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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Standard and non-standard existential predictions

Existential predictions are very rare in science. For this reason, when scientists can predict the existence of a new entity—a new planet, or a new particle, for example—we feel that something very exciting is going on! But what is going on is not only exciting—it is also very interesting from a philosophical point of view. These predictions raise various philosophically interesting puzzles, which relate to the epistemological, metaphysical, and methodological roles that mathematics can play in scientific representation. In this post, I want to present two main kinds of existential prediction in science and to sketch some philosophical problems related to them. However, I will not address these problems in this context. The interested reader may refer to Ginammi (2016) for a more detailed analysis and for a solution to these problems.1

Avoiding Reification. A new article on the Studies in History and Philosophy of Modern Physics!

I am pleased to announce that my new article Avoiding Reification will be published on the volume 53 of the Studies in History and Philosophy of Modern Physics (February 2016)!

In this article I have discussed and critically examined a very interesting case of existential prediction in particle physics: the prediction of the $\varOmega^-$ particle (a particle of the class of the spin-$\frac{3}{2}$ baryons). Existential predictions in science are always very thrilling, as you may imagine; but this prediction is even more interesting than usual because of the peculiar role that mathematics seems to play. Such a peculiar role raises a serious philosophical problem, since apparently we cannot justify it on the basis of standard methodological criteria. In this paper I discuss this problem and I offer a solution to it by offering a new logical reconstruction of the prediction of the $\varOmega^-$ particle, based on the representative and heuristic effectiveness that mathematics may exhibit under certain conditions.

Here is the abstract of the paper, just to give you an idea of the content:

According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal   mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the $\varOmega^-$ particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based on the employment of a highly controversial principle—what he calls the “reification principle”. Bangu himself takes this principle to be methodologically unjustifiable, but still indispensable to make the prediction logically sound. In the present paper I will offer a new reconstruction of the reasoning that led to this prediction. By means of this reconstruction, I will show that we do not need to postulate any “reificatory” role of mathematics in contemporary physics and I will contextually clarify the representative and heuristic role of mathematics in science.

Good read and happy new year to everybody!

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 Amazon.com! My contribution, Structure and Applicability, can be found in Part III, chapter 11 of the book.

This 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!

Conference: The Analysis of Theoretical Terms

Next week I will be in Munich for a conference on theoretical terms at the Munich Center for Mathematical Philosophy (MCMP). I will talk about the role of mathematics in a very interesting case of existential prediction in physics: the discovery of the omega minus particle. Here is the website of the conference.