Tuesday, 16 September 2014

What makes a mathematical proof beautiful?

(Cross-posted at NewAPPS)

In December, I will be presenting at the Aesthetics in Mathematics conference in Norwich. The title of my talk is Beauty, explanation, and persuasion in mathematical proofs, and to be honest at this point there is not much more to it than the title… However, the idea I will try to develop is that many, perhaps even most, of the features we associate with beauty in mathematical proofs can be subsumed to the ideal of explanatory persuasion, which I take to be the essence of mathematical proofs. 

As some readers may recall, in my current research I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? The general hypothesis is that most of the defining criteria for what counts as a mathematical proof – and in particular, a good mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case. (See also this recent edited volume on argumentation in mathematics.) Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, beauty may well play an important role, but its role will be subsumed to the ideal of explanatory persuasion.

There is a small but very interesting literature on the aesthetics of mathematical proof – see for example this 2005 paper by my former colleague James McAllister, and a more recent paper on Kant’s conception of beauty in mathematics applied to proof by Angela Breitenbach, one of the organizers of the meeting in Norwich. (If readers have additional literature suggestions, please share them in comments.) But perhaps the locus classicus for the discussion of what makes a mathematical proof beautiful is G. H. Hardy’s splendid A Mathematician’s Apology (a text that is itself very beautiful!). In it, Hardy identifies and discusses a number of features that should be present for a proof to be considered beautiful: seriousness, generality, depth, unexpectedness, inevitability, and economy. And so, one way for me to test my dialogical hypothesis would be to see whether it is possible to provide a dialogical rationale for each of these features that Hardy discusses. My prediction is that most of them can receive compelling dialogical explanations, but that there will be a residue of properties related to beauty in a mathematical proof that cannot be reduced to the ideal of explanatory persuasion. (What this residue will be I do not yet know).

As I mentioned, this is still very much work in progress, but for now I would like to sketch what a dialogical account of beauty in a mathematical demonstration might look like for a specific feature. Now, a fascinating desideratum for a mathematical proof, which has been discussed in detail recently by Detlefsen and Arana, is the ideal of purity:
Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. (Detlefsen & Arana 2011, 1)
A mathematical proof is said to be pure if it does not rely on concepts that are not present in the statement of the conclusion of the proof (the theorem). Many famous mathematical proofs are not pure in this sense, such as Wiles’ proof of Fermat’s Last Theorem, which utilizes incredibly sophisticated and complex mathematical machinery to prove a theorem the statement of which can be understood with knowledge of standard high school level mathematics. (The impurity of Wiles’ proof is one of the motivations often given to seek for alternative proofs of FLT, as described in this guest post by Colin McLarty.) Now, I take it to be fairly obvious that purity concerns can be readily understood as aesthetic concerns, in particular related to simplicity (which is one of the features widely associated with beauty). 

What would a dialogical account of the purity desideratum look like? Going back to the idea that the function of a proof is that of eliciting persuasion by means of understanding in an interlocutor (hence the stress on the explanatory dimension), it is clear that, in general, the less complex the mathematical machinery of a proof, the less it will demand of the interlocutor being persuaded in terms of cognitive investment. Moreover, if it relies on simpler machinery, the proof will most likely reach a larger audience, i.e. be persuasive for a larger number of people (those possessing mastery of the concepts used in it). In particular, a proof that only uses concepts already contained in the formulation of the theorem will be at least in theory comprehensible to anyone who can understand the statement of the conclusion. Thus, a pure proof maximizes its penetration among potential audiences, as it only excludes those who do not even grasp the statement of the theorem in the first place. In other words, purity sets the lower bound of cognitive sophistication required from an interlocutor precisely at the right place. (Naturally, I can also be convinced of the truth of a theorem even if I do not understand the proof myself, i.e. by relying on the expertise of the mathematical community as a whole.)

As I said, these are only tentative ideas at this point, so I look forward to feedback from readers. In particular, I would like to hear from practicing mathematicians their answers to the question in the title: what makes a mathematical proof beautiful? Do you agree with Hardy's list? (I could definitely use some input so as to render my investigation more in sync with actual practices!)


  1. I have a few comments. First, I don't think many professional mathematicians would agree 100% with your remarks about the advantages of pure proofs. I think it is necessary to add in an "all other things being equal" qualification. For instance, it often happens that both pure and impure proofs, and that the pure proof is long, messy and calculational, whereas the impure proof is short and conceptual (once you know the appropriate theory). In such a case, it is the impure proof that is regarded as the explanatory one. One might say that the transformation of the problem into something that looks a little different is in some sense revealing "what is truly going on", of which the original problem was a sort of epiphenomenon.

    But when it comes to your more general point, I wholeheartedly agree that beauty in mathematics should be analysed in terms of less aesthetic sounding concepts such as explanatory power. One such concept that I have written about in the past is memorability: I'd say that one condition that often applies to beautiful proofs is that they are based on one or two ideas from which the rest of the argument can be constructed fairly easily and standardly. This applies especially if the ideas are not obvious in advance but seem very natural with hindsight. (I realize that I've used another word, "natural", that itself is hard to explicate.)

    Of course, memorability is very closely related to explanatory persuasion, since if the basic idea of a proof is easy to hold in one's head, then one can indeed regard it as a convincing explanation.

    1. Yes, I suppose purity is more controversial than I had expected (on the basis of comments here and on Facebook). But ultimately, my main point is, as you put it, to argue that beauty in math is best analyzed as less 'aesthetic sounding' than one might think; I guess I'm too much of a pragmatist and thus tend to think that the ultimate goal of all things is to fulfill their function, and they are beautiful insofar as they do that (not in the sense of beauty an sich).
      Can you provide the reference to your work on memorability? It sounds like it's very much in the spirit of what I'm trying to argue with my dialogical conception of proofs.

  2. I'm not sure about purity as a criterium. I think the fundamental interconnectedness of many fields of mathematics yields its own beauty. As an example, consider the use of complex functions in number theory. The standard proofs of the Prime Number Theorem can be quite beautiful whereas the Erdos/Selberg "elementary" proof is a tour de force in the sense that it wasn't obvious that it could be done, but it won't win any prizes for beauty.

    1. Point taken! :) (See my comment above on how purity may be overrated.)

  3. Thanks, Catarina. I wanted to respond to the following point. "Now, I take it to be fairly obvious that purity concerns can be readily understood as aesthetic concerns, in particular related to simplicity (which is one of the features widely associated with beauty)." I think "aesthetic" as a category is a hard one to employ in understanding mathematical thought, because mathematicians use the term "aesthetic" rather loosely.

    Here's an example. In his book Elementary Methods In Number Theory (Springer GTM, 2000), the distinguished number theorist Melvyn Nathanson writes (pp. viii--ix) that "The theorems in this book are simple statements about integers, but the standard proofs require contour integration, modular functions, estimates of exponential sums, and other tools of complex analysis. This is not unfair. In mathematics, when we want to prove a theorem, we may use any method. The rule is "no holds barred." It is OK to use complex variables, algebraic geometry, cohomology theory, and the kitchen sink to obtain a proof. But once a theorem is proved, once we know that it is true, particularly if it is a simply stated and easily understood fact about the natural numbers, then we may want to find another proof, one that uses only "elementary arguments" from number theory. Elementary proofs are not better than other proofs, nor are they necessarily easy. Indeed, they are often technically difficult, but they do satisfy the aesthetic boundary condition that they use only arithmetic arguments."

    Nathanson thus identifies purity as a central concern (here cast as "elementarity", on which cf. a forthcoming paper of mine). He then calls purity an "aesthetic boundary condition". What does that mean here? He doesn't talk about beauty elsewhere in the passage; he mentions knowledge but no other attitudes or values. So why call it "aesthetic"? In my experience reading mathematicians, they use this term for any ~philosophical~ notion that they don't judge themselves to have understood well enough to explain, but that they judge to be important to mathematical thought. So epistemic, pragmatic, and what we'd label as genuinely aesthetic criteria are collected together under the label of "aesthetic".

    So I am hesitant to accept quickly that purity should be understood as merely an aesthetic concern, as opposed to an epistemic concern (our focus in the aforementioned article). Maybe it *can* be understood as an aesthetic concern, but I don't think that's its primary role in mathematical thought. We could think of it using our editorial hats: would our journal accept a new proof of an already-proved theorem if it were merely more beautiful than earlier proofs? Or would we demand that it provide some new understanding? I think the latter!

    But I'm uncomfortable with writing "merely more beautiful" there: why couldn't beautiful proofs provide for new understanding? But if they did, then the epistemic would remain central, as I indicated when discussing Nathanson.

    1. Thanks, Andy. In a sense, my appeal to purity was a bit of cheating: I want to argue that beauty is not the main thing, I pick purity, I claim it's related to aesthetic considerations, but then I go on to argue that the main attraction about purity belong to the cognitive/epistemic level... As you correctly pointed out, my crucial (and contentious) move is to associate purity with beauty (though I never said it is *only* about beauty). But I still think a plausible story can be told on how purity has at least an aesthetic dimension, even if it's not the whole story. Also, I'm thinking it's actually a good idea to bring up purity at the talk in Norwich, as clearly it generates a lot of interesting debate! :)

  4. Look at Brouwer's great proofs in topology 1910 to maybe 1915. which I guess are "pure" topology, but it is importantly difficult to say whether they are beautiful. Hopf captured exactly the experience I had before reading Hopf. At first sight Brouwer's proofs are incredibly beautiful: in being concise, insightful, persuasive. On a closer reading they are in Hopf's words "a bitter labor/eine sauere Arbeit." All trace of logical coherence evaporates. And yet, staring at them in dumb amazement long enough, you learn a vast amount. They are language-free proofs, though not at all in the sense of Brouwer's later articulation of intuitionism. They are brutally non-constructive even in places when it would be trivial to avoid, say excluded middle, if you had any desire at all to do so. In terms of the blog post, are they beautiful or painful? explanatory? or utterly unexplanatory? Do they bring understanding or force you to find your own? Hard to say.

    As Freudenthal says, the higher dimensional arguments are incomprehensible today and even Freudenthal could not see how Brouwer arrived at them without the later tools of homology. Is homology pure topology? A lot of people consider it modern folderol. Vick's book Homology Theory aims to put Brouwer's thoughts into homology. It is readable where Brouwer is not. But is it more or less beautiful than Brouwer's papers?

  5. An interesting counterpoint to purity is the idea of 'unexpectedness' that you quote Hardy has mentioning. There is often a certain pleasure in encountering a statement in domain X that is proven rather easily by making a quick excursion to domain Y for a key that slots in perfectly. Such a proof is unexpected, and therein lies the pleasure. But it is certainly impure according to the definition of purity you give.

  6. Some of the remarks above suggest to me that there may be different categories within which criteria of beauty are applied (whatever beauty amounts to here). Why can't one feel that one proof is beautiful in part because of its purity, while another borrows from another domain in a beautiful way? (Analogously, I feel that Ornette Coleman's 1959 quartet's recording of "Lonely Woman" is beautiful and heartbreakingly poignant, but I also feel that Gustav Leonhardt's 1973 recording of the Prelude in B-flat minor from Book 1 of Bach's Well-Tempered Clavier is beautiful and heartbreakingly poignant. These seem to be beautiful in different ways and for different reasons, and context might make one more than the other. James Brown's extended version of "Papa's Got a Brand New Bag" is beautiful in an entirely different way.)

  7. Coming from history of philosophy, I'm reminded of Hutcheson's remarks on beauty of theorems (An Inquiry into the Original of Our Ideas of Beaty & Virtue, Treatise I, Section III). For him, beauty is "Variety with Uniformity" and theorems, which exemplify this in bringing together an "infinite Multitude of Truths". I assume this would relate to "generality". He also highlights "clarity" (as compared to axioms) and "surprize" as a key factor in beauty of theorems.

  8. When you came to explain why the dialogical conception might explain why purity can make a proof beautiful, you didn't say what I thought you were going to say. That may well be because I'm not really familiar with your general project, but here's what I thought you'd say anyway.

    You said that you're thinking of the ultimate function of a proof as "convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case." I took it that your explanation focused mainly on the "convincing an interlocutor" part, but I think some of the motivation for purity may bear more on the "by showing why that is the case" part.

    The thought is that an impure proof can't really show why something is true, because e.g. Fermat's Last Theorem would be true even if, per impossibile, there were only natural numbers, and under those (impossible) circumstances it'd still be true for the same reasons it's actually true. And that means the explanation of why it's true should only reference the natural numbers. (I suppose that might also mean that if there isn't a pure proof then it's just true by accident, even if/though there's an impure proof out there.)

    I think that this fairly naive idea may well underestimate both how interconnected different fields are and just how necessary mathematical objects are, but insofar as it is thereby a bad idea, I think it's possible some of the apparent beauty of purity is an illusion.