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This is the meta thread for What advantage humans have over computers in mathematics?

It was closed, got reopened and reclosed, and reopened again.

In an effort to close the barn before the horse bolts, I create this post in case there is need for discussion.

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    $\begingroup$ My two cents: this is waaaaay too discussion-y. I think it would be great for a chat room discussion if people are interested in this, but the answers are just tissues of opinion and speculation and prognostication. (Don't get me wrong; I think it's a very interesting question, and I have opinions of my own. But to me it has all the appearance of idle conversation at a bar, or between bright math majors in a dorm room. Not a good use of MO, but people have other ideas apparently.) $\endgroup$ – Todd Trimble Mar 13 '16 at 18:45
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    $\begingroup$ I'm afraid I think these discussions don't seem to help much -- people end up having strong opinions one way or the other, and I'm not sure anyone changes anyone else's opinion. Why not just let people vote as they see fit? I think too that the question has just attracted a bunch of opinions, and I don't know what real answer is possible. But if enough people want it open, so be it. $\endgroup$ – Lucia Mar 13 '16 at 19:41
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    $\begingroup$ @Lucia the first point of this thread, and in this case my main motivation, is that if discussion should happen it can happen here rather than in the comments on the main post. Second, voting to open and close is not really intended as a majority vote; really it is a misnomer this is even called a "vote." Third, it did happen not rarely that users actually changed their mind, or if not at least found a compromise or understood each others opinion better after such a discussion. There are plenty of once controversial subjects that by now are simply resolved due to such discussion. $\endgroup$ – user9072 Mar 13 '16 at 20:02
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    $\begingroup$ Indeed, in more recent times we have the problem that there are some users that have moderation privileges but seem unaware of what is or at least once was considered standard etiquette related to the usage of these tools. Such as, as a rule votes to close are accompanied by an explanatory comment (if there is none already, and excepting egregious cases maybe) @Lucia $\endgroup$ – user9072 Mar 13 '16 at 20:05
  • $\begingroup$ What @ToddTrimble said. I didn't vote to close, but I'm not voting to re-open $\endgroup$ – Yemon Choi Mar 14 '16 at 1:03
  • $\begingroup$ Maybe a lot of people can discuss the issue but there are a few people some even MO-participants who can give good research-level scholarly answers based on experience in using computers for mathematics. $\endgroup$ – Gil Kalai Mar 15 '16 at 13:40
  • $\begingroup$ @GilKalai I'm in the middle of writing an opposite opinion, but there may be a chance of salvaging the question if it were more restricted in scope. I don't think anyone can really honestly, in a scholarly way, render expert opinion on the question of fundamental reasons. $\endgroup$ – Todd Trimble Mar 15 '16 at 13:45
  • $\begingroup$ you mean re-reopened? $\endgroup$ – tox123 Mar 18 '16 at 16:30
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    $\begingroup$ @tox123 I suppose one could express it like this. $\endgroup$ – user9072 Mar 18 '16 at 16:32
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In addition to my comment that the thread is too "discussion-y" (meaning too opinion or speculation driven), my feeling about this question is that there is probably no one in our community who can render an expert opinion on the matter based on demonstrable facts. It's all just speculation and "this is what I guess" about the matter, or even sentiments along the lines of "we shouldn't worry though, math will always be fun". And it could hardly be otherwise based on what little anybody really knows about human brains and AI (which I believe is beyond the scope of MathOverflow).

I think the problem is on how the question is put: "are there any fundamental reasons why a machine learning algorithm trained on a large database of formal proofs couldn't reach a level of skill that is comparable to humans?" I think that question is shooting much too high: no one can really say!! But if the question were to be reworded so as to emphasize more what actually has been accomplished and what is currently in the offing, on a concrete level, that would allow for more influx of expertise. So what is wrong with the question now is too much gazing into the future.

(My own opinion on the subject, for what little it may be worth, is that the role of human vision and even our kinesthetic sense in creating mathematics has hardly been touched on. There are huge swaths of geometry and low-dimensional topology, for example, that are just immensely difficult to cast into fully formal language, partly because the brain modules involved (the famous left/right brain dichotomy) are very different. I would aver that human vision endows human brains with a decisive advantage in certain respects for creating and communicating mathematics as we normally do it, at least for the foreseeable future, although I don't aver that this situation will never change. I'm actually hoping it does.)

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    $\begingroup$ In other words, the question itself is $\Sigma^0_1$ but not $\Delta^0_1$. Namely, if there is a positive answer, then at some point we will find it. But at no given time we are able to conclude a negative answer. $\endgroup$ – Asaf Karagila Mar 15 '16 at 14:38
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    $\begingroup$ Dear Todd, I agree that the question is hard to answer (but I like your own take on the matter ( in parenthesis ) ; I just not see why it couldn't be an answer on MO main rather than MO meta). But it is a very important question about mathematics, and quite a few people in the MO community worked on it. For example, Alexandre Eremenko, raised the interesting analogy with machine translation. The question of machine translation is certainly an important question about linguistics, and I'd expect there are quite a few MO linguistic questions devoted to it. $\endgroup$ – Gil Kalai Mar 15 '16 at 14:38
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    $\begingroup$ @AsafKaragila I think that's actually very well put -- thanks! $\endgroup$ – Todd Trimble Mar 15 '16 at 14:47
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    $\begingroup$ Todd, this is what happens when you're TA'ing a logic course about the incompleteness theorem and just happen to be in the midst of all the arithmetic hierarchy and $\Sigma^0_1$ predicates! :) $\endgroup$ – Asaf Karagila Mar 15 '16 at 14:50
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    $\begingroup$ @GilKalai As I have said, I agree that it's a very interesting question (and well worth considering), but I hope we can reach consensus that the question could be improved so that future answers could carry more substance and authority than the current ones do. (They are not bad per se; they just gaze into the future too much.) For example, Douglas Zare wrote a good meaty comment which looks like a potentially fine answer if the question were rewritten a bit. And thanks for your kind words, although my parenthetical comments are more opinionizing which I don't particularly promote here. $\endgroup$ – Todd Trimble Mar 15 '16 at 14:56
  • $\begingroup$ ams.org/notices/200210/comm-morin.pdf $\endgroup$ – Steve Huntsman Mar 16 '16 at 18:33
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    $\begingroup$ @SteveHuntsman That type of example is part of the reason why I included the word 'kinesthetic'. Bill Thurston also remarked once on the example of Morin and sphere eversion (More Mathematical People, pp. 337-338): "It's something most people have a great deal of trouble visualizing. In fact, I think that vision is somehow distracting to the spatial sense, because we have a spatial sense that is more than just vision. People associate it with vision, but it's not the same. If I close my eyes and imagine what this room is like, I will have a sense in my mind that there's a table here (cont.) $\endgroup$ – Todd Trimble Mar 16 '16 at 18:44
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    $\begingroup$ and something here and there. It will be a sense of the room that doesn't have much to do with perspective. It's hard -- that is, it takes a lot of training -- to go from a spatial image to a picture on paper. So these things are not necessarily stored in our minds in a visual sort of way. We translate what we see into a sense of space. If you think of it, you realize that if you imagine a table with four chairs around it, it doesn't matter whether you can see the seats of the chairs. You just know that they are there. It's kinesthetic as well as visual." $\endgroup$ – Todd Trimble Mar 16 '16 at 18:47

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