Mathematics is now so deeply embedded in philosophy that assumptions baked into mathematical models—like the idea that a perfectly rational agent can solve any logical problem instantly—are shaping theories of belief, preference, and justice, even though no actual human or computer can meet those standards. This project addresses a gap in how philosophers understand the tools they use. When a political philosopher invokes Arrow’s impossibility theorem, or an epistemologist uses Dutch book arguments, they rely on mathematical frameworks that carry hidden idealisations—such as logical omniscience—that may distort the conclusions. The researchers aim to develop a systematic method, called “reverse philosophy,” to identify exactly which mathematical and computational principles are required for a given philosophical argument to hold. They will test this method on three case studies: paradoxes of truth and vagueness, subjective probability in epistemology and artificial intelligence, and voting theory in political philosophy. This is fundamental, curiosity-driven research with no immediate practical application. However, by clarifying the hidden assumptions in philosophical arguments that underpin fields like AI reasoning, decision theory, and social choice, the work could eventually inform how we design voting systems, build more realistic models of rational agents, or assess the limits of automated reasoning—systems that quietly shape how societies make collective decisions.
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We live in an era in which mathematics is employed in almost every academic discipline. Despite its traditional focus, philosophy is by no means exempt from this trend. This project provides a critical framework for addressing the increasing role of mathematics and computation in several of its core areas. The entanglement of mathematics with philosophy arises in two principal ways. The first is when mathematics is used to formulate philosophical claims, such as when real numbers are used to measure credences (degrees of belief), or when partial orderings are used to represent agents’ preferences. The second is when mathematics is used to derive philosophical conclusions, such as the use of Dutch book theorems in epistemology, or impossibility theorems like Arrow’s or Sen’s in political philosophy. But the use of these mathematical frameworks sometimes induces idealizations. For example, the mathematical theory of probability tacitly assumes “logical omniscience”: ideally rational agents are assumed to be able, when given any sentence, to determine whether it is a logical truth or not. Depending on the logic in question, this is either an infeasibly hard problem to compute (what computer scientists call NP-complete) or outright impossible (as hard as Turing’s halting problem). Even highly idealized or “artificially intelligent” agents are subject to these limitations. Our overarching objective is to develop a unified methodology for understanding the mathematical turn in contemporary philosophy. We aim to achieve this by employing a group of related methods from mathematical logic and computer science clustered around the subject known as reverse mathematics. These tools will be used to isolate the mathematical and computational principles required to sustain various philosophical arguments and theories via a novel method we call “reverse philosophy”. We investigate this concretely by investigating three interlinked case studies of the application of mathematics in philosophy: paradoxes of truth and vagueness, subjective probability in epistemology and artificial intelligence, and voting theory as applied to political philosophy.
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