Research

Principal Interests

I have two longstanding research interests, namely

• Mutual influences between mathematics and philosophy throughout their histories

• Complexity of reasoning in mathematics (what it is, what causes it, how if at all we may measure it, why it matters and means of managing it)

Select Recent Publications

"Abstraction, Axiomatization & Rigor: Pasch and Hilbert", Festschrift for Hilary Putnam (G. Hellman and R. Cook eds.), Springer, to appear.

"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

"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.