2026/07/01 by Andrew Bacon
Psychology · Arts and Humanities · Mathematics · #Philosophy and Theoretical Science #Wittgensteinian philosophy and applications #History and Theory of Mathematics
paper · doi:10.1215/00318108-12453671
Could the truths of mathematics have been different than they in fact are? If so, which truths could have been different? Do the contingent mathematical facts supervene on physical facts, or are they free floating? This article investigates these questions within a framework of higher-order modal logic, drawing sometimes surprising connections between the necessity of arithmetic and analysis and other theses of modal metaphysics: the thesis that possibility in the broadest sense is governed by a logic of S5, that what is possible holds in some maximally specific possibility, and that every property can be rigidified. The investigation will distinguish sharply between platonic contingency—contingency about whether particular abstract platonic mathematical objects are arranged in a certain way (e.g., in the structure of the natural or real numbers)—from a deeper variety of structural contingency concerning what holds of objects whenever they are arranged in that way.