Work in progress
I am currently working on papers on the following topics: * Grounding in Bolzano’s : in this paper I look at Bolzano’s own mathematical practice (namely his famous 1817 proof of the Intermediate Value Theorem) to learn about his views on grounding or explanatory proofs in mathematics, and the value of these views for writing good proofs and for discovering truths. An important issue in this paper is the relation between what Bolzano calls “pure” mathematics (algebra, arithmetic, and analysis) on the one hand, and “concrete” mathematics, in particular geometry, on the other, and the way in which a proof of the IVT from a geometric truth is circular in Bolzano's view. I presented different versions of this paper on conferences.
Purely Analytic Proof* The analytic-synthetic distinction and grounding: in a recent paper, Bob Hale and Crispin Wright argued that in order to make sense of the analytic-synthetic distinction, this distinction should be connected to the notion of grounding. I will show in this paper that although Bolzano did not write this explicitly, this was exactly what he had in mind: every analytic truth is grounded in (or explained by) a corresponding synthetic truth. I will also show that in the whole of analytic and synthetic truths, a particular position is assigned to those truths that Bolzano calls logically analytic, which leads me to conclude that (contrary to what has been argued recently in Bolzanian literature) the formal laws of logic (i.e. logic in the narrow sense, so as we conceive of it nowadays) are for Bolzano of minor importance with respect to grounding or explanation. In this paper I will expand further on my Master’s thesis.
* Bolzano on imaginary numbers: half a century before Frege, Bolzano defined the natural numbers on the basis of a theory of collections. In turn, he defined the other number systems on the basis of natural numbers. As a consequence, all number systems (and in fact all mathematical objects) have to be consistent with (i.e. obey the laws of) the natural numbers. Since from these laws it follows that no number is negative when squared, in Bolzano’s view imaginary numbers do not exist (in the sense that mathematical objects exist). But at the same time, Bolzano acknowledges that imaginary numbers are highly usefull and help us to discover many mathematical truths. Is there any explanation for the usefulness of imaginary numbers in mathematics within Bolzano’s system? Are his views on this matter terribly outdated, or is there something to his distinction between real and imaginary mathematical concepts?
Talks and publications
Invited lectures (as speaker)
* “Bolzano, Axiomatic Sciences, and Computational Methods within Philosophical Research”, 3-hour lecture at a Graduate Research Seminar at McMaster University, Hamilton, Canada (Oct. 2013). * “Phil@Scale: Mid-term presentation”, Network Institute, VU Amsterdam (Feb. 2013, with Sanne Vrijenhoek). * “Digital Humanities and (the History of) Philosophy”, ICT and Research Seminar at VU University Amsterdam (Jan. 2013, with Hein van den Berg). * “My Vision on Studying Philosophy”, Opening Ceremony of the Academic Year, VU University Amsterdam (Sept. 2011). Invited lectures (delivered by others)
* “Computational Tools for the History of Philosophy”, Workshop “Idealism, Realism, Empiricism – Philosophical Debates around 1800”, University of Utrecht (Nov. 2013, joint talk, with Arianna Betti, speaker: Hein van den Berg). * “Bolzano and Impeccable Explanation: The State of the Art”, IHPST Paris (Nov. 2013, joint talk, speaker: Arianna Betti). * “Computational Tools for the History of Philosophy (or What Is This Digital Humanities Thing and What’s in It for Us?)”, EGSAMP Summer School, VU University Amsterdam (Aug. 2013, joint talk; speaker: Arianna Betti). * “Phil@Scale”, VU Mini-symposium on Causality, VU University Amsterdam (Mar. 2013, joint talk; speaker: Arianna Betti). Contributed talks (with selection of abstracts)
* “Grounding in Bolzano’s Purely Analytic Proof” OZSW Conference 2015 (Dec. 2015). * “Grounding in Bolzano’s Purely Analytic Proof”, Third Annual Meeting of the Association for the Philosophy of Mathematical Practice (Nov. 2015). * “Grounding in Bolzano’s Purely Analytic Proof”, MCMP Summer School in Mathematical Philosophy for Female Students (July 2015). * “Grounding in Bolzano’s Purely Analytic Proof”, Truths and Grounds Conference Ascona (May 2015). * “The Potential of Computational Methods in Philosophical Research”, OZSW Graduate Conference in Theoretical Philosophy, Radboud University Nijmegen (Apr. 2015). * “Bolzanian Analyticity and Proper Proofs”, Bolzano in Prague 2014 (June 2014). * “Phil@Scale: Computational Methods within Philosophy”, DHLU Symposium 2013, University of Luxembourg (Dec. 2013, with Arianna Betti). Publications
* “Phil@Scale: Computational Methods within Philosophy” (with Sanne Vrijenhoek, Stefan Schlobach, Arianna Betti), Transactions in Digital Humanities, Luxembourg, conditionally
accepted.
* “Bernard Bolzano”, APhEX: Portale Italiano di Filosofia Analitica, commissioned, due in December
(in Italian).
My Master's thesis on Bolzanian analyticity and computational methods for philosophical research can be found here.
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