Guanglong Luo

Enumerative inductive methods, which infer universal claims from a finite set of positive instances, are a cornerstone of empirical science and are also implicitly employed to justify belief in unproven mathematical conjectures like Goldbach’s Conjecture. This paper examines the scope and limits of this justificatory strategy. While I defend its use against simple “size-biased” skepticism for first-order mathematical claims, I argue that its application to justify the consistency of foundational axiomatic theories like Peano Arithmetic (PA) or Zermelo-Fraenkel set theory (ZF) is fundamentally problematic. The core of the argument is that evidence for consistency is not merely enumerative but also dimensional, crucially dependent on the length or complexity of the derivations surveyed. Drawing on proof theory, I demonstrate that inconsistencies can reside exclusively in proofs of lengths that transcend human practical and cognitive limits. Therefore, the inductive evidence from the absence of contradictions in all humanly surveyed proofs is probably weak and incomplete. Justified belief in consistency, I conclude, must be grounded not in enumerative induction alone, but in a holistic framework.

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, and we argue that all these strategies are untenable.

Recently, Kentaro Fujimoto raised a novel conservative argument against. deflationism regarding truth, presenting a potential challenge to this perspective. In this article, we aim to demonstrate that the effectiveness of his conservativeness argument greatly depends on its formalization. When employing more intuitive formalizations, the issue of non-conservativeness does not emerge. To substantiate this claim, we will offer two alternative formulations in response to Fujimoto's argument. © 2024 Elsevier B.V., All rights reserved.

In a recent article in this journal, Calemi challenges the Küng-Armstrong trilemma, a well-known objection to traditional class nominalism, by proposing a fusion of class nominalism with Zermelo-Fraenkel set theory (ZF). In this note, we argue that ZF-class nominalism faces significant challenges in the form of incompleteness and potential paradoxes stemming from Gödel’s incompleteness theorem. We will explore these issues in detail, highlighting the key implications for the viability of ZF-class nominalism as a philosophical position.

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, and we argue that all these strategies are untenable.