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What makes a mathematical formula beautiful?

attic-rats

Hiro Minato pointed me to this paper (hyped here) by Semir Zeki, John Romaya, Dionigi Benincasa, and Michael Atiyah on “The experience of mathematical beauty and its neural correlates,” who report:

We used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.

I wrote that I looked at the paper and I don’t believe it!

Minato replied:

I think what they did wasn’t good enough to answer or even approach the question (scientifically or otherwise). . . . Meanwhile, someone can probably study sociology or culture of mathematicians to understand why mathematicians want to describe some “good” mathematics beautiful, elegant, etc.

I agree. Mathematical beauty is a fascinating topic; I just don’t think they’re going to learn much via MRI scans. It just seems like too crude a tool, kinda like writing a bunch of formulas on paper, feeding these sheets of paper to lab rats, and then performing a chemical analysis of the poop. The connection between input and output is just too noisy and indirect.

This seems like a problem that could use the collaboration of mathematicians, psychologists, and historians or sociologists. And just think of how much sociologist time you could afford, using the money you saved from not running the MRI machine!

23 Comments

  1. Hernan Bruno says:

    The analogy to analysing the poop coming out of rats who were fed paper with mathematical formulas is wonderful and the best sentence I read this week (it’s only Wednesday). My brain is full activity just thinking of using it next time I review a paper with noisy data. If only I had a fMRI to prove it.

    • Rahul says:

      Didn’t someone correlate fMRI activity of a dead salmon with the images it was shown? Or was it a rat. Something like that.

      fMRI is a hotspot for crappy correlations.

      • Alex says:

        Similar to the recent report on potentially severely inflated error rates (http://blogs.discovermagazine.com/neuroskeptic/2016/07/07/false-positive-fmri-mainstream/), this was showing that certain methods of fMRI analysis lead to more false alarms than advertised. That doesn’t mean that it happens for every analysis method, or that any found ‘activation’ in a study is equally likely to be a false alarm.

        • Rahul says:

          What I’d love to have is a cheat-sheet to spot the potentially iffy stuff. Any ideas? What are the signs of a good vs fishing fMRI study?

          • Alex says:

            That’s a tough call. This is a pretty typical fMRI study as far as it goes; total N in the teens, canned options from a standard fMRI analysis software package, etc. So you can be skeptical of most fMRI studies (and that might not be a bad approach), but that could be throwing out perfectly valid results. For example, I don’t see any particular reason to doubt the major claim of this paper because it found activity in the same place that a number of related studies have previously. So you can assume that either the mOFC is particularly prone to giving a false signal in tasks where people are rating beauty, or you could say that if N studies all report the mOFC as being active in such a case (and other studies looking at something else don’t find mOFC activity!), then it is probably performing some function similar to what researchers are describing it as (beauty, or maybe pleasure or some other emotion-related evaluation).

            In terms of picking out fishing, I would say that some of the normal warning signs carry over. Do they do something known to be statistically invalid (e.g. http://www.edvul.com/voodoocorr.php)? Do they have a lot of potential contrasts that could give the described result but only mention one? If they have covariates, how much do the covariates matter and do they report all the options (e.g. forking paths)? Some things require a little more familiarity with fMRI research. Is this a crazy one-off study or does it fit with a larger (legitimate) literature, preferably with similar results from other neuropsych methods? Did they have to do something unusual to get significance (e.g. their result didn’t show up in a whole-brain analysis but does if they limit the statistics to some subset of the brain; this is sometimes reasonable but at least suggests that whatever they find is a pretty small effect)?

  2. But Andrew, without having read the paper, aren’t you missing some of the point? The point of the fMRI is not to figure out which formulas are beautiful, but rather to figure out whether *the experience* of observing a beautiful formula is mediated by the same kinds of brain processes as *the experience* of viewing say a beautiful painting or a beautiful photograph.

    From that perspective, fMRI is pretty much your only choice.

    Consider for example if viewing formulas that mathematicians called “beautiful” actually lit up brain areas associated with excitement or discovery of something unusual, which were distinct from say the areas lit up by paintings, sculptures, photographs, or hearing beautiful music. Certainly that wouldn’t tell us much about whether a particular formula was beautiful, but it would tell us something about whether “mathematical beauty” was really a similar brain state as “Viewing a Turner Painting”

    • To be clear, I am the one who hasn’t read the paper yet, not Andrew.

    • Andrew says:

      Daniel:

      I see your point, but I still think the measurements in this sort of experiment are too crude to get at anything useful here. To put it another way, even if FMRI is the only way to study this sort of thing, it might well be that nothing can be learned here because of the indirectness and noise in the measurements. (If you actually look at the study, it’s the usual cargo-cult imaging science: N=16, use of p-value peak voxel thresholds, multiple analyses giving may ways to claim success, etc.) To return to my kangaroo analogy, I think the signal-to-noise ratio is hopeless.

      Also, it’s not clear to me that brain imaging really is your only choice here. If you really want to study the experience of mathematical beauty, you could ask more informative survey questions about the experience (not simply rating formulas on a 5-point scale), you could do open-ended interviews with qualitative analysis, you could ask the mathematicians to discuss what in their view makes the formula beautiful, etc. If you wanted to be quantitative and reproducible you could do something like word-association questions.

      Lots of ways to go here, and it’s not clear at all to me that FMRI is the way to go. FMRI looks like science; it can impress funders, journal reviewers, and journalists; and its very expense can be seen as a feature, in that it’s Big Science. Also, what with forking paths and the presence of many large and interacting real signals, it’s easy to get statistically significant and interesting-looking results.

      One other thing, a slightly different point which I think is worth considering in any serious study of the topic: There is variation in mathematical beauty just as there is in the beauty of art, and these impressions are culturally dependent. It may be that every mathematician will find the Pythagorean theorem to be beautiful, and every art viewer will be charmed by Michelangelo’s David, but beyond this, all bets are off. For example, I find Picasso’s painting of the bathers to be beautiful, but I’m sure that many artistic conservatives at the time were not impressed. Meanwhile, I find modern art of the all-black canvas or drip-painting variety to be boring at best and ugly at worst, but I’m sure that others see beauty there. Or, to go to the mathematical domain, I see beauty in various statistical results that pure mathematicians might not appreciate. And when I was in the mathematical olympiad training, I remember our teachers finding beauty in various results in plane geometry that I just didn’t have the sophistication to appreciate.

      • Sure, especially without having read the article (yet), I’ll have to take your word for it about the noisy measurements etc.

        But, while I agree with you that there is culturally mediated variation in *what is considered beautiful* and you could get at different kinds of beauty within mathematics with questions about what the mathematicians thought, I don’t think that you get at causal processes in the brain without observing the brain.

        For example, suppose culturally everyone “knows” that Michelangelos David is “beautiful”, but actually 45% of people fail to light up the same regions of their brain as they do when viewing say landscape photographs or photographs of fashion models or Van Gogh paintings or a host of other things… so this would be really interesting, because it gets at the difference between what is expected for people to say, and what is actually happening.

        To me, the point of asking questions about mathematical beauty here is not to figure out what makes mathematical results beautiful, but rather to figure out whether the *kind of experience* that mathematicians have *in their brain* is really similar to the kind of experience that other people have which is commonly understood to be beauty.

        So, I’m totally against crappy expensive studies with statistical cargo-cultism, but I also think it’s a very interesting question to ask “what does experiencing beauty in mathematics mean for the brain, and how is it similar or different from experiencing beautiful paintings, sculptures, photographs, music, or the like? ” vs “which formulas are called beautiful and what are the given reasons?”

  3. matus says:

    I imagine an alternative study:

    What makes a neuroscientist design nonsense studies?

    We used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 neuroscientists when they viewed neuroscientific research proposals which they had individually rated as promising, indifferent or waste of time. Results showed that the experience of a promising study correlates parametrically with activity in the same part of the prefrontal cortex, associated with decisions to select nonsense in other studies – for instance decisions in studies where people were asked to discriminate non-sense words from meaningful words.

  4. Llewelyn says:

    I’m just envious that I don’t have an fMRI to examine actual real conditions — seems only researchers looking at mathematical beauty get them. We used to call many of the psychotropic drugs that had an impact on mood etc but about which the mechanism was mostly unknown “bucket chemistry”. The pretty pictures there were plots of how people taking them were in some other mental state measured by self report, mainly. Then there were all the masters students who would take a mental health condition (e.g., mood) and correlate measures of that against increasingly obtuse populations, almost never providing any real clinical impact due to low N and spurious stats. There probably is a study on how mathematical beauty impacts on mood as measured by the Kiribati Inventory of Clinical Something or other. Now we get pretty pictures where brain location and activity is viewed as evidence of some mood or other state; the brain just isn’t built like a computer with a bunch of different and unique parts but is far more plastic. So lots and lots of noisy data with the implication that there really is a rat involved. Why not go back to actually studying rats; they taught us a lot about most things in the past? Why not mathematical beauty?

  5. Alex Gamma says:

    There’s an interesting bit of background that helps put this study into context. The give-away is the word “neural correlate”. The term refers to the particular neural activity that is present whenever some particular mental state is present in a subject. This kind of research has grown out of attempts of neuroscience to finally tackle the most notorious problem in the life sciences and maybe all of sciences: consciousness. Especially since the 90s, the phenomenon of human consciousness in the sense of the having of states that have a certain “experiential quality” or that “feel a certain way to a subject” has seen a revival in the natural sciences after having been banned from much of 20th century psychology under the rule of behaviorism. This time around, also the neurosciences got involved, and thanks to efforts of philosophers such as David Chalmers, the neurosciences have started to take the issue of consciousness seriously, especially the “hard problem” of explaining how sth like a subjective experiential state could arise from physical brain processes.

    The neurosciences took a pragmatic approach and declared that if they just learned enough about the *neural correlates of conscious experience*, they might eventually gather enough information to build an explanatory bridge between the brain side and the consciousness side. And here we are now with a field of study hunting the neural correlates of all kinds of subjective states, including, as in this case, the experience of beauty. (Zeki, btw, is well-known in this field particular for his studies of vision in primates).

    This history explains why it’s the *experience* of mathematical beauty (and not other aspects of math formulae) that is of interest, and why it particularly needs the study of the *brain* (that’s what the whole approach is about). This does not necessarily invalidate the criticism that Andrew and others have raised, but it explains the particular focus of the study and choice of its methods, and places it into a larger research program (that, IMO, has a very worthy goal).

    • Rahul says:

      “Consciousness” seems the one problem everyone loves to discuss & debate about but in terms of actual empirical, hard-science progress have we gotten much? The neural correlates stuff seems like a lot of correlational antics.

      My impression is that our state of knowledge & experimental probes are not yet good enough to attack the question of consciousness. There’s a lot of other neurology to figure out before we can even dream of attacking consciousness in a meaningful, empirical way.

  6. Doug Davidson says:

    I don’t work with fMRI, but I have worked in cognitive neuroscience for a few years now. I have found this review to be helpful in thinking about what fMRI can tell you:

    http://www.annualreviews.org/doi/abs/10.1146/annurev-neuro-071013-014111

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