Category-Theoretic Foundations of Mathematics Workshop
The aim of this 2-day workshop is to provide a forum in which researchers from philosophy, mathematics, computer science, and allied disciplines can discuss the aims and significance of category-theoretic foundations of mathematics. The interdisciplinary character of this workshop provides a unique opportunity to discuss and deliberate upon what is specific to the success of category-theoretic foundations within the various disciplines.
Location: Room 1517, Social and Behavioral Sciences Gateway Building (SBSG). Building #214 on the interactive campus map (click here for printable version). See below for driving and parking directions.
Schedule (see below for titles and abstracts):
Saturday May 4
9:00- 9:30 Continental Breakfast
11:00-11:30. Coffee Break.
11:30-1:00. Samson Abramsky, Oxford
4:00-4:30. Coffee Break.
4:30-6:00. Olivia Caramello, Cambridge
Sunday May 5:
9:00-9:30. Continental Breakfast
9:30-11:00. John Baez, UC Riverside
11:00-11:30. Coffee Break.
11:30-1:00. Ralf Krömer, Siegen
This workshop is made possible through generous support provided by: the Department of Logic and Philosophy of Science at UC Irvine, the School of Social Sciences at UC Irvine, the Kurt Gödel Society, the University of Notre Dame, and the SPHERE lab at the University of Paris-Diderot.
The event will be preceded by a departmental colloquium talk on Friday May 3 by John Baez (UC Riverside). This event is one of a small cluster of recent category-theory related activities organized by the department of Department of Logic and Philosophy of Science. Others include Colin McLarty’s (Case Western) visit last spring as well as Jim Weatherall’s category-theory course this spring.
Driving & Parking Directions:
Find Parking: we recommend that all visitors park in the Social Sciences Parking Structure (SSPS) - enter structure from Campus Drive at Stanford. See the parking map.
Titles and Abstracts:
Speaker: Samson Abramsky, Oxford
Title: Category theory as a tool for making models
Abstract: I will discuss category theory in its aspect as a tool for mathematical modelling in the sciences. It has played an important part in computer science and some areas in theoretical physics for some decades, and is being used increasingly in areas ranging from quantum information and foundations to natural language semantics and game theory. I shall discuss how the conceptual and foundational features of category theory have a strong influence on its use in mathematical modelling and applications.
Speaker: John Baez, UC Riverside
Title: The Foundations of Applied Mathematics
Abstract: Suppose we take "applied mathematics" in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields. Suppose we look for mathematical structures that repeatedly appear in these diverse contexts - especially structures that aren't familiar to pure mathematicians. What do we find? The answers may give us some clues about the concepts that underlie the most applicable kinds of mathematics. We should not be surprised to find some category theory hiding here!
Speaker: Olivia Caramello, Cambridge
Title: Grothendieck toposes as unifying 'bridges' in Mathematics
Abstract: I will present a novel view of Grothendieck toposes as unifying spaces in Mathematics being able to effectively serve as 'bridges' for transferring concepts and results across distinct mathematical theories. This approach, first emerged in the context of my Ph.D. research, has already generated many applications into different mathematical fields, including Topology, Algebra, Geometry, Functional Analysis, Model Theory and Proof Theory, and the potential of this theory has just started to be explored. In the lecture, I will explain the fundamental principles that characterize my view of toposes as unifying 'bridges', and illustrate the technical usefulness of these methodologies by discussing a few selected applications.
Title: Category theory and set theory: examples of their interaction
Abstract: I will begin by reviewing basic arguments and counterarguments about either set theory or category theory as a possible foundational theory for mathematics. I will then devote the main part of my talk to a few examples of positive interaction between both theories in the field of logic. Two examples will be considered more specifically, the first one in connection with algebraic set theory, the second one in connection with sketch theory.
Title: Structuralist foundations for abstract mathematics
Abstract: Abstract: As Paul Benacerraf famously pointed out, set-theoretic foundations provide too much information for the working mathematician. Our talk is driven by the question: is it possible or desirable to develop a foundational framework that omits the irrelevant and distracting information? We consider one proposal motivated by the slogan that, "all mathematically relevant properties are structural." Unlike many previous structuralist proposals, this proposal is linguistic (syntactic) in nature. Namely, we present a language L together with an intended semantics, such that the following hold: (1) L is rich enough to be considered foundational; (2) L satisfies the appropriate adequacy conditions with respect to its intended semantics; and crucially (3) well-formed sentences in L are necessarily invariant under isomorphisms of the relevant type of structure. This proposal is strongly influenced by Michael Makkai’s "structuralist foundation of abstract mathematics," and his language FOLDS (first-order logic with dependent sorts).
Speaker: Ralf Krömer, Siegen
Title: "Psychological priority" vs. effects of training. The foundational debate on Category theory revisited
Abstract: The paper develops a particular answer to a standard objection (by Feferman) against uses of Category theory as a foundations of mathematics. This objection roughly says that concepts like "collection" and "operation" are "psychologically prior" to the concept of category. Our answer amounts to showing the relevance of mathematical training to the foundational debate on Category theory.
First, we briefly review this debate from the beginnings to Bénabou's 1985 proposal to use elementary portions of Category theory as a foundation of "naive" Category theory as a whole; we discuss both some of the relevant mathematical and metamathematical facts and a number of records of how workers in the field interpret these facts and their philosophical impact. The main thesis of the paper is that "psychological priority" (Feferman) is in the last analysis irrelevant for the problem of a "proper presentation" (Isbell) of Category theory; one needs to take into account the effects of training because without training one is not able to judge the "properness''.
In order to develop this claim more fully, a sidestep is made to the epistemology of C.S. Peirce, especially to his account of intuition and the role of consciousness in hierarchies of cognitions. What is particularly relevant for us in Peirce's anticartesian approach is his criticism of the role of the individual for epistemology. While traditional philosophy of the foundations of mathematics seems to seek a foundational stance accessible for every individual in isolation (and this is also implicit in Feferman's conception of psychological priority), Peirce stresses the importance of the presence of a community, of collective processes of regulation of work with concepts, and of learning in such a situation. On these grounds, a (still sketchy) comparison between hierarchies of cognitions in the sense of Peirce and hierarchies of mathematical concepts is presented. The Peircean conception of intuition is found to be relevant whenever the role played by mathematical training for mathematical intuition is stressed. This ultimately leads to a revision of the notion of foundations of mathematics.
Title: Some elements of the descent theory, from the point of view of Algebraic Geometry, with philosophical remarks.
Abstract: In the framework of CT descent theory is a characterization of morphisms (descent morphism) that constitute a theory of generalization and of extension concerning very different mathematical concepts and theoretical points of view.