# PhilMath Intersem 1.2010

## Theme: Foundations of Mathematics: What and Why?

### Description

The general aim of the seminar is to develop a better understanding of the following questions and plausible responses to them:

I. What might we reasonably want and ask of a foundation of mathematics?

II. What are the most important similarities and differences between the two major foundational alternatives today, set theoretical and category theoretical foundations?

III. What are the most significant comparative advantages and disadvantages of each of these two main approaches?

IV. What other reasonable and interesting conceptions of foundations are there, and what is known concerning them?

### When

14h–18h each Tuesday and Friday, from Tuesday, May 18th through Tuesday, June 22nd. Full day meeting 9h00–18h00, Friday, June 25th

### Where

The seminar will meet in the salle Klimt (# 366A), Batiment Condorcet, Grands Moulins campus, U of Paris 7-Diderot for all but one meeting (June 18th), when the seminar will meet in the salle Klee (# 454A) of the same building.

Video links to Notre Dame and Nancy 2 are planned for all meetings of the seminar.

### Featured Speakers

- Steve Awodey (Carnegie Mellon)
- Tim Bays (Notre Dame)
- John Bell (Western Ontario)
- Mic Detlefsen (Notre Dame)
- Solomon Feferman (Stanford)
- Curtis Franks (Notre Dame)
- Geoffrey Hellman (Minnesota)
- Luca Incurvati (Cambridge)
- Øystein Linnebo (Birkbeck)
- Giuseppe Longo (ENS)
- Jean-Pierre Marquis (Montreal)
- John Mayberry (Bristol)
- Colin McLarty (Case Western)
- Richard Pettigrew (Bristol)
- Michael Potter (Cambridge)
- Jean-Jacques Szczeciniarz (Paris 7)
- Gabriel Uzquiano (Oxford)

### Schedule

- May 18 - Mic Detlefsen ("General Background: The Critical Movement in 19th Century Foundations")
- May 21 - Øystein Linnebo and Richard Pettigrew ("Only up to isomorphism? Category theory and the foundations of mathematics")
- May 25 - Steve Awodey ("From Sets to Types to Categories to Sets") and Gabriel Uzquiano ("Foundations of set theory: between mathematics and philosophy")
- May 28 - John Mayberry ("The Foundations of Set Theory")
- June 1 - John Bell ("Russell's Paradox and Diagonalization in a Constructive Context")
- June 4 - Curtis Franks ("What set-theory and category theory can teach an anti-foundationalist")
- June 8 - Solomon Feferman ("Conceptual structuralism and the continuum")
- June 11 - Giuseppe Longo ("Foundations and the search of principles") and Luca Incurvati ("How to Be a Minimalist about Sets")
- June 15 - Jean-Pierre Marquis ("Sets and the CT")
- June 18 - Timothy Bays ("Some Remarks on the Foundations of Arithmetic")
- June 22 - Michael Potter ("How much impredicativity do we need (or want)?")
- June 25 - Mic Detlefsen ("Foundations of Mathematics---What and Why?"), Geoffrey Hellman ("What Can We Expect of a Foundational Framework for Mathematics?"), Colin McLarty ("Foundations as truths which organize mathematics"), Jean-Jacques Szczeciniarz ("Diagrammatic Activity in Category Theory: Is it Foundational?")

### Reading

**May 18 - Michael Detlefsen, "General Background: The Critical Movement in 19th Century Foundations"**

Suggested Reading:

- Keyser, C., “The Human Signiﬁcance of Mathematics”,
*Science*, New Series 42 (1915): 663–680. - Klein, F., Lectures I and VI of
*Lectures on the Development of Mathematics in the 19th Century*(English trans. of*Vorlesungen Über die Entwicklung der Mathematik im 19. Jahrhundert*) by M. Ackerman. Math Sci Press, Brookline, Massachussetts, 1979. - Kneebone, G., "The Critical Movement in Mathematics in the Nineteenth Century", ch. 5 of
*Mathematical Logic and the Foundations of Mathematics*, van Nostrand, London, 1963.

**May 21 - Øystein Linnebo and Richard Pettigrew, "Only up to isomorphism? Category theory and the foundations of mathematics"**

[handout - pdf]

Suggested Reading:

- Solomon Feferman, "Categorical Foundations and Foundations of Category Theory." In R. E. Butts and J. Hintikka, editors,
*Logic, Foundations of Mathematics and Computability Theory*, pages 149–169. Reidel, Dordrecht, 1977. - Geoffrey Hellman, "Does Category Theory Provide a Framework for Mathematical Structuralism?",
*Philosophia Mathematica*, 11(3):129–157, 2003. - Colin McLarty, "Exploring Categorical Structuralism",
*Philosophia Mathematica*, 12(3):37–53, 2004.

**May 25 - Steve Awodey, "From Sets to Types to Categories to Sets", and Gabriel Uzquiano, "Foundations of set theory: between mathematics and philosophy"**

Suggested Reading (S. Awodey):

- Awodey, S. "A brief introduction to algebraic set theory",
*Bulletin of Symbolic Logic*14(3): 281–298, 2008. - [handout - pdf]

Suggested Reading (G. Uzquiano):

- Shapiro, S. and Weir, A. (1999) "New V, ZF and Abstraction",
*Philosophia Mathematica*, 7(3): 293-321. - McGee, V. (1997) "How We Learn Mathematical Language",
*Philosophical Review*106(1): 35-68. - Uzquiano, G. "Bad Company Generalized",
*Synthese*. Vol. 170: 3. (September 2009) - Uzquiano, G. "Quantification without Domain", Bueno, O. and Linnebo, Ø. (eds.) (2009)
*New Waves in the Philosophy of Mathematics*, Palgrave.

**May 28 - John Mayberry, "The Foundations of Set Theory"**

[handout - pdf]

Suggested Reading:

- R.Dedekind. "Letter to Kefferstein". Translated by Hao Wang and Stephan Bauer-Mengelberg in
*From Frege to Gödel: A Source Book in Mathematical Logic, 1879 --1931*, edited by Jean van Heijenoort, Harvard University Press, Cambridge, 1971.

**June 4 - Curtis Franks, "What set-theory and category theory can teach an anti-foundationalist"**

Suggested Reading:

- "Set theory from Cantor to Cohen" by Akihiro Kanamori, from
*The Handbook of the Philosophy of Science*, edited by Andrew Irvine. 2007. [http://math.bu.edu/people/aki/16.pdf] - "From Foundations to Ludics" by Jean-Yves Girard, from
*The Bulletin of Symbolic Logic*, vol. 9(2). 2003. [http://iml.univ-mrs.fr/~girard/bsl.pdf.gz]

**June 8 - Solomon Feferman, "Conceptual structuralism and the continuum"**

Suggested Reading:

- "Conceptions of the continuum",
*Intellectica*51 (2009), 169-189. Also at http://math.stanford.edu/~feferman/papers.html, item #69

**June 11 - Giuseppe Longo, "Foundations and the search of principles", and Luca Incurvati "How to Be a Minimalist about Sets"**

Suggested Reading (G. Longo):

- Giuseppe Longo. Incomputability in Physics and Biology. Invited Lecture, Proceedings of Computability in Europe, Azores, Pt, June 30 - July 4, 2010. [ftp://ftp.di.ens.fr/pub/users/longo/CIM/incomput-phys-bio.pdf]
- Giuseppe Longo. Interfaces de l'incomplétude, pour "Les Mathématiques", Editions du CNRS, 2011. [ftp://ftp.di.ens.fr/pub/users/longo/PhilosophyAndCognition/Incompletude.pdf]

Suggested Reading (L. Incurvati):

- Oystein Linnebo, `Structuralism and the Notion of Dependence', in
*Philosophical Quarterly*, 2008, vol. 58: 59--79.

**June 15 - Jean-Pierre Marquis, "Sets and the CT"**

Suggested Reading:

- J-P Marquis, "Category Theory and the Foundations of Mathematics: Philosophical Excavations",
*Synthese*(103), 421-447, 1995. [pdf]

**June 18 - Timothy Bays, "Some Remarks on the Foundations of Arithmetic"**

Suggested Reading:

- T. Skolem, "Some Remarks on Axiomitized Set Theory," in J. van Heijenoort (ed.),
*From Frege to Gödel: A Source Book in Mathematical Logic, 1879--1931*, pp. 290--301.

**June 22 - Michael Potter, "How much impredicativity do we need (or want)?"**

Suggested Reading:

- M. Pottter,
*Set Theory and Its Philosophy: a Critical Introduction*, Oxford University Press, 2004, Chapters 2-4.

**June 25 - Mic Detlefsen, "Foundations of Mathematics---What and Why?", Geoffrey Hellman, "What Can We Expect of a Foundational Framework for Mathematics?", Colin McLarty, "Foundations as truths which organize mathematics", Jean-Jacques Szczeciniarz, "Diagrammatic Activity in Category Theory: Is it Foundational?"**

Suggested Reading (G. Hellman):

- G. Hellman, "Structuralism" from the
*Oxford Handbook for Philosophy of Mathematics and Logic*(pp. 536-562). Available as a pdf at: www.tc.umn.edu/~hellm001 - [handout - pdf]

Suggested Reading (C. McLarty):

- C. McLarty, "What Structuralism Achieves" [pdf]
- C. McLarty, "Introduction to Forthcoming Book" [pdf]

Abstract: We speak of "foundations of mathematics" in many senses, but all these senses get their interest from the fact as old as Euclid that large amounts of mathematical knowledge can be organized as deductive consequences of relatively simple axioms. A foundation for mathematics is a body of truths which organizes the practice of mathematics. I mean not merely truths which could in principle organize practice but which do in fact. Philosophers badly underestimate how rigorously the deductive practice of mathematics has been standardized since Hilbert, and how serious is the practical need for this rigorous organization. While we may learn a great deal from philosophical critiques, and we do learn a great deal from purely formalist investigations, this conception of foundations is committed to saying current mathematics is true.

Suggested Reading (M. Detlefsen):

- John Bell, "Category Theory and the Foundations of Mathematics",
*Brit. J. Phil. Sci.*32 (1981): 349--358 - Hermann Weyl, "Mathematics and Logic",
*American Mathematical Monthly*53 (1946): 2--13

### Participation

All interested persons are encouraged to attend and participate. If you have questions, please contact Mic Detlefsen at mdetlef1@nd.edu or Andrei Rodin at rodin@ens.fr.

### Student Participants

- Alexei Angelides (Stanford)
- Sylvain Cabanacq (ENS, Paris 7)
- Sara Confalonieri (Paris 7 & Paris 1)
- Davide Crippa (Paris 7 & Paris 1)
- Jules-Henri Greber (Nancy 2)
- Emmylou Haffner (Paris 7)
- Daniel Immerman (Notre Dame)
- Ramzi Kebaili (Paris 7 & Paris 1)
- Graham Leach-Krouse (Notre Dame)
- Amirouche Moktefi (Nancy 2)
- Julien Ross (Paris 7)
- Irina Starikova (Bristol)
- Anthony Strimple (Notre Dame)
- David Thomasette (Nancy 2)
- Iulian Toader (Notre Dame)
- Sean Walsh (Notre Dame)

INTERSEM 2010 is a cooperation of the University of Notre Dame, the Université de Paris 7–Diderot, the École Normale Superieure, the Ideals of Proof (IP) project (ANR), the Université de Nancy 2, the Université de Paris 1, the IHPST, the Henri Poincaré Archives and the Maison des Sciences de l'Homme Lorraine.

You can download a pdf presentation of INTERSEM 2010 by following this link.

##

Associated Seminar: "Category Theory and the Philosophy of Mathematics Today"

The aim of this seminar is (i) to cover more specific technical topics concerning Category Theory than there will be time to cover in Foundations of mathematics: What and Why? and (ii) to provide an opportunity for more extended philosophical discussion focusing on Category Theory.

### Where & When

All meetings will be from 9h–16h at the ENS (45 rue d’Ulm), salle “W” (staircase B, 3rd floor).

### Schedule

- May 17: R Guitart (Paris 7), C Houzel (CNRS)
- May 31: A Prouté (Paris 7), A Rodin (Paris 7 & RAS), J Salantin (Montpellier)
- June 14: J Bénabou (Paris 13), M Lachièze-Rey (CNRS & Paris 7), G Longo (CNRS & ENS), J Petitot (CREA), P Rosolini (Genoa)