How this is different from standard deductivism?:

 

            In that the view sketched above takes there to be real facts about logical consequence which combine with extra-mathematical concerns to fix mathematical practice it resembles deductivism about. I won’t worry about the question about whether this view counts as a kind of deductivism, but I do want to emphasize two differences between the view I mean to be advocating and the standard version of deductivism.

            First off, according to this standard view the initial characterization of a mathematical structure to be studied would be given by a choice of axioms in some formal system, and the consequences which mathematicians study would also be the formal consequences of these axioms within that system. This restriction of the way that a chosen structure can be characterized and what the consequences of such a characterization could be both runs into the well-known problem of incompleteness and fits awkwardly with some parts of mathematical practice. It has been proved that any suitably complicated set of axioms in a formal system will have intuitive mathematical consequences that aren’t consequences in this formal system. But even without this technical problem, the restrictive version of deductivism is an awkward fit with the parts of the history of mathematics where people have tried to formalize a pre-existing practice like arithmetic, or to choose axioms of set theory which match with intuitive ideas about collections.

            In contrast I want to suggest that structures whose nature mathematics aims to describe and reveal can be characterized in a richer, informal manner (whatever matches this practice, concept or physical structure) and the sense of ‘consequence’ in question is a matter of full analytic consequence (A is an analytic consequence of B if A couldn’t be true without B also being true). Thus, if there is anything which lets us recognize real finite-ness and the standard model of the integers as a standard model this can form part of the characterization of a mathematical structure to be studied. And the full higher level reasoning which lets us see that e.g. a certain equation can’t have any integer roots because we have specially constructed it only to have roots which are proofs that that equation has no proofs will thereby show us that it is a ‘consequence’ of the initial stipulation in this full sense of consequence.

            Secondly, this restricted or standard sense of deductivism can be self-undermining insofar as it is motivated by the intuition that there is no fact of the matter about mathematical statements whose truth is not determined by any presently accepted axioms. You might say, in bewilderment about the prospects of ever coming to a principled decision about the axiom of choice, that the real mathematical facts concern what is deducible from fixed sets of axioms, thus that it is up to math to discover what follows from ZF+C and ZF+~C but there is no further question of which is true. The problem with this is that if we allow that there are determinate facts about what is provable from a given bunch of axioms, and hence about which axioms are consistant then this has been shown to determine an answer to all questions of less than a certain complexity – including ones which are independent of our present axioms like choice. Thus deductivism does not provide a basis for saying that there is never fact of the matter about how we must choose axioms for set theory.

            In contrast, the theory I mean to advocate takes there to be a determinate fact of the matter about as large a fragment of mathematics as can be determined by linguistic/ meta-logical facts about which statements are consistent. In general the mathematical endeavor is a matter of working out what else would have to be true of a structure if it had various features. But in some cases these features are fixed not by explicit axioms, but rather by an intended application to some scientific or logical facts. Specifically we can use mathematics to describe logical notions of proof, consistency and consequence, and this intended application can then form part of the descriptions of mathematical structures, for which mathematics seeks to find the consequences. Our notion of finitude might be determined in part by the intended application that ‘proofs can be of any finite length’ – if we allow proofs of lengths corresponding to the ‘extra’ numbers in certain non-standard models of the integers we can derive proofs of contradiction from collections of sentences which are not contradictory. Similarly the behavior of ‘+’ is partly determined by intended application to describe logical truths about identity and disjunction.