#### 4th Nancy-Irvine FiLLM Workshop: Formalization in Logic, Language, and Mathematics

**SCHEDULE**

*Friday, May 17, 2024*

10:00-11:15 a.m. *Kai Wehmeier (UCI):* Compositionality as a constraint on logics

11:15-11:30 a.m. Break

11:30 a.m.-12:45 p.m. *Daniel Usma Gomez (Nancy):* Thomas Aquinas's Metaphysics of Geometry: a Medieval Insight on Mathematical Activity
(remote)

12:45-2:00 p.m. Catered Lunch

2:00-3:15 p.m. *Toby Meadows (UCI):* An Ordinary Language Response to a Structuralism Problem in Philosophy of Mathematics

3:15-3:30 p.m. Break

3:30-4:45 p.m. *Baptiste Mélès (Nancy):* Is Programming Different in Classic Chinese?

*Saturday, May 18, 2024*

10:00-11:15 a.m. *Jeremy Heis (UCI)*: Believing the Axioms (in 1788)

11:15-11:30 a.m. Break

11:30 a.m.-12:45 p.m. *Olivier Bruneau (Nancy):* Pedal Curves from 1650 to the Early 20th Century: A Heritage Mathematical Object

12:45-2:00 p.m. Catered Lunch

2:00-3:15 p.m. *Kenny Easwaran (UCI):* Scientific Realism in Mathematics: The Case of the Hyperreals (joint work with Henry
Towsner)

3:15-3:30 p.m. Break

3:30-4:45 p.m. *Andy Arana (Nancy):* Mathematical hygiene

*Abstracts:*

**Andy Arana (Nancy): Mathematical hygiene**

This talk aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way as the set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we briefly discuss two case studies. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it, a case of (im)purity in mathematics. Then, we compare Descartes' and Newton's mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas. We argue that these early modern normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political program: the development of a new bourgeois intellectual class. We then turn to the origins of Bourbaki and its context within the nationalisms of Germany and occupied France during the Second World War, wherein the politics of language and mathematics were both interrogated in parallel ways. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods open up a new area of inquiry at the borders of linguistics and the philosophy of mathematics.

**Olivier Bruneau (Nancy): Pedal curves from 1650 to the early 20th century: a heritage
mathematical object**

In my talk, I will present a long-term study (between 1650 and the beginning of the 20th century) of the history of a category of curves known as pedal curves. After a systematic exposition by Colin Maclaurin in 1720, these curves disappeared from the field of mathematics, only to return to prominence around 1850 in intermediate mathematical journals. I will show how, after this period of oblivion, pedal curves became part of the mathematical heritage, with a late reference to Maclaurin.

**Daniel Usma Gomez (Nancy): Thomas Aquinas's Metaphysics of Geometry: a Medieval Insight
on Mathematical Activity**

Scholars have recently attempted to reconstruct Aquinas's philosophy of mathematics from scattered remarks within his corpus (Svoboda and Sousedik, 2020; Jean W. Rioux, 2023). Besides difficulties regarding the coherence and systematicity of these views, such an attempt faces a major challenge vis-à-vis Aquinas's mathematical sources. The absence of proper mathematical examples in Aquinas's texts as well as the direct references to the Aristotelian corpus appear to suggest that the Medieval Master limited himself to Aristotle's examples. Accordingly, it would be natural to take Aquinas's claims as the speculations of a layman about a topic he did not master at all.

In this talk, I would like to adopt a new attitude with regard to Aquinas's philosophy of mathematics. I would like to suggest that a careful reading of Aquinas's texts provides valuable elements showing the internal coherence and systematicity of Aquinas's philosophy of mathematics as well as his, at least apparent, familiarity with mathematical texts. I will proceed in three stages. First, I will sketch an overview of contemporary readings of Aquinas's philosophy of mathematics in order to highlight the main notions structuring his views. Second, I will focus on two passages from mathematical texts whose structural parts seem to me to be described by those notions. Third, I will try to read these texts in parallel with Aquinas's main notions. My aim is to propose a charitable approach to Aquinas's philosophy of mathematics as a coherent and systematic view accounting for both mathematical knowledge and mathematical activity, rooted in 12th and 13th century debates about the status of mathematics more than on a naive reading of Aristotle's reaction to Plato's philosophy of mathematics.

**Jeremy Heis (UCI): Believing the Axioms (in 1788)**

(Abstract TBA)

**Toby Meadows (UCI): An ordinary language response to a structuralism problem in philosophy
of mathematics**

I don't know that much about structuralism in mathematics and, somewhat perversely, I'd like to keep it that way. I don't care whether structures are real or not. I can't imagine how we could definitively answer such a question and I can't see what good it would do even if we could. Don't get me started on boxes and diamonds. I guess this means it's open mic night at the phil math café.

Nonetheless from a practical perspective, I think I might be a structuralist. When I think about mathematical objects, I only care about the properties of those objects that are preserved by isomorphism. In essence, this is just an attitude. Joel David Hamkins has recently carved out this attitudinal approach to structuralism in mathematics and I think it's a winner. I think it accurately reflects the way practicing mathematicians think about mathematical structure right up to the point where they assiduously avoid questions of metaphysics and ontology (well at least the careful mathematicians do).

From my point of view, Hamkins' approach fully scratches any structuralist itches I might have, so there appears to be nothing left to say. But there is a minor worry lurking in the corner. What do we mean when we talk about the natural numbers? Sure, my attitude is such that I don't care to distinguish any particular version of the natural numbers from among its many isomorphic representatives. But how can I say, "the natural numbers," when I accept there are infinitely many equally deserving suitors to this title? Call this the "the" problem. I don't really care about it either. However, I'm going to offer what I think is an extremely hamfisted solution to this problem. I think it works and I also think it should be ignored in practice.

I should finally note that my ignorance means that I'm still in the process of finding out who's already said what I'm going to say, or something very close to it. So far, I've identified a very nice paper by Sy Friedman and Neil Barton, that uses a similar approach to make subtle comparisons between set theory and category theory. My talk will not be subtle.

**Baptiste Mélès (Nancy): Is Programming Different in Classic Chinese?**

Most programming languages inherit from English their alphabet, their lexicon and some syntactical features. Even "localized" languages, e.g. Scratch, do not change the situation, since they are literally translated from English. Would programming be different if it had been conceived in a language as different as Classic Chinese (文言 /wenyan/)? I will present the Wenyan programming language, its syntax and semantics, and explain some original syntactic and semantic features that have been suggested by the grammar of Classic Chinese.

References:

- Website: https://wy-lang.org/

- Source code: https://github.com/wenyan-lang/wenyan

- Software Heritage archive: https://archive.softwareheritage.org/browse/origin/directory/?origin_url=https://github.com/wenyan-lang/wenyan

**Kai Wehmeier (UCI): Compositionality as a constraint on logics**

From the very beginning of formal semantics as a linguistic discipline, the principle of compositionality has played a central role in the construction of systems of natural-language semantics. It says, roughly, that the meaning of a complex expression is uniquely determined by the meanings of its immediate constituents and the syntactic operation that created it. The semantics of formal logics are typically compositional by design; and even Hintikka's IF logic, whose original semantics is non-compositional, was later shown to possess a compositional semantics. Together with certain results as to the obtainability of compositional semantics, these observations might lead one to suspect that every formal logic possesses a compositional semantics. In this talk, I will show in what senses this is and is not the case.

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