This project investigates how mathematics impacts metaphysical realism-antirealism debates. Its goal is to develop a systematic account of the potentials and limits of using mathematics as a model for other a priori domains, thus responding to the rapidly increasing literature exploiting structural parallels between mathematics and domains like ethics, metaphysics, and religion.
While early Mesopotamian and Egyptian mathematics developed in response to the practical needs of settled civilizations, such as taxation, the measurement of plots of land, or the calculation of lunar calendars, philosophers soon assigned mathematics the additional role of paradigm for human reasoning and knowledge. Plato, for example, considered mathematics the highest form of knowledge, sharply distinguished from our uncertain beliefs about the empirical world. He thought that mathematical knowledge consists in knowledge of eternal ‘Forms’, perfect ideas that constitute ultimate reality, and he considered proficiency in mathematics essential for the acquisition of any knowledge at all. Similarly, Galilei saw mathematics as the ‘language of the Book of Nature’, Kant argued that mathematics provides essential insights into an entire class of human judgements, Cantor identified God with a particular class of transfinite numbers... In short: the idea that mathematics holds the key for understanding the fundamental structures of reality has, in different ways, dominated philosophy for centuries and also today many are convinced that the heuristic force of mathematics goes beyond its mere application in the natural sciences.
In current ontological, semantic, and epistemic debates that address these questions, mathematics frequently functions as a model for other a priori domains. Metaethicists in particular have started to use local structural parallels between mathematics and morality in order to corroborate particular metaethical views, but mathematics has also been argued to share relevant features with the domains of logic, modality, and religion. The mathematical analogies employed in those arguments share a common form: a local analogy between mathematics and another domain is identified, from which a global conclusion is drawn about one or both of the domains.
What is completely missing in these debates, however, is a discussion of
1. the plausibility of mathematics analogies in light of their mathematical background assumptions
2. how to develop an adequate methodology for analogical reasoning about a priori domains
3. the prospects of developing a unified framework for all a priori domains.
Only by answering these questions will it be possible to fully understand the evidential role mathematics has played for philosophers for centuries. This is the gap this project fills.
