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I want to discuss this question. It was closed, re-opened, then closed again. The close reason is "off-topic", and I want to ask if it really is off topic. The question itself basically asks Can we formalise the notion of "interesting" mathematics?

Two preliminary point: Firstly, I think that if you are going to participate in an open/close war then you should comment, or ensure that someone has commented for you (and if so upvote that comment to make it clear that you agree or something!). This has not happened here (10 users have voted to close, none commented). Secondly, I should make my stance clear: I believe that this is an interesting question, especially as I believe that "interesting" and "beautiful" are synonymous, and relevant to current trends (see, for example, here or this PhD thesis, and one can clearly see the benefits to the current trend of "automated theorem discovery" (this seems to do a reasonably survey, dated 2009)). The idea of explaining beauty in mathematics transfixes even the best of us. I do not know how to answer the question, nor do I know if it can be answered.

The case for it being "off-topic" seems to be the following.

There is no obvious answer. Specifically, a high-rep user said that they "don't see how this can be reasonably answered".

My interpretation of this argument is: "noone who has read the post knows how to answer it, and that includes a high-rep user!". I do not believe that this is a valid close reason.

Relevant:

An earlier meta discussion, here. (This is not really relevant to the discussion at hand, but does give background.)

My re-open request, here.

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    $\begingroup$ I think the question how to model aesthetics is no more on topic than the question how to model the social behavior of bees. In both cases, there might be wonderful solid answers possible. In both cses, these will not be purely mathematical answers. $\endgroup$ – Michael Greinecker Jun 19 '14 at 9:36
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    $\begingroup$ I think even modeling bee behavior is much more on topic because at least it is a fairly regular behavioral process that contains patterns. Modeling aesthetics is not just hard, it is hopeless. I can't imagine any measure of aesthetic opinion that isn't a poll, and that's not what we do here. $\endgroup$ – rschwieb Jun 19 '14 at 11:27
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    $\begingroup$ It would, indeed, be interesting to learn something about the aesthetics in math. Our tastes do vary, but, given the task of selecting the most aesthetically pleasing one from a list of proofs of the same fact, our votes would NOT be random. They say that beauty is in the eye of the beholder. We are not an exception to that rule, but then the question would become one of the psychology of math viewers/practitioners. Also, standards of beauty are affected by the surrounding culture. We aren't immune to that either, but then the question becomes sociological. $\endgroup$ – Jyrki Lahtonen Jun 19 '14 at 12:03
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    $\begingroup$ TL;DR; This question could lead to a vivid and interesting discussion, but I would rather participate in it holding a pint as opposed to a piece of chalk. $\endgroup$ – Jyrki Lahtonen Jun 19 '14 at 12:04
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    $\begingroup$ George Birkhoff was a top-notch mathematician. For a review of his attempt to model aesthetics, see ams.org/journals/bull/1934-40-01/S0002-9904-1934-05764-1/… $\endgroup$ – Gerry Myerson Jun 19 '14 at 13:11
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    $\begingroup$ @JyrkiLahtonen All the best mathematics begin their life over a pint of beer...(also, perhaps my point is that yes it sounds interesting, but it is more than that: someone will have to get to the chalk stage if we are to ever have a computer which can understand aesthetics. Has anyone got to the chalk stage yet? I don't know...) $\endgroup$ – user1729 Jun 19 '14 at 13:48
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    $\begingroup$ Oh, and @user1729PhD, congrats on the new degree. Welcome to the trade union. $\endgroup$ – Jyrki Lahtonen Jun 20 '14 at 21:23
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    $\begingroup$ From browsing what is google-able about Birkhoff's book, it comes across as an attempt to justify "scientifically"valuing traditional art over the then-exploding modern 20th century art. Based on his rules, a square is a 1.5; some triangle a 0.63, and you need to draw these "obvious" shapes into paintings as they are "naturally" structuring them; in poetry 2 of the 3 arguments increasing a poem's value are rhyme and alliteration, when the world was a few years away from The Waste Land, and Kandinsky was already in his 20s. $\endgroup$ – gnometorule Jun 23 '14 at 14:21
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    $\begingroup$ The core is a formula "O/C" - order divided by complexity. How this is scientific, and - only 4 years removed from "degenerate art" - doesn't leave a sour taste, I do not know. $\endgroup$ – gnometorule Jun 23 '14 at 14:21
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    $\begingroup$ The aesthetic beauty of shapes (curves and surfaces) can be expressed in mathematical terms, to some extent. But that's the mathematics of beauty, and the OP asked about the beauty of mathematics. Two entirely different things, in my view. $\endgroup$ – bubba Jun 28 '14 at 7:27
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    $\begingroup$ @bubba: OP did not, but claiming he did fits with the arguments which aren't any by those agreeing. Your baseless claim that "the beauty of shapes can be expressed mathematically" aside (to my ears, there's a self-satisfied tone in your comment; so I am just leaving a comment so it is not unopposed, but don't expect any change of mind or even introspection), I am commenting on the only example given by analogy that this is a worthwhile pursuit. $\endgroup$ – gnometorule Jun 30 '14 at 15:00
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    $\begingroup$ That only example is a "theory" by a great mathematician (who also happens to at least arguably be antisemitic) justifying traditionalist art in a way that it must have been recommended reading by Goering, as outlined above. A lot of people in math unhealthily put great mathematicians on a pedestal, but their great intellectual achievements don't mean that they're not flawed individuals. While I admire Von Neumann's work, I think his "first strike the Russians before they have the bomb" was not sound advice. And I am flabbergasted that people aren't appalled by the linked late-life work here. $\endgroup$ – gnometorule Jun 30 '14 at 15:00
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    $\begingroup$ @gnome, please, it should be possible to critique Birkhoff's work without bringing Goering into it. $\endgroup$ – Gerry Myerson Jul 1 '14 at 5:53
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    $\begingroup$ @GerryMyerson: I don't mean to have this come across as an indirect personal attack on you (as you posted the link) - you're the author of one of my favorite MSE answers ever, if only for that - but given B.'s conclusions in 1933 on what is "great art", and this, I find it hard to not take context into account. I'm from the town in which Hausdorff killed himself after failing to secure a Princeton transfer, and "art theory" from that time makes me wince. $\endgroup$ – gnometorule Jul 1 '14 at 14:41
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    $\begingroup$ This originally was my lowest voted question (negative votes) and is my top voted question and now my reputation is fluctuating wildly every several days with deletes and undeletes. I didn't expect it to be so controversial, especially since I just wanted to know how to solve a problem. While its been a wild ride, and I'm a bit disappointed that there isn't more work in this area (or even agreement if its possible) the argument around it has at least generated some avenues of investigation to help me solve my problem, specifically with automated theory formation. $\endgroup$ – dezakin Jul 3 '14 at 15:36
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I think that this question seems to be more of the type "I would like to have a conversation about ..." than of the type "Please answer this mathematical question".

The topic of "mathematical interest" or "mathematical beauty" of obviously of interest to mathematicians, but it is not a mathematical question. In that sense, the question is "off topic" for this site. If we were a general discussion site for mathematics, the question would certainly be on-topic.

How could the question be improved? The discussion could be trimmed some, and the question could be made more objective. For example: "What specific algorithms do existing automated theorem discovery programs use to filter their results?" is an objective question. But the existing question is "Is there more to be said about quantifying what is interesting and what isn't?". The former has objective answers; the latter asks for speculation.

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    $\begingroup$ I disagree that this question is of the form "I would like to have a conversation about...". The OP has taken the time to explain their thoughts, and you are suggesting that "the discussion could be trimmed some". I disagree with this entirely! For some questions this does make sense, but I don't see why here. It is not some random ramblings (nor is it perfect), but you seem to be implying that you would prefer the question to be simply "Can we quantify "interestingness" in mathematics?" I do not understand how this is better. $\endgroup$ – user1729 Jun 23 '14 at 8:31
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    $\begingroup$ Also, I disagree with your second paragraph for the same reason I disagree with (the sentiment behind) Asaf's comment. Your argument is simply "I do not know how to answer this question, nor do I see how it can be answered!". Again, I do not believe that this is a valid close reason, and, as Gerry Myerson has pointed out, a serious mathematician has had a crack at this problem. So I do not believe that this question can be simply disregarded like this. $\endgroup$ – user1729 Jun 23 '14 at 8:36
  • $\begingroup$ @user, I also suggested (in a comment on a since-deleted answer, alas) that the serious mathematician's work did not impress his colleagues, and seems to have led nowhere. $\endgroup$ – Gerry Myerson Jul 1 '14 at 5:58
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    $\begingroup$ @GerryMyerson Yeah, I read that comment. The point of my final sentence, about not simply disregarding the question, was to say that "someone good has looked into it, so we cannot simply say it is entirely worthless! without digging further." By "digging further" I mean actually reading what he wrote, or reading about his peers opinions of the work. I was not meaning to claim that Birkhoff's work was top-notch - I have never read it! $\endgroup$ – user1729 Jul 1 '14 at 11:10
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    $\begingroup$ I asked the question specifically because I wanted to solve a problem: How do I select mathematical statements to investigate with an automated theorem prover, and then how do I implement an algorithm to generate or select such statements. I have some ideas on how to do it now, and I wanted to know if there are better heuristics for it or even formal theories surrounding the notion or anything related to the notion. It seems strange that I should rephrase my question to hide my motive of searching for statements that are 'interesting.' $\endgroup$ – dezakin Jul 3 '14 at 15:17
  • $\begingroup$ @dezakin: As you probably have learned from the discussion in main and in Meta, there is no (known) good theoretical way to formalize what is "interesting". However, if you are a programmer, you may try a practical solution, say along the following lines. Consider for instance Euclidean geometry, which is one of the oldest fields in math, where automated theorem provers are available. Imitate these provers to write a program which would generate "random" theorems whose statements have lengh bounded by N. Then send the output to the "prover". Compile statistics: True/false theorems... $\endgroup$ – Moishe Kohan Jul 6 '14 at 2:52
  • $\begingroup$ ... as well as the running time it took the prover to prove random theorems. Then look for random theorems whose proofs are longest (they might be the most difficult one) and check them against the known Euclidean geometry theorems, compile the list of "new" theorems and ask a professional mathematician if they find their statements interesting and proofs difficult. $\endgroup$ – Moishe Kohan Jul 6 '14 at 2:59
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Questions about mathematics or what is considered "good" or "interesting" are really questions of Philosophy of Science.

These are important questions since the philosophy which dictates what we call "logical" or "correct" or "rigorous" has changed in the past few centuries. Even our notions of logic are an invention pretty much due to Aristotle and have many cultural biases.

I recommend he use Philosophy SE - he even tags it [philosophy] or we can attempt to migrate it there.
Unfortunately debates like these, run by philosophers rather than mathematicians, tend to have nothing to do with how math is actually practiced.

He mentions TPTP deals with automated theorem proving. This topic might even fly at CS StackExchange or even Theoretical CS StackExchange.

Over all the question is difficult to read and may not be good for here, but it is a good question and he should put it elsewhere.

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The Question you have championed seems to be something of a Rorschach test, in which you find great depth and some others (myself included) find a lack of sufficient context upon which a useful Math.SE Answer might be composed. No problem has been posed in my view.

When one asks on Math.SE for a notion to be "formalized", it implies there is a context in which that notion is used in mathematics. Efforts of this kind can be on-topic here. I recall a Question here about how the notion of generic point introduced by the Italian school of geometry was eventually formalized.

But throwing out the word "interestingness" as if it had some recognized application to doing mathematics, and requiring a "formalization" of that notion, seems misplaced, at least in the absence of even an illustrative example of how "interestingness" might be so used.

The one illustration that comes to mind is the so-called paradox of interesting numbers, which Wikipedia describes as "semi-humorous". If the Question were to ask specifically about the notion of "interestingness" entailed in this attempted paradox of self-reference, I think it would provide sufficient context for Answers to be written about which reasonable people could find mathematical merit.

That was not the case here. As a general observation "interestingness" seems to me primarily a subjective, even ephemeral notion. I can think of many other words bearing on the aesthetics of mathematics that might have parallel claims: usefulness, beauty, power, nonobviousness.

Interests wax and wane, even in the most highly devoted followers of mathematics, while mathematics promises truths that exist "for the ages". Throwing the word "interesting" around as if it has some formal usefulness does not capture my interest.

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  • $\begingroup$ Rather biting and somewhat from on top a high horse. Some slack to the Roark of this forum, please. $\endgroup$ – Yadnarav3 Jun 25 '14 at 1:45
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    $\begingroup$ @Yadnarav3: Some context: As one of the first five who voted to close the Question at issue, I am responding to user1729PhD's complaint that "none commented". I had tried to post a comment on the reopen thread but got an error (posting from a phone). The points above are not intended as a slight to "the Roark of this forum" but are simply my reasons for voting to close. $\endgroup$ – hardmath Jun 25 '14 at 3:34
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    $\begingroup$ @Yadnarav3 This answer seems constructive to me, it tries to analyze precisely what is wrong with the question with respect to this site. Shouldn't we consider this as a favor done to the OP? $\endgroup$ – Did Jun 29 '14 at 7:23
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    $\begingroup$ How is the notion of interestingness in numbers more worthy of merit than the notion of interestingness in mathematical statements? What I'm searching for in 'interestingness' is any heuristic for capturing interesting statements, or even interesting numbers, for sorting statements for investigation. While the notion of such a heuristic might not capture your interest, the generated statements would. It seems a bit strange to me that people would find merit in the destination, but not the vehicle. $\endgroup$ – dezakin Jul 3 '14 at 15:01
  • $\begingroup$ @dezakin: I do recall your post asked as well about heuristics, but above I'm responding to user1729's topic of formalizing mathematical "interestingness". My suggestion that that tying a Question to a particular application could mitigate being too broad or subjective may be of help to you in that other connection. There have been good Math.SE Questions about heuristics for approaching proofs and conjectures. $\endgroup$ – hardmath Jul 5 '14 at 21:18
  • $\begingroup$ @hardmath Fair enough then $\endgroup$ – Yadnarav3 Jul 11 '14 at 8:32

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