What the experience of mathematical beauty could imply

Back in September, 1992 Semir Zeki wrote an article for what was then a special issue of Scientific American called Mind and Brain. In it he described what was known about how the brain produces visual images.  I have referred back to the article many times because it highlights the philosophical implications of our current grasp of these processes.  Right below the title of the article was this remark:

In analyzing the distinct attributes of images, the brain invents a visual world.

Near the end he makes an important observation:

The past two decades have brought neurologists many marvelous discoveries about the visual brain.  Moreover, they have led to a powerful conceptual change in our view of what the visual brain does and how it accomplishes its functions.  It is no longer possible to divide the process of seeing from that of understanding, as neurologists once imagined, nor is it possible to separate the acquisition of visual knowledge from consciousness.  Indeed, consciousness is a property of the complex neural apparatus that the brain has developed to acquire knowledge.

(emphasis added)

Zeki’s investigation of the visual brain has lead to a significant amount of work on the neurobiology of aesthetics.  He heads the Institute of Neuroaesthetics at University College London.  VisLab, The Artificial Vision and Intelligent Systems Laboratory at the University of Parma, Italy, has contributed to the institute’s work.  Within an introduction to the institute’s purpose, and with respect to Vislab in particular, there is the following statement:

Over the past few years Vislab has contributed to neuroesthetics by exploring visual art in relation to the known physiology of the visual brain.
Underlying the approach are three suppositions:
•    that all visual art must obey the laws of the visual brain, whether in conception or in execution or in appreciation;
•    that visual art has an overall function which is an extension of the function of the visual brain, to acquire knowledge;
•    that artists are, in a sense, neurologists who study the capacities of the visual brain with techniques that are unique to them.

Very recently, Zeki co-authored a paper on The experience of mathematical beauty and its neural correlates.    Neuroscientist, John Paul Romaya;  physicist, Dionigi M. T. Benincasa; and mathematician, Michael Atiyah, were his co-authors.  The paper was published in the journal Frontiers on February 13.  Their study, was aimed at determining whether the beauty experienced in mathematics correlates with activity in the same part of the emotional brain (referred to as field A1 of the medial orbito-frontal cortex or mOFC) as the beauty derived from sensory or perceptually-based sources like visual art and music.  Their results showed that mathematical beauty was correlated with activity in this part of the emotional brain, which raises some interesting questions related what our experience of beauty is all about.

Unlike studies that looked at the neurobiology of musical or visual beauty, this study required the recruitment of individuals with a fairly advanced knowledge of mathematics.  And so, while it may be difficult to sort out, this effort, the author’s suggest,

…carried with it the promise of addressing a broader issue with implications for future studies of the neurobiology of beauty, namely the extent to which the experience of beauty is bound to that of “understanding.”

The study included 12 non-mathematical subjects, but the majority of these individuals indicated that they didn’t understand the equations and that they didn’t have an emotional response to an equation they may have found beautiful (despite the fact that some did rate particular equations as beautiful). Researchers were able to parse the components of the non-mathematicians’ judgment to some extent. Finding more intense activity in the brain’s visual areas for these subjects, confirmed their hunch that the beauty-rating from the non-mathematical participants was a judgment about the formal qualities of the equations – the forms displayed, their symmetries, etc.

The paper becomes even more interesting when the authors consider the implications of their work:

The experience of beauty derived from mathematical formulations represents the most extreme case of the experience of beauty that is dependent on learning and culture. The fact that the experience of mathematical beauty, like the experience of musical and visual beauty, correlates with activity in A1 of mOFC suggests that there is, neurobiologically, an abstract quality to beauty that is independent of culture and learning. But that there was an imperfect correlation between understanding and the experience of beauty and that activity in the mOFC cannot be accounted for by understanding but by the experience of beauty alone, raises issues of profound interest for the future. It leads to the capital question of whether beauty, even in so abstract an area as mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved.  (emphasis added)

And then a quote from a talk given by Paul Dirac in 1939 (one of the subjects of an earlier post of mine), where Dirac advices physicists to look first at promising mathematical ideas, and to consider beauty over simplicity.

There is no logical reason why the (method of mathematical reasoning should make progress in the study of natural phenomena) but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme. . . What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty… The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. . . We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty.

What I find very encouraging is that this paper suggests, in yet another way, a coupling of the body with its world that mathematics may yet have a hand in helping to reveal.

The Platonic tradition would emphasize that mathematical formulations are experienced as beautiful because they give insights into the fundamental structure of the universe (see Breitenbach, 2013). For Immanuel Kant, by contrast, the aesthetic experience is as well grounded in our own nature because, for him, “Aesthetic judgments may thus be regarded as expressions of our feeling that something makes sense to us” (Breitenbach, 2013). We believe that what “makes sense” to us is grounded in the workings of our brain, which has evolved within our physical environment…Hence the work we report here, as well as our previous work, highlights further the extent to which even future mathematical formulations may, by being based on beauty, reveal something about our brain on the one hand, and about the extent to which our brain organization reveals something about our universe on the other.

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