Conventionalism again

In Philosophical Investigations Wittgenstein himself writes about the temptation to interpret him as advocating conventionalism of some sort:

241. ―So you are saying that human agreement decides what is true and what is false?‖ – It is what human beings say that is true and false; and they agree in the language they use. That is not agreement in opinions but in form of life.

242. If language is to be a means of communication there must be agreement not only in definitions but also (queer as this may sound) in judgments. This seems to abolish logic, but does not do so. – It is one thing to describe methods of measurement, and another to obtain and state results of measurement. But what we call ―measuring‖ is partly determined by a certain constancy in results of measurement.

And in LFM, he is reported as asking us to

Consider Professor Hardy‘s article (―Mathematical Proof‖) and his remark that ―to mathematical propositions there corresponds – in some sense, however sophisticated – a reality‖ (LFM p. 239)¹⁴⁸,

… in regard of which ‗reality‘ Wittgenstein goes on to say

With regard to ―responsibility to reality‖: On the one hand you might say, ―This conclusion is responsible to certain axioms and certain rules.‖ This responsibility is based on our peculiar practice of using these rules. But then there is another question: as to whether such a system as a whole is responsible to anything.

(LFM . 242)

148 Hardy does not mention ‗correspondence‘ here, in fact, as Cora Diamond, the editor of LFM points out (ibid.). It is clear enough what is going on, though. Wittgenstein adds the parenthetical remark (ibid.), ‗The fact that he said it does not matter; what is important is that it is a thing which lots of people would like to say.‘ Elsewhere Hardy writes of ‗reality‘,

‗… I shall speak of ‗physical reality‘, and here again I shall be using the word in the ordinary sense. … I hardly suppose that, up to this point, any reader is likely to find trouble with my language, but now I am near to more difficult ground. For me, and I suppose for most mathematicians, there is another reality, which I will call

‗mathematical reality‘ … (Hardy 1992 pp. 122-123)

In this Hardy seems to anticipate Wittgenstein‘s ‗two ways‘ in which mathematics may be held to be ‗responsible‘ to reality in his (Hardy‘s) notion of different kinds of reality or different languages in use for physical and mathematical reality. He makes little of it, though, I suppose on account of his Platonism. In spite of protestations about differences, it does seem that Hardy trades on the sort of semantic homogeneity of which we have come to be suspicious.

As a first move towards unpacking here, consider the distinctions Wittgenstein makes in Philosophical Investigations §241/2 above: The distinctions are between what human beings say and the language human beings use

opinions and forms of life

results of measurement and methods of measurement

Wittgenstein is suggesting we notice a similarity between these distinctions.

Consider the first pair of conjuncts, for instance. A proposition – what I say – may be true or false, and is assessable for its truth. It makes sense without any further ado to ask about something I say – some proposition – whether or not it is true. Contrast with language itself, as a whole: there are many questions I might ask about the language I use to say something, but it does not ordinarily make sense to ask whether the language is true. (Note the claim is not that we cannot make sense of talk of language being true, but that we do not; this is intended as a description, once again, rather than part of, or a consequence of, a theory.) Language, we might say, is conventional, but this contrasts with what we say in language: we may agree on our language, but what we say in language is not dependent for its truth on our agreement.

For each of the pair of conjuncts above, the first of the two has a similar normative aspect. What human beings say may be true or false: likewise opinions may be correct or incorrect, as may results of measurement. The second of each of the two conjuncts, by contrast, characterises the frameworks within which the normative practice of assessing the first takes place. While it makes sense to ask, for instance, whether my opinion on some matter is correct or not (and similarly in the other cases), the question of whether the form of life I share with other humans is correct (and so on) does not arise. Here is one aspect of how Wittgenstein deals with what looks like a drive to relativism of some sort or other via what might appear to be – but is not – conventionalism.

There is a clear link here to what I explained, following Barry Stroud, as Wittgenstein‘s overview of the ‗hardness of the logical must‘ – of logical and mathematical necessity – in the previous chapter. In Philosophical Investigations

§242 (above) Wittgenstein writes of the agreement in judgements that he adverts to as a condition of the possibility of human language as seeming ‗to abolish logic‘.

That is, if human judgement determines what is or is not the case, or even more pertinently for our concerns, what is or is not necessarily the case, it seems that logic will lose its objectivity with its independence, as we saw Dummett and Putnam claiming. That this seeming abolition of logic is only apparent – a mere seeming –, however, as I claimed following Stroud‘s exegesis, is apparent once we note that the impossibility of its not being the case that p when p is necessary is based on the lack of sense we have given to its not being the case that p.

This lack of sense, now, can be seen in a particular light in its association with the move from the first to the second of each of the pairs of conjuncts I have highlighted

above; for example the sense we make within a system of measurement of considering whether such-and-such result of measurement is correct, contrasts with the lack of sense we allow to talk of whether a method or system of measurement is correct or incorrect. It is such lack of sense given or allowed which we saw above in chapter 4 to be deeply connected to the root of ascriptions of necessity. With this in mind let us add a line to the table of three conjuncts above taken from the distinction Wittgenstein makes in LFM between the two ways of considering mathematics to be

‗responsible‘ to reality.

The distinction is between mathematical propositions considered individually intra-mathematically

and

our practice of using the usual rules and axioms of

mathematics

Once again, the first conjunct here has a normative aspect; it makes sense to consider whether a particular mathematical proposition is responsible to ‗certain axioms and certain rules‘ – whether it is in accord with relevant parts of the rest of mathematics, in short. By contrast, again, the second of the two conjuncts characterises the framework within which such accord is assessed. We can mark the difference here by reference to the intra-mathematical aspect of talk of whether a certain mathematical conclusion is correct by contrast to extra-mathematical questions about whether ‗the system as a whole is responsible to anything‘.

So, in these terms, now, what is important for us is how the distinction Wittgenstein points up between normative practices and the frameworks within which such practices take place plays out in the case of the distinction between intra-mathematical propositions, assessed by our practice of using the usual rules and axioms of mathematics, and extra-mathematical questions about mathematics, such as whether the whole of mathematics is responsible to anything at all – or, by the same token, whether our practice of using the usual rules and axioms of mathematics is responsible to, or justified by, anything at all. I am going to argue that this distinction, and consequential differences between the sense we allow to such intra- and extra-mathematical propositions and questions, is key to solving the problem of mathematical applicability.

We do need to be a little careful here. In assimilating some aspects of the distinction Wittgenstein draws between what I am calling intra- and extra- mathematical questions to the distinctions in play in Philosophical Investigations §241-2, we should (once again) beware of thinking that what is on offer is some sort of overall or general theory about the difference between, say, frameworks and normative practices within them. The comparison between the two distinctions is intended, not as a generalisation part way to becoming a theory, but – as usual – in the hope of clarification, of arriving at a perspicuous overview of the philosophical terrain.