Exploring Logic and Mathematical Truths

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This chapter explores the distinctions between first order and second order logic, particularly focusing on the philosophical debates surrounding their implications. It delves into the nature of mathematical truths, discussing concepts such as quantifiers and the complexities of axiomatic systems in geometry and arithmetic. The conversation invites listeners to rethink foundational mathematical concepts and their philosophical underpinnings, particularly in light of Gödel's incompleteness theorems.

On a spectrum of philosophical topics, one might be tempted to put mathematics and morality on opposite ends. Math is one of the most pristine and rigorously-developed areas of human thought, while morality is notoriously contentious and resistant to consensus. But the more you dig into the depths, the more alike these two fields appear to be. Justin Clarke-Doane argues that they are very much alike indeed, especially when it comes to questions of “reality” and “objectivity” — but that they aren’t quite exactly analogous. We get a little bit into the weeds, but this is a case where close attention will pay off.

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Justin Clarke-Doane received his Ph.D. in philosophy from New York University. He is currently Associate Professor of philosophy at Columbia University, as well as an Honorary Research Fellow at the University of Birmingham and Adjunct Research Associate at Monash University. His book Morality and Mathematics was published in 2020.

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