Research
Principal Interests
My two longstanding research interests are
• The mutual influences mathematics and philosophy have had on each other throughout their histories
• The significance of complexity in mathematical reasoning: What is it, and what significant different forms does it take? What types of things significantly increase it? In what ways does complexity matter and why? What forms of complexity are measurable? When we can measure it, how may we best measure it? What forms of complexity are manageable? When we can manage it, how may we best manage it?
Select Recent Publications
``Formalism and Hilbert's understanding of consistency problems", proceedings of WoLLIC 24, to appear
"Abstraction, Axiomatization & Rigor: Pasch and Hilbert", Hilary Putnam on Logic and Mathematics (G. Hellman and R. Cook eds.), 161-178, Springer, 2018
"On the Motives for Proof Theory", in H. Wansing (ed.), Dag Prawitz on Proofs and Meaning, 137-164, Outstanding Contributions to Logic, Springer, 2015
"Gentzen's Anti-formalist Ideas", in Gentzen's Centenary: The Quest for Consistency, M. Rathjen and R. Kahle (eds.), Springer, 2015
"Duality, Epistemic Efficiency & Consistency", in G. Link (ed.), Formalism and Beyond, Logos, De Gruyter, 1-24, 2014
"Completeness and the Ends of Axiomatization", in Interpreting Gödel, J. Kennedy (ed.), 59-77, Cambridge University Press, 2014
"Freedom and Consistency", in Proceedings of the 12th Asian Logic Conference, J. Brendle, R. Downey, R. Goldblatt and B. Kim (eds.), 89-111, World Scientific, 2013
Select Recent Lectures
``Weyl on proof, intuition and knowledge'', invited lecture, Das Kontinuum: One hundred years later, congress honoring the centenary of the publication of Hermann Weyl's Das Kontinuum, School of Mathematics, University of Leeds, Sep 14, 2018
``Proof and Intuition in mathematics", invited lecture, Science's Voice of Reflection, University of Amsterdam, Sep 6, 2018
``Pasch, Hilbert and their different conceptions of rigor", invited lecture, joint meeting of the OCIE-HPM seminar and the SoCal HPL seminar, University of California-Riverside, March 19, 2018
"Hilbert's Approach to the Consistency Problem for Arithmetic", Western Illinois University, Dept of Mathematics, April 20, 2017
"Elements of Formalism", Charles University, Prague, conference on The Emergence of Structuralism and Formalism. Plenary lecture, June 2016
'Kreisel on formalism", IHφST, U of Paris 1. Conference on Intuitionism, Computation, and Proof: Selected themes from the research of G. Kreisel. Plenary lecture, June 2016.
"Bolzano's Critical Program", Inter-university Centre, Dubrovnik, Croatia. 43rd Annual Philosophy of Science Meeting. Invited lecture in Philosophy of Mathematics section. April 2016.
Institut de Mathématiques, U of Toulouse. Invited series of four lectures on "Gödel's incompleteness theorems and their applications." April 2016.
- Lecture 1: "Introduction & Background: Gödel's Statements of the Incompleteness Theorems"
- Lecture 2: "Generalizing Gödel's Second Theorem & the Stability Problem"
- Lecture 3: "Hilbert's Program: Axiomatic Method & the Consistency Problem"
- Lecture 4: "Hilbert's Program & Gödel's First Incompleteness Theorem"
"Duality and its Benefits", Colloquium, Dept of Philosophy, Kansas State University, November 2014.
"Peirce and the Necessity of Proof", invited lecture, Pragmatism and the Practical Turn in Philosophy of Science, annual meeting, Académie Internationale de Philosophie des Sciences, Pont-à-Mousson, France, September 2014.
"Mathematical Induction & the Knowledge it Represents'', invited lecture Perspectives on Induction symposium, sponsored by CSL-LICS (Logic in Computer Science) and the LC (Logic Colloquium), Vienna Summer of Logic, July 2014.
Institut de Mathématiques, University of Toulouse. Invited series of three lectures on "Axiomatic Method in the 19th and 20th Centuries", three lectures, March 2014.
- Lecture 1: The Axiomatic Method Old & New
- Lecture 2: Abstract Axiomatization & Its Benefits: Rigor & Freedom
- Lecture 3: Abstract Axiomatization & Its Benefits: Simplicity/Efficiency
"The Axiomatic Method & Its Completeness Ideals", Philosophy Department Colloquium, Case Western Reserve University, March 2013.
"Mathematical Freedom and Consistency", Philosophy Dept, University of Virginia, February 2013.
"Consistency Proofs & the Nature of Reasoning", French PhilMath Workshop IV, Paris, October 2012.
"Foundations & the Organization of Knowledge", invited talk Foundations of Mathematics: What are they, and what are they for?, Fitzwilliam College, Cambridge University, Cambridge, July 2012.
"On Axiomatic Methods & Their Ideals of Completeness", Bucharest Colloquium in Analytic Philosophy (BCAP), Bucharest, June 2012.