Sean Walsh, a graduate of Notre Dame’s departments of philosophy and mathematics, has been awarded a Kurt Gödel Research Prize Fellowship—one of the most prestigious honors in the field of logic.
The Kurt Gödel Society, named for the late mathematical logician and philosopher, awards the fellowships to support original research that carries on his legacy.
Walsh completed a joint doctoral degree in logic and foundations of mathematics in January 2011 under the direction of Michael Detlefsen, McMahon-Hank Professor of Philosophy, and Peter Cholak, professor of mathematics. He says he is “deeply humbled and grateful” to receive such recognition for his work on “The Limits of Arithmetical Definability.”
Currently a post-doctoral fellow in the philosophy department at Birkbeck, University of London, Walsh studies the philosophy of mathematics, mathematical logic, and the intersection of the two.
“Some parts of my research are pure philosophy—I ask and try to answer questions about the nature of mathematical objects and the possibility of mathematical knowledge,” Walsh says. “Other parts of my research are pure mathematics: I try to formulate conjectures and prove them, availing myself of known proof techniques from different parts of mathematics and different parts of mathematical logic. Perhaps what is most distinctive about my research is that I attempt to combine these two things.”
Walsh’s award-winning paper is representative of this type of interdisciplinary approach. Specifically, Walsh says he used tools from contemporary mathematical logic to gauge the strength of the logical resources needed to effect one traditional reduction of arithmetic to logic.
“Such a reduction played a central role in Gottlob Frege’s logicism,” he says, “one key idea of which was that mathematical knowledge does not require an appeal to intuition or other quasi-perceptual resources, as had been claimed by [Immanuel] Kant in his famous Critique of Pure Reason.”
In addition to having his paper published in the Annals of Pure and Applied Logic, Walsh will also give a lecture on April 28 at the 2011 New Trends in Logic conference in Vienna, Austria.
Gödel—who lectured on set theory at Notre Dame in the spring semester of 1939—is perhaps most well known for establishing the incompleteness of arithmetic and the consistency of the continuum hypothesis in set theory. Gödel would have been 105 this April 28.