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Is this question acceptable?: Logic in Mathematics and Philosophy

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    This thread is about question 62401 which currently has three votes to close.

    • CommentAuthormarkvs
    • CommentTimeApr 21st 2011
     
    It should be a Community Wiki. I voted to close because at least half of the question (about philosophy) has no mathematical substance.
    • CommentAuthorgilkalai
    • CommentTimeApr 21st 2011 edited
     
    I dont think this question should be a comminity wiki according to the CW rules. It is a specific question aimed to experts in mathematical logic about the relations between mathematical logic and logic as an area of philosophy. It can have a small number of good answers and it is not meant to create a sorted list of answers or to be a platform for people to express their opinion.

    Logic is an ancient area of philosophy which while extensively beein studied in Universities for centuries. Not much happened (unlike other areas of philosophy) from ancient times until the end of the 19th century. The work of Frege can be regarded both as the starting point for mathematical logic and as a turning point in logic as part of philosophy. In the first half of the 20th century there were close connections between the development of logic as an area of philosophy and the development of mathematical logic. In addition to its interest for mathematicians and philosophers logic became a central applied field in computer science. The question is about relations between logic as part of philosophy and mathematical logic mainly from the second half of the 20th century when its seems that connections between these two areas have weakened.

    I liked some of the answers and esmecially Timothy's. But I will still be happy to see a definite answer.
  4.  
    I think the question is fine as it is: the one criticism I see is that MO may simply not have the right experts that can answer the question meaningfully, but we won't know for sure until we ask, will we? I've known a few logicians who appeared to have a strong interest in current philosophy, so we may very well get some good answers.
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    For my part, I already expressed the extent of my concern: too broad. This is not the kind of question that is going to get a definitive answer; rather, the most useful answer I can think of is "Read this person's work, then that person's work, then..."

    I found it interesting that Professor Kalai responded to my claim that the question was sort of like asking about the connections between analysis and geometry by saying it was more like asking about the connections between mathematics and theoretical economics. In my opinion the latter comparison, while probably not as broad the one I suggested, is easily still too broad to make for a good question on a Q&A site.

    I take from this that we just have different tastes on the breadth issue. This is fine with me. Really fine -- I have not voted to close.

    • CommentAuthoran_mo_user
    • CommentTimeApr 21st 2011
     
    Regarding the issue of CW: leaving more fundamental questions aside, even only for practical reasons, I think it is better if the question is not CW. The only problem with this question I could see is that it could be spoiled by a flood of half-witty not overly informed/informative 'answers' (I believe to remember that initially, now deleted?, there was one such answer, basically just saying 'I believe no relation' or something of that quality). To keep this type of answers away non-CW has some value. At the moment, I find all answers interesting and (as far as I can tell, not being knowledegeable on this myself in any way) they were all written thoughtfully by people well-informed on the subject at hand.

    As long as the standard of answers stays like this I think the question is a good one. If for some unfortunate reason many 'joke-answers' should be given in the future then (but only then) a closure would in my opinion (unfortunately) be desirable.
    • CommentAuthorgilkalai
    • CommentTimeApr 21st 2011 edited
     
    Well, I tend to agree with Pete that even the question about the relation between mathematics and theoretical economics is too broad. But I do not think the question at hand is even as broad. I think the connections between philosophical logic and mathematical logic are much less broad. (It is an interesting question why.) This is something that cannot be learned from the "syntax of the question" but it seems to me true based on the answers so far. In any case, probably experts in mathematical logic are best to judge how board the question is. (On the other hand, I dont think the question is arbitrary as "what is the relations between large cardinals and PDE".)

    I think Mark's concern about mathematical substance is not without merit. (As a general rule I think we should allow questions of genuine interest to mathematicians and of academic substance.) If I understand what Mark is saying the claim is that the area of "philosophical logic" has no mathematical substance and half the question is about this area. The point it that for several decades
    around the turn of the 20th century philosophical logic and mathematical logic were closely related and were done to some extent by the same people (and had a lot of mathematical substance). Even if later much of the logic done by philosophers was of no mathematical substance I asked about those things in philosophical logic (if any) that did have mathematical substance. Also here experts in mathematical logic are best to judge.

    In any case, both the issue of breadth and the issue of mathematical substance depends more on what the answer to my question is rather than on the question itself.
    • CommentAuthormarkvs
    • CommentTimeApr 21st 2011
     
    @Gil: The problem with asking a non-math question (and half of your question is non-math) is that either you get a professional answer which you do not understand or you get a non-professional answer (starting with words "I do not know the material but will say something anyway") which you do not need. The economy question is a good example of such problem. If a good economist gives serious answer, you won't understand even the terminology. I am sure that philosophy is like that too. That is why I am against such questions. As for philosophy and logic, try reading Hegel's book which I think was called "Logic" or something like that. Without a special training the book is unreadable.
    • CommentAuthorgilkalai
    • CommentTimeApr 21st 2011
     
    Hi Mark, we can expect answers from a professional in mathematical-logic, we have quite a quite few experts in mathematical logic on MO. (It would be interesting to have an answer from a logician philosophers but I do not expect to find them at MO.) In a similar question about relations between mathematics and theoretical economics (or a more narrow similar question) I would expect here on MO an answer from a mathematician rather than from an economists.

    > If a good economist gives serious answer, you won't understand even the terminology.

    I dont think you are right, my experience is different. Did you try?

    > half of your question is non-math

    I dont understand this half-thing.
    The question is about the relation between a mathematical area A and a non mathematical area B. So to much extent you can think about it as A intersection B which is math.

    Probably you are right about Hagel. The question was not about Hagel's philosophy but about formal logic which was developed by logicians in philosophy in recent times (after both philosophers and mathematicians were involved in the crucial early stages of developing mathematical logic,) and its relation with mathematical logic.
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    In the interest of applying consistent standards to all users, I agree with Pete. From a typical user the question would quickly have been closed as too broad. I agree that the question is interesting, and it has also generated some good answers, but past meta discussions will affirm that neither of these is sufficient to keep a question open.

    • CommentAuthorgilkalai
    • CommentTimeApr 21st 2011
     
    Hi Qiaochu, the problems with Pete's claim about breath is first that it is a call an expert in mathematical logic should make (and Pete is not an expert in this field), and second that based on the information we have so far Pete is simply wrong and there are only handful of cases where modern mathematical logic interacts with modern philosophical logic. Perhaps it is more of an issue that Pete (and you) are not typical users than about me not being one. I agree with you that those small percent of MO users who also participate in meta discussions were more leaning in the past towards tough policies regarding closing of problems but there were also substantial number of other views (including mine) and the actual policy that was implemented was mixed. In anycase, the whole thing is somewhat unfriendly.
    • CommentAuthorEmerton
    • CommentTimeApr 22nd 2011
     

    I find the question interesting, and would like to see it stay open. I agree with Gil that there are mixed views on what should or shouldn't be closed (mine are reasonably close to his, I think), and in any case, this is a question with a mixture of technical and historical content, which is of interest to (at least some) research mathematicians.

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    Gil Kalai writes:

    Hi Qiaochu, the problems with Pete's claim about breath is first that it is a call an expert in mathematical logic should make (and Pete is not an expert in this field)

    Whoa, what's happening here? No need to make claims about who is or isn't an expert in whatever field. I mean, it turns out that Gil is not wrong -- I have only one published paper pertaining to mathematical logic and have taught only one course on it (more specifically model theory, on both counts) -- but I'm pretty sure that he can't see inside my head, and it's not necessary to try. We're both mathematicians: let's assume the best about each other.

    I disagree that you need to be an expert in field X or field Y to assert that comparing field X to Y is too broad a question for this site. The corollary to this is that assertions like this are more a matter of opinion than expertise. I think I have been open from the beginning that I have been stating my own opinion and taste.

    and second that based on the information we have so far Pete is simply wrong and there are only handful of cases where modern mathematical logic interacts with modern philosophical logic.

    Um, I actually didn't speculate on the cardinality of the intersection. (Also, the restriction to modernity -- specifically the last half of the 20th century onward -- is much more prominent in the latest version of the question than the original. That does help in narrowing the scope, actually.) My point was rather that to give a fair answer to such a question -- and equally, to evaluate anyone else's answer -- you have to pore over a vast quantity of information. As a case in point, one early answer was, almost literally, the statement "There is no connection, I think" and this was rightly deleted as being completely useless. Taking a lack of information as an answer to a question like this is in fact one of my concerns.

    Perhaps it is more of an issue that Pete (and you) are not typical users than about me not being one.

    I have no idea what this sentence means and I am pretty sure I don't want to know. We are all fine people and fine MO users, right?

    In any case, the whole thing is somewhat unfriendly.

    It has been my intention all along to be completely friendly. We can have friendly discussions about the scope of our site, I hope. If there is any way I can be friendlier, please let me know.

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    Dear Mr. Gil Kalai!

    Unfortunately, Your question was closed, so I have no other options besides to put my answer as a comment to meta discussione on the question.

    I cannot of course completely answer Your question. But I can give several concrete examples (very briefly) of some kinds of relation.

    First of all, the distinction between logic in philosophy and logic in mathematics (as fields) seems very conventional to me. For what we may think today as being in the domain of philosophical logic tomorrow may be part of mathematical logic. (As in fact was with some results.) Of course one simple distinction is that in mathematical logic we use formal or mathematical methods. But I think not so many people today are trying to solve problems of philosophical logic without formalizations.

    A new interesting field in philosophy is growing in this century often called Formal Philosophy (this term I believe has its origin from the title of Richard Montague's collected papers book). People of this field (among them R. Montague, H. Putnam, E. Zalta, D. Bonnay...) are trying to solve philosophical problems using formal logic. Particularly, and this is important, many of them try to formalize such kinds of reasoning as modality, induction, analogy, simplicity, naturality, generalization and so on as well as concrete philosophical theories (which are of course in the domain of philosophical logic) using formalisms and methods of mathematical logic.

    1. Works by Montague show that natural language is not SO MUCH different from formal languages, its syntax and semantics has strong structure. You may say that they are semiformal. That is why we may apply all the techniques and results from mathematical logic, in which specificaly mathematical languages are formalized and deeply studied. As a result we have a field of formal theory of natural language and its aspects (grammar, semantics, development and so on). You may see the broad range of topics presented on this conference: http://lacl.gforge.inria.fr/lacl-2011/appel.html. This opens the way for doing philosophy of language by formalizing the problems and answer them by mathematical proof.

    2. The work of Hilary Putnam presents attempts to tackle the problems of philosophy of mind and philosophy of language by comparing with formal models. To quote from his "Models and reality" paper:
    "In this paper I want to take up Skolem's arguments, not with the aim of refuting them, but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the "Lowenheim-Skolem paradox" is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy - the only resolution that I myself can see as making sense - has profound implications for the greate metaphysical dispute about realism which has always been the central dispute in the philosophy of language." See his collected papers.

    3. Edward Zalta in his Principia Metaphysica has formalization of general notions of abstract and concrete objects. In mathematics (today) the most basic objects are sets. In Leibniz' Monadology (which is pure philosophical theory) they are monads. Zalta formalizes the monads and does Monadology formally. As well as Plato's theory of forms, theory of meinongian objects, theory of situations, the theory of worlds, theory of times. Moreover, Zalta claims that his formalization enables to obtain new useful abstract notions (objects) automaticaly by mechanized theorem proving. See his home page.

    I am particularly concerned with so many comments of MOers which indicate complete incompetence in the question. (Not to mention the savage one by "Harry Gindi"). I am amazed by the fact that even in MO such irresponsibility is something normal. Hilbert, Godel, Tarski, Cohen would never have allowed themselves to such attitudes.

    We should have some serious (as MO, in respect of math) forum for formal philosophy, modern logic and so on. As I see it now this site is not the place for that.
    • CommentAuthorgilkalai
    • CommentTimeApr 22nd 2011
     
    Dear Sergei, thanks for the detailed and interesting answer.
    Dear Pete, you and Qiaochu are not typical in the (good) sense of being long-time highly involved (and highly successful) contributors. (Such an involvment enables, at times, speaking confidently on behalf of the site and having much influence, and this is, overall, very fine.) Also you are quite friendly....
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    @Sergei: Why was my comment savage? I was referencing something that happened a year and a half ago. =D!

    It's well known among the longtime readers of MO that I have an ongoing one-sided (he doesn't even know who I am) feud with Badiou.

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