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Sorry to be a curmudgeon, but this question is a question about philosophy, not a question about mathematics. I'm not saying it's a bad question, but it's just not a mathematical question (insofar as philosophy is not mathematics...).

Philosophical questions can certainly have useful and interesting mathematical "answers". But I am not sure whether the present question is likely to lead to such "answers".

The issue of philosophical questions arose a while back on MathOverflow. See this question on the Church-Turing thesis and this meta.MO thread.

Please discuss the appropriateness of philosophical questions on math.SE.

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    $\begingroup$ voted to close (we can tolerate some off-topic but this one is also subjective and argumentative — so should be closed); it's especially strange to see such question not in CW-mode $\endgroup$ – Grigory M Aug 11 '10 at 18:04
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    $\begingroup$ Please upvote my comment on the question if you think it should be community wiki. I don't want to do it instantly because people are providing thoughtful, interesting responses, but am willing to change it at 4 upvotes. Note that the decision to close or not is distinct from the decision to make it community wiki. Closed questions can still be voted on and grant reputation. $\endgroup$ – Larry Wang Aug 11 '10 at 18:08
  • $\begingroup$ related: discussion of CW policy $\endgroup$ – Grigory M Aug 11 '10 at 18:10
  • $\begingroup$ I should add that I am not necessarily calling for this question to be closed immediately. I think there should at least be some discussion here first. $\endgroup$ – Kevin H. Lin Aug 11 '10 at 18:11
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    $\begingroup$ @Kaestur the very idea of CW [in such cases] is that people shouldn't gain rep by relating vague philosophical ideas (as opposed to "giving precise answers to mathematical questions"); making question CW after people stopped answering it is pretty much pointless $\endgroup$ – Grigory M Aug 11 '10 at 18:14
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    $\begingroup$ @Grigory: Changing the question to Community Wiki will prevent further reputation change from voting, even on existing answers. The current top-voted answer is at +4/-0, so I don't think there is much harm done. The real rep issue has to do with such questions receiving hundreds of votes, spread out over months. $\endgroup$ – Larry Wang Aug 11 '10 at 18:17
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    $\begingroup$ @Kaestur, Grigory: I'm pretty sure that the ♦mod option for converting a question to CW automatically converts all answers on that question to CW and that rep already gained is preserved, but no further rep can be gained from those answers that are converted. $\endgroup$ – Isaac Aug 11 '10 at 18:43
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    $\begingroup$ Saw 4 votes, changed question to Community Wiki. $\endgroup$ – Larry Wang Aug 12 '10 at 13:24
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    $\begingroup$ Agreeing with Matt E and T..'s views on this question, I have cast the final vote to reopen. $\endgroup$ – Larry Wang Aug 12 '10 at 17:22
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I don't see why this question is being described as "philosophical". In fact, the question seems to be rather specific: given that mathematical arguments are rather long and complicated, and (possibly) understood by very few people, how much confidence can we have in them. Personally, I think that this is an interesting question, in fact an important one for working mathematicians who want to rely on other people's results, that admits thoughtful answers.

I find it easier to see why the question is regarded by some as subjective and argumentative, although I personally don't find it such, and have thus voted to reopen it.

Finally, I don't think that it is unreasonable for people to gain reputation for their answers to such a question. A thoughtful answer will likely reflect mathematical experience on the part of the answerer, and hence will merit the reputation it attracts through upvotes. (I should say that I don't feel strongly on this issue; I am mainly addressing it to point out that I don't think there is anything unfair about those who already answered the question earning reputation for their answers.)

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    $\begingroup$ Two points: (i) On SO it has been the case that this sort of question yields non-expert answers that attract lots of upvotes. That these questions should be CW is something that SO and MO have in common; (ii) I don't call this a philosophical question either, but it does indulge in needless, carelessly phrased epistemological remarks. Those are what I call philosophical time-wastebaskets. $\endgroup$ – Charles Stewart Aug 12 '10 at 17:15
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    $\begingroup$ If there are flaws in the rating system of postings or user reputation, they can be fixed directly without restricting what questions can be asked. E.g., have a second reputation score, weighted Google Pagerank-style, where upvotes from high rated posters count more, or multidimensional ratings of the postings (difficulty, originality, humor, etc). Upvotes for mediocre material also happens on MO; one user observes that "not knowing what Bernoulii numbers are will get you more reputation than carefully answering difficult technical questions". The wisdom of crowds often fails. $\endgroup$ – T.. Aug 12 '10 at 19:40
  • $\begingroup$ I agree that it is a very important and interesting question, especially the way that you frame it here and in your comment(s) to Pete's answer. I would find this question much more suitable for a Q&A site if it were rephrased to something like "How can we be more confident of a mathematical proof?". $\endgroup$ – Kevin H. Lin Aug 12 '10 at 20:00
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It's a good question that:

  • pertains to mathematics
  • is of obvious general interest
  • received a large number of views
  • led to thoughtful responses
  • elicited mathematical and not only "philosophical" content (sequential compactness!)
  • did not "lead to confrontation and argument" of any kind

I upvoted the question and voted to re-open.

Having reasonable questions closed capriciously --- without clear evidence of a problem --- and without opportunity for the poster to improve the material (if there is in fact a potential problem), can only annoy or discourage those who have taken the time to formulate and post their queries.

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  • $\begingroup$ I agree that it should not be so quickly closed. I don't see any particular harm in keeping it open for now. I apologize for inadvertently prompting its closure. :( $\endgroup$ – Kevin H. Lin Aug 12 '10 at 8:23
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    $\begingroup$ A "principle" that seems to have developed at MO is that a question is not necessarily a good/appropriate question just because it elicits good responses and good answers. $\endgroup$ – Kevin H. Lin Aug 12 '10 at 8:24
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    $\begingroup$ The reasons for that MO principle don't generalize to math.SE, which has (or will eventually have) a far larger, broader, heterogenous and more diffuse audience. $\endgroup$ – T.. Aug 12 '10 at 12:02
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    $\begingroup$ Also, since when did fundamental questions about mathematical knowledge become off-limits in a mathematics discussion forum ? If the question had cited some of the many published statements by famous core mathematicians (e.g., geometers and topologists, and not logic or set theory experts) on the question of how we come to trust a long proof, would that have been enough to classify the question as Real Math? $\endgroup$ – T.. Aug 12 '10 at 12:08
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    $\begingroup$ Provides a hand-wavy argument and ends with "Am I right?": this is a good question? Also note the movement between somes and alls. I call it poorly thought out and lazy. $\endgroup$ – Charles Stewart Aug 12 '10 at 12:14
  • $\begingroup$ Charles, the idea was clearly enough expressed that all readers and respondents understood the point: if a result (or theory) has long and recondite development, how do we "know" it's correct? This is not a novel question but some people may find it interesting to discuss nonetheless. I learned from some of the responses. $\endgroup$ – T.. Aug 12 '10 at 12:56
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    $\begingroup$ @T..: This is not a mathematics discussion forum, it is a Q&A site. Although I think this question is fine, there is good reason to discourage open-ended discussion-heavy questions (which has overlap with, but is not the same as philosophy questions) in general. $\endgroup$ – Larry Wang Aug 12 '10 at 13:18
  • $\begingroup$ This is certainly "a mathematics discussion forum", and the activity people pursue here is to "discuss" mathematics questions, even if that discussion is, exclusively, focused Q&A (which is not the case in practice). If you don't care for this usage, read my comments above with "Q&A" substituted for any use of terms like discuss or discussion, and there would be no difference in meaning. I think everyone contributing to the meta thread is aware of the restricted focus of the SE forums. $\endgroup$ – T.. Aug 12 '10 at 19:47
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I think that the question was partly philosophical -- how certain can we be in the things we believe we know is one of the classic epistemological questions. It is also inherently mathematical, and certainly the fact that I am a working research mathematician first and some guy who took a couple of philosophy classes in college second informs my response. [I will say that the question is not phrased in a way which evinces much mathematical or philosophical sophistication: it comes off as somewhat naive. Perhaps this was part of what made it problematic to those who voted to close.]

I am a bit surprised that everyone seems to agree that philosophical questions are a bad thing on a Q&A site about math, and that the term is being equated with the sufficient conditions for closure "subjective and argumentative".

I well understand that most contemporary research mathematicians have little to no knowledge of or interest in philosophical questions (or if they do, they keep it to themselves in roughly the same way that many people try to keep their religious beliefs out of "secular" conversations). For a site like MO, which is focused on the specific needs of research mathematicians, it is reasonable -- or at least, not a big target for surprise or outrage -- that questions with philosophical content are not especially valued. Frankly, at the moment MO has little content concerning the connections between mathematics and any adjacent fields -- even when theoretical physicists appear to talk math, the (understandable) cultural differences between math as done by mathematicians and math as done by physicists often intervene to make the physicists' contributions less than well received. Similarly I have seen almost no questions on the border of math and engineering, math and biology, math and economics, etc. (although there has been some sentiment on the meta site that MO should be, somehow, much more encouraging of such questions; no one has campaigned for math and philosophy that I can recall).

But the point of this site, as I understand it, is that it is supposed to have a broader appeal, and explicitly to cater to connections between mathematics and other fields. Are people really saying that there is no room here for mathematics and philosophy? I would find that to be rather disappointing.

Added: Sorry, once again I completed my reading of the links in the question only after crafting an answer. In the meta.MO thread that Kevin links to, [Andrew] Critch takes a position which is pretty similar to mine here: i.e., philosophy is not necessarily equal to discussiony and can be compatible with mathematics. But that was in October 2009 which is a loooong time ago in the history of MO.

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    $\begingroup$ Dear Pete, I think we have different takes on the question: to me, it is a more practical matter, along the lines of: how can we be confident in FLT, or Poincare, if the proofs are so long and recondite. The answer, to my mind, is: the proofs have been studied by many, have been rewritten in various ways which show a certain robustness to the arguments, giving us confidence in them, ... . I think the question is interesting for amateurs, who may be genuinely unsure how professional mathematicians decide these issues, and also unsure about exactly how many people are included in ... $\endgroup$ – Matt E Aug 12 '10 at 13:32
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    $\begingroup$ ... "the small group of experts" who understand developments such as FLT or Poincare. (The recent flap about $P \neq NP$ is another interesting example.) I think it is also interesting for developing mathematicians, because the methods that the community of mathematicians use to develop confidence in proofs (and the pitfalls that can and do happen in cases when difficult arguments aren't subjected to these methods) are (in my experience) quite similar to the methods that individual mathematicians use to scrutinize their own arguments, and it's helpful to make these explicit. $\endgroup$ – Matt E Aug 12 '10 at 13:35
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    $\begingroup$ @Matt: Never Be Sure, we thought that mathematics is the most secure branch of knowledge, but it isn't, we can never be 100% sure that a proof is always valid - all these speak against your charitable interpretation. $\endgroup$ – Charles Stewart Aug 12 '10 at 13:41
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I asked a pretty similar question back in the private beta, Philosophical time-wastebaskets. I don't see that any new issues have arisen since that inconclusive discussion.

My opinion:

  1. The problem is not philosophical content, but philosophical content that is presented in a manner that is distracting. There's a fun Alasdair Macintyre quote about how philosophy has the power not only to illuminate, but also to dazzle: that's what's happening here, unsophisticated bunny-rabbits being drawn into the glare of futile argumentation, rather than having the clarity of purpose to hop towards the useful answer at the other side of the road;
  2. We should have something in The FAQ about how to avoid writing questions with such distractions in them, how distracting questions are a waste of everyone's time, and how "subjective and argumentive" is a good reason for closing such questions, to give the questioner time to think about what the problem is, what they want to know, and how they can ask a question with direction.
  3. With this nice FAQ entry, we can point to the advice in the comment thread of the distracting question, vote to close, and point out that a usefully directed question can be reopened.
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  • $\begingroup$ Dear Charles, I think that I have a different (less philosophical) interpretation of the question; see my comment on Pete Clark's answer. $\endgroup$ – Matt E Aug 12 '10 at 13:38
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...And about philosophical questions in general, I can only quote corresponding MO.meta thread:

I agree that philosophical questions can be extremely interesting, and I like having discussions about philosophical topics, but I think they are generally very poorly suited to Math Overflow.

The framework that Math Overflow runs on is designed to prevent discussions. The answers to a question will get shuffled around as they are voted on, so any discussion is forced into the comments. The comments are limited to 600 characters and basically don't allow any formatting. This was specifically done in order to keep comments to the point. This isn't because discussions are bad; it's to make the framework really good for focused questions. Math Overflow is extremely good at handling "questions that have answers" (see this other post of mine), and that actually makes it bad for other things, like discussions.

which, IMO, stays true after substituting math.SE for Math Overflow.


While there is not much harm in the specific (or in almost any specific) case, policy of allowing such question could be, I'm afraid, toxic for the site.

And what is definitely toxic is allowing discussion questions in non-CW-mode, because it amplifies problems (see T's comment there) of rep system.

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    $\begingroup$ A good general point. In this specific case, these problems have not yet arisen. It's not a very discussion-y question; the answers are all self-contained, and little discussion exists in the comments. $\endgroup$ – Larry Wang Aug 11 '10 at 18:26
  • $\begingroup$ Grigory, see also my comment under Matt E's posting in this thread: one can weight the reputation rankings (algorithm can be Pagerank, for example) so that votes from high-rep users count more, and report this or other measures as secondary reputation scores. Many people would prefer to see the user list sorted in this fashion and it is very easy (in terms of programming) to add such a thing to the site, in the same way that some Badges are summary statistics of the upvotes. $\endgroup$ – T.. Aug 12 '10 at 19:57
  • $\begingroup$ @T Your ideas about changing rep system are certainly interesting, but I doubt they will be implemented in the near future. It doesn't mean we shouldn't consider them, but it does mean they can't solve any problem now. Anyway, reputation is not the problem with discussion questions — this one is easily fixed by CW mode. $\endgroup$ – Grigory M Aug 12 '10 at 20:41
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    $\begingroup$ Concerning right now, I think it is bad practice, in general and not just for this one question that has been re-opened, to close threads or in any other way restrict discussion, based on speculation that problems might occur, before there is evidence that they will develop. This includes the dreaded "argumentative and subjective" status. $\endgroup$ – T.. Aug 12 '10 at 21:04
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...And about the specific case of the question "Is It True that We Can Never Be Sure of Validity of a Mathematical Proof?". It is the question that was discussed by many first-class mathematicians, and with all due respect to math.SE users, I really doubt that any of us will be able to say something new and interesting about it.

It's pretty much like the question "what is the meaning of life" (or "what OS is better: Windows or Linux", if you will): asking it can create long (and some would say "interesting") discussion, but if you really care (as opposed to "want to spend some time") — better go to a library and then think. And if you really want to discuss it — discuss it iRL with friends. But it's not good topic for a forum — let alone for stackexchange-type Q&A site.

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  • $\begingroup$ Dear Grigory, I disagree with the sentiment of this post (as you might guess!). If we want to talk about the knowledge of the first-class mathematician in the sky, there will be very little that is new and interesting on either math.SE (or MO for that matter; in my observation, there is very little original research on MO). But if we instead want to talk about the actual audience of math.SE, which I expect will include many students who plan to become research mathematicians, as well as many amateurs perhaps somewhat removed from professional mathematics, then I believe that there are ... $\endgroup$ – Matt E Aug 12 '10 at 23:18
  • $\begingroup$ ... some valuable things that we can say about how we go about ascertaining the correctness of mathematical arguments. Also, I don't see that this question need elicit particularly more discussion than any other. (In my experience both here and on MO, lots of question elicit some back and forth in comments, from the most technical to the softest; I don't think that this particular questions will induce discussions that are beyond the capacity of the software to handle.) $\endgroup$ – Matt E Aug 12 '10 at 23:19
  • $\begingroup$ Dear Matt, the point is not that first-class mathematicians know more about this question but that they've wrote on this subject many times — one can just go and read it. Which is not the case for mathematical questions that are [or can be] asked here (while any first-class mathematicians indeed knows most [or maybe all] of the answers, I doubt that all the answers are written down in some easy to find place). $\endgroup$ – Grigory M Aug 13 '10 at 6:46
  • $\begingroup$ Dear Grigory, That's a good point, although I think it becomes more true if you give the question its more philosophical interpretation (following Pete, say), rather than my more practical interpretation. If that is the case, though, one could answer the question with some references to these writings. (I always find myself on the most liberal side of these closure discussions, though, so I understand if you're not convinced!) $\endgroup$ – Matt E Aug 13 '10 at 11:19
  • $\begingroup$ Dear Matt, while I indeed remain unconvinced, I'm inclined to agree that leaving this question open won't do much harm. $\endgroup$ – Grigory M Aug 13 '10 at 12:35
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    $\begingroup$ (Though I'm still wary of creating such a precedent.) $\endgroup$ – Grigory M Aug 13 '10 at 12:43
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I said this in the thread about improving the FAQ

The previous type of question is different than philosophical musings about existence of mathematical entity or interpretations of infinity though. I think this sort of question is difficult to cope with because often they are very mislead and the person asking the question can have their mind made up already which results in a lot of frustration for everyone.. but on the other hand they can be very fruitful if asked well - especially questions about pragmatics and motivation like "why is such and such defined in this way".

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