»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 2025

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OCIE Seminar in History and Philosophy of Mathematics

Spring 2025

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Logical structures arising from corpora

Speaker: Thomas Seiller (CNRS, Laboratoire d'Informatique de Paris-Nord)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

The impressive results obtained by generative language models witness that statistical information about a (large enough) corpus can be used to extract the structure of natural langage. As part of a general effort to investigate how to mathematically understand this process, Bradley, Gastaldi and Terilla have proposed a generalisation of the standard notion of formal concepts to incorporate quantitative information. This approach consists in moving from sets to functions over sets (using presheaves): the generalised formal concepts are thus obtained as so-called nuclei (fixed points) of an adjunction. In this talk I will explain how this generalisation of formal concepts turns out to coincide exactly with a construction of models of (fragments of) linear logic.

What Mathematical Explanation Need Not Be

(joint work with Elijah Chudnoff)

Speaker: Silvia De Toffoli (School for Advanced Studies IUSS Pavia)
Time: 4:45-6:15pm PST
Location: KC 153

Abstract:

Recent works in the philosophy of mathematical practice and mathematical education have challenged orthodox views of mathematical explanation by developing Understanding-ﬁrst accounts according to which mathematical explanation should be cashed out in terms of understanding. In this article, we explore two arguments that might have motivated this move: (i) the context-sensitivity argument and (ii) the inadequacy of knowing why argument. We show that although these arguments are derived from compelling observations, they ultimately rest on a misunderstanding of what explanation-ﬁrst accounts are committed to and an underestimation of the resources available to them. By clarifying the terms at play in the debate and distinguishing different objects of evaluation, we show that the insightful observations about practice and education made by challengers to the orthodoxy are in fact best accounted for within the traditional Explanation-ﬁrst framework.

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Speaker: Jacopo Emmenegger (University of Genoa)
Time: 4:00 - 6:50pm
Location: KC 153

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Infinity, Paradox, and the Fall of Hilbert’s Program

Speaker: Carl Posy (Hebrew University of Jerusalem)
Time: 9:00am-11:00am PST
Location: TBA

Abstract:

The tug between infinitary thought and constructivism has been a constant theme throughout the history of mathematics: Time and again, Platonistic infinitary methods (positing finitely inaccessible objects, unfulfillable tasks) have proven to be essential to mathematics; while empiricist thinkers have repeatedly striven to reconcile this fact with finitary (constructivist) constraints. In modern times, David Hilbert's formalist program is the most famous reconciliation attempt. Hilbert aimed to secure infinitary mathematics via finitely graspable formal systems. The program notoriously failed. Gödel’s theorems did it in.

The first part of my talk concerns that downfall. Using some known and new tools I’ll show that Hilbert’s program contains the seeds of its own downfall. Specifically, the central notion of a consistent and adequate formalism already presupposes ways of thinking that unleash infinity and thus clash with and even block his finitary goal.

But then I’ll point out that infinity itself is not so central a sticking point as Hilbert made it out to be. Those damning infinitary modes of thinking are in fact instances of principles that have nothing to do with infinity. Indeed, I will show that these principles underlie some decidedly finite paradoxes. An interesting observation on its own.

Having shown this, time permitting, I'll tell you about some consequences of that observation for understanding the conflict between Platonism and empiricism in particular and some other issues in the general metaphysics of mathematics

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Speaker: Greg Restall (University of St Andrews)
Time: 4:00 - 6:50pm
Location: KC 153

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Speaker: Vincent Jullien (University of Nantes)
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Location: KC 153

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Speaker: Otávio Bueno (University of Miami)
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Fall 2024

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Frege's Definition of Real Numbers is Consistent

Speaker: Marco Panza (Chapman University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

In the second volume of Grundgesetze (1903), Frege outlined an original definition of real numbers as ratios of magnitudes, and partially realized his plan, by presenting a formal definition of domains of magnitudes. The remaining part of the definition of reals, namely the definition of ratios over such domains, should have been presented in a third volume to be published later. This never happened since in the meantime Frege received Russell's unfamous letter informing him about the inconsistency of the formal system, presented in the first volume (1893), within which both his definition of natural numbers and that of domains of magnitudes are stated. Frege unsuccessfully tried to find a way out of the contradiction, then he abandoned his project. In my talk, I shall show that he could have gone ahead with his planned definition of real numbers by slightly modifying his definition of domains of magnitudes so as to avoid inconsistency. The changes he should have made for this purpose are so marginal and local that one might conclude that his envisaged definition is essentially consistent and that Frege was wrong in renouncing it.

The Gödel Translation: History and New Directions

Speaker: Guram Bezhanishvili (New Mexico State University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

The Gödel translation (1933) interprets the intuitionistic propositional calculus IPC as a fragment of modal logic. In 1940s, McKinsey and Tarski proved that this provides a faithful embedding of IPC into Lewis’ modal system S4. Their result was further generalized to other extensions of IPC. One of the culminations of this line of research is the celebrated Blok-Esakia theorem (1976) which establishes an isomorphism between extensions of IPC and Grz — the well-known extension of S4 introduced by Grzegorczyk (1967).

Things become more complicated when the propositional logics under consideration are replaced by their predicate extensions. As we will see, the Blok-Esakia isomorphism fails already for the one-variable fragments of these logics. This gives rise to a series of problems, which remain open to this day. The aim of this talk is to introduce the audience to this beautiful area of research. Towards the end, I also plan to discuss promising future directions.

A Possible Way to Read (and Interpret) Euclid's Elements

Speaker: Carlos Álvarez Jiménez (Universidad Nacional Autónoma de México)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

A classical way to introduce the main problems related to the Philosophy of Mathematics follows from the (classical) problem of the foundations of mathematics. In this talk, I do not intend to discuss directly this topic, but to suggest that an interesting contribution to this classical question may be answered by asking which are, if there are, the fundamental theorems in mathematics. This very wide question may be focused in the particular domain of Euclidean geometry and, particularly, concerning Euclid's reconstruction of plane geometry.

In this talk, I will suggest that Euclid's Books I and II of the Elements may be read in this way, as a geometric inquiry guided by some "fundamental theorems".

The Philosophical Significance of Gödel’s Dialectica Translation

Speaker: Stephen Mackereth (Dartmouth College)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

One of the most ambitious projects in the foundations of mathematics ever conceived was Hilbert’s Program in the 1920s. The aim of this Program was to give finitary consistency proofs for infinitary mathematics, thus putting infinitary mathematics on a more secure footing. There is a popular narrative that Hilbert’s Program was decisively refuted by Gödel’s incompleteness theorems in 1931. However, Hilbert’s school continued to work on consistency proofs for decades after the appearance of the incompleteness theorems. Moreover, Gödel himself, in his remarkable Dialectica paper of 1958, pursues a modified version of Hilbert’s Program: he presents a new, Hilbert-style consistency proof for arithmetic based on “an extension of the finitary standpoint,” and he clearly regards this proof as epistemologically significant. In this talk I shall discuss Gödel’s 1958 paper in some detail, with an eye towards the following questions. 1. Is there any truth to the claim that Hilbert’s Program was refuted by the incompleteness theorems? 2. What epistemological significance did Gödel ascribe to his Dialectica proof, and was he correct about this?

Stephen Mackereth - Handout 1 - OCIE Seminar 9.20.24

Stephen Mackereth - Handout 2 - OCIE Seminar 9.20.24

Interestingness in Mathematics

Speakers: Drew Moshier and Alexander Kurz (Chapman University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

AI (both symbolic and neural) is changing the way we do mathematics. Already now, cutting edge proofs by leading mathematicians are implemented in programming languages such as Lean and verified by type-checking algorithms. How far are we away from machines doing interesting mathematics independently and without human guidance?

Any answer to this question will depend on understanding what mathematics humans find interesing. This discussion, led by Alexander Kurz and Drew Moshier, will explore the question of interestingness in mathematics. In particular, we want to understand how to define proxies for interestingness based on existing archives of formal proofs.

From Constructive Mathematics and Quantum Mechanics to Fundamental Logic

Speaker: Wesley H. Holliday (UC Berkeley)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

Non-classical logics have been proposed in a number of domains, including constructive mathematics and quantum mechanics. In this talk, I will identify a base logic beneath some of these non-classical logics that I suggest has a certain fundamental status. I will give an introduction to the proof theory and semantics of this “Fundamental Logic.”

An associated paper is available at https://arxiv.org/abs/2207.06993.

Other references:

G. Birkhoff and J. von Neumann, “The Logic of Quantum Mechanics,” Annals of Mathematics, Vol. 37, 1936.

A.S. Troelstra and D. van Dalen, Constructivism in Mathematics, Vol. 1, North-Holland, 1988.

How can we make trustworthy AI?

Speaker: Mateja Jamnik (University of Cambridge)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

Not too long ago most headlines talked about our fear of AI. Today, AI is ubiquitous, and the conversation has moved on from whether we should use AI to how we can make the AI systems that we use in our daily lives trustworthy. In this talk I look at some key technical ingredients that help us build confidence and trust in using intelligent technology. I argue that intuitiveness, interaction, explainability and inclusion of human domain knowledge are essential in building this trust. I present some of the techniques and methods we are building for making AI systems that think and interact with humans in more intuitive and personalised ways, enabling humans to better understand the solutions produced by machines, and enabling machines to incorporate human domain knowledge in their reasoning and learning processes.

Can the necessity of mathematics be derived?

Speaker: Benjamin Faltesek (Chapman University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

Hannes Leitgeb (2020) has offered an argument for the necessity of mathematics. In particular, he gives a derivation using the modal system K of the statement that mathematical theorems are necessarily true. In this talk, I will first motivate the questions of whether mathematics is necessary, and if so, why. I will also defend the strategy of deriving the necessity of mathematics, on the principle that arguments can be explanatory. I will then set out Leitgeb's premises and reasoning. Some key assumptions are that all mathematical theorems are (actually or simply) true and that they can always be translated into ZFC. Although the argument has deficiencies, I will defend Leitgeb's suggestion that the truths of set theory are necessary due to the necessity of identity and the Axiom of Extensionality.

A Reassessment of Gödel’s Doctrine: The Necessity of Infinity

Speaker: Patrick Ryan (Chapman University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

In his landmark 1931 paper, Gödel demonstrated the existence of finitary statements that required infinitary resources to prove them. This led him to postulate what Solomon Feferman called Gödel’s Doctrine, namely, that “the unlimited transfinite iteration of the powerset operation is necessary to account for finitary mathematics.” This claim garnered further support over the course of the 20th century because of the production of various other “finitary independence” results. Nonetheless, proof theoretic work by Feferman and others showed that these finitary results could be proved using relatively weak systems, e.g., predicatively justifiable systems, thereby challenging Gödel’s Doctrine. In this talk, I would like to argue that, though the technical results of Feferman and others are unimpeachable, their philosophical significance is overstated. That is, even if Gödel’s Doctrine is dubious when we understand "necessary" to mean "proof theoretically necessary," it can be vindicated when we think of other senses in which strong infinitary resources might be necessary for mathematics. This is done by investigating a fascinating collection of finitary statements that possess multiple proofs employing both infinitary and finitary resources. I consider how an analysis of such results can inform debates in the philosophy of mathematics, especially discussions of purity, content, and explanation. In particular, if a finitary theorem τ has a perfectly cogent, finitary proof, why then provide an infinitary proof of τ , a proof involving principles of an ostensibly different sort? What is gained? Do such infinitary proofs play an explanatory role? Is there then a sense in which infinity is necessary? I conclude by indicating some promising directions for future research.

How to Benefit from Uncertainty: An Introduction to Paraconsistent Bayesian Update

Speakers: Walter Carnielli and Juliana Bueno-Soler (CLE/FT- University of Campinas, Brazil, and Chapman University, USA)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

One of the main questions in Bayesian thinking is how we adjust our beliefs when new evidence comes along. At the heart of this process are relationships between evidence E and hypotheses H, which bring up key ideas like plausibility, confirmation, and acceptability. However, these relationships can get tricky.

In this talk, we’ll show how paraconsistent and paracomplete logics can help solve some of these challenges in Bayesian reasoning. Using a new approach to probability based on the Logic of Evidence and Truth (LET_F), which is designed to handle both evidence for or against a judgement—even when it's incomplete or contradictory—we offer a way to measure how much evidence supports a given statement. We’ll look at some examples showing how paraconsistent and paracomplete Bayesian approaches can effectively handle contradictions in reasoning.

We think these ideas could impact not just the philosophy of science but also fields like Artificial Intelligence, probabilistic networks, and other new models where handling uncertainty and contradictions is crucial.

Reference:

W. A. Carnielli and J. Bueno-Soler
Where the truth lies: a paraconsistent approach to Bayesian epistemology.
Studia Logica, to appear.
Pre-print available Cambridge Open Engage Logica.

https://www.cambridge.org/engage/coe/article-details/6526998c45aaa5fdbbc54fd2

The Turing Test as a View From Somewhere:
Hilbert, Wittgenstein and Turing on ‘Surveyability’

Speaker: Juliet Floyd (Boston University)
Time: 4:00 - 6:50pm
Location: KC 153

Abstract:

Recent debates about the philosophical status of formalization and mechanization of proof may be illuminated by considering the mutual impact Wittgenstein and Turing had on one another around issues concerning the evolution of notations in symbolic logic. When Wittgenstein remarked in 1937 that ‘a proof must be surveyable’ he was reworking ideas of Frege, Hilbert and Turing. “Surveyability” for Wittgenstein was neither a verificationist requirement nor a refutation of the claim that all proofs must have corresponding formal proofs, much less a refutation of logicism. Instead, it placed front and center what mathematicians do, i.e., it explores what logicism comes to in an everyday sense. The idea -- consonant with certain trends in so-called “philosophy of mathematical practice”, including recent work by Kennedy on “formalism freeness”, and Floyd’s on “everyday phraseology” -- is not to provide or ask for a “foundation” for mathematics in any ordinary sense, but rather to take a pragmatic and mathematically flexible approach to the very idea of “foundations”.

In 1939 Wittgenstein and Turing discussed these ideas in Wittgenstein’s Cambridge lectures on the foundations of mathematics, sparking some of Turing’s subsequent work on types. The relevant ideas here draw out new ways of looking at Turing’s 1936 paper, as well as his more speculative writings in the late 1940s about “intelligent machinery” and his 1950 “Turing Test”.

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?

CONTACT US

For more information about the OCIE Seminar series, please contact:

Lisa Beesley
Graduate Program Coordinator – Math, Philosophy, and Physics
beesley@chapman.edu

Marco Panza
Provisional Program Director
panza@chapman.edu
(714) 997-5021

»OCIE Seminar in History and Philosophy of Mathematics

Spring 2025

January 24, 2025: Thomas Seiller

OCIE Seminar in History and Philosophy of Mathematics

Spring 2025

January 24, 2025: Thomas Seiller

Logical structures arising from corpora

February 7, 2025: Silvia De Toffoli (4:45-6:15pm PST)

What Mathematical Explanation Need Not Be

February 21, 2025: Jacopo Emmenegger

Title: TBA

February 28, 2025: Carl Posy (9:00am-11:00am PST)

Infinity, Paradox, and the Fall of Hilbert’s Program

March 7, 2025: Greg Restall

Title: TBA

March 14, 2025: Vincent Jullien

Title: TBA

April 4, 2025: Speaker TBA

Title: TBA

April 11, 2025: Otávio Bueno

Title: TBA

April 18, 2025: Speaker TBA

Title: TBA

April 25, 2025: Speaker TBA

Title TBA

May 2, 2025: Speaker TBA

Title TBA

May 9, 2025: Speaker TBA

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May 16, 2025: Speaker TBA

Title TBA

Fall 2024

August 30, 2024: Marco Panza

Frege's Definition of Real Numbers is Consistent

September 6, 2024: Guram Bezhanishvili

The Gödel Translation: History and New Directions

September 13, 2024: Carlos Álvarez Jiménez

A Possible Way to Read (and Interpret) Euclid's Elements

September 20, 2024: Stephen Mackereth

The Philosophical Significance of Gödel’s Dialectica Translation

September 27, 2024: Drew Moshier and Alexander Kurz

Interestingness in Mathematics

October 4, 2024: Wesley H. Holliday

From Constructive Mathematics and Quantum Mechanics to Fundamental Logic

October 18, 2024: Mateja Jamnik

How can we make trustworthy AI?

October 25, 2024: Benjamin Faltesek

Can the necessity of mathematics be derived?

November 1, 2024: Patrick Ryan

A Reassessment of Gödel’s Doctrine: The Necessity of Infinity

November 8, 2024: Walter Carnielli and Juliana Bueno-Soler

How to Benefit from Uncertainty: An Introduction to Paraconsistent Bayesian Update

November 15, 2024: Juliet Floyd

The Turing Test as a View From Somewhere:Hilbert, Wittgenstein and Turing on ‘Surveyability’

Spring 2024

February 23, 2024: Paul Levy

The price of mathematical scepticism

March 15, 2024: Alan Baker

Three Kinds of Topological Explanation in Science

March 29, 2024: Roy T. Cook

Queering Consequence: A Framework for Liberatory Logics

April 5, 2024: Karen Parshall

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

April 12, 2024 (3:15pm): Giuseppe Rosolini

Remarks about Cantor's Theorem

April 12, 2024 (5:00pm): Mirna Džamonja

Revised GCH or would Cantor have understood

May 3, 2024: Carl Posy

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

May 10, 2024: Jonathan Weinberger

Formalization of ∞-category theory

Fall 2023

September 1, 2023: Marco Panza

The Conundrum of the Fourth Postulate

September 8, 2023: Daniele Struppa

Superoscillations: The Main Ideas and Results

September 22, 2023: Ahmed Sebbar

The Congruent Number Problem

September 29, 2023: Erich Reck

Dedekind Abstraction: Towards a Logical Reconstruction and Defense

The Turing Test as a View From Somewhere:
Hilbert, Wittgenstein and Turing on ‘Surveyability’

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