Axiomatizations of arithmetic and the first-order/second-order divide

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s (Philos Top 17(2):69–90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus second-order logic divide. The descriptive approach is illustrated by Dedekind’s ‘discovery’ of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Frege’s Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects

Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views

An interactive approach to proof-theoretic semantics

In truth-functional semantics for propositional logics, categoricity and compositionality are unproblematic. This is not the case for proof-theoretic semantics, where failures of both occur for the semantics determined by monological entailment structures for classical and intuitionistic logic. This is problematic for inferentialists, where the meaning of logical constants is supposed to be determined by their rules. Recent attempts to overcome these issues have primarily considered symmetric entailment structures, but these are tricky to interpret. Here, I instead consider an entailment structure that combines provability with the dual notion of disproof (or refutation). This is interpreted as a dialogue structure between the roles of prover and denier, where an assertion of a statement involves a commitment to its defence, and a denial of the statement involves a commitment to its challenge. The interaction between the two is constitutive of a proof-theoretic semantics capable of dealing with the above issues

Objectivity Sans Intelligibility. Hermann Weyl's Symbolic Constructivism

A new form of skepticism is described, which holds that objectivity and understanding are incompossible ideals of modern science. This is attributed to Weyl, hence its name: Weylean skepticism. Two general defeat strategies are then proposed, one of which is rejected as a failure

Categoricity

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a syste