» OCIE Seminar in History and Philosophy of Mathematics

The Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics takes place at Chapman University as its main host, and is co-organized together with researchers from UC Riverside, CSU San Bernardino, and Pitzer College. It also occasionally integrates the Chapman University D.Sc. program in Math, Philosophy and Physics as its Graduate Colloquium.

The seminars are held in hybrid format on the Chapman University campus in the Keck Center, home of Schmid College of Science and Technology, in room KC 153 (no. 30 on the Campus map, at the intersection of Walnut Ave. and Center St.), or on Zoom.

Spring 2024

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The price of mathematical scepticism

Speaker: Paul Levy (University of Birmingham)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

I argue that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. Thus scepticism about mathematical reality comes at a price.

Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.

A key part of the talk is the “bivalence questionnaire”, designed to help us think about our intuitions and beliefs. By examining the different ways to answer it, we are led to a spectrum of positions, dubbed ultrafinitism, finitism, countabilism, sequentialism, particularism and totalism. Though they are very different, each maintains the belief alignment that I advocate.

Based on a paper in Philosophia Mathematica: https://academic.oup.com/philmat/article/30/3/283/6622013

Three Kinds of Topological Explanation in Science

Speaker: Alan Baker (Swarthmore College)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

The increasing prevalence of network models in science, especially when dealing with systems of high complexity, has led philosophers to take an interest in the notion of topological explanation. Network models work by mapping a physical phenomenon onto a mathematical graph (network) and then analyzing topological properties of the network such as node connectivity, clustering, and 'small world' features. How does topological explanation relate to scientific explanation more generally? Is topological explanation similar to other types of mathematical explanation in science?

In this talk I propose a classification of topological explanation in science into three basic kinds. The first kind involves cases where the target phenomenon literally possesses the topological properties in question (for example, the early stages of a developing embryo). The second kind comprises the network modeling approach described above, in which the target phenomenon maps onto a mathematical graph, and the topological analysis pertains directly to the graph and only indirectly to the target phenomenon. The third kind involves cases in which data is collected about the target phenomenon, resulting in a multi-dimensional array of data points, and topological analysis is performed on this array. I use this threefold classification to draw some more general philosophical conclusions concerning how applied mathematics works in actual scientific practice.

Queering Consequence: A Framework for Liberatory Logics

Speaker: Roy T. Cook (University of Minnesota)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

We will begin with a brief summary of Robin Dembroff's analysis of the concept "genderqueer" as a gender critical kind, using Dembroff's account to formulate a framework for logically revisionary gender critical kinds (i.e., gender critical kinds whose resistance to the gender binary requires rejection of the dominant, presumably classical, logic). After developing this framework, we will use the framework to taxonomize a number of extant attempts to argue for logical revision based on feminist concerns (including Val Plumwood, Gillian Russell, Maureen Eckert, Ashley Tauchert, Rory Collins and Becca Kosten), using the framework to tease out important commonalities and differences between the various approaches. Finally, we will look at some novel, purely technical questions that are raised if one wants to develop a thoroughgoing account of gender critical logical kinds.

Topology in 1930s America: A Tale of Two "Camps"

Speaker: Karen Parshall (University of Virginia)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

By the 1930s, two rival "camps" of topology had evolved in the United States: point-set topology animated by Robert L. Moore at the University of Texas and combinatorial (or algebraic) topology fostered by Oswald Veblen and, especially, Solomon Lefschetz at Princeton. This talk will sketch the contours of the American topological landscape in the decade before the outbreak of World War II and highlight the differences and divisions between the adherents of the two approaches as they jockeyed for influence within the American mathematical research community.

Remarks about Cantor's Theorem

Speaker: Giuseppe Rosolini (University of Genoa)
Time: 3:15 - 4:45pm
Location: KC 153

Abstract:

In 1890, Cantor produced his famed theorem about the infinity of infinite cardinalities with the declared intention to give a new, very abstract proof of his other theorem of 1874 about the two distinct cardinalities of the set of natural numbers and of the set of real numbers. And certainly one proof is strikingly different from the other. The earlier result was just a technical lemma to obtain a result about transcendental numbers – a renowned theorem in the mathematical circles, though rarely associated with Cantor.

We present the two proofs in contemporary mathematical language, and show how Cantor's second approach applies directly to several other paradoxical situations.

Revised GCH or would Cantor have understood

Speaker: Mirna Džamonja (CNRS and Université de Paris-Cité)
Time: 5:00 - 6:30pm
Location: KC 153

Abstract:

The Continuum Hypothesis states that every infinite subset of the real line is bijective either with the set of natural numbers or with the set of real numbers. It was famously postulated by Cantor in 1878 and is known to be Independent of the usual axioms of set theory. Solution or acceptance of the non-solution to this and to the related Generalised Continuum Hypothesis have been a subject of constant interest in set theory. Which of these are philosophically acceptable? And which ones would meet the, it seems to us reasonable, requirement that Cantor should understand the answer to his own question? We’ll make a tour of some of the contemporary thoughts on this subject.

Intuitionism and Model Theory: Two Ways of Thought in the Foundations of Mathematics?

Speaker: Carl Posy (Hebrew University of Jerusalem)
Time: 9:00-11:00am
Location: AF 209A

Abstract:

Brouwerian intuitionism and contemporary classical model theory are generally, and rightly thought to be opposed approaches to the foundations of mathematics. At the very least, Brouwer rejected the formalism and inherent infinitary attitude that lie at the heart of model theory. However, I will point to a paradox in intuitionistic mathematics whose solution requires attributing to intuitionism some ways of thought indigenous to model theory. Having made this point, I will turn to two questions:

(a) What are structural similarities that engender that apparent methodological overlap?

(b) What then is the deep difference between these two approaches to the foundations of mathematics?

If there is time, I will suggest that this discussion bears on some broader issues both inside and beyond the foundations of mathematics.

Formalization of ∞-category theory

Speaker: Jonathan Weinberger (Johns Hopkins University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

The field of category theory has served as an excellent tool to unify and translate between constructions and theorems across different subfields in mathematics, but also computer science and physics.

Informally, a category is a structure that models composition (e.g. of functions, transformations, processes, or algorithms). In many settings (e.g. algebraic geometry, quantum field theory, and homotopy theory) composition is not given by a well-defined function but rather up to higher-dimensional topological data. This gives rise to the notion of ∞-category, an infinite-dimensional structure.

I will introduce the main elements of a formal language to reason about ∞-categories in a "synthetic" way. The language is an extension of homotopy type theory à la Voevodsky and Awodey–Warren. This constitutes an alternative, arguably more slick foundational system than set theory. Moreover, the new proof assistant Rzk implements this formal language. If time permits, I'll outline the ideas and computer formalization of the Yoneda Lemma, the fundamental theorem of category theory. This result turns out to be easier to prove for ∞-categories in the synthetic setting than for 1-categories in classical set theory. The material is based on joint work with Buchholtz, Gratzer, Kudasov, and Riehl.

Fall 2023

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The Conundrum of the Fourth Postulate

Speaker: Marco Panza, Chapman University

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: The talk will present a by-product of a common work with J. Gil Férez (Chapman), A. Naibo (Paris 1), A. Moshier (Chapman), J.M. Salanskis (Paris Nanterre). It tackles a classical problem at the intersection of mathematics, its history and the philosophical relexion on it and its foundation: is Euclid's fourth postulate ("all right angles are equal to one another") provable within Euclid's (constructive) setting?

After having discussed the nature of the problem, and some of its historical aspects, the talk offers a positive and original answer.

Superoscillations: The Main Ideas and Results

Speaker: Daniele Struppa, Chapman University

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: Superoscillations are a research topic that engages many faculty at Chapman University. In mathematics, faculty like D. Alpay, A. Sebbar, and myself, and in physics faculty like Y. Aharonov, R. Bunyi, J. Howell, A. Jordan, J. Tollaksen, to mention a few. We can say that Chapman is one of the international hubs for superoscillations research, and we have collectively written upwards of 50 peer-reviewed papers on this topic.

But what are superoscillations? In this talk, I will try to stay away from the technical aspects and will try to convey in a manner accessible to non-specialists what superoscillations are, why they are interesting and possibly surprising, and why they matter. For those interested, I will provide a reasonably updated bibliography on the topic.

The Congruent Number Problem

Speaker: Ahmed Sebbar, Chapman University

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: A positive integer is called a congruent number if it is the area of some right triangle with rational side-lengths. For example, 5, 6, 7 are congruent numbers since they are area of right triangles with side lengths (3/2, 20/3, 41/6), (3, 4, 5), and (35/12, 24/5, 337/60), respectively. Furthermore 5 is the first congruent number. The congruent numbers problem has a thousand years of history and has closed relation to the Birch and Swinnerton-Dyer (BSD, for short) conjecture. The BSD conjecture is one of seven millennium problems listed by the Clay Mathematical Institute.

In this presentation we will try to explore, as simply as we can, the long path from a plane geometry question to one of the deepest and most difficult problems in Mathematics.

Dedekind Abstraction: Towards a Logical Reconstruction and Defense

Speaker: Erich Reck, University of California, Riverside

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: In recent philosophy of mathematics, a variety of structuralist positions have been introduced, most prominently the eliminative version of Geoffrey Hellman and the non-eliminative ones of Stuart Shapiro, Charles Parsons, and Ed Zalta, respectively. In past publications, I have attributed another version of structuralism to Richard Dedekind, both in the more practice-oriented sense of "methodological structuralism" and as a distinctive "abstractionist" form of non-eliminative structuralism. In this talk, I will come back to the latter. Doing so is meant to capture a form of reification of mathematical entities that one can find in mathematical practice more broadly, also beyond Dedekind; and it will take the form of a logical reconstruction and defense of "Dedekind abstraction." This has been attempted before, especially in a paper by Ø. Linnebo & R. Pettigrew. My goal is to present a reconstruction that avoids a main limitation of their approach and that has other advantages as well. (The talk is based on an ongoing collaboration with Ø. Linnebo.)

Counting Arithmetic

Speaker: John Mumma, California State University, San Bernardino

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: Counting is central in the everyday concept of numbers, but it is at best peripheral in the two major foundational approaches to arithmetic that we have inherited from Dedekind and Frege. I present in my talk a new foundational approach in which it is central. The approach is based on an account of counting that is different from the standard one in terms of one-to-one correspondence. I first present this account, and then move on to the formal details of the first order theory of arithmetic developed from it. Termed CA for counting arithmetic, the theory interprets arithmetical equations as equivalences rather than identities and secures generality through the pigeonhole principle rather than mathematical induction.

What do Extended Wigner’s Friend arguments tell us about Copenhagenish Interpretations of Quantum Mechanics?

Speaker: Matthew Leifer, Chapman University

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: Wigner’s Friend is a dramatization of the measurement problem in quantum mechanics, in which a “friend” makes a measurement in a sealed laboratory while Wigner, who is outside the laboratory, treats the whole system, including his friend, as an isolated physical system, obeying the Schrödinger equation. Several recent works have studied extended Wigner’s Friend arguments where the Wigner’s Friend scenario is combined with other thought experiments, usually involving quantum entanglement and nonlocality. These arguments claim to teach us something new about the possible interpretations of quantum mechanics. In this talk, I will review three of these arguments and explain their significance for Copenhagenish interpretations of quantum mechanics, i.e. modern interpretations that draw inspiration from or bear a family resemblance to the ideas of Bohr, Heisenberg et. al. Along the way, I will give a precise definition of Copenhagenish interpretations and explain how they differ from other views they are sometimes conflated with. The Extended Wigner’s Friend arguments intend to show that a Copenhagenish interpretation must be (ontologically) perspectival. I argue that they fail to do so, but they do show that non-perspectival Copenhagenish interpretations are committed to the existence of relative frequencies that cannot be predicted by quantum mechanics and cannot be empirically verified.

A review article I wrote on Extended Wigner’s Friends with David Schmid and Yìlè Yīng is available as a preprint at https://arxiv.org/abs/2308.16220

Definition and Content in Frege's Logic

Speaker: Robert May, University of California, Davis

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: In this talk, presenting joint work with Rachel Boddy, we take a theoretical perspective to Frege’s central notions, such that they are to be understood relative to the theoretical role they play in the logical system. The central notions to be discussed are centered on Frege’s canons of proper definition, and the role definition plays in Frege’s theoretical account of mathematics. Our gaze will thus be on the core definition of his logicist program, the definition of the cardinal number of a concept (Definition Z of Grundgesetze). In this context we discuss, among others matters, the conservativity of definitions, its relation to Frege’s Proof of Referentiality in Grundgesetze, how definitions can be mathematically expressive, and how Frege sees the relation of concepts and extensions as essential to defining core theoretical concepts. This latter move, however, is fatal, leading to Russell’s Paradox.

Aristotle’s Criticism of Democritus’ Continuum

Speaker: Giovanna Giardina, University of Catania

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: In my presentation, I will focus on Democritus’ concept of the ‘continuum’ as investigated, interpreted, and criticised by one of its most authoritative sources, namely Aristotle. The reason for this choice is that Democritus played a pivotal role in the development of Aristotle’s continuum theory. My discussion will consist of four main points. First, I will demonstrate that the Presocratic philosophers had an intuitive understanding of the ‘continuum’, which they employed without providing a formal definition. Secondly, I will show that in De generatione et corruptione I 8, Aristotle simultaneously examines the Eleatics and Atomists because the similarities and differences he identifies in the theories of these philosophers are crucial for the formulation of his own theory of the continuum. Thirdly, I will elucidate why Democritus considered natural bodies and motion to be continuous. Fourthly, I will also highlight the mistakes Aristotle attributed to Democritus and present Aristotle’s solution.

**Comparing Nicole Oresme Concept of curvitas with Bernhard Riemann's Definition of Curvature**

Speaker: Bogdan Suceavă, CSU Fullerton

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: In a paper published in 1952, J. L. Coolidge points out that "the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme." Coolidge also comments: "Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?" This question is the starting point of our reflections, as Orseme's works should be read in their historical context, the historical period referred to as Aristotle's Recovery, in the Western European 14^th century. By examining Oresme's 'Tractatus de configurationibus qualitatum et motuum,' written between 1351 and 1355, we point out that the first definition of curvature anticipates several evolutions of the concept developed later on in Riemannian geometry. The fact that the concept of curvature served in the 14th century to study psychological representations could be rather surprising, but Nicole Oresme was motivated to view the matters in such ways by the scholarly (and political) context of his time.

The Structure of Residuated Lattices

Speaker: Peter Jipsen, Chapman University

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: This talk provides a brief introduction to what we have learned about the structure of residuated lattices in the last 25 years. It aims to be accessible to graduate students in Philosophy and Physics. Starting from motivating examples related to groups, relation algebras and Heyting algebras, we will consider structural results about MV-algebra, generalized BL-algebras and Kripke frames for residuated lattices. If time permits, we also describe recent work with Melissa Sugimoto and José Gil-Férez about the structure of locally integral involutive residuated lattices.

Mathematical Structuralism: Weyl's Positions

Speaker: Wilfried Sieg, Carnegie Mellon University

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: The structuralism of modern mathematics found one of its earliest and clearest expressions in Dedekind’s work, for example, in the essay Was sind und was sollen die Zahlen? and a contemporaneous letter to Keferstein. Dedekind rejected (Kantian) intuitions for grounding and developing mathematics. Weyl, when turning to Brouwer’s intuitionism, was highly critical of Dedekind and Hilbert’s view on structural axiomatics and the associated abstract conception of proof. That criticism came to light also in discussions with Emmy Noether in the early 1930s. However, in a manuscript for a talk written around 1953, Weyl sketched a balanced view of structuralist and constructive approaches. That perspective will be compared to that of Bernays on arithmetically grounded methodological frames.

Finally, this talk will describe a dramatic expansion of methodological frames grounded in a concept of inductively generated accessible domains; the latter can be characterized category theoretically as initial algebras of certain endomorphisms.

What Does (Non)-Absoluteness of Observed Events Mean?

Speaker: Emily Adlam

Time: 4:00 - 6:50pm

Location: KC 153

Abstract: Recently there have emerged an assortment of theorems relating to the 'absoluteness of emerged events,' and these results have sometimes been used to argue that quantum mechanics may involve some kind of metaphysically radical non-absoluteness, such as relationalism or perspectivalism. However, in this talk I will argue that a close examination of these theorems fails to convincingly support such possibilities. I will also argue that these theorems taken together suggest interesting possibilities for a different kind of relational approach in which dynamical states are relativized whilst observed events are absolute, and I will show that although something like 'retrocausality' might be needed to make such an approach work, this would be a very special kind of retrocausality which would evade a number of common objections against retrocausality.

The OCIE Seminar on Friday, January 12 is a special session and full-day workshop in celebration of World Logic Day.

Speakers:

David Waszek (École Normale Supérieure, Paris): Boole's Philosophy of Algebra and its Reception

Deborah Kent (University of St. Andrews): Mostly in the Zone: Mathematics and 19th-Century Eclipse Expeditions

Kati Kish Bar-On (Massachusetts Institute of Technology): Mathematics and Society Reunited: Social Aspects of Brouwer's Intuitionism

Ken Saito (Yokkaichi University; Osaka Prefecture University, emeritus): Why and How Can a Line be Moved?

» OCIE Seminar in History and Philosophy of Mathematics

Spring 2024

February 23, 2024

The price of mathematical scepticism

March 15, 2024

Three Kinds of Topological Explanation in Science

March 29, 2024

Queering Consequence: A Framework for Liberatory Logics

April 5, 2024

Topology in 1930s America: A Tale of Two "Camps"

April 12, 2024 (3:15pm)

Remarks about Cantor's Theorem

April 12, 2024 (5:00pm)

Revised GCH or would Cantor have understood

May 3, 2024

Intuitionism and Model Theory: Two Ways of Thought in the Foundations of Mathematics?

May 10, 2024

Formalization of ∞-category theory

Fall 2023

September 1, 2023

The Conundrum of the Fourth Postulate

September 8, 2023

Superoscillations: The Main Ideas and Results

September 22, 2023

The Congruent Number Problem

September 29, 2023

Dedekind Abstraction: Towards a Logical Reconstruction and Defense

October 6, 2023

Counting Arithmetic

October 13, 2023

What do Extended Wigner’s Friend arguments tell us about Copenhagenish Interpretations of Quantum Mechanics?

October 20, 2023

Definition and Content in Frege's Logic

October 27, 2023

Aristotle’s Criticism of Democritus’ Continuum

November 3, 2023

Comparing Nicole Oresme Concept of curvitas with Bernhard Riemann's Definition of Curvature

November 10, 2023

The Structure of Residuated Lattices

November 17, 2023

Mathematical Structuralism: Weyl's Positions

December 1, 2023

What Does (Non)-Absoluteness of Observed Events Mean?

January 12, 2024

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**Comparing Nicole Oresme Concept of curvitas with Bernhard Riemann's Definition of Curvature**