Carnap’s thesis of pluralism in mathematics is quite radical. We are told that “any postulates and any rules of inference [may] be chosen arbitrarily”; for example, the question of whether the Principle of Selection (that is, the Axiom of Choice (AC)) should be admitted is “purely one of expedience” (Logical Syntax of Language); more generally,
The [logico-mathematical sentences] are, from the point of view of material interpretation, expedients for the purpose of operating with the [descriptive sentences]. Thus, in laying down [a logico-mathematical sentence] as a primitive sentence, only usefulness for this purpose is to be taken into consideration.
So the pluralism is quite broad – it extends to AC and even to ∏01-sentences. There are problems in maintaining ∏01-pluralism. One cannot, on pain of inconsistency, think that statements about consistency are not “mere matters of expedience” without thinking that ∏01-statements generally are not mere “matters of expedience”. The question of whether a given ∏01-sentence holds is not a mere matter of expedience; rather, such questions fall within the provenance of theoretical reason. One reason is that in adopting a ∏01-sentence one could always be struck by a counter-example. Other reasons have to do with the clarity of our conception of the natural numbers and with our experience to date with that structure. On this basis, for no sentence of first-order arithmetic is the question of whether it holds a mere matter of experience. Certainly this is the default view from which one must be moved.
What does Carnap have to say that will sway us from the default view, and lead us to embrace his radical form of pluralism? In approaching this question it is important to bear in mind that there are two general interpretations of Carnap. According to the first interpretation – the substantive – Carnap is really trying to argue for the pluralist conception. According to the second interpretation – the non-substantive – he is merely trying to persuade us of it, that is, to show that of all the options it is most “expedient”.
The most obvious approach to securing pluralism is to appeal to the work on analyticity and content. For if mathematical truths are without content and, moreover, this claim can be maintained with respect to an arbitrary mathematical system, then one could argue that even apparently incompatible systems have null content and hence are really compatible (since there is no contentual-conflict).
Now, in order for this to secure radical pluralism, Carnap would have to first secure his claim that mathematical truths are without content. But, he has not done so. Instead, he has merely provided us with a piece of technical machinery that can be used to articulate any one of a number of views concerning mathematical content and he has adjusted the parameters so as to articulate his particular view. So he has not secured the thesis of radical pluralism. Thus, on the substantive interpretation, Carnap has failed to achieve his end.
This leaves us with the non-substantive interpretation. There are a number of problems that arise for this version of Carnap. To begin with, Carnap’s technical machinery is not even suitable for articulating his thesis of radical pluralism since (using either the definition of analyticity for Language I or Language II) there is no metalanguage in which one can say that two apparently incompatible systems S1 and S2 both have null content and hence are really contentually compatible. To fix ideas, consider a paradigm case of an apparent conflict that we should like to dissolve by saying that there is no contentual-conflict: Let S1 = PA + φ and S2 = PA + ¬φ, where φ is any arithmetical sentence, and let the metatheory be MA = ZFC. The trouble is that on the approach to Language I, although in MT (metatheory) we can prove that each system is ω-complete (which is a start since we wish to say that each system has null content), we can also prove that one has null content while the other has total content (that is, in ω-logic, every sentence of arithmetic is a consequence). So, we cannot, within MT articulate the idea that there is no contentual-conflict. The approach to Language II involves a complementary problem. To see this note that while a strong logic like ω-logic is something that one can apply to a formal system, a truth definition is something that applies to a language (in our modern sense). Thus, on this approach, in MT the definition of analyticity given for S1 and S2 is the same (since the two systems are couched in the same language). So, although in MT we can say that S1 and S2 do not have a contentual-conflict this is only because we have given a deviant definition of analyticity, one that is blind to the fact that in a very straightforward sense φ is analytic in S1 while ¬φ is analytic in S2.
Now, although Carnap’s machinery is not adequate to articulate the thesis of radical pluralism in a given metatheory, under certain circumstances he can attempt to articulate the thesis by changing the metatheory. For example, let S1 = PA + Con(ZF + AD) and S2 = PA + ¬Con(ZF + AD) and suppose we wish to articulate both the idea that the two systems have null content and the idea that Con(ZF + AD) is analytic in S1 while ¬Con(ZF + AD) is analytic in S2. No single metatheory (on either of Carnap’s approaches) can do this. But it turns out that because of the kind of assessment sensitivity, there are two metatheories MT1 and MT2 such that in MT1 we can say both that S1 has null content and that Con(ZF + AD) is analytic in S1, while in MT2 we can say both that S2 has null content and that ¬Con(ZF + AD) is analytic in S2. But, of course, this is simply because (any such metatheory) MT1 proves Con(ZF+AD) and (any such metatheory) MT2 proves ¬Con(ZF+AD). So we have done no more than reflect the difference between the systems in the metatheories. Thus, although Carnap does not have a way of articulating his radical pluralism (in a given metalanguage), he certainly has a way of manifesting it (by making corresponding changes in his metatheories).
As a final retreat Carnap might say that he is not trying to persuade us of a thesis that (concerning a collection of systems) can be articulated in a given framework but rather is trying to persuade us to adopt a thorough radical pluralism as a “way of life”. He has certainly shown us how we can make the requisite adjustments in our metatheory so as to consistently manifest radical pluralism. But does this amount to more than an algorithm for begging the question? Has Carnap shown us that there is no question to beg? He has not said anything persuasive in favour of embracing a thorough radical pluralism as the “most expedient” of the options. The trouble with Carnap’s entire approach is that the question of pluralism has been detached from actual developments in mathematics.