Research

Principal Interests

I have two longstanding research interests, namely

• Mutual influences between mathematics and philosophy throughout their histories

• Complexities of reasoning (what they are, what causes them, why they matter, how to manage them)

Select Recent Publications

"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, to appear, 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.

"Dedekind Against Intuition: Rigor, Scope and the Motives of his Logicism", in Logic and Knowledge, C. Cellucci, E. Grosholtz and E. Ippoliti (eds.), 205-221, Cambridge Scholars Publications, 2011.

"Purity of Methods", with Andrew Arana, Philosophers' Imprint 11:2 (2011): 1-20.

"Discovery, Invention and Realism: Gödel and others on the Reality of Concepts", in Mathematics & Its Signicance, John Polkinghorne (ed.), 73-96, Oxford University Press, 2011.

"Rigor, Re-proof and Bolzano's Critical Program", in Construction, P-E Bour et al. (eds.), 171-184, King's College Publications, 2010.

Select Recent Lectures

"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. 

"Axiomatic Method in the 19th and 20th Centuries", three lectures, Institut de Mathématiques, Universite Paul Sabatier, Toulouse, France, 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 Department Colloquium, University of Virginia, February  2013.

"Consistency Proofs & the Nature of Reasoning", French PhilMath Workshop IV, October 2012.

"Foundations & the Organization of Knowledge", invited talk Foundations of Mathematics: What are they, and what are they for?, Cambridge University, July 2012.

"On Axiomatic Methods & Their Ideals of Completeness", Bucharest Colloquium in Analytic Philosophy (BCAP), June 2012.

"Cantorian Freedom", keynote lecture, Twelfth Asian Logic Conference & Meeting of the Australasian Association of Logic (AAL), Victoria University of Wellington, Wellington, NZ, December 2011.

"Axiomatization, Formalization and Completeness", two lectures, Collège de France, October 2011.

  •      Lecture 1: Varieties of Axiomatization & Their Goals
  •      Lecture 2: Descriptive & Declarative  Axiomatization & Their Completeness Conditions

"Axiomatization, Epistemic Compression and Randomness", workshop on Randomness, Templeton Foundation, Calvin College, August 2011.

"Complexities of Proof", PhilMath Intersem 2011, Complexities of Proof, June 2011.

"Freedom in Mathematics", IREM (Institut de Recherche sur l'Enseignmement des Mathématiques de Paris 7), Seminaire d'Epistemologie et Histoire des Idees Mathematiques, May 2011.

"Freedom in Mathematics", Colloquium, Department of Philosophy, Carnegie Mellon University, April 2011.

"Impossibility, Permanence & the Axiom of Solvability", Impossibility Proofs in Mathematics, Université de Paris 7, PhilMath Workshop, REHSEIS-SPHERE, February 2011.