at 3:48 PM on February 21

posted by Maias at 3:48 PM on February 21

> *I'm going to have to give... an anti-shout-out to #28, 3*^{2} + 4^{2}= 5^{2}.

>*I wish to join Elementary Penguin in the anti-shoutout to 3*^{2} + 4^{2}= 5^{2}.

Not just you. Per their supplementary data Table 1 (xls), nearly everyone dinged it on the -5 to 5 rating scale:*I'm still a little miffed that the section of my Ph.D. thesis ... was what they picked for the "ugly" picture, though.*

Well, that's 'cuz it*is* ugly. ;)

But while Ramanujan's power series for 1/pi (Eqn 14) did indeed have the lowest mean rating, it also had the highest variance. In fact, the sum of its positive scores exceeded that of Eqn 28.*Some* people clearly appreciated it, escabeche! (De gustibus non est disputandum &c &c.)

As for me: after seeing the list, I understand why Dirac had a distaste for stat mech. Stirling's approximation is*not* pretty. (Especially when it's written like that. $\ln(n!)=n\ln(n)-n+O(\ln(n))$ seems a bit less offensive; phrasing is not unimportant in matters of aesthetics.)

>*I want to know if analysts and algebraists find different things beautiful/elegant.*

And physicists! I suspect there might be some divergence. (*cough*renormalization*cough*)

posted by Westringia F. at 4:10 PM on February 21 [2 favorites]

>

Not just you. Per their supplementary data Table 1 (xls), nearly everyone dinged it on the -5 to 5 rating scale:

rating: -5 -4 -3 -2 -1 0 1 2 3 4 5 N (of 16): 2 1 3 1 1 2 3 0 2 0 0>

Well, that's 'cuz it

But while Ramanujan's power series for 1/pi (Eqn 14) did indeed have the lowest mean rating, it also had the highest variance. In fact, the sum of its positive scores exceeded that of Eqn 28.

As for me: after seeing the list, I understand why Dirac had a distaste for stat mech. Stirling's approximation is

>

And physicists! I suspect there might be some divergence. (*cough*renormalization*cough*)

posted by Westringia F. at 4:10 PM on February 21 [2 favorites]

I know everyone's a critic but the fundamental theorem of calculus isn't the first thing I would think of as being mathematically beautiful.

showing that stokes theorem is just the FTC is nice because you see that something which in vector calculus looks complicated and somewhat exotic is actually just something mundane, i.e. the FTC.

it's conceptual art that way I guess.

posted by ennui.bz at 4:17 PM on February 21

This discussion reminds me of my favorite proof.

In the process of proving his first incompleteness theorem Gödel constructed a logical statement which, in plain English, said "This sentence is not provable in ZFC."

Löb's sentence is "This sentence*is* provable in ZFC."

If this sentence is provable it's true, and if it's not provable it's false, so there's no incompleteness concerns here, but is it provable?

Assume no. If there is no proof in ZFC of Löb's sentence, consider the formal system consisting of all the axioms of ZFC and ~"This sentence is provable in ZFC." Since, by assumption, we know there is no proof of Löb's sentence in ZFC, we know that it is impossible to contradict one of the axioms of this system. Therefore the new system must be consistent. But Gödel's second incompleteness thereom says that any axiomatic system which includes all the axioms of ZFC is incapable of proving it's own consistency. A contradiction! Therefore Löb's sentence must be provable! Notice that we don't actually have a proof of it, we just know a proof must exist.

I am not doing this argument justice, since I am trying to recall it from long-lost memory without the benefit of any notes. But when this was presented in graduate seminar by a professor, all of my fellow grad students broke out in laughter at the punchline.

posted by Elementary Penguin at 4:21 PM on February 21 [1 favorite]

In the process of proving his first incompleteness theorem Gödel constructed a logical statement which, in plain English, said "This sentence is not provable in ZFC."

Löb's sentence is "This sentence

If this sentence is provable it's true, and if it's not provable it's false, so there's no incompleteness concerns here, but is it provable?

Assume no. If there is no proof in ZFC of Löb's sentence, consider the formal system consisting of all the axioms of ZFC and ~"This sentence is provable in ZFC." Since, by assumption, we know there is no proof of Löb's sentence in ZFC, we know that it is impossible to contradict one of the axioms of this system. Therefore the new system must be consistent. But Gödel's second incompleteness thereom says that any axiomatic system which includes all the axioms of ZFC is incapable of proving it's own consistency. A contradiction! Therefore Löb's sentence must be provable! Notice that we don't actually have a proof of it, we just know a proof must exist.

I am not doing this argument justice, since I am trying to recall it from long-lost memory without the benefit of any notes. But when this was presented in graduate seminar by a professor, all of my fellow grad students broke out in laughter at the punchline.

posted by Elementary Penguin at 4:21 PM on February 21 [1 favorite]

This reminds me of one of my favorite short stories, full text available online:

A New Golden Age

by Rudy Rucker

*“It’s like music,” I repeated. Lady Vickers looked at me uncomprehendingly. Pale British features beneath wavy red hair, a long nose with a ripple in it.*

“You can’t hear mathematics,” she stated. “It’s just squiggles in some great dusty book.” Everyone else around the small table was eating. White soup again.

I laid down my spoon. “Look at it this way. When I read a math paper it’s no different than a musician reading a score. In each case the pleasure comes from the play of patterns, the harmonies and contrasts.” The meat platter was going around the table now, and I speared a cutlet.

I salted it heavily and bit into the hot, greasy meat with pleasure. The food was second-rate, but it was free. The prospect of unemployment had done wonders for my appetite..

posted by charlie don't surf at 4:48 PM on February 21 [1 favorite]

A New Golden Age

by Rudy Rucker

“You can’t hear mathematics,” she stated. “It’s just squiggles in some great dusty book.” Everyone else around the small table was eating. White soup again.

I laid down my spoon. “Look at it this way. When I read a math paper it’s no different than a musician reading a score. In each case the pleasure comes from the play of patterns, the harmonies and contrasts.” The meat platter was going around the table now, and I speared a cutlet.

I salted it heavily and bit into the hot, greasy meat with pleasure. The food was second-rate, but it was free. The prospect of unemployment had done wonders for my appetite..

posted by charlie don't surf at 4:48 PM on February 21 [1 favorite]

The real diminishing comes from reducing both experiences to a neural firing pattern that we can measure with the incredibly crude tools we have now. It's just a silly fMRI result that the press will rush to overinterpret.

I think it's interesting anyway, not because it makes me think "mathematics is just as beautiful as art or music", but because it makes me think "art and music are wonderful because they make the kind of beauty you can find in mathematics accessible to anyone who can see or hear".

The link on "mathematical beauty" goes straight to the difference between math and art:

Equations aren't beautiful on their own. What's beautiful is the entire structure you've built up in your mind that enables you to see what the equation really means and why it is true.

That takes a level of dedication to mathematics that's never going to be accessible to everyone. I spent my teenage years deeply immersed in that mystical world of intense abstract beauty. But I suffered from not being able to share that beauty with other people. I surprised everyone that knew me and deeply disappointed my thesis adviser when I abandoned mathematics, because I felt like I didn't have what it takes to be a professional mathematician. I wanted to live in the world, not in my head.

Decades later, I still miss having access to that experience. I know I will never again feel the way I did when I could really see and understand Galois Theory, just like I'll never have again the feeling of the first time I was in love. Art and music are like a blurred reflection of the purity of mathematical beauty, distorted by viewing that beauty through the imperfections of being human. I can still get a glimpse of that feeling sometimes by just being in nature and suddenly seeing a part of how its rhythms work.

Still, there's something about mathematical beauty that for me will always be beyond anything else that experience has to offer.

posted by fuzz at 5:40 PM on February 21 [11 favorites]

orly_owl.jpg

posted by odinsdream at 7:24 PM on February 21

posted by odinsdream at 7:24 PM on February 21

What happens when you put a dead fish in a fMRI and expose it to the beauty of math?

posted by humanfont at 9:19 PM on February 21 [1 favorite]

posted by humanfont at 9:19 PM on February 21 [1 favorite]

The mathematicians respond just like they do to Poisson's Equation.

posted by benito.strauss at 10:58 PM on February 21 [2 favorites]

posted by benito.strauss at 10:58 PM on February 21 [2 favorites]

In other fMRI news: Brain Scans Show Striking Similarities Between Dogs and Humans

posted by homunculus at 12:26 AM on February 22

posted by homunculus at 12:26 AM on February 22

I took Calculus 1 last semester it was amazing. It was the first time where a math class made me feel like I had a better understanding of the world around me. It was fun.

Now I'm in Calculus 2 and I feel like I'm seeing how the sausage is made.

posted by TrialByMedia at 6:53 AM on February 22 [4 favorites]

Now I'm in Calculus 2 and I feel like I'm seeing how the sausage is made.

posted by TrialByMedia at 6:53 AM on February 22 [4 favorites]

Calculus 2 is basically Algebraic Manipulation Bullet Hell. Sorry!

posted by Elementary Penguin at 7:07 AM on February 22 [3 favorites]

posted by Elementary Penguin at 7:07 AM on February 22 [3 favorites]

> That takes a level of dedication to mathematics that's never going to be accessible to

> everyone. I spent my teenage years deeply immersed in that mystical world of intense

> abstract beauty. But I suffered from not being able to share that beauty with other people.

It's just too dam' bad there's no way to perform math for non-mathematicians, as there is to perform music for non-readers.

> Now I'm in Calculus 2 and I feel like I'm seeing how the sausage is made.

It was a disappointment so profound it actually made me angry when I had learned the methods of working out first and second derivatives and wanted to go on and learn the method of working out integrals and they said "Nope, sorry, no such thing." It was like there was an obvious symmetry that ought to exist and the universe had gotten lazy and failed to provide it. Nothing remotely comparable, not even when they explained GoTo and I'm like "OK, I get it, now explain ComeFrom. How do you do that?" and they looked at me blankly.

posted by jfuller at 12:34 PM on February 22

> everyone. I spent my teenage years deeply immersed in that mystical world of intense

> abstract beauty. But I suffered from not being able to share that beauty with other people.

It's just too dam' bad there's no way to perform math for non-mathematicians, as there is to perform music for non-readers.

> Now I'm in Calculus 2 and I feel like I'm seeing how the sausage is made.

It was a disappointment so profound it actually made me angry when I had learned the methods of working out first and second derivatives and wanted to go on and learn the method of working out integrals and they said "Nope, sorry, no such thing." It was like there was an obvious symmetry that ought to exist and the universe had gotten lazy and failed to provide it. Nothing remotely comparable, not even when they explained GoTo and I'm like "OK, I get it, now explain ComeFrom. How do you do that?" and they looked at me blankly.

posted by jfuller at 12:34 PM on February 22

> OK, I get it, now explain ComeFrom.

Unfortunately, once you put the car through the trash compactor, there's no easy way to turn it back into a car. We lost information. We could build a new one from the raw materials, though.

posted by I-Write-Essays at 1:25 PM on February 22

Unfortunately, once you put the car through the trash compactor, there's no easy way to turn it back into a car. We lost information. We could build a new one from the raw materials, though.

posted by I-Write-Essays at 1:25 PM on February 22

I remember being bugged by the same thing, but I've learned a heuristic: whenever my aesthetic intuitions are frustrated by a mathematical structure, this just means that I need to think hard about the problem until I see how it is in fact elegant and beautiful and then I correct my aesthetic intuitions.

For instance, the way the sin and cosine functions are taught in high school is "sin and cos are these two ratios who live in triangles, here are some acronyms for remembering how the compute them if you happen to run into a triangle" which is totally unsatisfying, and I was bugged by those functions until I got a taste of complex analysis, and then blammo --

As for the asymmetry between integration and derivation, from a certain perspective it's inevitable -- Imagine an ant crawling along a function reporting its derivative. All the ant needs to know is the difference between adjacent points on the function. All the calculations are local to arbitrary points on the function itself. But if the ant wants to take an integral, it needs to keep track of where it started on the function, and all the values of the function along the way, so this calculation is not inherently local to an arbitrary point on the function. If you think about it this way, integration and derivation are thus subject to intrinsically different constraints, and ought not the be symmetrical overall. Ants on curves.

posted by serif at 5:57 PM on February 22 [1 favorite]

Also, that fMRI paper is an embarrassment, although I admit it's rare to see so many well-worn "shitty fMRI research" tropes all at once.

posted by serif at 6:05 PM on February 22

posted by serif at 6:05 PM on February 22

So what you're saying is, if you stuck a neuroscientist in an fMRI and showed them this study, it would register the same as a mathematician looking at 3^{2}+4^{2}=5^{2}?

posted by Elementary Penguin at 6:14 PM on February 22

posted by Elementary Penguin at 6:14 PM on February 22

it would be more like showing a mathematician 80081355 on a calculator

posted by serif at 3:05 AM on February 23 [1 favorite]

posted by serif at 3:05 AM on February 23 [1 favorite]

> so this calculation is not inherently local to an arbitrary point on the function.

I do see that, I do. What hurt my head to the point where I gave up thinking about it was considering the problem like this:

Think of all the well defined plane figures for which the area is easy to find. A circle? Easy-peasy, give me r and I'll give you the area. Rectangle? Length and width, easy-peasy. Triangle? easy-peasy. Ellipse? Easy-peasy. There are clean, simple expressions for the area of all of these.

Now what about the area of this figure here, defined by y = f(x) in the domain from x = a to x = b? [Say f(x) stays above the x axis for this whole domain, just for simplicity.] For this bounded figure I have one easy to find side: the length on the x axis is (b - a). I have a second easy to find side: the height above a is f(a). I have a third easy to find side: the height above b is f(b).

Then there's the wiggly part at the top, defined as f(x) while x varies from a to b.**What is it that's lacking, what's so imperfect about specifying the figure's upper border that way that it ruins the project of finding a clean expression for the area of the figure?**

The moment of insight never came. That's when I knew that, no matter that I knew the secret (*) of doing well in math courses, I was not a mathematician.

(*) Nb, for them as don't know, the secret is*don't cut class; pay attention in class; do the assignments; anything you don't grok about an assignment, ask about it next class*. Highest marks guaranteed. But alas it will not make you a mathematician.

posted by jfuller at 6:17 AM on February 24

I do see that, I do. What hurt my head to the point where I gave up thinking about it was considering the problem like this:

Think of all the well defined plane figures for which the area is easy to find. A circle? Easy-peasy, give me r and I'll give you the area. Rectangle? Length and width, easy-peasy. Triangle? easy-peasy. Ellipse? Easy-peasy. There are clean, simple expressions for the area of all of these.

Now what about the area of this figure here, defined by y = f(x) in the domain from x = a to x = b? [Say f(x) stays above the x axis for this whole domain, just for simplicity.] For this bounded figure I have one easy to find side: the length on the x axis is (b - a). I have a second easy to find side: the height above a is f(a). I have a third easy to find side: the height above b is f(b).

Then there's the wiggly part at the top, defined as f(x) while x varies from a to b.

The moment of insight never came. That's when I knew that, no matter that I knew the secret (*) of doing well in math courses, I was not a mathematician.

(*) Nb, for them as don't know, the secret is

posted by jfuller at 6:17 AM on February 24

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posted by escabeche at 2:10 PM on February 21 [6 favorites]