The desire to be puzzled

Here is a very interesting post on possible alternatives to Cantor’s Transfinite Numbers Theory! Was Cantor’s notion of infinite really “unavoidable”?
The topic is particularly interesting when considered within the general framework of mathematical development. How does mathematics evolve? What “forces” mathematics to take some roads instead of other roads? And how do these choices influence the way in which we model and understand the natural world?


A prominent professor in the philosophy of mathematics once told me that the key to writing an attractive philosophy paper is to present the reader with a puzzle. “Give me a puzzle, and I’ll be interested”, he said. As I was surrounded by mathematicians and philosophers of mathematics which were steadily exchanging puzzles, I had no doubt that he was right: mathematicians and philosophers of mathematics like puzzles. But then, mightn’t it be the case that this fondness of puzzles influences much more than just our judgment of a philosophy paper (and our conversations over dinner)? Here’s a crazy idea – or maybe not so crazy – does our desire to be puzzled affect our judgement of a certain foundational mathematical theory?

The foundational mathematical theory which I have in mind is, of course, Cantor’s transfinite set theory. Given its general acceptance nowadays, it is easy to forget that in order to generalize arithmetic from the finite to…

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Yitang Zahng and prime numbers

On the New Yorker, an interesting article on Yitang Zhang, the mathematician who in 2013 proved a theorem in number theory according to which there are infinitely many pairs of prime numbers that differ by 70 million or less. The theorem may not sound really interesting, but actually it is the first proof to establish the existence of a finite bound for prime gaps, resolving a weak form of the twin prime conjecture.

Here you can find the article. Good reading!

From Logic to Practice. Italian Studies in the Philosophy of Mathematics

I’m happy to announce that the new book From Logic to Practice is now available on! My contribution, Structure and Applicability, can be found in Part III, chapter 11 of the book.

From Logic to PracticeThis book — it is written in the back cover — brings together young researchers from a variety of fields within mathematics, philosophy and logic. It discusses questions that arise in their work, as well as themes and reactions that appear to be similar in different contexts. The book shows that a fairly intensive activity in the philosophy of mathematics is underway, due on the one hand to the disillusionment with respect to traditional answers, on the other to exciting new features of present day mathematics. The book explains how the problem of applicability once again plays a central role in the development of mathematics. It examines how new languages different from the logical ones (mostly figural), are recognized as valid and experimented with and how unifying concepts (structure, category, set) are in competition for those who look at this form of unification. It further shows that traditional philosophies, such as constructivism, while still lively, are no longer only philosophies, but guidelines for research. Finally, the book demonstrates that the search for and validation of new axioms is analyzed with a blend of mathematical historical, philosophical, psychological considerations.​

Let me express my gratitude to Gabriele Lolli, Marco Panza and Giorgio Venturi, whose initiative and perseverance made this work possible.

FilMat international conference

The new-born FilMat italian network for the philosophy of mathematics announces its first international conference on the topic “Philosophy of Mathematics: objectivity, cognition, and proof”. The conference will take place at San Raffaele University, Milan, on 29-31 May 2014.

The FilMat — of which I am proud to be a member — was created few months ago by a group of Italian scholars in philosophy of mathematics , originally met at the Scuola Normale Superiore in Pisa at the 2012 conference “Philosophy of Mathematics: from Logic to Practice”, and aims to foster the gathering of scholars working either in Italy or abroad on the philosophy of mathematics and strictly related fields, with special attention to those at early stages of their careers.

Here you can find the FilMat’s website. Here is the link to the call for abstracts for the 2014 international conference.

Are numbers sets?

One of the milestones of contemporary philosophy of mathematics is Benacerraf‘s 1965 article “What Numbers Could Not Be“.1 There he offers a compelling argument according to which numbers cannot be considered as sets — namely, they cannot be metaphysically identified with sets.

The argument is quite simple: if numbers were sets, we should be able to find a unique progression of sets with which numbers can be identified. But this is apparently impossible: there is a lot of ω-series that can serve as well for the aim. For example, we can adopt von Neumann’s series, and say that 0=\emptyset, 1=\{\emptyset\}, 2=\{\emptyset,\{\emptyset\}\}, and so on, where the successor function S is defined by S(x)=x \cup \{x\}. Or we can adopt Zermelo’s series, and say that 0=\emptyset, 1=\{\emptyset\}, 2=\{\{\emptyset\}\}, and so on, where the successor function S is defined by S(x)=\{x\}. Now, the problem is: is 3=\{\{\{\emptyset\}\}\} or is 3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}? Benacerraf presents then the example of two children, Ernie and John. The first learned that von Neumann’s ordinals are the natural numbers, while the latter that Zermelo’s ordinals are the natural numbers. Now, they will be easily able to learn arithmetic set theoretically via the above constructions, and they will agree on any arithmetical theorem, except that for Ernie it is true that 3 \in 17, while for John it is false!

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Happy Easter!

Happy Easter!

Are you enjoying your Easter egg? Well, have you ever tried to draw an egg? Have you ever wondered about its shape? It is not a circle, it is not an oval — what is that? It’s an egg-curve! Ok, but how can we draw it and how can we define it in mathematical terms? Here you can find a lot of different ways to do that!

But maybe you should just enjoy your chocolate eggs and don’t think too much…