I have two aims in my current project in philosophy of mathematics, using episodes from the history of number theory as my case studies. One is to understand better the nature of ampliative reasoning in mathematics, reasoning that adds content as it solves problems. The other is to improve our philosophical account of the historicity of mathematical knowledge. Discussions with Hourya Sinaceur, Karine Chemla, David Rabouin, Valérie Debuiche and Claude Imbert in Paris; with Herbert Breger in Hannover and Berlin; with Carlo Cellucci and Emiliano Ippoliti in Rome; with Donald Gillies and Brendan Larvor in London; and with Dirk Schlimm, Danielle Macbeth, Andy Arana and Norma Goethe on this side of the Atlantic, have played an important role in the evolution of my thought since I wrote *Representation and Productive Ambiguity in Mathematics and the Sciences* (Oxford University Press, 2007).

In serious mathematical research, problem-solving, illuminating proof and good explanation typically enlarge knowledge, going beyond what is currently accepted, and given in the formulation of the problem. This fact about research has been obscured by the assumptions of logicist philosophers of mathematics, who pretend, or hope, that mathematical reasoning can be translated into predicate logic (perhaps with set theory added), and then located within the closed box of an axiomatic system, proved from the first principles by deductive logic. But, first, this is how classroom or textbook problem-solving proceeds, not mathematical research. Second, mathematical reasoning is carried out in highly specific languages, both symbolic and iconic, created for solving certain kinds of problems: one-way translation (into predicate logic, for example) diminishes the expressive, explanatory and exploratory power of those languages. Third, sometimes translation enhances the expressive, explanatory and expressive power of mathematical languages, but only when the disparate languages are retained: juxtaposed, superposed, and brought into novel rational relation by natural language. It is thus philosophically rewarding to examine how the growth of mathematical knowledge in fact occurs, in detail, by studying the historical record or working with mathematicians whose research is in the process of enlarging our knowledge. I will use specific episodes from the history of algebraic number theory to show how problem-solving adds significant content to mathematical knowledge.

Analysis is the search for conditions of the intelligibility of existing things in mathematics (as Leibniz defines it), or for the conditions of the solvability of a problem (as Pappus defines it). It is either way an ampliative process, which increases knowledge as it proceeds. From an analysis, an argument can be reconstructed, as when Andrew Wiles (building on the results of many other mathematicians) finally wrote up the results of his seven years-long search in the 130 page full dress proof in the May, 1995 issue of *Annals of Mathematics*. Analysis begins with a problem to be solved, on the basis of which one formulates a hypothesis; the hypothesis is another problem which, if solved, would constitute a sufficient condition for the solution of the original problem. To address the hypothesis, however, requires making further hypotheses, and this "upwards" path of hypotheses must moreover be evaluated and developed in relation to existing mathematical knowledge, some of it available in the format of textbook exposition, and some of it available only as the incompletely articulated "know-how" of contemporary mathematicians. Indeed, some of the pertinent knowledge will remain to be forged, since the pathway of hypotheses sometimes snakes between, sometimes bridges, more than one domain of mathematical research (some of which may be axiomatized and some not), or when the demands of the proof underway throw parts of existing knowledge into question.

The philosophy of mathematics is in the process of renegotiating its relation to the history of mathematics; I am happy to be on the Directive Committee of the Association for the Philosophy of Mathematical Practice, and look forward to our upcoming meetings in Helsinki (August 2015) and Paris (November 2015). But this relationship must be undergirded by a philosophy of history that does not reduce the narrative aspect of history to the forms of argument used by logicians and natural scientists. History is primarily narrative because human action is, and therefore no physicalist-reductionist account of human action can succeed. So philosophy must acknowledge and retain a narrative dimension, since it concerns processes of enlightenment, the analytic search for conditions of intelligibility. Indeed, the very notion of a problem in mathematics is historical, and this claim stems from taking as central and irreducible (for example) the narrative of Andrew Wiles's analysis that led to the proof of Fermat's Last Theorem. If mathematical problems have an irreducibly historical dimension, so too do theorems (which represent solved problems), as well as methods and systems (which represent families of solved problems): the logical articulation of a theory cannot be divorced from its origins in history. This claim does not presume to pass off mathematics as a series of contingencies, but it does indicate in a critical spirit why we should not try to totalize mathematical history in a formal theory.

This project was supported by a Research in Paris 2011 Grant (Senior Level), for work as a Senior Foreign Researcher at REHSEIS / SPHERE, UMR 7219 CNRS and University of Paris Denis Diderot – Paris 7, September 2011-January 2012 ($22,500), as well as an NEH Fellowship, for work on a book on the philosophy of mathematics, 2004-2005, also in Paris with REHSEIS. ($24,000).

This research has resulted in the following publications:

*Logic and Knowledge*. Co-edited with Carlo Cellucci and Emiliano Ippoliti. Cambridge Scholars Publishing, 2011. The Introduction is written jointly by the three editors, and the collection of eighteen essays includes my "Logic, Mathematics, Heterogeneity."*Representation and Productive Ambiguity in Mathematics and the Sciences*. Oxford University Press, 2007. This book has been given substantive, positive reviews in*Philosophia Mathematica*,*Review of Metaphysics*,*Isis*,*British Journal for Philosophy of Science, Studia Leibitiana, Philosophical Reviews*(Notre Dame, on line journal), the*Mathematical Intelligencer*,*Metascience*,*History and Philosophy of Log*ic, and*O nó dó problema ocidental*(a philosophical blog in Portuguese).- "Leibnizian Analysis, Canonical Objects, and Generalization," in
*Handbook on Generality*, ed. Karine Chemla, David Rabouin, and Renaud Chorley. Forthcoming in 2015. - "Leibniz, Locke and Cassirer: Abstraction and Analysis,"
*Leibniz and Analysis, Studia Leibnitiana*, Band 45, Heft 1 (2014), Special issue on Analysis edited by H. Breger and Wen Chao Li, pp. 97-108. - "Fermat's Last Theorem and the Logicians,"
*From a Heuristic Point of View: Essays in Honor of Carlo Cellucci*, E. Ippoliti, ed. Cambridge Scholars Publishing, 2014, pp. 147-162. - "Teaching the Complex Numbers: What History and Philosophy of Mathematics Suggest,"
*Journal of Humanistic Mathematics*, Vol. 3, No. 1. (On-line) - "Philosophy of Mathematics and Philosophy of History,"
*Paradigmi: Revista di critica filosofia*. Special issue on philosophy of mathematics, ed. C. Cellucci, Vol. 29, No. 1 (2011), pp. 13-28. - "Logic, Mathematics, Heterogeneity," with a discussion by Valeria Giardino,
*Logic and Knowledge*, E. Grosholz, C. Cellucci and E. Ippoliti, eds., Cambridge Scholars Publishing, 2011, pp. 305-319. - "Locke et Leibniz: Forme et expérience," (tr. Gaetan Pegny), in
*Locke et Leibniz: Deux styles de rationalité*, M. de Gaudemar and P. Hamou, eds., Europaea Memoria Series, Series I, Vol. 84, Georg Olms, 2010, pp. 93-108. - "Leibniz on Mathematics and Representation: Knowledge through the Integration of Irreducible Diversity,"
*The Philosophy of the Young Leibniz*, M. Kulstad, M. Laerke, and D. Snyder, eds.,*Studia Leibnitiana*Sonderheft 35, Franz Steiner Verlag, 2009, pp. 95-110. - "Simone de Beauvoir and Practical Deliberation,"
*PMLA*, Vol. 124, No. 1 (January 2009), pp. 199-205. - "Productive Ambiguity in Leibniz's Representation of Infinitesimals,"
*Infinitesimal Differences: Controversies between Leibniz and his Contemporaries*, U. Goldenbaum and D. Jessup, eds., Walter de Gruyter, 2008, pp. 153-170. - "Locke, Leibniz and Hume on Form and Experience,"
*Leibniz: What Kind of Rationalist?*, ed. M. Dascal, Springer Academic Publications, 2008, pp. 167-182. - "How to Say True Things in Algebraic Topology,"
*Demonstrative and Non-demonstrative Reasoning in Mathematics and Natural Science*," eds. P. Pecere and C. Cellucci, Edizioni dell' Università degli Studi di Cassino, 2006, 27-54. - "Constructive Ambiguity in Mathematical Reasoning,"
*Mathematical Reasoning and Heuristics*, eds. D. Gillies and C. Cellucci, King's College Publications, 2005, pp. 1-23. - "Jules Vuillemin's
*La Philosophie de l'algèbre*: The Philosophical Uses of Mathematics,"*Philosophie des mathématiques et théorie de la connaissance: L'Oeuvre de Jules Vuillemin*, R. Rashed and P. Pellegrin, eds. Paris: Albert Blanchard, 2005, pp. 253-270. - "Symmetry," review essay of
*Love and Math: The Heart of Hidden Reality*, by Edward Frenkel (New York: Basic Books, 2013),*Journal of Humanistic Mathematics*(published online February 2015). - Review essay of
*Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method*, by Carlo Cellucci(Springer, 2013),*Philosophia Mathematica*(published online September 2014). - Review essay of William Byers,
*How Mathematicians Think, Using Ambiguity, Contradiction, and Paradox to Create Mathematics*(Princeton University Press, 2007),*The Mathematical Intelligencer*, Vol. 36, issue 3 (September 2014), pp. 84-87. - "Great Circles: The Analysis of a Concept in Mathematics and Poetry," Special issue on poetry, ed. S. Glaz,
*Journal of Mathematics and the Arts*, Vol. 8 / 1-2, 2014, pp. 24-30. - "Against One-Sidedness," review essay of Edward Skidelsky,
*Ernst Cassirer, The Last Philosopher of Culture*(Princeton University Press, 2007),*Hudson Review*, Vol. LXII, No. 4 (Winter 2010), pp. 691-98. Reprinted in Spanish translation by Cinthia García Soria,*La Gaceta*No. 492 (Dec. 2011), pp. 6-8; Fondo de Cultura Económica, Mexico City, Mexico. - Review essay of Joseph Mazur,
*Euclid in the Rainforest*, Pi Press, 2005,*The Mathematical Intelligencer*, Vol. 28, No. 2 (2006), pp. 84-88. - Review essay of Karine Chemla et Guo Shuchun, eds.,
*Les Neuf Chapitres: Le Classique mathématique de la Chine ancienne et ses commentaires*, Dunod, 2004,*Gazette des Mathématiciens*, No. 105 (July 2005), pp. 49-56. - Review of Chikara Sasaki,
*Descartes's Mathematical Thought*, Kluwer, 2003,*Philosophia Mathematica*(III) 13 (2005).

And I have given the following presentations on these topics:

- "The Specificity of Mathematical Language and the Uses of Ambiguity," 15
^{th}Congress of Logic, Methodology and Philosophy of Science, Helsinki, August 2015. - "Theory Reduction and Mathematical Meaning: Gödel Numbering, Logic and Number Theory," CEPERC, University of Aix, March 2015.
- "Algebraic Number Theory and the Complex Plane," IREM, Poincaré Institute, Paris, March 2015.
- "Number Theory and the Complex Plane," Second Joint International Meeting of the Israel Mathematical Union and the American Mathematical Society, Panel on Recent Trends in the History and Philosophy of Mathematics, Tel Aviv, June 2014.
- "Great Circles: The Analysis of a Concept in Mathematics and Poetry," Workshop on Creative Writing in Mathematics and Science, Banff International Research Station for Mathematical Innovation and Discovery, Banff, Canada, November 2013.
- "Reducibility and Meaning: Logic and Number Theory," Midwest Philosophy of Mathematics Workshop 2013, Champaign – Urbana, Illinois, October 2013.
- "Ampliative Reasoning: Explanation in Number Theory," Conference on Mathematical Values, Mathematical Cultures Project, London, September 2013.
- "Problem Reduction in Algebraic and Analytic Number Theory," Mathematics and Philosophy (19
^{th}and 20^{th}c.) Working Group, SPHERE, University of Paris Denis Diderot – Paris, December 2012. - "Philosophy of Mathematics and Philosophy of History: Wiles' Proof of Fermat's Last Theorem," Distinguished Visitor Program, Haverford College, February 2012.
- "Fermat's Last Theorem and the Logicians," Weeklong workshop on Explicit versus Tacit Knowledge in Mathematics, Mathematisches Forschungsinstitut Oberwolfach, Germany, January 2012. I presented a revised version of the paper to the STS Seminar, University College London, January 2012.
- "Fermat's Last Theorem and the Number Theorists," Institut Poincaré, Paris, January 2012.
- "Leibniz, Locke, and Cassirer: Abstraction and Analysis," Conference on Analysis as a Mathematical Method: Leibniz and Precursors," Leibniz Endowed Professorship (Wenchao Li), Leibniz Universität Hannover, Germany, May 2011.
- "Logic, Mathematics, and Heterogeneity," Conference on Logic and Knowledge, University of Rome La Sapienza, June 2010.
- "Abstraction vs. Generality: Cassirer's
*Substanz und Funktion*," REHSEIS / University of Paris Diderot – Paris 7, March 2009. - "Teaching the Complex Numbers: What History and Philosophy of Mathematics Suggest," University of Paris Diderot – Paris 7, March 2009.