Determinacy and Non-Classicality in Arithmetic

The mathematician`s self-image is often portrayed as that of a theoretician studying a domain of immutable truths, a domain devoid of ambiguity or indeterminacy. To give a well-known example, take Goldbach`s Conjecture, the statement that every even number greater than two is the sum of two prime numbers. Although it remains open at present, orthodoxy has it that the Conjecture has a determinate truth-value: that there is a fact of the matter about whether the Conjecture is true. The opposite view, that the Conjecture is true relative to some intended ways of interpreting our mathematical theories but also false relative to others, would strike most has misguided. And what goes for Goldbach`s Conjecture is also said to go for any other sentence in the language of mathematics. But is this picture sustainable in light of twentieth-century mathematics and philosophy? It seems not. The work of several logicians has shown just how pervasive so-called independent mathematical sentences are. These are sentences - sometimes of a very simple form, similar to Goldbach`s Conjecture - such that neither they nor their negations are provable from our most common mathematical theories alone. Standard logical results seem to suggest that independent sentences are true relative to some intended ways of interpreting our mathematical theories and false relative to others, yielding cases of indeterminacy. Faced with independence results, philosophers of mathematics divide themselves into two broad camps: some double down on the mathematician`s self-image and argue that independence only reveals that our theories do not describe mathematical truth with enough precision, whilst others renounce orthodoxy and accept that some mathematical sentences simply lack determinate truth-values. We think that progress on questions about determinacy has been hampered by researchers` strict reliance on classical logic. That is, questions about determinacy have, on the whole, been studied within a single logical framework - with its particular rules, languages, and other formal properties - which, as we see it, has had the effect of obscuring important mathematical structures together with the philosophical information that these structures provide. In this project, we systematically develop the distinctive dialectical move of analyzing traditional theorems and arguments through the lens of non-classical logic. We contend that the application of non-classical logics lends support to the thesis of mathematical indeterminacy, evincing an earlier conjecture found in the literature to the effect that the present confidence in arithmetical determinacy is merely contingent on present mathematical techniques. Furthermore, our results pave the way for new theories of arithmetic and their philosophy, and afford original perspectives on old philosophical problems about truth and reference.

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