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This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs appropriate for high school level.)

The background is this: we routinely distinguish between proofs that explain and proofs that demonstrate. This distinction has been around at least since Aristotle's time, but it is an open question, for instance in contemporary philosophy of mathematics, what explanation really means. One recent suggestion has been that explanation might be related to beauty. It seems reasonable that explanatory proofs are nicer than non-explanatory ones in SOME way, but are they necessarily more beautiful? And similarly, is beauty necessary for explanation? It seems a good way to attempt to answer these question is to look at a bunch of good examples. And it seems a good way to get examples is to ask mathematicians, which is why I post the question here.

To clarify: the question is not at all to discuss the nature of explanation or beauty (if you want to discuss, I can give you my email address and we can chat offline). The purpose to is collect some good mathematical examples that help understand the relation between beauty and explanation in mathematics.

Examples that would be relevant: beautiful proofs that are not explanatory, explanatory proofs that are not beautiful.

Thanks in advance.

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    $\begingroup$ +1 just for specifying that this question is meant to collect examples, not indulge in discussion or soap-boxing $\endgroup$
    – Yemon Choi
    Commented Jul 24, 2013 at 6:24
  • $\begingroup$ There is a 2016 paper on this topic that seems to have been inspired by the OP's work. $\endgroup$
    – Timothy Chow
    Commented Nov 5, 2019 at 23:32

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A proof that many people say they find beautiful, but in my view is not at all explanatory, is Zagier's one-sentence proof of the sum of two squares theorem.

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    $\begingroup$ Zagier can take it, so: I have spent more time than I should thinking about various proofs of F2ST. Zagier's proof is probably my least favorite: it seems showoffy rather than explanatory, and I don't find that beautiful. Optimizing an argument to have as few instances of "." as possible seems pretty weird to me. $\endgroup$
    – Pete L. Clark
    Commented Jul 26, 2013 at 9:28
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    $\begingroup$ @PeteL.Clark: As you can probably tell, from my weasel-y use of "many people say", I myself am not among those people. One thing that bugs me is that despite being one sentence long, I'm never going to remember how that proof goes (or I'd only remember if I studied a longer proof where the underlying explanations saw the light of day). $\endgroup$
    – Todd Trimble
    Commented Jul 27, 2013 at 21:02
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    $\begingroup$ It is not a one sentence proof, I've never understood why one would say or think that. The decoding is part of the proof. $\endgroup$
    – Andrés E. Caicedo
    Commented Aug 3, 2013 at 1:56
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    $\begingroup$ @AndresCaicedo Well, what can I say? That what Zagier calls it, and what others call it as well. Maybe for him it's a kind of gestalt and can stand on its own, but I'm like you in that I have to spin it out into something longer before it looks like a proof to me. $\endgroup$
    – Todd Trimble
    Commented Aug 3, 2013 at 2:08
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    $\begingroup$ And it is not Zagier's proof. He took it from Heath-Brown. Heath-Brown took it from Uspensky. $\endgroup$
    – Alexey Ustinov
    Commented Apr 9, 2015 at 13:39
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I'd say that the proof of the four-colour theorem (particularly the "first generation" proof of Appel and Haken) is explanatory (we see that the source of four-colourability is the presence of unavoidable subconfigurations in any planar graph which are all reducible, in that the four-colourability of any planar graph containing such a configuration can be deduced from the four-colourability of a smaller graph) but not beautiful. (See for instance the Notices article at http://www.ams.org/notices/199807/thomas.pdf for a description of the proof strategy.)

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    $\begingroup$ This is a good example, but to play Devil's advocate for a moment, the argument basically comes down to discharging and Kempe chains and both of those are kinda beautiful (if old hat by the time of Appel and Haken). $\endgroup$
    – Noah Snyder
    Commented Jul 24, 2013 at 19:18
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    $\begingroup$ In general I think arguments that involve a lot of casework tend not to be beautiful but could still be explanatory. $\endgroup$
    – Qiaochu Yuan
    Commented Jul 25, 2013 at 8:13
  • $\begingroup$ Perhaps I have badly misunderstood the premise of the question, but I don't find that proof very explanatory either. Allow me to illustrate: "Q: Why are all maps four-colorable? A: Because all maps can be simplified to one of these four-colorable cases. Q: Why are all of those cases four-colorable? A: Because they are." $\endgroup$
    – Kevin
    Commented Apr 13, 2022 at 6:21
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1) "There is no simple group of order $n$" (for various composite values of $n$ in the interval $[50,200] \setminus \{60,168\}$ or so). These arguments are explanatory but not beautiful. They seem very ad hoc and futile in the sense that there are obviously plenty of larger values of $n$ for which one will not succeed in proving the result in the same way. To my taste under/graduate algebra courses ask too many of this type of question: it's not beautiful and it's not practical, since by the way we know all the orders of finite simple groups!

2) The classification of finite simple groups is (it seems; I am no expert) neither explanatory nor beautiful. The first generation proof was, it turned out, not even within two hundred pages of being formally complete. This is an impressive list of negatives for what everyone agrees is one of the most important theorems of all time. It seems that the goal of the third generation proof is to be more explanatory.

3) The theory of the $26$ sporadic simple groups is beautiful to an almost ridiculous degree. There seems to be plenty of room for it to be more explanatory...which is beautiful.

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    $\begingroup$ I've always thought the point of some of these exercises is precisely to emphasize how ad hoc they are, so one understands why the elaborate theory that leads to classification would be desirable. $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 26, 2013 at 16:02
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    $\begingroup$ @AndresCaicedo: I would disagree. For me, the point of the exercises is to demonstrate the power of elementary group theory (in particular, the Sylow theorems) to answer a wide variety of such questions. Of course, the ad hoc nature of the arguments may inspire one to want to develop a more systematic and general theory, but the basic tools are a powerful starting point. $\endgroup$
    – Eric Wofsey
    Commented Aug 3, 2013 at 2:56
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    $\begingroup$ @EricWofsey And that may well be the intention of the author of the exercises, I agree. All I am saying, when I think about them, when I discuss such results in lecture, etc, I personally do not think of them this way. $\endgroup$
    – Andrés E. Caicedo
    Commented Aug 3, 2013 at 3:07
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I find Furstenburg's proof of van der Waerden's theorem using the Stone-Čech compactification of the natural numbers beautiful but not so explanatory.

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There are some theorems where it is clear that a straightforward but tedious computation can establish them, and where it is also fairly clear that there is little hope of eliminating the computation. "Checkers is a draw" is a rather extreme example. Few people would call the proof beautiful. On the other hand, I would consider the proof explanatory, because what better explanation is there that there is no winning strategy than an explicit recipe for countering every winning attempt? To deny that the proof is explanatory would, I think, be to implicitly demand more explanation, and it seems clear to me that there is not going to be any way to get a better explanation than just giving the strategy. (Parts of the proof could perhaps be made more conceptual, but there is always going to be a large computational residuum.)

In short, sometimes the best explanation of a fact is simply that that's how a certain tedious calculation turns out.

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    $\begingroup$ Though I agree, there is something that bothers me here. Let me try to illustrate it using a different example: Many results in, say, analytic number theory, assert all $n$ have some property. Perhaps this was conjectured based on copious numerical evidence. (Cont.) $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 25, 2013 at 6:34
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    $\begingroup$ A proof is produced that very large $n$s have the required property. The proof explains why this is so, but only applies to large values of $n$, say $n>M$. We then complete the proof by extending our numerical computations, until we reach $M$. But a proof like this, though explanatory, seems lacking, as a "true explanation" should perhaps address the small numbers (with something other than a table), since they were what made us suspect the conjecture to begin with. Yes, what is lacking is a more conceptual argument. Somehow it seems unsatisfactory to call the current proof explanatory. $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 25, 2013 at 6:37
  • $\begingroup$ (A different issue is whether such a proof is not beautiful. I am not addressing this here.) $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 25, 2013 at 6:38
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    $\begingroup$ I think that in your number-theoretic example, it's still reasonable to hold out hope that the computational residuum can be eliminated by a conceptual advance. I intentionally picked an example where there are heuristic reasons, based on computational complexity, for believing that there is no purely conceptual argument, if for no other reason than that any proof must be too long to be conceptual. $\endgroup$
    – Timothy Chow
    Commented Jul 25, 2013 at 6:48
  • $\begingroup$ Concrete examples of what Andrés E. Caicedo is referring to include Helfgott's proof of the ternary Goldbach conjecture and Catalan's conjecture. In both cases, reducing the size of the finite computation involved significant conceptual advances. Arguably, therefore, the earlier arguments were not entirely "explanatory." $\endgroup$
    – Timothy Chow
    Commented Aug 22, 2023 at 16:04
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I'm not sure this question is appropriate for MO, but: I find the usual proof of the Recursion Theorem beautiful but not explanatory. (See my answer to another MO question: Are there proofs that you feel you did not "understand" for a long time?)

The other direction is a little bit trickier: (it is my opinion that) the fact that a proof explains something well - if it does - makes it beautiful, at least to some degree. "Beauty" in mathematics, to me, is a fundamentally emotional property; it's not just the simplicity of the proof, or the cleverness of the argument, but my reaction upon reading through it that determines what sort of beauty it possesses for me, and explanatory proofs automatically score high in this regard. So I'm not sure a truly explanatory, but ugly proof exists.

The closest I can think of for this category are theorems asserting that such-and-such technical construction has some desired properties - often, these theorems are perfectly clear yet unremarkable, and the beauty of the construction comes in how it is applied or the idea behind it in the first place, not in the proof that it is what it ought to be. As an example, consider Kleene's proof that the statement "Turing machine $e$ on input $k$ halts" is a $\Sigma^0_1$ formula in the language of arithmetic. The proof of this is completely explanatory, but ugly. The real beauty is in the realization that arithmetic can express this kind of statement efficiently, which is at least plausible as soon as the question is brought up at first. But this kind of example feels pretty unconvincing.

LARGELY UNRELATED EDIT: Re: the Recursion Theorem, there are more explanatory proofs. It can be gotten as a corollary of Lawvere's Fixed Point Theorem (see Lawvere's fixed point theorem and the Recursion Theorem); staying within computability theory, Adam Day has given an absolutely wonderful argument (http://sigmaone.wordpress.com/2013/04/19/the-recursion-theorem/). A rule of thumb I tend to believe: all beautiful proofs can be "unpacked" to reveal an explanatory proof which, while maybe not as beautiful, is still beautiful in the same way.

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  • $\begingroup$ Perhaps also many of the priority argument constructions in computably theory, such as the usual construction of incomparable c.e. Turing degrees, are further examples of messy proofs of beautiful and explanatory facts. $\endgroup$
    – Joel David Hamkins
    Commented Jul 24, 2013 at 12:44
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    $\begingroup$ In general, certainly (and in particular, I'd list the Thickness Lemma as one which is perfectly explanatory but ugly); although personally I think Friedberg-Muchnik is gorgeous. One quibble, though, is that I'm not sure how much explanatory power these arguments really have - whenever I see an infinite-injury proof, I'm left thinking that there really ought to be a forcing-like metatheorem for organizing the ideas, and that without it I'm only understanding a part of why the theorem holds. $\endgroup$
    – Noah Schweber
    Commented Jul 24, 2013 at 12:56
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    $\begingroup$ Noah, on that note---since I also regard many priority argument constructions as essentially similar to forcing arguments---one could regard many complex forcing arguments as further instances. $\endgroup$
    – Joel David Hamkins
    Commented Jul 24, 2013 at 14:03
  • $\begingroup$ I'm not sure I agree that all beautiful proofs can be unpacked into an explanatory proof. Some proofs are beautiful because they are amazingly short and slick, and their brevity relies on lucky coincidences that don't generalize. (I'm trying to think of an example but don't have one at my fingertips.) In such cases I don't think that there's an "explanatory" proof buried inside. $\endgroup$
    – Timothy Chow
    Commented Jul 25, 2013 at 6:40
  • $\begingroup$ That's why I phrased it as "a rule of thumb I tend to believe." It's certainly not true all the time, but it is frequently true; and at least for me, it's a useful heuristic. $\endgroup$
    – Noah Schweber
    Commented Jul 25, 2013 at 12:34
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I think Fourier analysis generates a lot of interesting examples of this phenomenon; it is an extremely powerful tool, but it's not always easy to see what the tool is actually doing.

A good case study is the classical isoperimetric inequality. There is a gorgeous and short proof using Fourier series which you can teach to an undergraduate. But the concept of length and area sort of disappear into the Fourier coefficients well before the key inequality, and so it takes a bit of effort to see the geometry.

Compare this to, say, Euler's proof using the calculus of variations. It is very direct: it attacks the problem of minimizing the area functional over the space of curves with a given length head on, and the argument doesn't really use anything outside of the standard set of tools for similar optimization problems. But Euler's original proof had a crucial gap involving the problem of determining when a critical point is a global minimizer, and it took centuries to get it right.

So we see a sharp contrast between elegance and simplicity (the Fourier series argument) and direct, explanatory power (calculus of variations).

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    $\begingroup$ (None of my remarks should be interpreted as disparaging the beauty of the calculus of variations, by the way.) $\endgroup$
    – Paul Siegel
    Commented Sep 24, 2015 at 15:59
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I would suggest that the proof of Burnside using group characters that a group of order $p^{a}q^{b}$ ($p$,$q$ primes) is solvable is beautiful, but not really explanatory, from a group theoretical point of view. The same can be probably be said for Frobenius's characterization of finite permutation groups in which no non-identity element fixes more than one point. While there are now "explanatory" proofs of the first result using purely group-theoretic methods,(which are arguably less beautiful) there is presently no purely group-theoretic proof of the second (although Terry Tao has recently found an alternative proof using character theory of certain commutative algebras, which can be seen on his blog).

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I think proving compactness of logic from completeness is explanatory but not beautiful. The proof with ultraproducts is more beautiful.

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    $\begingroup$ Not sure I agree. The former proof is mysteriously elegant and doesn't at all reveal the nature of the model of the infinitely many sentences. So it's not really explanatory at all; conversely, isn't mysteriousness beautiful? $\endgroup$
    – Ryan Reich
    Commented Jul 24, 2013 at 16:04
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David Hilbert's proof of the transcendence of $e$ and $\pi$ is extremely elegant, and totally mysterious.

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    $\begingroup$ I'm not sure what proof you're referring to. Could you add a link? $\endgroup$
    – Noah Snyder
    Commented Jul 24, 2013 at 19:20
  • $\begingroup$ @Noah, I am away now from any references. Thus let me mention from memory a proof on $e$ and $\pi$ in a single-volume textbook by William LeVeque (he had an earlier one in 2-volumes, and unfortunately one more consisting of a single volume--sorry). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jul 24, 2013 at 20:01
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Bombieri's proof of the analog of the Riemann hypothesis for a curve C over a finite field is beautiful, in that it uses little more than the Riemann-Roch theorem for curves. But it seems less enlightening than the proof of Weil using the Jacobian of C or the proof of Grothendieck using Riemann-Roch for CxC, even though these require a great deal more background.

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In the game of 'chomp', the first player always has a winning strategy, by a simple and (I think) beautiful argument (he can 'steal' a possible winning strategy from the second player). However, as far as I know, apart from special cases, there is no general known way of describing the winning strategy (which would maybe count as more explanatory)

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Applications of Zorn's lemma, like the basis existence theorem, the existence of ultrafilters or the existence of a well-ordering may count. These proofs are short and beautiful but they remain (at least for me) somehow mysterious.

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    $\begingroup$ I don't see anything beautiful about them... $\endgroup$
    – Eric Wofsey
    Commented Aug 3, 2013 at 2:51
  • $\begingroup$ Dear Eric, you use the right idea (Zorn) on that richt structure and than You get it for free. For me as a lazy guy this is beauty. Very Best 9i $\endgroup$
    – Jörg Neunhäuserer
    Commented Aug 6, 2013 at 12:22
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Since this thread has some new activity: I've always found Rosenblum's proof of Fuglede's theorem (apply Liouville's theorem to the appropriate operator-valued function) extremely elegant, but rather mystifying and non-explanatory.

It wasn't until much later that I saw a proof using the spectral theorem, as presented in Halmos's article What Does the Spectral Theorem Say? (American Mathematical Monthly, Vol. 70, No. 3 (Mar., 1963), pp. 241–247) which I feel gives a better feeling for why the result is true.

(If I recall correctly: apparently Fuglede's original proof used the spectral theorem, but I haven't ever looked it up.)

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