On this page I will tell you a bit about my experiences with and interest in philosophy, mathematics, and Bolzano. At the bottom of this page you can find a list of my research interests, my work in progress, topics I’d like to work on in the future, and my full c.v. .
Philosophy & me
I did a Bachelor’s and Master’s in philosophy at the VU University Amsterdam in The Netherlands. Although I was interested in virtually every topic that was taught, right from the start I have been attracted by the manner of reasoning in analytic philosophy. This became my area of specialization.
I wrote a Bachelor’s thesis on the concept of analyticity (more specifically, on the account of analyticity that the philosopher of language/linguist Jerrold J. Katz presents in his book Cogitations). Analyticity was also central in my master thesis: here I investigated the highly original account of analyticity that was developed by the Czech polymath Bernard Bolzano, and the role that analytic truths have in proper science in Bolzano’s view. Another way to phrase the central question of my Master’s thesis is: what is in Bolzano’s view the explanatory value of analytic truths?
During my Master’s I have been working together with master student in Artificial Intelligence Sanne Vrijenhoek in a Network Institute’s Academy Assistants Project called Phil@Scale under the supervision of Arianna Betti. The goal of this project was to develop a computational tool to aid philosophers in doing text-based research. The project resulted in a computational tool based on text-mining, called SalVe. The central question of my Master’s thesis has been the test-case for SalVe, and an evaluation of SalVe in particular, and the potential of computational methods for philosophy in general, was part of my thesis. The project and its results showed that with computationally very basic methods, philosophers can gain a lot in speed and thoroughness of their text-based research.
After my Master’s at the VU, I have been at the Departement of Philosophy of the University of Notre Dame in South Bend, USA, as a visiting graduate student, and as a research assistant. Here, I fell in love with philosophy of mathematics.
At this moment I am a PhD student in Philosophy at the Scuola Normale Superiore in Pisa, Italy. I finished the course requirements and am working on my dissertation. The title of my dissertation is: Paradoxes of the Applied Infinite: Infinite Idealizations in Physics. The central question of the thesis is: (why) do we need mathematical infinity to understand the finite world?
In my dissertation, I analyze the currently debated problems concerning infinite idealizations in physics from the viewpoint of the philosophy of mathematics. What is taken to be problematic in this debate is – roughly – that physicists in their calculations often let certain parameters go to infinity (e.g. the number of components in the `thermodynamic limit’ of statistical mechanics), while our best scientific theories (the atomic theory of matter) tell us that physical systems are not actually infinite. Such techniques technically work well, but are philosophically problematic. In the debate on infinite idealizations, the classical mathematical framework which lies at the basis of the physical theories which give rise to these philosophical problems is never questioned – even though this framework is in itself already philosophically problematic, especially when applied to physics. My project is to clarify the extent to and manner in which the problems surrounding infinite idealizations in physics are caused by the classical mathematical framework and its notion of infinity, for example by means of considering what these problems will look like in a different mathematical framework based on a different notion of infinity (i.e. different from the now classical, cantorian, notion, thus for example constructive mathematical systems, or the theory of numerosities). As I see it, for a better understanding of the philosophical problems raised by physical theories, we need to understand to which extent they are relative to a mathematical framework.
Math & me
My interest in mathematics was sparked during my stay at the University of Notre Dame. There, I followed courses in philosophy of mathematics, in mathematical logic, mathematical modal logic, general proof writing, and real analysis. Having no significant background in mathematics (I basically only did high-school level “girls math” – that was how the math teacher used to call it), I was mostly surprised by the creative aspect of mathematics: how we can put certain mathematical concepts together and create something in an abstract realm; and how in this process certainly not “anything goes”, but nevertheless there is a lot of freedom.
My philosophical research is connected to mathematics in two main ways. First, in my experience philosophical reasoning (at least within the kind of philosophy that I like to do) is (or: should be) essentially mathematical reasoning: as I see it, both consist in clarifying concepts (or objects, things), seeing which concepts are consistent with each other, and recognizing what follows if you put certain concepts together in a certain way. Second, mathematics is the central (and my favorite) topic of my research. I find mathematics fascinating because of its pure and a priori character, what it tells us (and does not tell us) about human reasoning, and about how we experience and try to understand the world.
Bolzano & me
I have been introduced to the works of Bolzano during my Master’s, thanks to Arianna Betti. And as it seems to be the case for everybody who studied Bolzano: he got under my skin. I find Bolzano a fascinating thinker, because somehow he manages to have both a coherent framework in which he gives many, many topics a place and explanation, and at the same time to give many philosophical problems an honest and surprisingly thorough analysis. His writings are full of clever insights and resourceful inventions, and his style is marvelously clear. I think Lakatos was absolutely right in characterizing Bolzano as ‘the clearest logical mind of his century’. But there is something else with Bolzano: somehow his admirable personality shines through in his writings. He seems always concerned with the good of mankind, and even, so to say, with the good of his opponents, because he takes every opportunity to credit another thinker. As I see it, Bolzano offers a valuable lesson both in thorough and creative thinking, and in being a good person and a good academic.
Further, I have the good fortune that virtually all of the topics that I am interested in, were already discussed by Bolzano. Therefore, his works are for me often a good starting point and source of inspiration.
My research interests include:
- foundations of mathematics
- foundations of physics
- infinities and infinitesimals
- infinite idealizations in physics
- history of mathematics (in particular, of analysis and of set-theory)
- mathematical logic and its development
- constructive mathematics
- the (ontological) nature of mathematical concepts and truths
- methods of teaching/ learning mathematics
- grounding, or explanatory proofs, in contrast with non-explanatory proofs (primarily in mathematics)
- the notion(s) of logical consequence
- the benefits and limits of logical reasoning
- the analytic-synthetic distinction
- the works of Bernard Bolzano
- the works of Gottlob Frege
- the Classical Model of Science
- Digital Humanities, and its potential for philosophical research
Work in progress:
- Grounding in Bolzano’s Purely Analytic Proof: 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. An important issue in this paper is the relation between what Bolzano calls “pure” mathematics (algebra, arithmetic, and analysis), and “concrete” mathematics, in particular geometry, and the way in which a proof of the IVT from a geometric truth is circular. I presented different versions of this paper on conferences.
- 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 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 irrelevant to grounding or explanation. In this paper I will expand further on my Master’s thesis.
- Bolzano on the infinite: Bolzano is often credited for having anticipated Cantor’s transfinite theory. This because allegedly, he accepted (1) actually infinite collections, and (in the end) (2) one-to-one correspondence between the elements of collections as a sufficient criterion for equality of size. In this paper, I challenge claim (1) by supporting a reading of Bolzano not as platonism, but as a (in terms of Paola Cantù) semantic objectivism; I challenge claim (2) by showing that, as conjectured by Paolo Mancosu, Bolzano accepted one-to-one correspondence as a sufficient criterion for equality of sets only in arithmetical, and not in geometrical contexts.
And then there are the following topics, mostly totally unrelated to what I am doing now, that I am interested in and I would like to work on in the future:
- Philosophy of animal rights;
- Philosophy of technology;
- Free will versus determinism;
- Philosophy of psychology (I am particularly interested in the origin and justification (if there is any) of the fundamental concepts in psychology);
- Building a “big system” (virtually everybody in philosophy is working on the millimeter nowadays, I’d like to work on a big framework à la Kant, Hegel, etc. Just because nobody does it anymore, because it is exciting, and because I wanna see what my determinist intuitions about the real world imply on other levels).
You can find my full c.v. here.
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