# Mathematics for Philosophers (of Physics)

## Rough dependencies of physical theories on mathematical theories

• Classical Mechanics depends on: Differential Geometry and Sympletic Geometry.

• Special Relativity depends on: Linear Algebra and Differential Calculus

• General Relativity depends on: Differential Geometry

• Elementary (non-relativistic) Quantum Mechanics depends on: Linear Algebra is sufficient to grasp the essence of the measurement problem, of quantum nonlocality, and of the simplest "no hidden variables" proofs. To deal with realistic systems (e.g. particles with position degree of freedom) you also need Functional Analysis.

• Quantum Field Theory (mainstream approach) depends on: If a theory must have a formulation in known mathematics, then quantum field theory is not yet really a theory. But the practitioners of the "art" of quantum field theory use a number of mathematical tools such as group representation theory, ordinary and partial differential equations and the calculus of variations.

• Quantum Field Theory (algebraic approach) depends on: Operator Algebras, and for certain topics (e.g. superselection theory) Group Representation Theory and Category Theory. QFT on curved spacetime also requires Differential Geometry.

## Rough dependencies of mathematical theories on mathematical theories

By "A depends on B", we here mean that you need to know B before you can learn A.

• Differential Geometry depends on: Multivariable Calculus, and preferably Real Analysis. Point-Set Topology would also be very helpful (since topological and metric concepts are central in Differential Geometry)

• Operator Algebras depends on: Functional Analysis, Abstract Algebra, and Measure Theory. You also need a passing acquaintance with Complex Analysis; and for some topics (e.g. modular theory) you should have mastery of Complex Analysis.

• Functional Analysis depends on: Real Analysis, Linear Algebra, and Point-Set Topology. You also need a passing acquaintance with Complex Analysis.

• The beauty and utility of Category Theory only becomes apparent against the backdrop of a number of other mathematical fields such as Abstract Algebra, Algebraic Topology, and Algebraic Geometry.

• Topos theory depends on category theory.

## Books by field

The following is a list of some of the books that I have found most useful for developing a set of tools to work on philosophy of physics.

### General

Geroch, Robert, Mathematical physics. University of Chicago Press, Chicago, IL (1985) ISBN: 0226288617; 0226288625. [The title is misleading. It might have been called: "The minimum abstract mathematical prerequisites needed to work in contemporary mathematical physics." It surveys several branches of abstract mathematics including Vector Spaces, Groups, Manifolds, Topological Spaces, etc..]

Schechter, Eric, Handbook of analysis and its foundations. Academic Press Inc., San Diego, CA (1997) ISBN: 0126227608 [Don't try to read from front to back. Instead, use as a reference work. This book contains materials on mathematical structures not included in Geroch, such as topological groups, vector measures, convex sets, and set theory.]

### Linear Algebra

Jänich. Linear Algebra or the original German: Lineare Algebra doi:10.1007/978-3-540-75501-2

Axler. Linear algebra done right. ISBN 0387982582

Halmos. Linear algebra problem book. ISBN 0883853221

### Abstract Algebra (groups, rings, modules, etc..)

Aluffi. Algebra: Chapter 0

Hungerford. Algebra ISBN 0387905189

Mac Lane and Birkhoff. Algebra

### Point-Set Topology

Kelley, John L., General topology. Springer-Verlag, New York (1975) ISBN: 3540901256

Munkres, James, Topology. Prentice Hall, Englewood Cliffs, N.J. (2000) ISBN: 0131816292

Steen, Lynn Arthur and Seebach, Jr., J. Arthur, Counterexamples in topology. Dover Publications Inc., Mineola, NY (1995) ISBN: 048668735X

### Differential Geometry

Lee. Introduction to Smooth Manifolds

Munkres, James R., Analysis on manifolds. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA (1991) ISBN: 0201510359

Wald. General Relativity. ISBN 0226870332

### Real and Complex Analysis

Rudin, Walter, Real and complex analysis. McGraw-Hill Book Co., New York (1987) ISBN: 0070542341

Needham, Tristan, Visual complex analysis. The Clarendon Press Oxford University Press, New York (1997) ISBN: 0198534477

### Measure Theory

Halmos, Paul R., Measure theory. Springer, New York, N. Y. (1974) ISBN: 0387900888

David Fremlin, Measure theory.

### Functional Analysis

Kadison, Richard and Ringrose, John, Fundamentals of the theory of operator algebras. AMS, Providence, RI (1997) ISBN: 0821808192 [Chapters 1 through 3]

Jordan, Thomas, Linear operators for quantum mechanics. Dover (2006).

### Operator Algebras

Start with Kadison and Ringrose (see under Functional Analysis). Aspire to Takesaki's three volume masterpiece (ISBN: 354042248X, 354042914X, 3540429131)

### Category Theory

Awodey. Category Theory.

Borceux. Handbook of categorical algebra

Mac Lane. Categories for the working mathematician.

### Topos Theory

Mac Lane and Moerdijk. Sheaves in geometry and logic.

Author: Hans Halvorson
Updated: Nov 10, 2010