2) assertability vs. truth and golfarb’s problem about p or not p

I have claimed that what I’m trying to do is (a la Wittgenstein) not trying to interpret mathematical practice in the sense of discovering what mathematical statements really mean but rather trying to account for it. This distinction may seem needlessly subtle, so now I want to mention an interesting problem told to me by Warren Goldfarb which shows that it’s not.

My general impulse is to compare mathematical practice to a kind of communal make-believe (rather like what a cynical person might say about theology). We pretend that there are certain kinds of objects and learn how to govern what claims we make about these objects either by some rules which we are directly taught or generally by logic. In grade school we are taught by rote, without hearing any arguments, what to say (or rather, what procedures to employ to get to what to say) about which imagined objects stand in which imagined relations to which other imagined or concrete objects. In later math classes we are given explicit statements which we are to assent to and we learn to make other claims about what’s going on in the mathematical realm by applying logic to the statements we’ve heard.

So, overall, the rules governing mathematical assertions are something like this. If a statement is one of the axioms which your math book told you, or one of the ones you were trained to produce in grade-school, or is a logical consequence of either of these then you can assert it, otherwise you can’t. And here’s where Goldfarb’s problem comes in provided we make two assumptions: 1) logical consequence matches with classical logic in the sense of allowing arguments from ‘p or ~p’ for any statement 2) the rules above state what mathematical sentences mean. If the theory just sketched about the assertability conditions for mathematical statements is used to give an interpretation i.e. truth conditions for mathematical statements then we get something like this. A mathematical statement written as ‘P’ is true iff P is a logical consequence of the mathematical axioms or gradeschool productions just mentioned. Now there are surely some mathematical statements S which aren’t decided one way or another by this body of axioms. So ‘S’ won’t be true on the above theory, and neither will ‘~ S’. But, given our assumption that logical consequence is aptly described by classical logic as above ‘S or ~S’ is a logical consequence of anything, so it’s a consequence of the axioms so that statement will come out to be true. Thus we have a contradiction: the disjunction ‘S or ~ S’ is supposed to be true while both of its disjuncts are supposed to be false.

So here’s a place where I think it’s crucial that the theory I’m giving is an account of mathematical practice, rather than an attempted interpretation of mathematical statements. I’m trying to describe when it’s right to assert certain claims, not give truth conditions for mathematical statements in terms of something else. In the case mentioned above there’s nothing contradictory in saying that we should assert ‘p or ~p’, but that we shouldn’t assert either disjunct. In fact that’s just what we do with mathematical conjectures. Goldfarb’s problem only arises if we take this attempted description of mathematical assertability conditions for a description of truth conditions.

Hopefully, this is already clear but I want to mention a little analogy to dispel any lingering suspicion. Suppose someone wants me to explain our practice of talking about the weather, and particular our claims about when it’s raining. I might say that people do things like look out windows, listen to weather reports, look for rain on people who have just come indoors and when they have seen certain things they then (should) say that it is raining. These assertability conditions (you should say its raining if you see drops outside etc.) are obviously different from the truth conditions of the sentence ‘it is raining’. It might well be correct for a person to assert that its raining when in fact it isn’t (because they’ve been duped by circumstances) or to assert that it isn’t raining when it is.

There is, of course, a relationship between the idea of assertability conditions which we get from studying the practice surrounding a certain sentence, and it’s truth conditions. Normally, (I don’t know if there are any exceptions) the truth conditions of a sentence have to be the kind of things which it’s assertability conditions are evidence for. So, e.g. one of the ways that we know the truth conditions of ‘its raining’ have to do with water falling from the sky is that the circumstances in which we consider it right to assert this sentence are all ones in which we have evidence that water is falling from the sky. So, though I’m trying to stress here that what I want to understand is the norms governing mathematical assertion, this might shed light on truth conditions also. Though, on the other hand, there are some cases (like saying ‘checkmate’ or ‘punch buggy blue’) where we have rules about when you can say X but we don’t have an idea of X having anything like truth conditions, or of what it would take for X to be true or false.