# 2021 Midwest PhilMath Workshop

## The 2021 Midwest PhilMath Workshop will take place via Zoom on **Tuesdays in November**, 2021.

Please find the program below.

For information about the Zoom link, please contact Paddy Blanchette: blanchette.1@nd.edu

### Program

#### November 2

- 5pm: Sam Levey, Øystein Linnebo, Stewart Shapiro: “Potential Infinity and Theology”

Abstract: Beginning with Aristotle, major philosophers and mathematicians rejected the notion of the actual infinite, in favor of the potentially infinite. Typically, this involves a notion of potential existence. It is natural to explicate this potentiality in terms of modal logic. Potentiality, understood modally, is acting here as a kind of guard-rail against paradox---initially the well-known Zeno paradoxes. The nineteenth and early twentieth centuries saw a definitive change in orientation towards the infinite, culminating in the pioneering work of Georg Cantor. At least on the surface, however, Cantor was not entirely consistent in his rejection of the potential infinite. Sometimes he ascribed to the so-called "absolutely infinite", or what he dubbed "inconsistent multitudes" (e.g., the ordinals), features closely analogous to those of the potentially infinite. The theme of this paper is to see how well the modal conception of potential infinity stands up when a Judeo-Christian deity enters the picture. If God "sees'' (or otherwise comprehends) all the points there are in a given line, or all of the ordinals, etc., then these points and ordinals (etc.) exist even before their generation takes place. Our goal is to articulate the relevant modality in a way that it becomes plausible that even for God, there is no index that has all of the relevant arguments.

- 6pm: Owain Griffin: “The Problem of Isomorphic Structures

Abstract: In McCarty (2015) an argument is made to undermine the ante-rem structuralist account of mathematics. At the conclusion of his paper McCarty poses the question of whether the same argument `might not prove just as powerful when directed against the set-theoretic structuralists'. Through this paper I answer the question firstly by developing McCarty's original argument, and then showing that set-theoretic structuralism does not fall victim to the challenge. I therefore conclude that the answer to McCarty's question is negative: set-theoretic structuralism is not effectively undermined.

#### November 9

- 5pm: Xinhe Wu “A Theory of Boolean Valued Models”

Abstract: In this paper we will study Boolean-valued models for first-order languages, in which the value range can be any complete Boolean algebra instead of just the Boolean algebra 2. Although Boolean-valued models are used extensively in set theory, Boolean-valued models for arbitrary first-order languages, as a subject on its own, have not been as well-studied. The goal of this paper is to take the initiative to develop a robust and detailed theory of Boolean-valued models that parallels that of traditional two-valued models.

#### November 16

- 5pm Rachel Boddy and Robert May: “Definition and the Proof of Referentiality”

Abstract: In *Grundgesetze*, Frege attempted to demonstrate that his logical language, the Begriffsschrift is a fully referential language. Although Frege’s proof of referentiality fails (Russell’s Paradox), Frege’s reasons for requiring referentiality remain of interest, and these reasons are our topic. We argue that Frege’s core purpose was to legitimize the use of definitions, and accordingly the proof must be considered in the context of Frege’s broader concern with canons of proper definition, that is, definitions that are scientifically useful. We start from the observation that the sections of *Grundgesetze* where the proof of referentiality is located are placed by Frege in the Table of Contents under the heading “Definitions”. This encompasses §§26-33, labelled “General remarks” on definitions, which are placed just before the sections containing the definitions of arithmetical notions. Building on this, we explore how and why Frege saw the proof of referentiality as essential to the justification of definitions.

- 6pm: Stanley Chang and Sean Ebels-Duggan: “On Unexpected Quantity”

Abstract: Mathematicians ask classificatory questions, of the form: what are the various kinds of Xs, and how many are there of each kind? Sometimes the answers to these questions are surprising, especially when put in contrast. For example, there is no way (so far) to classify the simplest numbers, the primes: any list of "kinds" of prime numbers leaves infinitely many exceptions. But there is a way to classify the simple finite groups leaving only finitely many exceptions. And how many exceptions are there? Exactly 26. This last point is surprising, since it seems to come out of nowhere. Our presentation is an exploration of why we find results such as these surprising. To some degree, it is because we lack explanatory proofs for why these quantities are exactly as they are. One question this raises is why we should expect such explanatory proofs. Is this expectation just a Humean habit? We think (and hope) not. We also consider whether the absence of explanation fully explains why we find such results surprising.

#### November 23

- 5pm: Tom Colclough: “Components of Arithmetic Theory Acceptance”

There is widespread disagreement among foundational standpoints in the philosophy of mathematics as to what it means to "accept" a formal theory. Recently there has been some discussion around this idea for arithmetic theories, couched in terms of the implicit commitment thesis (ICT): accepting a mathematical system T implicitly commits one to additional resources not immediately available in T. In this way, what it means to accept an arithmetic theory is explained in terms of implicit commitment to those additional resources. Dean [2015] argues that on this conception, certain strong readings of the ICT are incompatible with some foundational standpoints. In particular: Tait's finitism and Isaacson's first-orderism. These examples are said to cast doubt on the ICT for arithmetic theories. Nicolai & Piazza [2019] argue that these foundational standpoints are nonetheless compatible with the ICT. They argue that when accepting a theory T, one is necessarily committed to a fixed set of principles extending T that express minimal soundness requirements for T. They also argue that there is a variable component of acceptance of a theory T concerning attitudes towards extending induction schema present in T. I argue that if accepting a theory T is spelled out in terms of certain implicit semantic and schematic commitments, then a fixed semantic commitment and a variable commitment concerning induction schema is not required to render well-established foundational positions like Isaacson's first-orderism compatible with the ICT. In fact, a better reading of theory acceptance is to regard both kinds of commitment as variable.

- 6pm: Bokai Yao “What is Set Theory with Urelements?”

It is natural and useful to quantify over non-sets, i.e., urelements, in the standard set theory. ZFCU is ZFC set theory modified to allow a proper class of urelements. We will show that several ZFC theorems, such as the reflection principle and the collection principle, hold in ZFCU just in case there are “enough” sets of urelements. Using this equivalence, we prove that the Los theorem holds for the internal ultrapowers of a model of ZFCU just in case the model satisfies the collection principle. However, when the urelements do not form a set, ZFCU has little control over how many sets of urelements there are. As a result, there is a hierarchy of principles, including the reflection principle and the collection principle, that are independent of ZFCU. A natural question then arises: What should be an adequate axiomatization of ZFC with urelements?

*Times are U.S. Eastern Time Zone, i.e. New York time.*

*Notice that this changes from EDT to EST (i.e., our clocks ‘fall back’ an hour) on Nov 7.*

*Prior to Nov 7, our times are UTC-4; after Nov 7 our times are UTC-5.*

### Program Committee:

· Paddy Blanchette: blanchette.1@nd.edu

· Jc Beall: jbeall@nd.edu

· Tim Bays: bays.5@nd.edu

· Curtis Franks: cfranks@nd.edu