I defend the “settist” view that set theory can be done consistently without any form of distinction between sets and “classes” (by whatever name), if we think clearly about belief and the expression of belief—and this, furthermore, entirely within classical logic. Standard arguments against settism in classical logic are seen to fail because they assume, falsely, that expressing commitment to a set theory is something that must be done in a meaningful language, the semantics of which requires, on pain of Russellian paradox, a more powerful set theory. I explore the consequences of this response to the standard argument against “classical logic settism” for our notion of belief, and argue that what is revealed is that representationalist theories of belief cannot be right as long as it is possible to believe that every set is self-identical.

Contemporary mathematical theories are generally thought to be consistent. But it hasn’t always been this way; there have been times in the history of mathematics when the consistency of various mathematical theories has been called into question. And some theories, such as naïve set theory and (arguably) the early calculus, were shown to be inconsistent. In this paper I will consider some of the philosophical issues arising from inconsistent mathematical theories.

I distinguish two different views on what makes a proof of a theorem ‘pure’, firstly by characterizing them abstractly, and secondly by showing that in practice the views differ on what proofs qualify as pure.

It was shown by Frege that four of the five axioms of Peano can be regarded as analytical truths; and it was shown by Russell that the remaining axiom cannot be regarded as being analytically true or even as being analytically false, that this axiom thus is to be regarded as a synthetic statement. In using the concept of apriority in the sense of Reichenbach, it can be shown that this synthetic axiom is to be regarded as an apriorical truth within the usual background theory of measuring theories, which are used not as generalizations of empirical results but as— not moreover provable— preconditions of receiving measuring results and of ordering these results. Furthermore, the systems of numbers, starting with the natural numbers, are developed in a way such that the pre-rational numbers— but not the rational ones— turn out to be those ones which are used in performing measurements according to such theories, while the pre-real numbers— but not the real ones— then turn out to be those ones which are used in using such measuring theories together with their background theories for purely theoretical reasons.

The interaction between intuitions about inference, and the normative constraints that logical principles applied to mechanically-recognizable derivations impose on (informal) inference, is explored. These intuitions are evaluated in a clear testcase: informal mathe­matical proof. It is argued that formal derivations are not the source of our intuitions of validity, and indeed, neither is the semantic recognition of validity, either as construed model-theoretically, or as driven by the subject-matter such inferences are directed towards. Rather, psychologically-engrained inference-packages (often opportunistically used by mathematicians) are the source of our sense of validity. Formal derivations, or the semantic construal of such, are after-the-fact norms imposed on our inference practices.

This paper focuses on an argument against the existence of mathematical objects called the “Makes No Difference Argument” (MND). Roughly, MND claims that whether or not mathematical objects exist makes no difference, and that therefore, we have no reason to believe in them. The paper analyzes four different versions of MND for their merits. It concludes that the defender of the existence of mathematical objects (the mathematical Platonist) does have some retorts to the first three versions of MND, but that no adequate reply is possible to the fourth, and most crucial, version of MND. That version argues that mathematical objects make no difference to our epistemic processes: they play no role in the process of obtaining mathematical knowledge.