Thursday, April 1, 2010

Mathematical models and philosophical progress.

One of the most attractive features of The Reasoner, the montly zine out of Kent, is the interview this month with Hannes Leitgeb. One passage caught my attention:

"Hannes Leitgeib: Overall, and ultimately, mathematical methods are necessary for philosophical progress, yes. But of course there can be points in a philosophical argumentation at which there is no payoff applying such methods. And while I do not think that there is any area of philosophy that is ‘beyond mathematical methods’, in some areas they do not pay off as yet because these areas are not quite developed enough. Or that’s at least the diagnosis of a mathematical philosopher!"

Let's grant for the sake of argument that mathematical methods have a distinguished record of progress. (In the interview Leitgeib does not offer a historical argument for the claim, but surely we can point to the history of analytic philosophy with some satisfaction.) Let's also grant that all areas of philosophy can benefit from mathematical methods.
But what could the (mathematical???) argument be that mathematical methods are *necessary* for philosophical progress? What to make of un-mathematical philosophy; is all the progress achieved without mathematical methods merely apparent?

And...let's accept that philosophical progress by mathematical methods ought to be understood in terms of "clarity" (as Leitgeib seems to suggest in the interview). Ought we to accept that it is cost-free?
Here are some possible costs within philosophy (I created the list while thinking of the role of Bayesianism as an aid to understanding scientific practice in the fields I am familiar with):
1. Focus on tractable machinery and toy-examples (at expense of complexity)
2. Training in technical skill at expense of good judgment
3. Inflated expectations from technique rather than learning how to ask right questions (or the making of distinctions)
4. Focus on producing 'results' rather than insight
5. Focus on the model and not the messy world


  1. Eric,
    Yes, I agree with you that mathematical methods have costs as well as benefits. We see this also in many sciences (biology, economics, physics) where some initial progress with mathematical models leads some people to go overboard and extend things too far. This is also my impression with similar strategies in philosophy -- they can definitely help you, but don't forget their limitations!

  2. As Kripke said, "There's no mathematical substitute for philosophy."

  3. Mathematical models in philosophy, just as everywhere else, are just that: models. As such, the main question (which is a philosophical, not a mathematical, question) is what is to count as criteria of adequacy for mathematical models with respect to philosophical questions. Given that it is not immediately obvious that in philosophy we could have something analogous to the empirical data that keep things 'real' in other fields such as physics, biology, economics etc., a methodological discussion of uses of mathematical models in philosophy will have to be rather specific. E.g., do 'intuitions' in philosophy play the role that empirical data play elsewhere? A thorough account of the methodology of formal methods in philosophy is still to be written, I think.

  4. Catarina,
    We agree that the methodology of formal methods in philosophy still needs to be written. (Weren't you going to do that?)
    The input into such models will be very context specific don't you think? For example, last week I heard a nice talk by Stephan Hartmann on his Bayesian modeling of science--in his model the 'data' would be provided by estimates of (or about) scientific experts. (In principle, to be collected by survey data, for example, or some representative agent; in practice, provided by the modeler.) Of course, in other context philosophical judgment/intuition may play a larger role (etc).

  5. Hi Eric,
    Working on it! :)
    Stephan has very interesting ideas on the topic. In conversation, he suggested that there are basically two kinds of attitudes when using formal (mathematical) methods in philosophy: the mathematician's attitude and the physicist's attitude. The latter is roughly what I have described above (and most certainly my own): formal models (in philosophy and elsewhere) only give you approximations, and as such can be 'trusted' only to a certain point, also needing constant revision.
    There is though an aspect of formal methods that do make them particularly useful (everywhere and in philosophy in particular), which is their potential for countering our naturally conservative doxastic and epistemic tendencies. That's also a significant part of my story about the epistemology of formal methods. Suggestions to the effect that they 'enhance clarity' etc are of course not entirely false, but to my mind they only scratch the surface of what's really going on.

  6. Hi Catarina,
    Yeah, I picked on Leitgeib because of highly rhetorical character of his claims. But I don't want to deny some 'clarity' can be gained by models, but I agree with you that
    this should only be the start of the story. I like how you emphasize the potential for countering doxastic/epistemic tendencies. But...once trained on a model these same tendencies may well become fairly conservative again. (So, I would focus a lot on institutional/social framework in which models are used.) For example, these days in economics graduate students learn a fairly narrow range of models and this can skew judgment. (In physics the danger(s) of settling on a single paradigm may be much smaller because there are, after all, some epistemic and evidential differences between background theory in physics and economics.)
    One thing I worry about in studying the social sciences (but I think the problem can generalize to formal philosophy) is that mathematical models encourage a search for numbers. Success at finding a reliable stream of data can encourage a false sense of solidity. In the recent use of models many supremely confident economists assumed they understood risk while there really was only uncertainty.

  7. Dear Eric, Chris, Greg, Catarina,

    many thanks for your interesting comments on my interview in the Reasoner.

    I agree with a lot of what you say, let me just add a couple of remarks:

    - Eric: What you grant after the quote from the interview is pretty much what I intended to say there. The remaining worries you had were

    (i) "But what could the (mathematical???) argument be that mathematical methods are *necessary* for philosophical progress? What to make of un-mathematical philosophy; is all the progress achieved without mathematical methods merely apparent?"


    (ii) "Ought we to accept that it [philosophical progress by mathematical methods] is cost-free?"

    Concerning (i), that is why I added the qualifications "Overall, and ultimately" to my claim that mathematical methods are necessary for philosophical progress. Roughly: For almost all areas/topics in philosophy, the application of mathematical methods will be necessary in the long run in order to make progress. I did not argue for this in the interview, but of course from what I did say it does not follow either that an argument for this thesis would have to be mathematical itself (as you suggest by your "mathematical???" above), nor does the existence of "un-mathematical philosophy" at this point of time contradict the thesis.

    About (ii): I did not claim that the application of mathematical methods in philosophy is "cost-free". I completely agree with you (and Chris) that it is not, but that should be obvious, right? One could not do theoretical physics nor research in artificial intelligence nor medical research without the help of matematical methods (whether it is functional analysis or logic or statistics that gets used), but in all of them the mathematical methods in question can be abused and their application can mislead people in some ways. But that's the fate of just any method or tool. In fact, you could enumerate the entries of your list of "possible costs" just as well for any of those areas. (I am not so happy, though, about your contrasting "results" and "insight" in point 4: I do not think that you meant it that way, but it makes it sound as if there were some sort of almost-mystical insight that one could get from philosophy for which the activity of establishing results by proof from some philosophical assumptions would be counterproductive.)

    I am sure my claims in the interview were "highly rhetorical"--it was an interview, and indeed a short one, after all.

    - As far as Greg's quote from Kripke is concerned, of course no one is aiming at finding a "mathematical substitute for philosophy": it is philosophy that we want to do, not mathematics. Nor is the aim of applying mathematical methods in physics, AI, medicine, etc. to give mathematical substitutes for physics, AI, medicine, etc. Again I am stating the obvious here, and of course Greg is using logical-mathematical methods in his work himself, just as Kripke does.

    - Catarina: Yes, the methodology of mathematical methods in philosophy of a tricky business. Stephan and I have plans to write a joint paper on this, and we should all discuss this matter in detail at some point. Clarity--good old Carnapian explication, that is--is one of the components, but certainly not the whole story, as you say.

    (And Eric: It's really not important at all, but it's "Leitgeb" rather than "Leitgeib".)

    Thanks again for the discussion, Hannes

  8. Hannes,
    Thanks for the clarifications. Well, as it turns out you will be chairing the session where I will be speaking next week in Tilburg, so whether you want it or not, you'll be exposed to some of my ideas in the very near future :). If there is time, it will be great to discuss these issues in more detail over there.

    I think your angle on the issue is very much informed by your knowledge of what goes on in economics, a field virtually unknown to me. Concerning formal methods in philosophy, more generally, I think the search for 'numbers' does not play such a significant role. In the case of philosophy, I think the real issue is, as I already said, what is to count as conditions of adequacy for a given model which purports to be a model of a given phenomenon. But perhaps there are indeed many 'real issues', and it's good that we split the work a bit of course :).

  9. Catarina,

    It will be a pleasure to chair your session :-) See you in Tilburg.

  10. Apologies to Hannes for misspelling his last name! (I got your last name right the first time, but it has been down hill from there.) Mea Maxima Culpa!

    I love Carnapian explication. But surely explication is not identical to the application of mathematical models in philosophy? In fact, much recent action in formal epistemology does not look like explication at all; it looks more like application of formal technique.

    I once asked Putnam why if he was so enamored by Carnap when he was young, he did not embark on program of explication. Putnam's disarming answer was that "it's very difficult to pull off."

    Nevertheless, I would still like to know what the claim that "For almost all areas/topics in philosophy, the application of mathematical methods will be necessary in the long run in order to make progress" amounts to (and also what an argument on its behalf could be). I have no doubt that *sometimes* mathematical methods can contribute to progress (whatever that is), but why think it is necessary? (And how much of it is necessary?) Contemporary analytic metaphysics has recovered (and reinterpreted) important Scholastic distinctions; many view those distinctions as progress. (Some, especially the modal ones, can be captured by formal models, but the models were not necessary for the rediscovery.)

    One reason to take claims like Hannes' seriously is that attitudes that may lie behind them carry over into research funding opportunities & decisions in competitive research grant environments. The grant winners rarely highlight the ("obvious") costs in their applications!

    This is not an idle concern: if "the application of mathematical methods" is "necessary in the long run in order to make progress" then it becomes tempting to re-direct scarce resources in that direction. (What bureaucrat can resist the promise of "progress"?) European grant agencies love that kind of talk.

  11. One further point, Hannes, I don't want to rule out that there is "some sort of almost-mystical insight that one could get from philosophy for which the activity of establishing results by proof from some philosophical assumptions would be counterproductive" after a point. I take it Plato thought that the mathematical approach would be useful in training for such an experience, but that at some point mathematical proof had to be left behind.

    If one does not like such mysticism, one can also think of (for example, situationist) views in ethics in which formal methods may be thought of as a hindrance at getting at the moral truth.

  12. Eric,

    Here are some quick remarks on your replies:


- Yes, explication is not identical to the application of mathematical models in philosophy: what I said above meant to relate explication to the search for clarity. But in fact many of the paradigm cases of explication do involve mathematical methods crucially: just think of Tarski's logical/set-theoretic explication of truth or the modern probabilistic explications of confirmation (the latter of which is also a paradigm case of formal epistemology, in contrast with what you say). But I totally agree that explication in a strict sense of the word is very difficult to pull off; especially, if one restricted the method of explication just to explicit definition, it would definitely become impossible in most cases.

    - I won't try to argue here for my thesis that for almost all areas/topics in philosophy the application of mathematical methods will be necessary in the long run in order to make progress; hopefully, Stephan's and my future article will be helpful in this respect, and other people are working on related stuff. Contemporary analytic metaphysics, which you mention, very often does rely on mathematical methods: modal and conditional logic and their possible worlds semantics; mereology; probabilistic theories of causation; theories of chance and propensity; criteria of identity and abstraction principles; and more. I do not agree with your claim that the modern formal models for modalities were not necessary for the philosophical progress on metaphysical modalities and their applications in philosophy--I think they were.

    - If European grant agencies love talk of progress and of mathematical methods in philosophy, then of course I am all in favour of it :-)

    - Finally, ethics: there may be views in ethics in which formal methods may be thought of as a "hindrance at getting at the moral truth" (if there is such). But there is also terrific work done in ethics and value theory in which mathematical methods are omnipresent--just have a look at Wlodek Rabinowicz's wonderful work!

  13. Hannes, thanks for the follow up. I look forward to that article you and Stephan are developing. It would be nice if you could explain why 80 years after the Vienna Circle manifesto philosophy should still be restricted to a single (mathematical model) approach. (Again, this is not to deny that mathematical models can be important and very useful in all domains of philosophy, but I do not wish to reduce ethics to decision theory.)

    I deny there is an analogue in confirmation theory to Tarski-style explication of truth. I certainly would reject the Bayesian approach (if that's what you have in mind) that status. (For a far better articulation of my concerns than I ever could provide, see John Norton's “Challenges to Bayesian Confirmation Theory,”)

    Moreover, while confirmation theory is very interesting (and certainly should continue to be investigated) it is not very revealing for understanding the practice(s) and evidential status of paradigmatic case of knowledge, viz. natural science(s).

    I agree with your colleague, James Ladyman, that claims about the significance of mereology are often overblown.

    Of the list you mention I am unfamiliar with mathematical models of criteria of identity and abstraction principles--what do you have in mind?

    I think you might be right that some of the formal models were necessary for progress in study of modality but they also led to many of its extravagancies.

  14. It seems to me that there is substantial agreement between the different parties in this debate. We all agree that the application of formal methods in philosophy can deliver (has delivered!) crucial insight and promote ‘philosophical progress’; moreover, we all agree that it comes at a cost and that formulating the 'right' formal models is far from trivial or straightforward. Still, I think the burden of proof at this point is on Hannes, as he seems to be defending an indispensability claim: the application of formal methods is *necessary* for progress in philosophy. Those who oppose this claim can comfortably accept many instances of actual philosophical progress promoted by the application of formal methods, but for them it is sufficient to point out a few cases where there seems to have been real philosophical progress without the application of formal methods to counter the claim.

    I am myself extremely sympathetic to the use of formal methods in philosophy (in fact, most of my research up to now has been either doing it or talking about it!), but the suggestion that what is crucial about formal methods is the clarity they provide does not seem to me to be particularly convincing if it is intended to support the ‘indispensability claim’. Briefly put, formal models are neither necessary nor sufficient for philosophical clarity. I do think that there is something really special and unique about formal methods in general (along the lines of what I have suggested above), but from the looks of it what Hannes has in mind is something different, and I’d love to hear more about it! (Either in Tilburg or from the forthcoming Leitgeb & Hartmann paper :-)).

  15. Eric,

    I agree with you that more than just mathematical methods will be important for philosophy in the future. My claim was that mathematical methods will be among the crucial ones, but others can be so as well.

    About confirmation and explication: Have a look at Patrick Maher's article at which is forthcoming in the Journal of Philosophical Logic. This is the abstract: "Inductive probability is the logical concept of probability in ordinary language. It is vague but it can be explicated by defining a clear and precise concept that can serve some of the same purposes. This paper presents a general method for doing such an explication and then a particular explication due to Carnap. Common criticisms of Carnap’s inductive logic are examined; it is shown that most of them are spurious and the others are not fundamental." Accordingly, in his article at the final sentence is: "The predicate `C` is a good explicatum for confirmation because it is similar to its explicandum and theoretically fruitful. This predicate was defined in terms of probability. In that sense, probability captures the logic of scientific confirmation." I am not saying any of this is beyond doubt, but then again good and successful explications need not be the last words on their subject matter anyway, as they may well get replaced by better explications in the future (just as Tarski's definition of truth was not the last word on truth either).

    I share James' and your worries about some application cases of mereology in the recent literature, but there are positive examples, too.

    Concerning mathematical models of criteria of identity and abstraction principles: With respect to the former, I had in mind, for instance, Tim Williamson's Identity and Discrimination, Oxford: Basil Blackwell, 1990, his related "Criteria of identity and the Axiom of Choice", The Journal of Philosophy 83/7 (1986): 380-394, and my colleague Leon Horsten's "Impredicative Identity Criteria", Philosophy and Phenomenological Research 80 (2010), 411-439. On abstraction, see for example Kit Fine's The Limits of Abstraction, Oxford: Oxford University Press, 2002, and Gabriel Uzquiano's and (my former colleague's) Oystein Linnebo's "Which Abstraction Principles Are Acceptable? Some Limitative Results", The British Journal for the Philosophy of Science 60/2 (2009), 239-252.


    As you suggest, I do make an indispensability claim, clarity is part of the game but not all of it, and I am perfectly happy to carry the burden of proof :-)

    This exchange was a lot of fun: thanks very much!

  16. Hannes, if you do make the indispensability claim, doesn't your position seems to presuppose that the exclusive aim (and progress) of philosophy is the discovery of either functional relations among magnitudes [or among (functional) relations] or the promotion of (Carnapian) conceptual engineering or explication. I don't want to reduce philosophy to such approaches, although I am no enemy of any of these projects. But I think philosophy can be more than that: at minimum it is the (second order, self-conscious) reflection on all kinds of activities important to human beings. It also has a non-trivial roles of speaking truth to power and general reflection on the human condition at any given time and age. I do not wish to farm out these latter activities to journalists and novelists alone. But if we wish to maintain these additional roles we need to cultivate humane learning and experiences along side formal technique. Philosophy remains special because we can, in principle, integrate reflection on the sciences and humanities.

  17. Hannes, I am busy reading Maher. I am a bit puzzled why this is called Carnapian explication. In Maher the explicandum is derived from "ordinary language" while in Carnapain explication the explicandum comes, I thought, from science (or some artificial language). (Maybe we should call this project 'vulgar explication'?)

  18. I've included this post in the latest Philosophers' Carnival. I hope that's ok!

  19. Eric,

    if I remember correctly, at the beginning of the Logical Foundations of Probability Carnap says that the explicandum may belong to ordinary language or to some scientific language. Of course, in the case of "confirmation" it does not matter much, since it is used both in everday life and in science. But it is certainly true that Carnap was interested primarily in scientific languages and in their metatheory.

    In the background of our discussion there might well be differences concerning the extent, role, and aims of philosophy in general. I am tempted to count some of the activities that you classify as philosophical as not belonging to philosophy as an academic discipline, as long as they remain on the level of reflection on which journalists or novelists work. For me, philosophers differ from the latter not just because they may integrate reflection on both the sciences and the humanities: Much more importantly, they are different in view of the methods by which they carry out their reflective activity. And mathematical methods constitute, or will constitute, an essential part of the philosopher's toolbox of methods.

  20. Hannes, your memory is better than mine! (Mine was probably biased because I think it is a mistake to try to explicate ordinary terms--in my view this opens Carnap up to the most serious Quine-ian criticisms.)

    On the differences of the aims of philosophy: I worry that your project(s) end up making philosophy either subordinate to the bureaucratic state or un-illuminating of science (or both). But these matters go beyond your comments in the Reasoner, so I think we have indeed reached the end of our exchange here. Thank you for engaging.


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