Monday, May 14, 2018

Fixing Frege

Frege ("Logic in Mathematics"):
Definitions proper must be distinguished from elucidations [Erläuterungen]. In the first stages of any discipline we cannot avoid the use of ordinary words. But these words are, for the most part, not really appropriate for scientific purposes, because they are not precise enough and fluctuate in their use. Science needs technical terms that have precise and fixed Bedeutungen, and in order to come to an understanding about these Bedeutungen and exclude possible misunderstandings, we provide elucidations. Of course in so doing we have again to use ordinary words, and these may display defects similar to those which the elucidations are intended to remove. So it seems that we shall then have to provide further elucidations. Theoretically one will never really achieve one’s goal in this way. In practice, however, we do manage to come to an understanding about the Bedeutungen of words. Of course we have to be able to count on a meeting of minds, on others’ guessing what we have in mind. 
Also Frege (The Foundations of Arithmetic):
At first, indeed, [Mill] seems to mean to base the science, like Leibniz, on definitions, since he defines the individual numbers in the same way as Leibniz; but this spark of sound sense is no sooner lit than it is extinguished, thanks to his preconception that all knowledge is empirical. He informs us, in fact, that these definitions are not definitions in the logical sense; not only do they fix the meaning of a term, but they also assert along with it an observed matter of fact. But what in the world can be the observed fact, or the physical fact (to use another of Mill’s expressions), which is asserted in the definition of the number 777,864? Of all the whole wealth of physical facts in his apocalypse, Mill names for us only a solitary one, the one which he holds is asserted in the definition of the number 3. It consists, according to him, in this, that collections of objects exist, which while they impress upon the senses thus, \, may be separated into two parts, thus, (.. .). What a mercy, then, that not everything in the world is nailed down; for if it were, we should not be able to bring off this separation, and 2+1 would not be 3! What a pity that Mill did not also illustrate the physical facts underlying the numbers 0 and 1!
I'm no philosopher of mathematics, but Frege's criticism of Mill does seem on target. Mill goes wrong by imagining too limited a set of examples and by thinking of the one example he does consider as if it were fixed. But that is not the real world, and somehow we do manage to get by even though not everything is nailed down. But then Frege seems to want to pin down the meanings of words, despite conceding that in practice this is not really necessary and that in theory it is impossible.

Mill makes what, after reading Frege's criticism of it, seems like a stupid mistake. Frege's problem is of a different kind. There is something wrong with what he wants. He sees the problems himself, but still, apparently, goes on wanting the same thing. So pointing out the problems won't help at all. We might say he needs a kind of therapy, although this won't be regular psycho-therapy. Nor does it at all follow that therapy is what Mill needs.

7 comments:

  1. I'm not seeing the connection you try to draw between the Mill criticism and Frege's remarks about elucidations. The problem with Mill (according to Frege) is that he assumes the empirical fact that objects can be collected together and separated apart, and claims that this is what gives us knowledge of arithmetic; this has nothing to do with whether words are "pinned down" in their meanings or not, which is a metaphor. The Mill bit you bolded seems to me an unfair joke on Frege's part; if Mill complained back that our "mathematics" would indeed be quite different if nothing was capable of motion, it seems to me he'd have a point.

    Frege wanting mathematicians to be precise in their terminology (and especially to stop overloading their operators, like Boole did) seems to me a reasonable request after the 19th century spent quite a lot of time being confused over what a "function" was (because they didn't properly distinguish a function from its graph before Weierstrass). The first passage you quote strikes me as an ordinary defense of the use of technical terminology. The "theoretical" possibility of never reaching an end to elucidations is also not a real worry, as Frege's own practice shows: not all words of ordinary language are equally likely to cause misunderstandings, so using less troublesome words (like "some" and "all") to elucidate more troublesome words (like "successor") lets us avoid some practical troubles. The two bits you bolded are thus not in tension, as you seem to suggest: terms of ordinary language are (in some cases) too fluid for scientific purposes, but by elucidating troublesome terms by means of less-fluid terms, we can manage to produce rigorous proofs. (Think of the difference between the proof Socrates walks the slave-boy through in the Meno and a proof in Euclid.)

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    1. I'm not seeing the connection you try to draw between the Mill criticism and Frege's remarks about elucidations.

      One passage reminded me of the other, that's about all really.

      this has nothing to do with whether words are "pinned down" in their meanings or not, which is a metaphor

      Yes, you're right

      if Mill complained back that our "mathematics" would indeed be quite different if nothing was capable of motion, it seems to me he'd have a point.

      Yes, but I think there's more to Frege's point than would be answered by this

      The "theoretical" possibility of never reaching an end to elucidations is also not a real worry

      You're probably right. If it were something that someone really took seriously, though, then perhaps my remarks about something like therapy would apply.

      So what's my point? I might not have a good one, for reasons you've explained. But my thinking was a) here are two passages from Frege that bear a superficial resemblance, b) read a certain way, one of them suggests both a place for something like therapy in philosophy and a reason for philosophers to attend to practice (both Wittgensteinian themes, of course), and c) even if one grants that there is a place for such therapy in philosophy, it does not follow that this is the only tool that philosophers need (as perhaps some Wittgensteinians might be tempted to think). All of this being meant in a "it seems to me right now that maybe ..., what do you think?" spirit.

      Thanks for your reply, which is more thoughtful than the original post.

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  2. You put it, "Frege's problem is of a different kind. There is something wrong with what he wants. He sees the problems himself, but still, apparently, goes on wanting the same thing. So pointing out the problems won't help at all." and I think this is right. And I am tempted to think of a fly voluntarily entering a fly bottle. However, the first thing your criticism suggested to me was something Wittgenstein said in various places in different formulations specifically about the difference between the intellect and the will. In The Big Typescript he phrases it this way:

    "What makes a subject difficult to understand - if it is significant, important - is not that some special instruction about abstruse things is necessary to understand it. Rather it is the contrast between the understanding of the subject and what most people *want* to see. Because of this the very things that are most obvious can become the most difficult to understand. What has to be overcome is not difficulty of the intellect but of the will."

    I think Frege gets credit for seeing his problem, but seeing is clearly not enough. You have to *want* different things for any chance of escaping the fly bottle, and that does seem to be more a matter of the will.....

    I have recently stuck my nose back in Remarks on the Foundations of Mathematics, and a bit into Lectures on the Foundations of Mathematics, and I just love Wittgenstein's examples that ignore the picture of mathematics as something otherworldly and instead turn to the understanding that the certainty we have is wrapped up in a practice, and that the practice can be challenged in various empirical ways.

    One example I liked was this"

    "If 2 and 2 apples add up to only 3 apples, i.e. if there are 3 apples there after I have put down two and again two, I don’t say: “So after all 2 + 2 are not always 4”; but “Somehow one must have gone”." [RFM I§157]

    Severin Schroeder explains this as well as I ever could in Mathematics and Forms of Life, in the Nordic Wittgenstein Review Special Issue 2015 • Wittgenstein and Forms of Life. He puts it,

    "That is to say, when on the basis of my experience with nuts and apples etc. I put forward the sum ‘2 + 2 = 4’ it is not presented as an empirical generalisation, but as a rule or norm of representation, which means that it is made immune from empirical falsification. We will insist on it even where experience seems to contradict it: in
    that case we declare our experience as inaccurate. We say that something else must have happened that we didn’t see. However, even though no conflicting experience can falsify a mathematical proposition, repeated conflicting experiences can undermine its usefulness. If putting together pairs of apples frequently resulted in more or fewer than 4 apples, we would have to say that our arithmetic was not applicable to apples; as, in fact, it
    is not applicable to drops of water (one and one make one) or measures of different liquids (one quart of alcohol and one quart of water yield only 1.8 quarts of vodka)."

    Wittgenstein spent a lot of time thinking about issues like these, and Mathematics was a fertile ground for exposing some of our common detours in better understanding. Being clear on the difference between things that function empirically for us and things that do not, in fact, seems a central theme in his later work. On Certainty is essentially that problem in a nutshell. Mathematics just holds up a picture that makes separating those roles perplexing, until we come to better grips with whether we are *discovering* mathematical truths in some mathematical space or are *inventing* practices, mathematical practices, that have more or less useful applications in the outside world.

    Not sure if I put that as well as I needed, but that is what your ruminations had me thinking :)

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    1. Thanks. Yes, the idea about difficulties of the will is exactly what I was thinking about. And I like those examples from mathematics. The one about water and alcohol is especially interesting. Presumably it's not just that someone steals some of the alcohol. :)

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    2. You put that very well, indeed.

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  3. Funnily enough, I recently finished a post on what I think would constitute a therapy for Frege: http://metaphorhacker.net/2018/05/therapy-for-frege-a-brief-outline-of-the-theory-of-everything/.

    Reading this remind me to post it. I just changed the title and added a quick section at the start.

    The core argument is that the acceptance of the impossibility of perfect reference must be a starting point. And that its implications are everywhere in our daily practice rather than just theoretical possibilities.

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    1. Thanks for this. Looks like something productive has come from this post after all.

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