Can "interesting-ness" be quantified?

And how does it relate to what mathematicians mean by "beauty"?

“…balance, complexity, body, harmony, intensity, nuance, and structure.” Seen on a table at the Centre International de Rencontres Mathématiques in Marseille

It is more important that a proposition be interesting than that it be true; the importance of truth is that it adds to interest.

(Alfred North Whitehead, Process and Reality, Part III, Ch. VI, Sect. ii)

This sentence is the starting point for an article by the mid-20th century philosopher W. T. Stace, entitled Interestingness (without the hyphen).¹ Stace begins his discussion by insisting on the audacity of Whitehead’s assertion:

It runs counter to all our common thinking about truth. It might even appear shocking to some people… even if one had had the wit to think of it, one probably would not have had the courage to say it.

Bearing in mind that Whitehead was also a mathematician, or at least a part-time logician, his claim is more likely to appear commonplace than shocking. Every mathematician who has given any thought to automated theorem proving knows that it is trivial to program a machine to prove random true theorems and is well aware that the real difficulty is to generate something interesting. Many of the participants in the NAS workshop alluded to this challenge one way or another, but already in 1958, when Newell and Simon published their famous list of overly optimistic predictions for the brand new field of AI, the one about mathematics specified

That within ten years a digital computer will discover and prove an important new mathematical theorem.

Today’s AI researchers would be satisfied with “interesting” rather than “important,” and some of them are talking about how this desirable quality might be quantified. Stace would be skeptical:

I want to be very careful to make it clear that I do not mean to imply that the quality of interestingness is an objective quality of some propositions in the sort of way in which some philosophers have apparently thought that goodness is an objective quality which certain things have. That may be the case for all I know, though I do not think it myself.

                                                 Attempting to quantify beauty

Circumstances that were at one time within my control have forced me to review the theory of beauty developed in Kant’s Critique of Judgment (or of the Power of Judgment [Urteilskraft]). This is the text also known as the “third Critique” and it is the one I have found the most impenetrable, to the point that I never managed to read more than the first third. Among many other ideas that I mainly find incomprehensible (for reasons closely related to the theme of this newsletter), Kant’s aesthetic theory implies that it is a category mistake,² for reasons that I will elaborate below, to try to find an objective measure of "beauty."

In a passage that many of our non-mathematician friends have read and quoted with a mixture of approval and perplexity, G. H. Hardy wrote about what he saw as the central importance of “beauty” for mathematics:

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. (Hardy, A Mathematician’s Apology)

Hardy didn’t go so far as to propose to quantify beauty but he did point to several “‘purely aesthetic’ qualities” — unexpectedness, inevitability, economy — as characteristic of good proofs. Twelve years ago a literary magazine asked me to write a critical essay about Hardy’s theory of mathematical beauty.³ The essay was published and was incorporated as half of one chapter of my book Mathematics without Apologies. In the course of writing the essay and the book I learned that there was a discipline called “aesthetics of mathematics” that was represented by a substantial and serious literature in scholarly journals as well as at least two serious books, respectively edited by Nathalie Sinclair, David Pimm, and William Higginson,⁴ and written by Ulianov Montano.⁵ In the introduction to their book, Sinclair and Pimm report on George David Birkhoff’s 1933 attempt to mathematize aesthetics — to do what Kant’s third Critique claimed was impossible:

Birkhoff admitted that the aesthetic feeling was ‘‘intuitive’’ and ‘‘sui generis,’’ but held nevertheless that the attributes upon which aesthetic values depend are accessible to measurement. He proposed three main variables constituting typical aesthetic experiences: the complexity of the object (C), the feeling of value or aesthetic measure (M) and the property of harmony, symmetry or order (O). With the following equation, M = O/C, he presented his hypothesis that the aesthetic measure be determined ‘‘by the density of order relations in the aesthetic object’’ (1933b/ 1956, p. 2186). He also provided equations that could define both the variables O and C more formally.⁶

Sinclair and Pimm note that “Birkhoff’s formula never gained much currency in the world of art criticism, nor in the world of mathematics” and speculate that

perhaps artists and mathematicians alike were unimpressed by Birkhoff’s formula for its tacit presumption that aesthetic value can be measured in some absolute way (regardless of personal, social or cultural styles), based on a set of accurate rules.

This alludes to a part of the Kantian argument, but their book mentions Kant’s aesthetic theory only briefly, in connection with his controversial claim that aesthetic judgments must be disinterested. In Montano’s book Kant’s name appears only four times, in each case recalling Kant’s contention that aesthetic judgments are necessarily subjective. It’s this latter contention that is relevant to the topic of this newsletter, specifically with regard to interesting-ness as an aesthetic judgment.⁷ In an earlier post I quoted Tim Gowers to the effect that it is

[d]ifficult to see how the notion of a mathematical community could survive [the] development [of computers able to] judge what is interesting and worth proving…

a development Gowers sees as inevitable because "[m]athematical interest is sufficiently objective." More recently I’ve learned that at least one eminent AI researcher, possibly picking up where Gowers left off, is thinking actively about how in the countably infinite haystack of all true theorems⁸ one can quantitatively pick out the interesting ones.

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             Two versions of AI, two approaches to quantifying interesting-ness

If your aim is to teach machines to recognize when a mathematical statement or proof is interesting, you have (at least) two very different options. The approach based on Symbolic AI (also known as top-down AI or GOFAI — Good old-fashioned AI) presumes that the characteristics that determine whether or not a mathematical text is interesting can be defined in advance — by a formula like Birkhoff’s involving detectable features. Then the machine’s task is to measure these features in existing texts, and then to imitate them in the texts it generates. Gowers is working in the GOFAI paradigm and, although he hasn’t explained why he thinks mathematical interest is objective, presumably this is the approach he would favor.

The alternative — deep learning, or bottom-up AI, is the one behind the machines that learn to play Chess and Go, as well as ChatGPT and the other generative AI platforms, and is undoubtedly the route the engineers would choose. One can imagine training a deep neural network to identify the common features among the items human mathematicians consider interesting. In principle this could be based on citation indices or Amazon-like star ratings. Or one could work directly with human built-in neural networks. Stace writes:

…I think it will be worth while to study this quality of interestingness, or rather the state of mind of a person who is interested in something. …This state of mind of being interested is a state of great mental excitement and absorption. …There is intense mental activity.⁹

Today’s computational theory of mind assumes that “state of mind” can be measured objectively. Our professional organizations like the American Mathematical Society could convince their members to attach electrodes to their heads and submit to functional MRI scans to determine what kinds of mathematics light up the relevant brain regions¹⁰.

The serious problem with such proposals is that “humans alone” will not suffice to expand the existing mathematical corpus by the five orders of magnitude¹¹ needed for effective training. This seems to mean that 99.999% of the data used for training will be based on machines’ preexisting aesthetic preferences.

On the other hand, it may just be that those who wish to train the AI have a unjustified bias toward what is interesting to human beings. If they drop this bias then it shouldn’t matter if the machines follow their own silicon fantasies. The next two sections explain why this may be the most sensible solution.

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                 Thinking about mathematical interesting-ness with Kant’s help

The heading of this section is taken from the book Thinking with Kant’s Critique of Judgment by philosopher and literary scholar Michel Chaouli. Unlike the third Critique itself, Chaouli’s book is one I managed to read from start to finish. It’s a book that acknowledges the strangeness of Kant’s formulations and helps the reader to see their originality and internal coherence; I recommend it highly. Chaouli’s aim is not to bring the reader to accept Kant’s positions but rather to see them as a legitimate way to think about beauty.

As far as quantifying beauty is concerned, Kant is categorical:

"There can ... be no rule in accordance with which someone could be compelled to acknowledge something as beautiful", Kant writes. It is a strong statement: I will not find something beautiful just because … it accords with some supposedly universal rule of beauty.¹²

Chaouli’s conclusion is that Kant would have seen Gowers’s difficulty in imagining the survival of the mathematical community, after interesting-ness is quantified, is misplaced:

…Kant's theory urges us to think of aesthetic pleasure in different way, one not amenable to being measured by sensors, MRI machines, surveys, or big data.

It’s hard to judge whether Kant or Gowers is right without exploring Kant’s “different way” and whether it might capture something, that by its nature resists measurement, about what makes mathematics interesting. Chaouli helps us to see that as well.

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                                          Will robots find you interesting?

The title of Chaouli’s first chapter, from which I am taking most of these quotations, is “Pleasure.” This suggests that for Kant, a machine capable of detecting beauty, or an aesthetic surrogate such as interesting-ness, would have to know something about pleasure. Alternatively, if interesting-ness is objective or quantifiable then pleasure must be objective or quantifiable as well.

The robot pleasure scenarios that spring to mind all arise from deep in the uncanny valley. I am thoroughly convinced that Silicon Valley has its own dark and uncanny corners, that some of my colleagues are in a hurry to explore, where quantified and objective plans are being hatched for a future synthesis of human and mechanical pleasure. If such an aesthetic experience is to be a Kantian synthesis, however, it will be poor material for formal verification: it

is not of [the] kind [that can] …crush the particular and contingent … since it does not determine anything in any one way. There are no laws or theorems governing beauty and aesthetic feeling; nothing controls the particular from above. It is entirely serendipitous. ¹³

Indeed, Kant points out that the “faculty of aesthetic judgment has been given the very name of ‘taste’” and insists that it is personal — “I try the dish with my tongue and my palate, and on that basis (not on the basis of general principles) do I make my judgment.”¹⁴ An MIT Technology Review that just came out this week expresses skepticism, along Kantian lines, about the future of AI aesthetics:

AI systems lack a fundamentally crucial skill for creating good art: taste. They still don’t understand what humans deem good or bad.

But to whom would it matter, assuming Gowers is right about the grim future prospects of the human mathematical community, if the machines, or the future offspring of a human-robot synthesis, don’t care about what humans find interesting?

1. W. T. Stace, “Interestingness,” Philosophy, Vol. 19, No. 74 (Nov., 1944), pp. 233-241. I thank my Columbia colleague, the philosopher Justin Clarke Doane, for finding this and other references relevant to the present essay.

2. Rest assured that I cleared this with a professional philosopher before making this claim.

3. This was the occasion of one of my failed attempts to read Kant’s third Critique.

4. Mathematics and the Aesthetic, New York: Springer (CMS Books in Mathematics #25, 2006).

5. Explaining Beauty in Mathematics: an Aesthetic Theory of Mathematics. Cham, Switzerland: Springer (Synthèse library #370, 2014).

6. Sinclair and Pimm, in Mathematics and the Aesthetic, op. cit.

7. The attentive reader will have noted the anticlimactic slippage from beauty to interesting-ness. This is not mainly because the online Stanford Encyclopedia of Philosophy has a solid entry entitled Beauty but apparently none that treats interesting-ness as a philosophical topic. It’s rather because the latter is the value judgment that a few computer-minded colleagues see as ripe for quantification; and as a value judgment interesting-ness is closer to beauty than to truth or goodness, its invariable rivals.

I maintain that the passages in Kant’s third Critique that deny an objective character to beauty apply essentially unchanged to interesting-ness.

Justin Clarke Doane found a discussion of this very topic on math.stackexchange, initiated by “user1729,” who went so far as to say “I believe that "interesting" and "beautiful" are synonymous, and relevant to current trends.” In contrast, the only really interesting [sic!] idea I found in Stace’s article, most of which was rather bland and conventional, is his claim that

Interestingness has the same relation to concepts which beauty has to percepts.… This difference is the difference between the intellectual and the aesthetic.

This could be true; but then he seems to be denying that mathematics can be beautiful.

8. For example, most mathematicians would struggle to see the interest in “If 1 = 1, then 1 = 1.”

9. Stace, op. cit.

10. A choice favored by some neuro-aestheticists — assuming we can conflate interesting-ness with beauty — is the medial orbito-frontal cortex. To quote Chapter 10 of my Mathematics without Apologies:

Researchers at London's Wellcome Laboratory… claim to have demonstrated that this correspondence is similar for auditory and visual beauty, and that therefore "there is a faculty of beauty that is not dependent on the modality through which it is conveyed."

11. See the discussion of Kontorovich’s presentation in this earlier post.

12. Quotations from Chapter 1 of M. Chaouli, Thinking with Kant’s Critique of Judgment, Harvard University Press (2017).

13. I have taken the liberty of substituting “it” for Chaouli’s “the ‘subjective universality’ of aesthetic experience.” For Kant, a genuine aesthetic experience is necessarily subjective but is also universal as expressed in the following quotations from the third Critique:

one solicits assent from everyone (Kant, §19)

and

this claim to universal validity so essentially belongs to a judgment by which we declare something to be beautiful that without thinking this it would never occur to anyone to use this expression. (Kant, § 8) .

If Kant were to come back, he would have to admit that, according to his system, “everyone” must include robots. That doesn’t mean the robots would agree.

14. Quoted in Chaouli, p. 17.