Wednesday, December 28, 2011

Loose connections (a principled take)

Fiction is like a spider's web, attached ever so lightly perhaps, but still attached to life at all four corners. Often the attachment is scarcely perceptible. -- Virginia Woolf

(This quote occupies a rather prominent place in the chambers of my mind; my interests coincide fairly well with the category of "things that are attached to life at all four corners.")

Michael Dummett has died; I was convinced I had blogged about him but I can't find the post. He was an interesting character -- analytic philosopher, Tarot expert, and more -- but I knew him primarily through a 1975 essay on "The Philosophical Basis of Intuitionistic Logic" that was in a book of philosophy-of-math readings. It appealed very strongly to me because Dummett's views ratified my biased belief that all the interesting problems in analytic philosophy could be restated as problems in the philosophy of mathematics. A few highlights (I realize that this is old hat for a lot of people, but it was my first exposure to late Wittgenstein -- late W. is a difficult writer and I needed someone to Dummett down for me): here he is describing the sort of theory of meaning an intuitionist could hold --

A model of meaning is a model of understanding, i.e. a representation of what it is that is known when an individual knows the meaning. Now knowledge of the meaning of a particular symbol or expression is frequently verbalisable knowledge, that is, knowledge which consists in the ability to state the rules in accordance with which the expression or symbol is used or the way in which it may be replaced by an equivalent expression or sequence of symbols. But to suppose that, in general, a knowledge of meaning consisted in verbalisable knowledge would involve an infinite regress: if a grasp of the meaning of an expression consisted, in general, in the ability to state its meaning, then it would be impossible for anyone to learn a language who was not already equipped with a fairly extensive language. Hence that knowledge which, in general, constitutes the understanding of the language of mathematics must be implicit knowledge. Implicit knowledge cannot, however, meaningfully be ascribed to someone unless it is possible to say in what the manifestation of that knowledge consists: there must be an observable difference between the behaviour or capacities of someone who is said to have that knowledge and someone who is said to lack it. Hence it follows, once more, that a grasp of the meaning of a mathematical statement must, in general, consist of a capacity to use that statement in a certain way, or to respond in a certain way to its use by others.
This essay was the first thing I'd read pointing out the similarity between Quine's philosophy of language and Hilbert's chronologically earlier -- earlier, btw, than Woolf -- philosophy of mathematics (both "holistic" in some sense -- see below -- and both of which must be rejected, as Dummett argues, by the "intuitionist" view).
it is not that a statement or even a theory has, as it were, a primal meaning which then gets modified by the interconnections that are established with other statements and other theories; rather, its meaning simply consists in the place which it occupies in the complicated network which constitutes the totality of our linguistic practices. [...] Frequently such a holistic view. is modified to the extent of admitting a class of observation statements which can be regarded as more or less directly registering our immediate experience, and hence as each carrying a determinate individual content. These observation statements lie, in Quine’s famous image of language, at the periphery of the articulated structure formed by all the sentences of our language, where alone experience impinges. [...]

For Hilbert, a definite individual content, according to which they may be individually judged as correct or incorrect, may legitimately be ascribed only to a very narrow range of statements of elementary number theory [sg. Hilbert was talking about operations like addition of integers etc. which one can "verify" with reference to collections of carrots, sticks, and the like]:  these correspond to the observation statements of the holistic conception of language. All other statements of mathematics are devoid of such a content, and serve only as auxiliaries, though psychologically indispensable auxiliaries, to the recognition as correct of the finitistic statements which alone are individually meaningful.
(The connection with Virginia Woolf should be obvious.) Dummett goes on to suggest that if one rejects this sort of holistic view -- for which there are various reasons, Godel's incompleteness theorems not least among them -- one might find a path to a revisionist theory of meaning. I'm not going to bother with the argument here, interesting as it is (some of it also amusingly echoes his prescriptivist views on grammar in Grammar and Style (1993)); just one more amusing snippet:
The [conventional notion of mathematical truth] does not provide for inflections of tense or mood of the predicate ‘is true’: it has been introduced only as a predicate as devoid of tense as are all ordinary mathematical predicates; but its role in our language does not reveal why such inflections of tense or even of mood should be forbidden.
Finally I should mention that the Hilbert-Woolf-Quine metaphor comes up rather widely in discussions of mathematics; not just its validity but its value, as in this remark of Michael Atiyah's (quoted by Gowers, who is explicitly a Hilbertian formalist, though Atiyah probably isn't):
the ultimate justi cation for doing mathematics is intimately related with its overall unity. If we grant that, on purely utilitarian grounds, mathematics justi es itself by some of its applications, then the whole of mathematics acquires a rationale provided it remains a connected whole. 
(I have always tended to hold something approaching this view about physics, but it is far less popular in the physics community -- partly because the appeal to direct applications is easier, partly because a fair number of physicists believe it is obvious that new theories supersede old ones rather than adding to them, a view that I have never been in complete sympathy with.)

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