Juliette Kennedy's new book reminds us that some have claimed that the existence of so-called deviant encodings or numeral systems raises doubts about the wide acceptance of "Church's thesis" understood as the identification of computability with recursiveness for numerical functions. It is argued that if responses to such doubts, by Stewart Shapiro and others, in the literature do not already suffice to dispel such doubts, attention to careful discussions, by Charles Parsons and others, of ontological issues in philosophy of mathematics should be more than enough to do so.
According to realist or 'non-eliminative' versions of mathematical structuralism, mathematical objects are merely positions in structures, ontologically dependent on them. This raises questions about identity statements linking positions across distinct structures, such as 'the natural number 2 is identical to the real number 2'. I develop a novel Aristotelian account on which structures ontologically depend on their corresponding systems. On this in re version of structuralism, cross-structural identities are false, while the expressions flanking the identity sign are referentially indeterminate, even if isomorphism suffices for the numerical identity of structures.
Conceptual engineering is a philosophical activity that involves reframing or refining concepts to suit specific scientific, philosophical, or social contexts. Originally applied to ethical concepts, it has expanded to formal ones. Not all concepts can be engineered. Some are fixed points that preserve the intended conceptual structures of their frameworks; some are natural fixed points playing the same role across frameworks. Conceptual fixed points have been studied in formal contexts. Concepts 'truth', 'existence', 'quantifier' resist engineering. This paper explores the concept of computation as explained by the Church-Turing thesis and argues that it constitutes a natural conceptual fixed point.
Consistency appears to be an essential feature of mainstream mathematics: practices deemed inconsistent are either rejected as incorrect, or treated as outsiders. In this paper I analyze the role of consistency in mathematics by looking at deviations from consistency: I argue that attempts to ground the distinction between mainstream and inconsistent mathematics cannot succeed without incorporating some kind of practice-level commitment to inconsistency, and propose a notion of inconsistent practice which clarifies, by way of contrast, how consistency functions in the mainstream. This exemplifies a general methodology for better understanding core features of mainstream practices through contrast with deviating practices.
I investigate the modal commitments of the various conceptions of arithmetic potentialism that arise from the models of arithmetic by taking them as realms of feasibility with respect to their natural extension concepts, such as end extensions and arbitrary extensions, thereby shedding light on the range of differing philosophical positions available for arithmetic potentialism. The main analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account.
This paper focuses on a lesser known version of abstractionism - called Arbitrary Logicism - which combines a deflationist account of abstraction and recent proposals about arbitrary reference. It makes explicit the relation of this proposal to canonical versions of Fregean abstractionist programs. In particular, Arbitrary Logicism will be tested against the epistemological claims of the abstractionist tradition, within which it emerges as an interesting alternative to Neologism and a direct heir to Frege's Logicism.
Structures are ubiquitous in mathematics. But how should they be understood? Modelists claim they are model-theoretic structures. This thesis can be read in two ways: as a claim about what structures refer to, or about how we conceptualize them. Objects-modelism, developed by Button and Walsh, pursues the first; the second leads to concepts-modelism, which remains underexplored. In this paper we develop and defend a version of concepts-modelism, cognitive modelism, drawing on Carey’s theory of conceptual development, and we show how it addresses the challenges Button and Walsh pose for a conceptual account of mathematical structures.
Accepting some form of potentialist set theory promises to help us solve puzzles about the intended height of the hierarchy of sets. However, philosophers have developed two different schools of potentialist set theory: minimalist and dependence-based approaches. In this paper, I will argue that minimalist formulations of potentialism have some important advantages over dependence-based formulations.
I shall explore various senses in which ultrafinitism fruitfully engages with the potentialist perspective in mathematics. For example, every model $ M $ of the theory of finite arithmetic - arithmetic with a largest number, in which addition and multiplication are merely partial functions - is bi-interpretable with a strictly taller such model $ M<^>{+} $, in which the arithmetic of the prior numbers becomes fully defined. By iterating this construction, we find a deep connection between the models of finite arithmetic and the theory of bounded induction I$ \Delta_{0} $. More generally, ultrafinitist ideas emerge in the potentialist system of all models of arithmetic under end-extension.
