• Hoo
    I've wrestled with the meaning of mathematics for about 5 years now. I'm just starting a PhD program in math, but it was philosophy of math that called my attention to math proper in the first place. I've always loved Wittgenstein, so I've naturally wrestled with his philosophy of math. Recently I bumped into this piece by Ray Monk, which really moved me. I find it plausible as an interpretation. I'm not endorsing or opposing this view. I just find it worth talking about: http://sammelpunkt.philo.at:8080/435/1/08-1-95.TXT

    Here's a sample:
    Moreover, the attitudes he is propagating are the exact reversal of
    the ones which predominate in the debate to which his remarks are
    addressed: the debate about the logical foundations of mathematics.
    Anyone capable of understanding the issues on which his remarks are
    intended to shed light must, it seems, be predisposed to take the
    alternative view. If we did not find it deeply perplexing that, though
    it works in practice, the reasoning on which the differential calculus
    is based is not logically sound; if we did not think Russell's Paradox
    a serious matter; if we could not see any serious motivation behind
    Hilbert's formalisation programme; if we did not care about logical
    consistency; if we did not think the liar paradox worthy of our
    attention, is it likely that we would ever have shown any interest in
    the problem of providing logical foundations for mathematics? And if
    we have shown an interest in these things, is it likely that we should
    be impressed by being told that the fear of contradiction is a
    superstition?, that the reasoning behind Russell's Paradox is all
    bosh? that the liar paradox is a useless game? that set theory might
    just as naturally be interpreted as a joke as a serious piece of
    mathematics, and that the problem of the infinitesimal was, all along,
    simply a matter of succumbing to the picture of 'very tiny things'?

    Wittgenstein's philosophy of mathematics is, then, doubly paradoxical:
    not only is he attempting to contribute to an area of discussion in
    which attitudes are held which he does not hold, and in which
    techniques of argument are expected which he does not use, his
    contribution is precisely to attack those attitudes and techniques.
    Nothing more Quixotic could be imagined. It is simply impossible, I
    believe, to interpret Wittgenstein's remarks as constituting a serious
    philosophy of mathematics in the sense that logicism, formalism,
    intuitionism and strict-finitism are serious philosophies of
    mathematics - i.e., serious attempts to make sense of the subject
    studied by professional mathematicians. And this is for the strong and
    simple reason that Wittgenstein does not take that subject seriously.
    Indeed, as I have said, he hardly seems to believe it exists. The only
    activity that might deserve the name 'pure mathematics' that emerges
    from his 'description' is the construction of calculi for either use
    or amusement; that is, an activity that is either indistinguishable
    from applied mathematics or else is a frivolous pastime that has
    nothing to do with science."
  • mcdoodle
    I've always loved Wittgenstein, so I've naturally wrestled with his philosophy of math. Recently I bumped into this piece by Ray Monk, which really moved me. I find it plausible as an interpretation.Hoo

    I'm not a mathematician, but I've been doing some re-reading/study of P.I. over the summer and thinking a lot about Wittgenstein. My interest is in philosophy of language. Witt is of course a challenge to any philosophy project - one of my current tutors gets fierce at the mention of his supposed 'anti-philosophy'. It does feels as if the 'game' notion is at the heart of both how Witt looks at natural language and at mathematics. What game are we in? What use is it? How is it used? Whenever the game is in play, rules emerge. Where do these rules come from? Do they matter?

    I am at the point in philosophy of language in thinking that much of the talk about truth or 'truth conditions' that is at the heart of the traditional enterprise is 'bosh', as Witt might have put it. If it means something, it means something about the desire of the secondary workers at this coalface - the philosophers of linguistics in my case, the philosophers of mathematics in yours - to find a primary justification for why they do things the way they do.

    I don't know exactly what to do with this feeling. I think I am an 'engineer' as Monk puts it of language, and there are quite different engineering ways of writing about language that I might know about, because I've written and thought about writing and worked in language.

    But it means starting from a different place from where the traditional analytic approach to phil of language starts. Maybe you're in the same boat.
  • Hoo

    I like the "engineer" metaphor. It's one of my favorites. The "generalized engineer" seems to capture my current "summary" of the human mind. We solve problems, satisfy desires. Foundational quests are probably related to the "god shaped hole." Then we also desire to minimize cognitive dissonance.

    I very much sympathize with this: "much of the talk about truth or 'truth conditions' that is at the heart of the traditional enterprise is 'bosh'" I wouldn't call it nonsense, but I do feel that I've had a "vision" of the futility of trying to dominate language from the outside. I want to say that "ordinary language" is the genuine metalanguage, but philosophers of a different persuasion (the very ones I'd be try to share this "insight" with) can't help but try to assimilate it in the exactly the sort of bad-math object language which is (as object language) dependent on this OrdLang. By "object language" I mean a normalized discourse, a type of communication with fixed rules (consensus) about what is "true" or "reasonable" therein. But we are always already communicating more or less successfully as we try to hammer out these nakedly secondary rules and this ideal instrument (the right system of terms of inferences) for trapping Truth. We are always already knee-deep in the meta-language, that language build and check the ideal properties of the object language we often think we need. I want to say that we are always "in fact" largely improvising, getting by on proverbs and rules-of-thumb. Generations come and go. The ideal consensus remains anything but achieved. Meaning-by-fiat is ignored away from the narrow game (futile or boring to most) of "fixing" what mostly ain't broke. Some non-philosophical problems are about language, surely, but even these are perhaps better addressed by comedians and novelists. Myths, sensual and emotional, seem to lie at the heart of the drives that employ thought. If we imagine a mind without a heart, then whether we wouldn't care whether a proposition was true or false (prudent or reckless, productive or counter-productive). But yes I get the sense that we are on the same page somehow.
  • mcdoodle
    My feeling is that language is something that happens between people in many different language-communities (including soliloquy). We bring various capacities to bear - our vocal arrangements, our brains, our gestures - to express, try to understand or communicate with each other. These may well all be activities for which we can formulate both (a) rules we find we enact by whether we like it or not, and (b) human-imposed rules, or modifications of a-rules, that make the process work more to our satisfaction.

    Then grammarians' rules, dictionaries, meta-languages, T-sentence concepts are all type (b) rules. But Chomskyan generative grammar and certain rules of dialogue would be (a) type rules.

    To me exchange of 'truths' is just one of the things such language-communities might regard as what they're doing through language.

    And in all this we are then, as you say, largely improvising most of the time.

    This could all be bosh, of course, I'm just thinking about how I see it.

    I can see how mathematical language could fit my schema. We find we just do count, subtract, divide, multiply. Elaborations make this work better. But I may be on the wrong tack for math. Witt for instance thinks that regarding sets as fundamental is just silly, whereas that's so much part of my psyche now I can't dislodge it to look at it afresh. (I wanted to refer to 'truth-telling' as a subset of language a little earlier!)
  • Hoo

    I relate to everything you've said, even about sets. I've been doing math all day, actually. And it's sets, functions, and logic. I don't intuitively think of the integers as sets (except when I'm trying to), but I do find it natural to speak of sets of numbers and sets of sets of numbers and so on. (I'm actually doing an independent study in axiomatic set theory this semester, too.) More on topic, I do think "worldly math" has an empirical foundation. Calculus earned our trust before it was made rigorous. We would continue to use it even if some contradiction were found in ZFC, for instance. So the quest for foundations seems aesthetically driven, which I can respect. I cooked up a construction of the real numbers this summer that I haven't seen elsewhere. It absolutely felt like I was sculpting.
  • wuliheron
    What Wittgenstein could not know at the time is he was describing a systems logic. A universal recursion in the law of identity, or paradox of existence such as quantum mechanics suggests, means that the foundations of classical logic must be built upon an analog systems logic that allows the law of identity and its own logic to inevitably go down the nearest rabbit hole or toilet of your personal preference. Its what everyone from Google to the US government wants and is fundamental to a Theory of Everything.

    Theoretically it would reflect the organization of the human brain as well which can be described as a distributed gain amplifier incorporating Bayesian probabilities vanishing into indeterminacy. You could say within a paradox of existence everything would organize around what's missing from this picture because yin-yang dynamics apply and what is true must inevitably transform into B.S. because everything is context dependent, but knowing that allows the systems logic to express everything as both juxtapositions and bandwidth issues. It would mean things like Mach's Conjecture are merely nature begging the question because our classical mathematics are formulated for beauty at the exclusion of humor and what is required are Intuitionistic mathematics that are brand new and about four times as complex.

    Such a systems logic should resemble a striking classical appearing fractal dragon equation within a larger and more subtle Mandelbrot pattern. I'm writing a book on the subject with this first book being extrapolated from the Tao Te Ching to represent the fractal dragon and its sequel would be the I-Ching that provides the Mandelbrot. Reconciling the two using metaphoric logic is a sort of top-down approach to the problem that must make both more and less sense out of more bottom-up metaphysically oriented classical logic. That's the way more holistic analog approaches work because they must reconcile any content with the greater context and require broader foundations.

    In more classical terms it means mathematics and everything that exists can be considered to express a conservation of creativity and efficiency. Both a black hole and the neurons of our brains can convey any mass, energy, and information with the highest efficiency possible for anything their size, allowing them to transform their extreme efficiency into greater creative output allowing mathematics to become a pragmatic affair along with quantum mechanics that only has demonstrable meaning in specific contexts.
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