PHIL 2505 Contentualness & Formalism (Spring 2026)

Instructor: Doug Blue

Email: doug.blue[at]pitt.edu

Meetings: Tuesdays 10am-12:30pm in CL 1008B

Zoom link: Email me

Description

David Hilbert once described “the question of relations between contentualness (Inhaltlichkeit) and formalism in mathematics and logic” as among the most difficult epistemological problems with scientific significance. This seminar will take up that question directly.

Our central text will be Juliette Kennedy’s Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness, through which we will explore the epistemological advantages of formalization in mathematics and the phenomenon of “formalism freeness.” Along the way, we will consider questions such as:

How does formalization contribute to knowledge acquisition?
What is the meaning of mathematical language?
What counts as a natural formal theory?
To what extent are formal systems adequate models of mathematical practice?

We will reflect on the broader implications of these issues for the philosophy of mathematical practice, traditional philosophy of mathematics, and linguistic philosophy, both ideal and ordinary.

Course structure

The course will consist of four units.

Rigor
The relation of mathematical logic to mathematics
Intension and interpretation
Formalism freeness

Requirements

Auditors and participants taking the course for credit may be asked to present material.

Participants taking the course for credit are expected to write a term paper.

1/13 Introduction

Background reading:

Burgess, Rigor and Structure, chapters 1 & 2
Avigad, Mathematics and the formal turn

1/20 Proof

Reading:

Rav, Why do we prove theorems?
Azzouni, Why do informal proofs conform to formal norms?
Azzouni, The derivation-indicator view of mathematical practice.
Rav, A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices.

1/27 Proof

Reading:

Tanswell, A problem with the dependence of informal proofs on formal proofs.
Azzouni, The algorithmic-device view of informal rigorous mathematical proof.

2/3 Virtues of informal proof

Reading:

Dawson, Why do mathematicians reprove theorems?
Hersh, Some proposals for reviving the philosophy of mathematics.
Jaffe and Quinn, “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics.
Thurston, On proof and progress in mathematics.

2/10 Logic and natural language

Reading:

Glanzberg, Logical consequence and natural language
Glanzberg, Models, model theory, and modeling
Sagi, Considerations on logical consequence and natural language
Sagi, Logicality in natural language.

2/17 Logic as modeling

Reading:

Cook, Logic-as-modeling: a new perspective on formalization, PhD thesis.
Cook, Vagueness and mathematical precision.

2/24 The question of logic

Speaker: Seoyeon Park

Reading:

Kripke, The question of logic.
Williamson, Accepting a logic, accepting a theory.

3/3 Kreisel

Reading:

Kreisel, Mathematical logic: what has it done for the philosophy of mathematics?
Kreisel, Mathematical logic: Tool and object lesson for science

3/10 Spring break

3/17 No meeting

3/24 Informal rigor

Reading:

Kreisel, Informal rigour and completeness proofs
Kreisel, Church’s thesis and the ideal of informal rigour

Deep dive:

Dean and Kurokawa, On the methodology of informal rigour: set theory, semantics, and intuitionism

3/31 Formalism Freeness

Kennedy, Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness, chapters 1 & 2

Recommended:

Baldwin, Model Theory and the Philosophy of Mathematical Practice, Part I & chapter 14

4/7 Formalism Freeness

Speakers: Stephen Perry and Philip Sink

Kennedy, Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness, chapters 3 & 4

4/14 Juliette Kennedy

Kennedy, Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness, chapters 5-7

Recommended:

Baldwin, Model Theory and the Philosophy of Mathematical Practice, Part II

4/21 Gil Sagi

Recommended:

Cohnitz & Nicolai, “How to adopt a logic” (dialectica 2021)
Birman, “The adoption problem and the epistemology of logic” (Mind 2024)
Devitt and Roberts, “Changing our logic: a Quinean perspective” (Mind 2024)
Besson, “Knowledge of generality and the possibility of deductive inference” (Inference and Consciousness, eds: Timothy Chan and Anders Nes, 2019).

Gil solved philosophy. It was crazy.

References

Unfinished.

Books

Baldwin, Model Theory and the Philosophy of Mathematical Practice.
Burgess, Rigor and Structure.
Kennedy, Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness.
Rorty (editor), The Linguistic Turn: Essays in Philosophical Method.

Hilbert and the contentual

Blanchette, The Frege-Hilbert controversy.
Hilbert, On the infinite.
Hilbert, Axiomatic thinking.
Mancosu, From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s.

Rigor

Avigad, Understanding proofs.
Avigad, Understanding, formal verification, and the philosophy of mathematics.
Avigad, The mechanization of mathematics.
Avigad, Relibability of mathematical inference.
Avigad, Mathematics and the formal turn.
Azzouni, How and why mathematics is unique as social practice.
Azzouni, Why do informal proofs conform to formal norms?
Azzouni, Does reason evolve? (Does the reasoning in mathematics evolve?)
Azzouni, The algorithmic-device view of informal rigorous mathematical proof.
Burgess & De Toffoli, What is mathematical rigor?
De Toffoli, Reconciling rigor and intuition.
De Toffoli, Proofs for a price: Tomorrow’s ultra-rigorous mathematical culture.
Feferman, And so on…: Reasoning with infinite diagrams.
Granville, Proof in the time of machine.
Hamami, Mathematical rigor and proof.
Harris, Automation compels mathematicians to reflect on our values.
Larvor, How to think about informal proofs.
Larvor, Why the naive derivation recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge.
Paseau, What’s the point of complete rigour?
Pelc, Why do we believe theorems?
Rav, Why do we prove theorems?
Rav, A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices.
Tanswell, A problem with the dependence of informal proofs on formal proofs.
Tanswell, Mathematical Rigour and Informal Proof.
Thurston, On proof and progress in mathematics
Wagner, Mathematical consensus: a research program.
Weatherall & Wolfson, Correctness, artificial intelligence, and the epistemic value of mathematical proof.

Mathematical logic and practice

Baker, Non-deductive methods in mathematics.
Cheng, Mathematics, morally.
De Toffoli and Mancosu, Philosophy of mathematical practice.
Hofweber, Formal tools and the philosophy of mathematics
Kreisel, Mathematical logic: what has it done for the philosophy of mathematics?
Kreisel, Mathematical logic: Tool and object lesson for science
Macintyre, The mathematical significance of proof theory.
Polya, Mathematics and Plausible Reasoning
Post, Absolutely unsolvable problems and relatively undecidable propositions–account of an anticipation.

Computers and mathematics

Avigad, Is mathematics obsolete?.
Buzzard, Mathematical reasoning and the computer.
Commelin, Abstract formalities (video).
De Millo, Lipton, and Perlis. Social processes and proofs of theorems and programs.
Hales, Mathematics in the age of the Turing machine.
Hales, Big conjectures (video).
Kontorovich, The shape of mathematics to come.
Macbeth, Algorithm and abstraction in formal mathematics.
Mackenzie, Mechanizing proof.
Tao, The story of Erdös problem 126.
Wayne, Three ways formally verified code can go wrong.
Wu et al., Autoformalization with large language models.

Logic in mathematics

Kaye, Review of Craig Smorynski, Logical Number Theory I, An Introduction.
Poonen, Barry Mazur’s work relating logic and arithmetic geometry.
Poonen, Undecidability in number theory.
Poonen, Undecidable problems: a sampler.

Logic as modeling

Barwise, Noun Phrases, Generalized Quantifiers and Anaphora
Cook, Logic-as-modeling: a new perspective on formalization, PhD thesis.
Cook, Vagueness and mathematical precision.
Kripke, The question of logic.

Intension and interpretation

Feferman, Arithmetization of metamathematics in a general setting.
Lindstrom, Aspects of incompleteness.

Natural theories and linearity

Friedman, FOM: Does Mathematics Need New Axioms?
Hamkins, Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength
Walsh, On the hierarchy of natural theories.