Carnap introduced his notion of explication to arrive at concepts that are precise enough to progress science. This means that one has to start from less precise concepts---an understanding of explication inapplicable to mathematics. We show that within mathematics explications are widespread. We discuss examples such as topological spaces, prime numbers, and Imre Lakatos's case-study of the polyhedron. We argue that such explications are needed for the modern mathematical endeavour which takes foundational work serious. E.g., automated theorem proving necessitates the expansion of functions and introduction of conventions which are kept. There also are foundational aspects that might make some type system desirable, also enforcing the distinction between addition on different numbers. Such cases appear to be basic, but they either render advanced topics false or enforce being explicit about several of the implicit assumptions.

We argue that these developments correlate with arguments of productive ambiguity within mathematics. Carnap sees potential explications as \emph{proposals} which are not themselves truth-apt. However, as Carnapian explication is not applicable to such cases, we argue that these mathematical case-studies also motivate a different notion of explication which covers the mathematical cases by allowing precise mathematical concepts to be explicated.

Logic is often considered theoretically neutral, imposing no significant constraints on the world. In this talk, we explore how this purported neutrality can be understood, focusing on a proposal Beall has done. He suggests that logic should be viewed as the universal closure relation of theories. Under this view, a logic that just closes truths gathered by extralogical theories is superior to a logic independently proving logical truths. The main thesis of the talk is that the absence of logical truths can be interpreted as a hallmark of logic's neutrality. We present this proposal, examine its complexities, and ultimately argue that this account of neutrality is tenable when approached through a meta-linguistic framework.

Some philosophers of mathematics have argued that mathematics “cannot be regarded as a priori” (Ferreirós 2016, 310). The argument points to mathematical practices, and in particular to cases in which mathematicians successfully justify the truth of a mathematical proposition. It is argued that these practices often fail to meet the rigorous standards of apriorism.

For example, Lakatos (1976ab) argues that mathematical justifications are tentative, fallible, and sometimes even plainly empirical. Putnam (1979, 64) claims that mathematicians often use “quasi-empirical” justifications that are not so different from those used in the sciences. More recently, such claims have been echoed in historical studies (Ferreirós 2016), in questions about the admissibility of probabilistic and computational proofs (Fallis 1997; McEvoy 2007), and in the distinction between proofs and simil-proofs (De Toffoli 2021).

In this talk I will argue that these attempts to establish aposteriorism are mistaken about the a priori/a posteriori distinction. Mathematical practices cannot give us the kind of evidence needed to settle whether mathematics is a priori or a posteriori. I will argue that doing so requires modal evidence, whereas practices can only give us factual evidence.

Enumerative induction in mathematics—the idea that sampling evidence can yield high credence or even knowledge of mathematical statements—was long dismissed, particularly by Frege. However, it has recently regained prominence. This paper aims to move the debate about its viability forward in several ways. First, it clarifies what exact epistemic good enumerative induction might secure. It then critically evaluates three recent arguments for enumerative induction’s viability. Based on this evaluation, the paper proposes a new criterion for explaining why enumerative induction can, in some settings, surmount inductive scepticism. Finally, a case study on Goldbach’s Conjecture illustrates when enumerative induction can meet this criterion.