The question itself seems to pass the usual filters but for some reason this one quickly degenerated into a situation where nobody listens to one another. I would like to hear what community members think of this one.

Except for the downvotes, I am seeing very little negativity and in my view blatant disrespect seems absent from the commentary.
I do think people are talking at cross purposes, and the original poster should probably be enlightened with a semester in foundations, but I prefer to stop short of recommending that and instead giving the answer I gave. I think each voice is representing a reasonable (if not fully informed) take on the question (including mine), which to me will have nothing but very individual opinions and no "right" answer. Of course if ad hominem starts, call in the mods.

Gerhard "Teetering On The Philosophical Brink" Paseman, 2013.05.25

I will agree to using some provocation recently, and I should recognize that not everyone reads as I do. I respect Todd Trimble's opinion, and while I will apologize for offending, I will not apologize for the provocation. I think Todd deserves some clarification, but I can only say that my intent is to provoke nonoffensive thought; if Todd finds the following paragraphs offensive as well, I can only apologize in advance.

Summoning respect for the original poster, I think the question suffers from some misguidance. It suggests that all of mathematics can be reduced or expressed in a formal symbol-based language which is axiomatized in much the same way that Zermelo-Fraenkel set theory is done in first order logic. Even if the original poster comes out and says something like "I didn't mean first order! Seventh order will do for me.", I still find that limiting as what mathematicians do isn't always encodable into such a language. As researchers delve into category theory and other ways of developing, expressing, and using other forms of logic, one may find set theory a definite hindrance in such endeavours. It is good to have a place to start in one's studies, and I would be among the first to recommend set theory if one wanted to deal with set-based structures and other subjects where set theory has been effective, but I would not suggest the same starting point for everyone, and this is implicit in my response to the question.

In my commentary exchange with The User, I suspect the same sort of misguidance, but of a different order. Hilbert's program, and the response of several mathematicians, philosophers, and other scientists towards resolving that program, has led to a notion that mathematical endeavours can be relegated to a formalistic process. There is some comfort and satisfaction that have been gleaned from the notion that one can take ones ideas, and justifications, encode them into a symbolic language, and in principle feed them into a machine and have the machine spit out "Yes, you have a valid statement and proof." or "Error at line number 153: you broke this rule in your encoding." . However, I balk at the suggestion that all of mathematics, especially the mathematics to come, can be so reduced. It suggests to me a self-imposed limitation in how ideas and justifications can be represented.

Some of the most intriguing ideas I encountered in graduate school were alternative ways of expressing mathematics. One involved infinitary diagrams, and the other was some variant on bicategories (I probably have that detail wrong). Instead of using some string of symbols to encode ideas and justifications, an alternate structure was used for a statement. I don't know if a mechanism of proof in either way meant finding a substructure or a particular pattern in the given structure, or if a correspondence to part of a larger structure was intended as the justification. I can imagine where such activities can have meaning and be called mathematics.

An intriguing thought experiment: bring Leonhard Euler to the present day, and ask him to present one of his results in the context of foundations chosen by the original poster, along with a sufficient degree of formalism as well as one or two assistants to help with the encoding. He might well wonder why such was needed to do mathematics, when the statements he wrote are self evident, with all the ideas there. (He might also wonder why we gave up using a perfectly good language like Latin.) It would not surprise me if he rejected the formalism that the original poster and The User seem to promote as essential to mathematics.

I can imagine a situation where the present form of doing mathematics would be replaced by an activity of structure building, or of finding a pattern or configuration inside some larger structure, and exhibiting such as an alternate to a proof in some formal system in a symbolic language. I don't call for such a revolution today, but in such a new era of investigation, I can see where the phrase "Proofs: Crutches For The Unimaginative" can be apt: it refers to a method of doing mathematics that lacks some of the creativity and imagination that are available in this new era. The phrase is intended to suggest that mathematics is and will remain more than our attempts to define it concisely, found it compactly, or practice it symbolically.

(This is just part of one way to counter The User's notion that proofs are essential to mathematics. There are other ways to counter it as well. Further, my intention is not to remove the essential character of proof from The User's notion of mathematics; (s)he is welcome to keep it. I am suggesting that proof does not enjoy that position in other notions of mathematics.)

Gerhard "Off Convention, Not Off Balance" Paseman, 2013.05.25

I could have indeed suggested Archimedes, or some Babylonian scribe. How about Poincare?
I like the example using Euler, but not from the standpoint of rigor though. I wanted to work in the remark about Latin too. All of them did mathematics, and they did it differently.
Further, they did it without the benefit/stricture of the foundational system and standard of symbolic manipulation we currently enjoy/suffer.

I am not suggesting rigor be abandoned either, or minimized. I am suggesting that it will not always look as it does now, and that it is possible proofs will be replaced by something else.

Gerhard "You Can Be The Change" Paseman, 2013.05.25

Thanks for your elaboration, Gerhard. There is an awful lot one could say in response to what you have just written, but here at meta is probably not the right place to get into a protracted discussion. It does seem, echoing what you said yourself in an earlier comment, that there is some talking at cross-purposes. Possibly it would do good to point readers of your MO answer and related MO comments to what you have written here.

There are some intriguing points of your comment that -- if I understand them correctly in their intended spirit! -- I can find agreement with. You probably anticipated that the brief passing mention of bicategories would catch my eye, and you'd be correct in that. Specifically, there is the wonderful and powerful method of string diagrams (whose precise expression was given by Joyal and Street) that provides rapid-fire methods for calculating in bicategories, that those with experience would accept as fully convincing, encapsulating in a flash what would require many lines of calculation in a more traditional style of proof (including proofs by appeal to commutative diagrams). If this is the type of example you had in mind when you wrote

finding a pattern or configuration inside some larger structure, and exhibiting such as an alternate to a proof in some formal system in a symbolic language

then I can find agreement in spirit, but -- I'd still call such calculations "proofs". Such proofs are also recognized by the cognoscenti as fully formalizable in the fuddy-duddy first-order theory of bicategories. And I think this last sentence is an important point, and is one illustration of the essential deep purpose of reductionism in mathematics. (Just last night I was reading Saunders Mac Lane's eloquent reply to Freeman Dyson in the New York Review of Books, <a href="http://www.nybooks.com/articles/archives/1995/oct/05/a-matter-of-temperament/?pagination=false">here</a>, on the purposes of such reductionism in mathematics. I might recommend this to your attention as well.)

To circumvent further misunderstandings, it might help to specify clearly and carefully what you mean when you say "proof". I don't mean that you should do this here! But if you have further discussion with The User, this could have the welcome effect of making your provocations seem less off balance. (I also suspect The User is more sophisticated, or at least less prone to 'misguidance', than you seem to be giving that user credit for. A more careful and mathematically nuanced discussion on your part, if you have the time and interest, might elicit the same from him/her.)

Finally, to indulge in some provocation of my own: I find something deeply ironic in Gerhard "Proofs: Crutches For The Unimaginative" Paseman. Might it be you who is considering proofs in a limited and unimaginative way? But, I might immediately withdraw from that by remembering that I normally have a very high opinion of you as well, and also by seeking solace in your

I can imagine a situation where the present form of doing mathematics would be replaced by an activity of structure building

since this is actually how modern conceptions of proof appear: as constructions! Here I would draw your attention to the fantastic comment by Andrej Bauer (http://mathoverflow.net/questions/127889/is-rigour-just-a-ritual-that-most-mathematicians-wish-to-get-rid-of-if-they-could/130125#130125), and ask you, if you haven't done so already, to read it five or ten times and take the message deeply into your heart. And just for kicks, try substituting "Numbers: Crutches For The Unimaginative" and see how that would sound. :-)

Peace.

Thomas, you sound overly crankish in that comment of yours. In particular the reference to the Skolem paradox.

It's very easy to confuse theory and meta-theory, and it's probably one of the larger pitfalls that people can step into when wading through the bogs of intricacies in set theory and logic. Being conclusive about things (even in such an implicit way) does not paint anyone positively in the minds of others. Especially when you're calling out for a mature audience, many of which that have mathematics (and in particular foundations of mathematics) close to their hearts.

Max,

Some year and a half ago I ran into an economics Ph.D. student who did his undergrad in math & economics. When he heard I was a set theory student he began asking me "foundational questions", but he wasn't at all interested in an answer. He was interested in showing his superiority over a set theorist. The discussion quickly deteriorated into me trying to explain something and him replying "So what's <something>?" without paying any attention or showing the slightest interest in my actual answer.

People who want to ask provocative questions must listen to others, otherwise they are no better than the cranks we keep banishing from the site. If you want to complain about how you don't understand a particular topic, that's fine, but unless you are going to be receptive to the likely possibility that you are wrong, and listen to what others have to say, there is little to no point in posting on this site, or on any other community which respects itself.

Max,

My point is that if you want to ask provocative questions, of any sort, then you need to be:

1. Extra careful in their formulation, as not to offend those that you want to answer. It's easy to misunderstand Godel's incompleteness theorems, but there is a huge difference between asking "What am I missing here?" and "Is Godel wrong because ...?". Where the former shows some humility and respect to the generations of mathematicians that examined the proofs from top and bottom, the latter implies that the person asking the question is better than those people because they are all wrong.

2. Very open to the fact that it is far more likely scenario that you are wrong/misunderstanding the concept, than to that your "thought provoking question" is going to wake everyone up from the Cantorian/Godelian/otherwise nightmare that has been haunting mathematics for the past who knows how many years.

Thomas' comment on the meta makes him sound as if all those hundreds (if not thousands) of extremely capable people that studied set theory in the past 91 years are wrong, and blind and it's quite insulting. I was ambivalent regarding closing/reopening the question in the topic, but I can assure you that with the current attitude I find it quite worthless and will vote to re-close it if it were to reopen (unless I'll see an improvement from Thomas' side).

Why your answer downvoted? Probably because it's not really an answer but more of "Ah! I agree!!! Look at the question I asked which proves that incompleteness is a disease in the foundations of mathematics!" sort of a comment. I didn't downvote, but I can understand those who did.

Because MathOverflow is not your local department's afternoon tea. It's not meant for philosophical and elaborate discussions about what is better or worse. The goal of MathOverflow is to provide a platform for mathematicians to ask (relatively) concrete questions with (relatively) concrete answers about their mathematical works. It is true that sometimes big list, or soft questions appear, but those are usually made in good taste and are closed otherwise.

Asking a question to invite an open-ended discussion which is unlikely to convince any of the parties taking sides in it is by definition a non-constructive question which invites closing (and possibly deletion) votes.

As for my subjective point of view, yes. It is my subjective point of view that Thomas made a condescending comments here, but it is the responsibility of whoever poses the provocative question to avoid hurting the feelings of other people, not the other way around. Thomas failed in that task, which signals me (and probably other people too) that it is not a well-phrased question.

I find your comments in this thread to be almost unreadable, Max1. If you don't want to be treated like a troll, please act in a way that indicates you want to be treated seriously.

Is Max1 writing some kind of hermetic poetry? I also suggest that this thread be closed.

>Because MathOverflow is not your local department's afternoon tea. It's not meant for philosophical and elaborate discussions about what is better or worse. The goal of MathOverflow is to provide a platform for mathematicians to ask (relatively) concrete questions with (relatively) concrete answers about their mathematical works.

Well, you know, at least for me it is "an afternoon tea". For better or for worse, I tend to comment on whatever interests me, be it a mathematical, or a philosophical thing both here and there. You are free to prune the unwelcome responses afterwards, but if I start feeling like I need to exercise an internal censorship beyond normal levels (the levels of the afternoon department tea, that is), I'll just go somewhere else. :)

I know :). So I usually abstain from asking such things myself, but that's where I draw the line. Anyway, I don't think we disagree too much, just our ways to express things in words are quite different...