# Workshop: Variations on Philosophy of Mathematics

Master in Philosophy

23 Febbraio 2024

Foundational positions in the philosophy of mathematics are numerous, ranging from the more nominalist to the structuralist and passing by the neologicist. By bringing together speakers with different views on mathematics and its foundations, this workshop aims to constructively defend and critique these main views.

Date: **13 – 14 April 2024**

Venue:

**Room A-31**, Red Building

USI West Campus, Via Buffi 13

Lugano, Switzerland

Speakers:

**Francesca Boccuni** (Vita-Salute San Raffaele)**Kit Fine** (NYU)**Balthasar Grabmayr** (Tübingen)**Mary Leng **(York)**Andrea Sereni** (Pavia)**Achille Varzi **(Columbia)

Programme:

**Saturday 13th:**

13.15 - 13.30 Welcome

13:30 - 15:00 Francesca Boccuni and Andrea Sereni:* The Conqueror’s Arrogance: Neo-logicism, Caesar Problem and Frege’s Constraint*

15.00 - 15.10 Coffee break

15.10 - 16.40 Balthasar Grabmayr: *The Structuralist Turn in (Meta-)Mathematics*

16:40 - 16:50 Coffee break

16.50 - 18.20 Mary Leng: *Mathematically Natural Kinds*

**Sunday 14th:**

09:00 - 10:30 Achille Varzi: *Do we Really need the Reals?*

10:30 - 10:40 Coffee break

10:40 - 12:10 Kit Fine: *Some Reflections*

Abstracts:

*The Conqueror’s Arrogance: Neo-logicism, Caesar Problem and Frege’s Constraint. * **Francesca Boccuni** and **Andrea Sereni**

Against structuralist accounts of arithmetic, neologicists argue that only a definition of natural numbers based on Hume's Principle – unlike a structuralist stipulation of Peano Axioms – meets Frege's Constraint (FC): this requires that a good definition of the concept of number encodes an explanation of its general applicability. According to neologicists, FC is met only if the concept of number is a concept of individual objects. A threat for HP as a definition is the Caesar Problem (CP), and a solution to CP is needed for the concept of number to be a (pure) sortal concept. We will discuss the interdependency between the neologicist solution to CP and the claim that HP meets FC. We will suggest that the neologicist solution to CP makes HP an "arrogant", and hence unsuitable, definition, according to neologicists' own distinctions. But if a proper solution to CP isn't offered, the ability of HP to meet FC is jeopardized, and so is its primacy over rival axiomatic definitions.

*The Structuralist Turn in (Meta-)Mathematics. ***Balthasar Grabmayr**

The emergence of metamathematical investigations at the turn of the 20th century brought with it the introduction of formal languages as objects of mathematical inquiry. While Hilbert conceived of formal expressions as strictly spatial objects, namely, as strings of symbols, Gödel’s celebrated technique of arithmetisation showed that strings can be replaced by numbers or sets as the objects of metamathematics. As I will show, this technical innovation has led to a structuralist turn in metamathematics. According to the structuralist view, the subject matter of metamathematics are syntax structures, which can be exemplified in multiple ways. In this talk, I will isolate a weak structuralist tenet that underlies most (if not all) strands of structuralism. I will then argue that this tenet fails in the case of metamathemtics. I will conclude that structuralism, at least in its current form, is not a tenable view for all of mathematics.

*Mathematically Natural Kinds*. **Mary Leng**

According to mathematical fictionalists, there are no mathematical objects. Mathematicians are engaged in uncovering what is 'true in the story (or, perhaps better, stories)' of mathematics, where 'the story' is either given by a background set theory as an ultimate foundational theory within which all of the rest of mathematics can live, or alternatively any coherent axioms are viewed as setting the scene for their own mathematical 'story', with the story of elliptic geometry, for example, as being given by the theory's axioms and their consequences. In order to fit intuitions about truth in mathematics outrunning provability, many fictionalists will insist that mathematical stories can be told using second-order axioms, and insist on a semantic, rather than deductive, consequence (making use of primitive modality rather than set theoretic models to elaborate on this notion of consequence). But with all this in place, it would appear that fictionalists ought to be liberal pluralists about mathematical stories: metaphysically-speaking, any coherent system of axioms is as good as any other in telling a story about how mathematical objects could be (though perhaps some stories will be of more interest than others). Indeed, fictionalism here seems to be in step with a general trend in the philosophy of mathematics towards embracing some form of mathematical pluralism (as seen e.g. in Balaguer's full blooded Platonism and Shapiro's ante rem structuralism). This paper will consider whether fictionalists have to be pluralists, or whether it is possible - and even desirable - within a fictionalist approach to recognise sources of mathematical objectivity that might vindicate some mathematical theories over others. In particular I will consider whether fictionalists could (and indeed should) embrace Putnam's indispensability argument as an argument for taking some mathematical concepts to be objectively vindicated by their use in empirical science, or Maddy's considerations of mathematical depth as an argument for taking some mathematical concepts to be objectively vindicated by their use in mathematics.

Organiser:

Léon Probst (USI)

An event of the Institute of Philosophy (ISFI)

For any query: [email protected]