Mathematics and Metaphilosophy
by Justin Clarke-Doane

In Mathematics and Metaphilosophy, Justin Clarke-Doane discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the challenge to explain the (defeasible) justification of our mathematical beliefs ('the justificatory challenge'), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the challenge to explain their reliability ('the reliability challenge'), arises to the extent that we could have easily had different beliefs. The Element shows that mathematical facts are not, in general, empirically accessible, contra Quine, and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is unmysterious. The Element concludes with a complementary 'pluralism' about modality, logic and normative theory, highlighting its surprising implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap's pragmatism, transposed to the key of realism.

About the Author

Justin Clarke-Doane is an Associate Professor of Philosophy at Columbia University. His work centers on metaphysical and epistemological problems surrounding apparently a priori, or 'armchair', inquiry, like mathematical, logical, moral and modal inquiry. He is the author of Morality and Mathematics (Oxford University Press, 2020), Mathematics and Metaphilosophy (Cambridge University Press, 2022), and various articles. The Philosopher's Annual selected his articles as among the "ten best in philosophy" in 2013 and 2015. In addition to his primary areas of research, he has written on the mind-body problem, the philosophy of physics, and (with Kathryn Tabb) free will and psychopathology.

About the Speakers

David Z. Albert is the Frederick E. Woodbridge Professor of Philosophy at Columbia University. He earned his B.S. in Physics from Columbia College in 1976 and his Ph.D. in Theoretical Physics from Rockefeller University in 1981. Professor Albert is the author of Quantum Mechanics and Experience and Time and Chance, and has published numerous articles on quantum mechanics, primarily in the Physical Review.

Jenann Ismael is the William H. Miller III Professor of Philosophy at Johns Hopkins University. Previously, she held the position of Professor of Philosophy at Columbia University and was an affiliate of the Zuckerman Institute. She also taught at Stanford University from 1996 to 1998 and at the University of Arizona from 1998 to 2018. Her areas of specialization encompass Philosophy of Physics, Metaphysics, Philosophy of Science, Cognitive Science, and the Philosophy of Mind.

Christopher A.B. Peacocke is the Johnsonian Professor of Philosophy at Columbia University. His books include Sense and Content (Oxford University Press, 1983), Thoughts: An Essay on Content (Blackwell Publishing, 1986), and A Study of Concepts (Massachusetts Institute of Technology Press, 1992). He is known for his work in philosophy of mind and epistemology. His recent publications, in the field of epistemology, have defended a version of rationalism.

Jeannette M. Wing is Executive Vice President for Research and Professor of Computer Science at Columbia University. She previously served as Avanessians Director of the Data Science Institute. Her current research interests are in trustworthy AI. Her areas of research expertise include security and privacy, formal methods, programming languages, and distributed and concurrent systems. Wing’s seminal essay, titled “Computational Thinking,” was published more than a fifteen years ago and is credited with helping to establish the centrality of computer science to problem-solving in all other disciplines.