Outline of topics covered

Meeting 1: August 29

Introduction

• Traditional concerns in the philosophy of mathematics
• More general epistemological issues
• Theories of "understanding"
• The use of historical case studies
• Finding inspiration in Kant, Husserl, Wittgenstein

Overview of structural developments in mathematics

• Group actions in the nineteenth century
• The algebraization of the foundations of algebraic geometry (Weil c. 1940)
• Global vs. local analysis in algebraic topology
• Serre, Grothendieck, and algebraic geometry over arbitrary fields

Overview of Galois theory

• Classical problems (solution of quintic, nonconstructibility with circles and lines)
• Abel, Galois, Kronecker, Dedekind, Weber, Artin

Background on Husserl

Meeting 2: September 5

Algebra review

Meeting 3: September 12

Husserl

• Ways in which Husserl differs from the Frege-Russell tradition
• The progression from Philosophy of Arithmetic to Logical Investigations to Ideas
• Some basic notions in P of A : "psychic / cognitive act", "representation", "contents"
• Husserl's account of number: collective combination, multiplicity, number, "something", "one"

Galois theory

• Abel's 1824 paper and his summary of 1826
• Background on the theory of equations
• From the additive to the multiplicative conception
• Symmetric functions and the Lagrange resolvant

Meeting 4: September 19

Kant (thanks to Jeremy Heis)

• A priori / a posteriori distinction
• Analytic / synthetic distinction
• Syntheticity of mathematical judgments
• Intuitions vs. concepts
• Mathematical vs. philosophical method (counter to Leibniz's view)
• Example: Euclid I 32
• The logical vs. phenomenological readings

Abel summary

• The normal form
• Components are rational functions of the roots
• Effects of permutations on algebraic formulas
• Invariant theory (Jordan, Hilbert, ...)
• Newton: elementary symmetric polynomials and invariants under Sn
• Cauchy: invariants under An
• The form of a polynomial that takes exactly five values
• The final step: the impossibility proof

Meeting 5: September 26

Modern Galois theory and the quintic

• Irreducible polynomials over Q, the fundamental theorem of algebra, and splitting fields
• Group actions: orbits, stabilizers, transitivity, conjugate orbits and stabilizers, Lagrange's theorem
• Splitting extensions and their symmetries
• The fundamental theorem of Galois theory

Meeting 6: October 3

Husserl and The Philosophy of Arithmetic

• Recap: phenomenological analysis characterized (anti-naturalistic, ontological questions bracketed)
• PA's analysis of number
• Discussion of the analysis of "multiplicity"
• The notion of "contents"

Modern Galois theory

• Correspondence between subfields of a field extension and subgroups of itsGalois group
• The maps from a field to the set of automorphisms that fix it, and from a group to its fixed field
• Stabilizers of translates of an element are conjugate groups
• Group fixing the field generated by an orbit is a normal subgroup
• Example: splitting field of x^2 - 2 over Q
• Conclusion: a polynomial is solvable by radicals if and only if its splitting field is contained in a radical extension

Meeting 7: October 10

Husserl

• The "transcendental turn": phenomenological epoche, eidetic abstraction, and the transcendental ego
• Outline of Logical Investigations
• H's critique of psychologism
• The real and ideal contents of a psychological act
• H's view of logic and the concept of science

Kant (thanks to Jeremy Heis)

• Discussion of the construction of concepts ("exhibit a priori the intuition", "have taken account only of the action of constructing the concept", "universal in the particular", "determined under certain general conditions of constrruction", "individual corresponds only as its schema")
• Friedman's analysis (diagrams needed to warrant the infinite; diagrams act as "free variables" in an argument")
• Critique of Friedman's analysis
• Discussion of the schemata (A140-143/B179-182)
• Concepts, schemata, and a priori intuition of space and time

Meeting 8: October 17

Galois theory

• Review of Galois correspondence; normal subgroups correspond to stable subfields
• Subgroup structure of S5
• Transitive subgroups of S5
• Possible factorization behavior of an irreducible quintic

Meeting 9: October 24

Husserl

• Review of the method of phenomenological analysis and the view of logic
• Expressions vs. signs
• Role of language
• Distinction between meaning intention and meaning fulfillment
• Some intensions are never fulfilled; what about a priori unfulfillable intentions?
• Distinction between real and ideal meanings
• The "inkpot" example in Investigation VI
• Meaning intention and meaning fulfillment w.r.t. mathematical expressions

Galois theory

• Subgroup structure of M20
• Analysis of the field structure obtained by adding roots of an irreducible quintic
• Solvable by radicals implies the Galois group is solvable
• The converse

Meeting 10: October 31

The philosophical problem: how do individuals share responsibility and come to agreement on mathematical arguments and results?

• Descartes's answer (Rules: agreement taken for granted; Meditations: individualism taken to its limit, deity invoked)
• Leibniz's answer: individual monads, pre-established harmony
• Kant's answer
• Frege's answer: logic (= reduction to a standard form) / truth, objects, concepts
• Frege's Begriffschrift, and the static view of concepts; the resulting focus on objects and truth (e.g. in modern set theory)
• Husserl's answer (and the transcendental turn)

Galois theory

• The fundamental theorem of symmetric polynomials
• Solutions to quadratic and cubic equations

Meeting 11: November 7

Galois theory

• Lagrange's analysis of the cubic
• Lagrange resolvents
• Galois resolvents

Witttgenstein

• Philosophy of mathematics on the Fregean analysis
• Finding a place for mathematical thought: agency, studied via rules (requirements) and their control of human agency
• Need to account for certainty ("hardness of the logical must"), irreducibility to Fregean standpoint

Meeting 12: November 14

Galois theory

• Recap: Galois resolvents
• The Galois group
• Galois group over an extension of the field of known quantities
• Solvable by radicals implies group is solvable

Wittgenstein

• Descartes's "invidualism": mind / world distinction, no basis for "agreement"
• Kant: humans can think together on space and time; representations don't have content in isolation; agreement follows from common structure
• Kripkenstein: undermines "togetherness" claims; concern with perfect agreement, "in principle," down to most basic levels
• Wittgenstein: agreement isan empirical fact, hence subject to limitations, in contexts that are not sharp
• Positive program: representation use has to admit limitations; focus on improvements in representation use
• How does W save mathematics from empiricism? "Taking" something to be a rule; hardening a theorem into a rule

Meeting 13: November 21

Galois theory

• Recap: Galois resolvents
• Galois' proof that the Galois group really is a group
• Moves: from particular to more general theorems; using particular representations to get general results; eliminating representations

Dedekind

• Foundational works, algebraic number theory, theory of fields
• Methods: set-theoretic language, nonconstructive methods, axiomatic characterization, emphasis on mappings

Logical systematization

• Axiomatic theories and structures
• Separate formulations -> common property -> elaborate language for common axiomatization and proof
• Economy of shared proof, separation of shared / non-shared properties

Extraction of invariants

• E.g. group invariants in topology, fundamental group of a space

Meeting 14: November 28

Totalities

• Definite collection vs. concept
• Examples: domains of rationality, finite groups of permutations
• Ancient universal quantifier: exclude singular cases
• Tarski semantics: exceptionless universal quantifiers
• Further 19th century examples: "real number," "complex number," "function"

Dedekind

• Definition of the real numbers (and requisite properties)
• Comparison with Kroneckarian view
• "Fundamental characteristics" vs. calculations
• Definition of the field generated by elements alpha_1, ..., alpha_n

Meeting 15: December 5

Dedekind

• Discussion of Dedekind's definition of a field generated by elements
• Dedekind's definition of the Galois group (compared to Galois's definition)
• Dedekind's lemma
• One direction: if Pi is a group of n automorphisms of L over K, and K is the fixed field, then [K:L] <= n
• The Dedekind / Artin proof
• Galois's Proof

Response and indifference

• Grothendieck's schemes (generalization of algebraic varieties)
• Wedderburn's theorem (finite divison rings are commutative)
• The beginnings of a proof of Wedderburn's theorem
• Easy corollary: there is no three-dimensional noncommutative algebra over R
• Hamilton's lack of conceptual resources
• "Respndif," as facilitated by algebraic methods

Final discussion

• "Merelifiction"
• 19th century debates over the role of concepts
• Mathematical and philosophical strands of the seminar