The constructive mathematics developed by Bishop in Foundations of Constructive Analysis succeeded in gaining the attention of mathematicians, but discussions of its underlying philosophy are still rare in the literature. Commentators seem to conclude, from Bishop’s rejection of choice sequences and his severe criticism of Brouwerian intuitionism, that he is not an intuitionist–broadly understood as someone who maintains that mathematics is a mental creation, mathematics is meaningful and eludes formalization, mathematical objects are mind-dependent constructions given in intuition, and mathematical truths (...) are experienceable. This paper develops and defends an intuitionistic interpretation of Bishop’s philosophical views. (shrink)

Non-well-founded set theories allow set-theoretic exotica that standard ZFC will not allow, such as a set that has itself as its sole member. We can distinguish plenitudinous non-well-founded set theories, such as Boffa set theory, that allow infinitely many such sets, from restrictive theories, such as Finsler-Aczel or AFA, that allow exactly one. Plenitudinous non-well-founded set theories face a puzzle: nothing seems to explain the identity or distinctness of various of the sets they countenance. In this paper I aim to (...) sharpen this puzzle, make clear who it does and does not apply to and, ultimately, to argue in favor of a plenitudinous theory like Boffa. (shrink)

According to the iterative conception of set, sets are formed in stages. According to the purely iterative conception of set, sets are formed by iterated application of a set-of operation. The cumulative hierarchy is a mathematical realization of the iterative conception of set. A mathematical realization of the purely iterative conception can be found in Peter Aczel’s type-theoretic model of constructive set theory. I will explain Aczel’s model construction in a way that presupposes no previous familiarity with the theories on (...) which it is based. (shrink)

An essay review of Gila Sher's *Logical Consequence*.

In his groundbreaking book, Thin Objects, Linnebo (2018) argues for an account of neo-Fregean abstraction principles and thin existence that does not rely on analyticity or conceptual rules. It instead relies on a metaphysical notion he calls “sufficiency”. In this short discussion, I defend the analytic or conceptual rule account of thin existence.

Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.

This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having (...) explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion. (shrink)

Informally rigorous mathematical reasoning is relevant. So too should be the premises to the conclusions of formal proofs that regiment it. The rule Ex Falso Quodlibet induces spectacular irrelevance. We therefore drop it. The resulting systems of Core Logic $ \mathbb{C}$ and Classical Core Logic $ \mathbb{C}^{+}$ can formalize all the informally rigorous reasoning in constructive and classical mathematics respectively. We effect a revised match-up between deducibility in Classical Core Logic and a new notion of relevant logical consequence. It matches (...) better the deducibility relation of Classical Core Logic than does the Tarskian notion of consequence. It is implosive, not explosive. (shrink)

In Intuition et idéalités: Phénoménologie des objets mathématiques (2020), Dominique Pradelle questioned the nature of mathematical knowledge–the status of math.

Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and prove that (...) it fails to characterize the domain uniquely. Thus, we conclude, there is no easy road to impredicative definabilism. (shrink)

Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst (...) affording the theoretical resources for a defence of Boffa set theory as a faithful description of set-theoretic reality. (shrink)

Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions’ lack of empirical factual content. In order to (...) see how this is the case, we will examine what it is to apply mathematics internally and describe examples. (shrink)

The paper introduces the notion of size of countable sets, which preserves the Part-Whole Principle. The sizes of the natural and the rational numbers, their subsets, unions, and Cartesian products are algorithmically enumerable as sequences of natural numbers. The method is similar to that of Numerosity Theory, but in comparison it is motivated by Bolzano’s concept of infinite series, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree, but some differ, (...) such as the size of rational numbers. However, set sizes are only partially, not linearly, ordered. Quid pro quo. (shrink)

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. (...) The final outcome is a taxonomy that should help researchers navigate the rich landscape of modal set theories. (shrink)