**The concept of infinity has many applications throughout history in the fields of mathematics, philosophy, logic, and probability. New perspectives on the concept of infinity have brought together Professor Paolo Mancosu (Professor of Philosophy at UC Berkeley) and Professor Marco Panza (Director of research a****t****the CNRS; IHPST, CNRS and Univ. of Paris 1 Panthéon-****Sorbonne and Professor at Chapman University). In this interview they discuss how they came to research this topic, what they have learned from it, and how the France-Berkeley Fund has supported them.**

**Can you both introduce yourselves and talk about your academic journeys that led you to this project? **

PM: I am Paolo Mancosu and I am a professor in the philosophy department at UC Berkeley. My areas of work are mathematical logic and the history and philosophy of logic and mathematics. Throughout my career (Milan, Stanford, Oxford, Yale, UC Berkeley), I have been engaged with the challenges that mathematical infinity poses to philosophy. This takes very different forms: can infinite objects exist? If so, how can we have access to them? How can reasoning with infinite objects be grounded in an epistemologically satisfactory way? These issues have been central to the foundations of mathematics, for instance in Bolzano, Frege, Cantor, Dedekind and Hilbert. The project itself focuses on some very recent mathematical results that justify new ways of counting the size of infinite sets.

MP: I am Marco Panza and I am a research director at the CNRS (IHPST, CNRS and Univ. of Paris 1 Panthéon-Sorbonne) and a professor at Chapman University (Orange, CA). I’m a historian and philosopher of mathematics and logic. I previously held appointments in Geneva (Philosophy), Nantes (Mathematics), Mexico City (UNAM, Mathematics), Barcelona (UPF, Humanities), University Paris Diderot (REHSEIS and Dept. of History and Philosophy of Sciences). In the first part of my career, up to my Habilitation in 2000, my research mostly focused on the history of the infinitesimal calculus in the seventeenth and the eighteenth centuries. Thus, the topic of the infinite – more specifically, the way mathematicians have managed to tame it by finite means – has always been at the center of my interests.

**What made you interested in the concept of infinity as it relates to the philosophy of mathematics and probability? **

PM: As mentioned above, the concept of infinity plays the role of a watershed for several foundational positions in logic and philosophy of mathematics. In the case of this project, I was interested in investigating some recent developments (the theory of numerosities) that allow one to assign the ‘size’ of infinite sets by respecting a condition called part-whole: if a collection A is strictly contained in a collection B then the size of A should be strictly smaller than that of B. The ordinary notion of size for infinite sets, due to Cantor, flouts this principle. The same mathematical theories have applications in probability theory and allow one to satisfy a property called ‘regularity’ according to which every possible event gets a non-zero probability. The standard theory of probability, due to Kolmogorov, flouts this principle (think of the fact that, according to Kolmogorov’s axioms, if one throws a dart at the real interval [0, 1] the probability of hitting any specific single point in the interval is 0; and yet every point in that interval is possibly the one which gets hit).

MP: My contribution to this project has focused on the role of the infinite in abstraction principles within a neo-Fregean perspective. Hume’s Principle gives a criterion for when two concepts are assigned the same number (and the criterion is a Cantorian style one-to-one correspondence among the objects that fall under the two concepts). Starting from Frege, Hume’s Principle has played a central role in second-order (non-set-theoretic) foundations of arithmetic. Hume’s Principle yields the existence of infinitely many objects (the natural numbers), but alternative routes to the foundations of second-order number theory are also available. For instance, the appeal to a finitary version of Hume’s Principle or to a finitary (and consistent) version of Frege’s Basic Law V, whose infinitary version is responsible for the contradiction in Frege’s theory. The extension of the Fregean approach to real numbers requires, by contrast, the use of genuinely infinitary principles, which can, however, be shown to be free from contradiction by working within a third-order setting.

**How and why did you decide to work together on your research? **

PM and MP: We go back a long way. We knew each other as students in the early 1980s in Milan. We have collaborated on many projects as we share and continue to share several research interests (the history of the infinitesimal calculus; modern and pre-modern foundations of mathematics; neo-logicism; and the philosophy of mathematical practice), and we are co-founders of the Association for the Philosophy of Mathematical Practice. It was natural to join forces also in this case.

**Can you explain a bit more in detail what your research focuses on?**

PM: When one extends counting from the finite to the infinite, two possibilities are immediately available. The first can be called part-whole: if a collection A is strictly contained in a collection B then the size of A should be strictly smaller than that of B. The second one-to-one correspondence: if the elements of A and those of B can be associated in a one-to-one way then A and B have the same size. This latter principle is enshrined in Cantor’s set theory (and Hume’s Principle, mentioned above) and according to it the natural numbers and the even numbers have the same size since we can associate with every natural number its double (0 → 0, 1 → 2, 2 → 4 etc.) and the association is one-to-one. But according to part-whole, there are fewer even numbers than natural numbers. Which is correct? Is there a principled way to decide which is the correct generalization of counting to infinity given that both principles coincide in the finite but diverge at infinity? I am interested in the philosophical, mathematical, and historical issues surrounding these notions.

MP: In some recent papers, Paolo has raised the “good company” objection to the neo-logicist version of Frege Arithmetic. The objection consists in showing that the role of Hume’s Principle within this theory can be played by infinitely many other principles, all equivalent to it in logical form. Each of these alternative principles agrees with Hume’s Principle on finite concepts but differs from it on the infinite ones. This raises a challenge to identifying Hume’s Principle as analytical, for the others would seem to qualify as analytical too. I suggest that the objection can be answered by changing the classical notion of analyticity and replacing it by a new one based on the notion of epistemic economy, a notion that I am developing in a book under preparation. This would single out Hume’s Principle as analytic on account of its small epistemic cost. By extending the approach to the real numbers, one discovers that Frege’s original perspective, consisting in defining these numbers as ratios of magnitudes, does in no way comply with epistemic economy. Nevertheless, an alternative approach (also suggested by Frege, as an aside to his definition of a domain of magnitudes) can be shown to satisfy the constraint of epistemic economy. The approach achieves a complete definition of the real numbers by adding to Hume’s Principle an additional abstraction principle inspired by Euclid’s definition of proportionality (Elements, def. V.5).

**What are the next steps within your research? **

PM: I have recently finished a book on mathematical infinity in the thirteenth century that connects the early discovery of the conflict between part-whole and one-to-one correspondence with some very recent mathematical theories in this area (non-standard analysis, numerosity theory). In addition, I am now engaged in proving some mathematical results connecting totality, regularity, and cardinality in probability theory.

MP: I’m finishing the book on epistemic economy mentioned above, and I have begun (with another colleague, A. Sereni) a new book on an original interpretation of Frege’s program for the foundation of arithmetic and real analysis. I’m also at the finishing stages of writing a third, pedagogical book (in collaboration with D.C. Struppa, a colleague at Chapman) on three theories of infinity in mathematics: projective geometry, infinitesimal calculus, and set theory. Finally, with G. R. Giardina, I’m also working on Aristotle’s notion of continuity, another topic closely connected with the history of infinity.

**What are the reasons that pushed you to apply for the FBF (France-Berkeley Fund)? **

PM and MP: Both of us had already been grantees of the program. In 1998, P.M. applied for the organization of a conference on Bernard Bolzano in Paris that brought together four specialists from UC Berkeley and four from Paris. The articles from that conference were published in the Revue d’histoire des sciences. In 2014 M.P. applied for the organization of a double workshop involving UC Davis and Paris 1 on the topic of ontological commitments in mathematics. Some of the papers presented at these workshops were published in different venues, including two books published by Springer.

**What has the FBF done for your research? What type of opportunities did it open for you? **

PM and MP: What we both like about the FBF is the flexibility it displays in considering the particular needs of the applicants. In this specific case, we needed support to complement the activities funded by Paolo’s Chaire d’Excellence InternationaleBlaise Pascal in Paris (2021-2022). What the funding has allowed us to do was to involve in the project two of Paolo’s Ph.D. students at UC Berkeley (Guillaume Massas and Patrick Ryan), whose dissertations engage with mathematical infinity. They came twice to Paris where they interacted, in the context of some workshops especially organized for the occasion, with the strong community of historians and philosophers of mathematics active in Paris. The benefit was mutual. The French community, including Marco, enjoyed Paolo’s presence in Paris and that of his students. Paolo’s presence in Paris allowed him to strengthen his network of contacts in France, learn from his colleagues, and to give international visibility to his work and that of his students.

**We thank both Professor Paolo Mancosu and Professor Marco Panza for taking the time to answer these questions and we wish them the best of luck on their research. You can learn more about their project here.**