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On the closing of: Does mathematics require axioms?

This question was first closed, and after changing the question, it was voted to be reopened again. I don't understand why it is now closed again. I opened this meta thread more because of principle reasons then that I desperate need for it to be reopened again. I just put quite some effort in this question, and also in rewriting the question after it was closed first time. I'm now quite frustrated that it just get closed without any further explanation.

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    $\begingroup$ I think this insta-closing without giving an explanation is really a bad habit. In particular, if (like in this case) it is evident that the OP put a lot of effort into the question. $\endgroup$ – azimut Apr 20 '13 at 10:04
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    $\begingroup$ @azimut: in this particular case, I think the closing reason explains it: questions that depend on opinion are not a very good fit for this site, because they tend to lead to drawn-out arguments and discussions rather than objective answers. Even the first "do infinite sets exist" question was borderline, in my opinion, for this site. Perhaps people voted to close when they saw a second, parallel question so soon. However, there are many questions about the paper that would be suitable, such as "What is the formal definition of an 'infinite set'?" and "Are the computable reals complete?". $\endgroup$ – Carl Mummert Apr 20 '13 at 13:07
  • $\begingroup$ @CarlMummert: Even if the closing reason does explains it, it would still be helpful to the OP if one of those voting to close gave a bit more detail in a comment. For example, if they had taken the time to write precisely what you just wrote. That would have been helpful, polite, and would have saved Kasper from putting substantial effort into an edit which ended with the question being closed again... $\endgroup$ – user1729 Apr 21 '13 at 15:15
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    $\begingroup$ Isn't the word "mathematics" plural so that the question should be "Do mathematics require axioms"? If so (not my native language, so I'm not 100% sure), that's a serious reason to close! $\endgroup$ – Julien Apr 21 '13 at 17:08
  • $\begingroup$ @azimut Zev gave a reason and that was exactly the reason I voted to close. $\endgroup$ – Michael Greinecker Apr 21 '13 at 17:48
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    $\begingroup$ @julien: Mathematics is definitely a singular noun. Also physics, economics, aerobatics, gymnastics... $\endgroup$ – TonyK Apr 22 '13 at 12:18
  • $\begingroup$ @TonyK I see, thanks... I should have googled first. According to this, it is plural but extensively used as if it were singular. $\endgroup$ – Julien Apr 22 '13 at 12:24
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    $\begingroup$ @julien: No, it really isn't plural! It might have been plural in the past, but not any more. As for that link, I doubt that "billiards" was ever a plural noun. $\endgroup$ – TonyK Apr 22 '13 at 12:45
  • $\begingroup$ That's a load of balls. (As in billiards :-).) $\endgroup$ – copper.hat Apr 29 '13 at 23:21
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(This answer is speculative. While I agree to some extent with what I write, I have not considered it enough to really believe or disbelieve it)

As the closing reason says,

this question will likely solicit debate, arguments, polling, or extended discussion.

The question isn't so much about a mathematical problem, but instead asking people to write essays on a potentially controversial topic.

While this is arguably already a poor fit for this site, you have the added problem of asking people to argue specifically with a well-known crankish essay, a task that is generally considered fruitless, even when it is brought up by someone other than the original author.

Furthermore, giving serious responses to it is arguably counter-productive to begin with, by implying the original essay actually merits a direct response -- and consequently, an open question asking for such responses would be harmful.

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  • $\begingroup$ I was writing essentially the same answer, but this was posted first. $\endgroup$ – Carl Mummert Apr 20 '13 at 0:46
  • $\begingroup$ I think this summarizes pretty well my vote to close. $\endgroup$ – Pedro Tamaroff Apr 20 '13 at 2:55
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    $\begingroup$ If it is considered harmful to directly answer some of Wildberger's arguments, and if the words "well known crankish essay" can be tossed about without a second thought, as if everyone knows he can be dismissed as a heretic, then that is a nearly complete proof of the strongest claim made in the essay: that foundations of mathematics functions more as a religious or political system than as a science. I found Wildberger's points to be mostly correct and reasonable, and expressed candidly but not crankishly. (Voted to reopen) $\endgroup$ – zyx Apr 20 '13 at 6:25
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    $\begingroup$ @zyx the crankyness and dishonesty of Wildberger was alread discussed on MSE here. $\endgroup$ – Michael Greinecker Apr 20 '13 at 8:10
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    $\begingroup$ @MichaelGreinecker: The formulation of your comment seems to indicate that you consider "the crankyness and dishonesty of Wildberger" an established fact. Really? In that thread I see a number of opinions. Most of them quite obviously based on not actually reading and thinking about the essay. $\endgroup$ – Martin Apr 20 '13 at 11:24
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    $\begingroup$ @zyx: Who's being religious or political?!?! Crackpottery has nothing to do with whether the speaker's views agree with the listener; crackpottery is about the attitude of the one speaking, and one of the most common characteristics is the unwillingness to actually understand what they attack so zealously; his treatment of the axiom of infinity is a particularly egregious example. (possibly he understands, but deliberately sets up a strawman; arguably that's worse than being a crackpot!) The essay starts reasonable-sounding, but it eventually winds up painting a rather insulting caricature. $\endgroup$ – user14972 Apr 20 '13 at 12:23
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    $\begingroup$ When people have already made judgement regarding something, then without new material to suggest questioning that judgement, there is no reason to do so. Your implication that one should forget everything one ever knew about the essay every time a new person brings it up so that you can analyze it from scratch rather than apply one's past judgement regarding the article is rather silly. $\endgroup$ – user14972 Apr 20 '13 at 12:27
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    $\begingroup$ I confess I find it hard to reconcile your judgement that it is "reasonable and expressed candidly" with the actual content of the article. At least, I find it difficult to do so without inferring that because you agree with the conclusions, you are only really paying attention to the parts you agree with and overlooking all of the problems with the essay, a habit that is distressingly common. :( The aforementioned treatment of the axiom of infinity is, IIRC, the most clear demonstration that Wildberger is not interested in treating ZFC reasonably. $\endgroup$ – user14972 Apr 20 '13 at 12:30
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    $\begingroup$ @Martin: I have read the essay, and I am extremely familiar with the actual, well-thought-out positions of constructive mathematicians whom I respect. I see two options about the essay: Wildberger has little idea what he is talking about, or he is making an elaborate joke by pretending to know little about what he is talking about. The clearest sign it is a crankish essay is that it deals with complex issues many other people have written about, but fails to mention any previous work. $\endgroup$ – Carl Mummert Apr 20 '13 at 12:37
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    $\begingroup$ @Carl Mummert I did see that you read it and yours is the only answer I voted on. I am not one to defend or subscribe to Wildberger's opinions, as you can judge by the topics I write about on this forum. However, I object to taking the thread as sufficient reason to denigrate Wildberger publicly as a "crank" or "dishonest". // All in all, this only supports the notion that math.SE is not the proper venue for discussing the merits or demerits of this or any other confrontational article on foundations. $\endgroup$ – Martin Apr 20 '13 at 12:51
  • $\begingroup$ @Martin: I agree. $\endgroup$ – Carl Mummert Apr 20 '13 at 13:00
  • $\begingroup$ @Hurkyl I do understand that my question may not fit a Q/A model. But as [this][1] topic isn't closed, I concluded that such questions do fit the Q/A model of MSE. I think that, or both should be open, or both should be closed. [1]: math.stackexchange.com/questions/356264/… $\endgroup$ – Kasper Apr 20 '13 at 13:04
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    $\begingroup$ @Kasper: as I wrote above, I thought that the "do infinite sets exist" question also had a good chance of being closed. Perhaps people gave the benefit of a doubt for one question, but not for a second question about that paper so soon after the first. It is sometimes hard to tell whether a question is objective enough for the taste of the overall community here. However, if you ask questions about the mathematics, rather than about the philosophy, I think those questions are likely to be answered and unlikely to be closed. $\endgroup$ – Carl Mummert Apr 20 '13 at 13:11
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    $\begingroup$ It would be nice to see an example, and not only the claim that one exists, of Wildberger being dishonest or "unwilling to actually understand what he attacks so zealously". The points about the axiom of infinity include some venerable objections to completed infinities, the often-made observation that set theoretically important distinctions do not appear in practice (see the paragraph discussing nonmeasurable sets), and other things less frequently said that do not of themselves signal crackedness. Not citing earlier work (@Carl) is a sign of an opinion piece, not crankery. $\endgroup$ – zyx Apr 21 '13 at 1:09
  • $\begingroup$ @zyx: you could see a separate answer I wrote at math.stackexchange.com/a/356640/630 for a few points about the essay. However, I do not argue he is being dishonest, only that if we take the essay charitably we have to view it as written with a voice that is intentionally naive. $\endgroup$ – Carl Mummert Apr 21 '13 at 2:26
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I am sort of torn on this. By the letter of the [close reason] I guess the question should be closed, especially given the situation in the comments below Asaf's answer. But we are also not bound to always follow the rules precisely and exactly at every turn.

I do believe that Kasper asked the question in good faith, and was not looking to simply have a soapbox from which to promote his views (and even admits to being agnostic about the issues in the question itself). I feel that if this question, which at its heart appears more philosophical than most questions currently tagged , is closed, then it's almost saying that any philosophical question which does not strictly adhere to the majority opinion of mathematical philosophy would suffer the same fate.

But I also feel that the origin of the question greatly influenced its closure. I doubt, for example, that a question about the merits of the intuitionistic standpoint of mathematics, using Heyting's Intuitionism: An Introduction as a reference, would have received the same treatment.

Perhaps a quite severe edit to the question (or an entirely new question!) can be made along the following lines to make it more acceptable:

  1. Crop out virtually all of the quote from Wildberger's article. The first paragraph, with perhaps scattered quotes, would be sufficient to explain his standpoint. (Of course, retain a link to the article in question, and mention that some answers might be direct responses to the article in question.)
  2. Ask the question in the title: "Does mathematics require axioms?" But be somewhat more exact. Wildberger seems to see, for example, the axioms of group theory as definitional in nature, and doesn't appear to have problems with these. He seems to be against what one may term foundational axioms for the whole of mathematics (and certainly against the use of the axioms of ZFC in this foundational role).
  3. Offer Wildberger as an example of someone that does not see the need for these sorts of axioms, and mention that his is (appear to be) a minority opinion in the mathematical community. But keep the question focused on the role of axioms within mathematics, and not a response to a particular article.

These changes certainly won't please everyone, but I think they will have the effect of neutralising the question, making it less likely to solicit heated debate.

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    $\begingroup$ The comments to my answer are certainly confusing. Apparently writing to someone that their answer is "shitty" and "crappy" constitutes of a constructive criticism! I always knew I dislike constructive stuff, but now more than ever :-) $\endgroup$ – Asaf Karagila Apr 20 '13 at 14:11
  • $\begingroup$ +1 I would vote to open under such changes. One should certainly add the philosophy-tag though. $\endgroup$ – Michael Greinecker Apr 20 '13 at 14:54
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On more a constructive tack one might ask about the alternatives to axiomatic approaches to mathematical reasoning. For examle, @DougSpoonwood commented on an Answer of mine that natural deduction is an example of doing logic (propositional or predicate) without logical axioms.

In the case of natural deduction one makes a tradeoff between having only one rule of inference (modus ponens) and standard logical axioms versus no (purely logical) axioms and multiple rules of inference. This doesn't immediately translate into formulation of mathematical theories without axioms (at least not in any systematic way).

Would such an answer or even framing of the question be interesting? I don't know. But it would certainly be a step back from our two weapons, hand-waving, vagueness, and pontification. No, that's three. I'll begin again...

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