Daniel Waxman

This paper investigates the procedure of conceiving of a model of mathematical structures, where this involves a distinctive kind of visual mathematical thinking. I argue that (i) conceiving of structures in this way is best understood as an exercise of the imagination (and not, as many philosophers would contend, an exercise of rational intuition); (ii) once the relevant kind of imaginative capacity is clarified, it becomes apparent that we can in fact conceive of infinite mathematical structures; and (iii) by doing so, we obtain justification in the consistency or coherence of certain mathematical theories.

A striking fact about mathematics is that it is, in a certain sense, free or unconstrained in its practice: mathematicians are free, in their capacity as mathematicians, to work within any theory provided that minimal standards of logical coherence are satisfied. How can this feature of mathematics be explained? I argue that contemporary views in the philosophy of mathematics are unable to provide a satisfying explanation. I present a novel account of mathematical truth, drawing on the doctrine of alethic pluralism, according to which (to a first approximation) mathematical truth is realized by the property of coherence. I conclude by showing that this view provides a surprisingly fruitful framework for resolving resolving issues of mathematical disagreement and the question of new set theoretic axioms.

In 1936, Gerhard Gentzen famously gave a proof of the consistency of Peano arithmetic. There is no disputing that Gentzen provided us with a mathematically valid argument. This paper addresses the distinct question of whether Gentzen's result is properly viewed as a proof in the epistemic sense: an argument that can be used to obtain or enhance justification in its conclusion. Although Gentzen himself believed that he had provided a “real vindication” of Peano arithmetic, many subsequent mathematicians and philosophers have disagreed, on the basis that the proof is epistemically circular or otherwise inert. After gently sketching the outlines of Gentzen's proof, I investigate whether there is any epistemically stable foundational framework on which the proof is informative. In light of this discussion, I argue that the truth lies somewhere in between the claims of Gentzen and his critics: although the proof is indeed epistemically non-trivial, it falls short of constituting a real vindication of the consistency of Peano arithmetic.

In Mathematical Truth, Paul Benacerraf famously raised a dilemma for the philosophy of mathematics: very roughly, that mathematics needs a reasonable semantics and a reasonable epistemology but that it's impossible to see how these needs can be simultaneously satisfied. In this paper I argue that (i) Benacerraf's semantic constraint is in fact ambiguous between a stronger and a weaker reading; (ii) only the stronger reading is incompatible with the idea that mathematical truth is an epistemically tractable property; and (iii) the stronger reading is under-motivated.

Many who have considered the subject believe that our best mathematical theories (e.g. arithmetic, analysis, and set theory) are consistent. But, in light of Gödel's incompleteness theorems and for other reasons, it is not at all obvious how this attitude can be rationally defended. In this paper I consider one influential argument (schema) for the consistency of mathematical theories, going via the notion of truth. Roughly, it runs as follows: the axioms of the theory are true, the inference rules of logic preserve truth, so all of the theorems are true; but if the theory were inconsistent, it would have false theorems; so it must be consistent. I argue that this truth-theoretic argument is, on many plausible views of mathematics, epistemically useless: in the sense much discussed in recent epistemology, it is incapable of transmitting justification from its premises to its conclusion.