Freedom and Necessity
It is commonly argued that the real problem with a conventionalist philosophy of mathematics is that you cannot base the necessity of a mathematical truth on an arbitrary linguistic practice. The most common criticism levelled against conventionalism is that mathematical truth must be sempitemal and universal: properties that outstrip the reach of conventions, which are rooted to the decisions of a speaker or community. In a single sentence Donald Harward dismisses the entire conventionalist approach for failing to see that '''2 + 2 = 4" does not mean that "here and now, 2 + 2 - 4" or "in general contexts 2 + 2 = 4".' Hence the conventionalist simply misconstrues 'what the mathematician is most interested in: the necessity of mathematical propositions'.! It was stressed in Chapter 7 that it was indeed part of Wittgenstein's ultimate objective to drive out the metaphysical concept of necessity from the normative domain of mathematics, but only in order to make room for the logical certainty which characterises mathematical propositions qua 'rules of syntax'. But Wittgenstein certainly did not think that the rule '2 + 2 = 4' means 'here and now, 2 + 2 = 4' or 'in general contexts, 2 + 2 = 4'. The former would be like saying that the rule only applies at this instant, while the latter treats mathematical propositions as statistical generalisations. Yet neither did he hold that '2 + 2 = 4' means 'in all possible worlds at all possible times 2 + 2 = 4'; both this and its denial are equally absurd. (Just as it would be to argue that 'here and now it is true that the bishop moves diagonally in chess', or to turn to possible world semantics in order to specify when and where this is true.) These misconceptions are caused by the failure to grasp that to say that 2 + 2 must equal 4 is to use this rule as a standard of correct repre-
The feeling behind Harward's criticism is that the conventionalist contends that a mathematical proposition only expresses either a speaker's or a community's decision to use symbols in a certain way. He fails to distinguish, however, between the argument that a convention is what Lazerowitz calls a 'verbal proposition'2 (i.e. that a mathematical proposition describes the manner in which a speaker or community uses those symbols) and Wittgenstein's normative interpretation that a mathematical proposition stipulates the use of those concepts. Harward is led into his harsh verdict because, as the above formulations indicate, he conflates these two different conceptions. Thus he concludes that 'Where the conventionalist erred is in thinking that because the rules were applicable in different ways of life or contexts that the necessity of a mathematical proposition would be dependent on that context.'3 Were we concerned with the 'descriptivist' version then it would indeed be the case that 2 + 2 might not equal 4 for different communities with alternative 'verbal propositions'. The fact that this is unintelligible - since it overlooks that the meaning of these signs would thus be radically different from what we understand - illustrates the incoherence of the descriptivist interpretation. But Harward explicitly refers to the normative conception here, which renders his objection equally inexplicable; for in that case we simply could not describe anyone who denied that 2 + 2 == 4 as operating with our arithmetical rules (WWK 177). As Gasking explains, we could only understand a Martian translation of 2 + 2 = 4 as such if the Martians used this mathematical proposition in exactly the same manner as we do; i.e. applied the same rules of addition. 4 But then, as Gasking notes, this still leaves open the problem - which presumably is what Harward was trying to develop - of explaining how we are to reconcile the freedom which we enjoy in the construction of rules of grammar with what the platonist regards as the necessity governing the construction as well as the application of mathematical truths.