## Math I Ought to Learn

*23 Dec 2020 09:40*

Some of you may have had occasion to run into mathematicians and to wonder therefore how they got that way...

---Tom Lehrer, "The Great Lobachevsky"

I know so little math for someone in my position that frankly I sometimes feel like a fraud. And much of what I do know is the half-wrong physicist's version.

Abstract algebra (beyond group theory): universal algebra, category theory, lattice theory, Galois lattices. Functional analysis (for real, not just the rudiments needed for probability and Markov processes). Algebraic geometry (for algebraic statistics). Matroids.

*See also:*
Basis Selection in Function Decomposition;
Cellular Automata;
Computation, Automata, Languages;
Dynamics;
Economics;
Ergodic Theory;
Gödel's Theorem;
Graph Limits and Infinite Exchangeable Arrays;
Information Theory;
"Math Methods";
Mathematical Logic;
Optimization;
Physics;
Probability Theory;
Statistics;
Operator Semigroups

- Recommended, big picture, non-technical:
- Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking
- G. H. Hardy, A Mathematician's Apology
- Mark Kac and Stanislaw M. Ulam, Mathematics and Logic
- George Polya, How to Solve It
- David Ruelle

- Recommended, big picture, technical:
- A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev (eds.), Mathematics: Its Content, Methods and Meaning [Everything you ever wanted to know about math, but were afraid a Soviet professor would tell you. Now available in a cheap, one-volume Dover edition.]
- Timothy Gowers (ed.), The Princeton Companion to Mathematics
- Neil Gershenfeld, The Nature of Mathematical Modeling
- Terence Tao, Structure and Randomness: Pages from Year One of a Mathematical Blog [Or you could just read the blog. My review: Obstacles and Tricks]

- Recommended, "how they got that way" (i.e., history and philosophy):
- J. L. Berggren, Episodes in the Mathematics of Medieval Islam
- J. L. Heilbron, Geometry Civilized: History, Culture, and Technique [Review: Construction of the Rhodian Shore, with Straightedge and Compass]
- George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics
- Philip Kitcher, The Nature of Mathematical Knowledge
[I'm not sure I quite agree with Kitcher's views about the nature of
mathematical reality, but I very much like, and agree with, his views about how
mathematics develops, and his sketch of how mathematics
*could be*an empirical science. (But I think his account fails when it comes to logic.) Review by Ian Hacking] - Hilary Putnam, "Mathematics without Foundations", The Journal
of Philosophy
**64**(1967): 5--22 [JSTOR. Thanks to Steve Laniel for pointing this out to me.] - William P. Thurston, "On proof and progress in mathematics", arxiv:math.HO/9404236 [Comments by Jordan Ellenberg]

- Recommended, close-ups:
- Paul Alexandroff, Elementary Concepts of Topology [With an introduction by Hilbert!]
- V. I. Arnol'd
- Mathematical Methods of Classical Mechanics
- Catastrophe Theory
- Ordinary Differential Equations

- Paul R. Halmos, "What Does the Spectral Theorem Say?",
American Mathematical Monthly
**70**(1963): 241--247 [JSTOR] - Mark Kac
- Enigmas of Chance [His autobiography; not very
profound, but an excellent view into the mind a modern mathematician. Kac was
*very*good, and his pedagogy, at least in writing, flawless, but (to use his own terms, which he doesn't apply to himself) he was a mere garden-variety genius, someone who thinks like your or I would, if only we were much smarter, instead of a freak like Feynman or von Neumann. It's a lot easier to learn from garden-variety geniuses.] - Probability and Related Topics in Physical Sciences
- Statistical Independence in Probability, Analysis and Number Theory
- Integration in Function Spaces

- Enigmas of Chance [His autobiography; not very
profound, but an excellent view into the mind a modern mathematician. Kac was
- A. N. Kolmogorov and S. V. Fomin, Introduction to Real
Analysis [An extremely good introduction not just to real analysis, but
also to the elements of complex and functional analysis, and of measure theory.
It also demands a level of mathematical sophistication which makes it
*deeply*unlikely that anyone who reads it is seeing real analysis for the first time.] - John McCleary, A First Course in Topology: Invariance and Dimension [Mini-review]
- Cristopher Moore and Stephan Mertens, The Nature of Computation [Cris and Stephan were kind enough to let me read this in manuscript; it's magnificent. Review: Intellects Vast and Warm and Sympathetic]
- W. V. O. Quine, Mathematical Logic [Rashly reviewed]
- Frigyes Riesz and Béla Sz.-Nagy, Functional Analysis
- Bertrand Russell, Introduction to Mathematical Philosophy [Was it Quine who called this the Principilla? If not, it should have been. Read this before Quine.]
- Bernard F. Schutz, Geometrical Methods of Mathematical Physics [Review]
- Michael Spivak, Calculus on Manifolds
- Ian Stewart and David Tall, Complex Analysis, the Hitchhiker's Guide to the Plane
- Sylvanus P. Thompson, Calculus Made Easy
["Considering how many fools can calculate, it is surprising that other fools
think it is difficult... What one fool can do, another can." Ignores all
sorts of subtleties about limits, which makes it excellent for
*learing*calculus. Analysis can come later.] - John von Neumann
- Norbert Wiener

- Not
- Paulus Gerdes, Marx Demystifies Calculus; translated
by Beatrice Lumpkin (from Karl Marx arrancar o veu misterioso a
matematica; I was puzzled about what language this was, but a
correspondent helpfully tells me it's Portuguese). Minneapolis: MEP
Publications, 1985, as vol. 16 of Studies in Marxism. [Collects
and expounds Marx's writings on mathematics and dialectics, for the benefit of
students confused by bourgeois explanations of differentiation and integration.
No I am
*not*making this up, I found it myself in Doe Memorial Library at Berkeley, with my own two hands I turned the pages.]

*exactly*recommended:

- To read, popular and miscellanea:
- Richard Courant and Herbert Robbins, What is Mathematics?
- Philip Davis
- Thomas Gray in Copenhagen: In Which the Philosopher Cat Meets the Ghost of Hans Christian Andersen
- The Thread: a Mathematical Yarn

- Anatolii Fomenko, Mathematical Impressions
- Caroll V. Newsom, Mathematical Discourses: The Heart of Mathematical Science
- Hugo Steinhaus, Mathematical Snapshots
- Ian Stewart
- Nature's Numbers: the Unreal Reality of Mathematical Imagination
- The Problems of Mathematics

- To read, history and philosophy:
- Andrew Aberdein, "The Uses of Argument in Mathematics", math.HO/0504090
- Archimedes, Works
- Marcia Ascher, Mathematics Elsewhere: An Exploration of Ideas Across Cultures
- Jody Azzouni, Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences
- Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings
- Alexandre V. Borovik, Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice
- David M. Bressoud, Calculus Reordered: A History of the Big Ideas [Review in MAA Reviews]
- Florian Cajori, Mathematics in Liberal Education
- Jean-Luc Chabert (ed.), A History of Algorithms: From the Pebble to the Microchip
- W. K. Clifford, Common Sense of the Exact Sciences
- Leo Corry, Modern Algebra and the Rise of Mathematical Structures
- Tobias Dantzig, Henri Poincaré: Critic of Crisis
- E. J. Dijksterhuis, Archimedes
- Jose Ferreiros
- Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics
- Mathematical Knowledge and the Interplay of Practices [Not clear how this improves on Kitcher's Nature of Mathematical Knowledge (if it does)]

- Michael Fitzgerald and Ioan James, The Mind of the Mathematician
- Charles Coulston Gillispie, Pierre-Simon Laplace, 1749--1827: A Life in Exact Science
- Roger Hart, The Chinese Roots of Linear Algebra
- Luke Hodgkin, A History of Mathematics: From Mesopotamia to Modernity
- Jens Hoyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin
- Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History
- Martin H. Krieger, Doing Mathematics: Convention, Subject, Calculation, Analogy
- Penelope Maddy, Naturalism in Mathematics
- Reviel Netz, The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations
- Kim Plofker, Mathematics in India
- Polya, Patterns of Plausible Inference
- Constance Reid
- Courant
- Hilbert
- Julia
- A Long Way from Euclid

- Robert Rynasiewicz, Shane Steinert-Threlkeld and Vivek Suri , "Mathematical Existence De-Platonized: Introducing Objects of Supposition in the Arts and Sciences", phil-sci/5345
- Sanford L. Segal, Mathematicians under the Nazis
- Reinhard Siegmund-Schultze, Mathematicians Fleeing from Nazi Germany: Individual Fates and Global Impact
- Stanislaw Ulam, Adventures of a Mathematician
- V. S. Varadarajan, Algebra in Ancient and Modern Times
- Ferdinand Verhulst, Henri Poincaré: Impatient Genius
- Benjamin Wardhaugh, Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World

- To read, pedagogical (see also math methods):
- Axler, Linear Algebra Done Right
- Richard Beals, Analysis: An Introduction
- Dennis S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas
- Adam Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction
- David A. Brannan, Matthew F. Esplen and Jeremy J. Gray, Geometry
- Victor Bryant, Yet Another Introduction to Analysis
- Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms
- J. Scott Carter, How Surfaces Intersect in Space: An Introduction to Topology
- D. Choimet and H. Queffélec, Twelve Landmarks of Twentieth-Century Analysis
- Richard Courant, Introduction to Calculus and Analysis
- Ebbinghaus, Hermes, Hirzebruch, Koecher, Remmert, Mainzer, Neukirch and Presetel, Numbers
- Robert Geroch, Mathematical Physics ["Really, it all becomes much clearer once you start using category theory! Wait, don't run away! Why are you all looking at me like that? Doesn't anyone believe me?" (Not an actual quote from what, about half-way through, proves to be a very nice book.)]
- Larry Gonick
- The Cartoon Guide to Calculus
- The Cartoon Guide to Algebra

- Jürgen Jost
- Partial Differential Equations
- Postmodern Analysis
- Riemannian Geometry and Geometric Analysis

- David W. Kammler, A First Course in Fourier Analysis
- James T. Kinard and Alex Kozulin, Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom
- T. W. Korner, Fourier Analysis
- Ian Bradley and Ronald L. Meek, Matrices and Society: Matrix Algebra and Its Applications in the Social Sciences
- Tristan Needham
- Visual Complex Analysis
- Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts

- L. A. Pars, Introduction to the Calculus of Variations
- L. Ridgway Scott, Numerical Analysis
- Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis

- To read, technical:
- Ravi P. Agarwal, Fixed Point Theory and Applications
- Artin, Modern Algebra
- Keith Ball, "An Elementary Introduction to Modern Convex Geometry"
- Alexander Barvinok, A Course in Convexity
- Marcel Berger, Geometry Revealed [Review by Danny Yee]
- Vasile Berinde, Iterative Approximation of Fixed Points
- Birkhoff, Lattice Theory
- Birkhoff and MacLane, A Survey of Modern Algebra
- Salomon Bochner, Harmonic Analysis and the Theory of Probability
- Jonathan Borwein and David Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century [Review in American Scientist]
- Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery [Review in American Scientist]
- Burris and Sankappanavar, A Course in Universal Algebra
- Bernd Carl, Entropy, Compactness, and the Approximation of Operators
- Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick, Analysis, Manifolds and Physics
- Paul M. Cohn, Universal Algebra
- Thierry Coquand and Henri Lombardi, "A logical approach to abstract
algebra", Mathematics Structures
in Computer Science
**16**(2006): 885--900 - B. A. Davey and H. A. Priestly, Introduction to Lattices and Order
- Reinhard Diestel, Graph Theory
- Xavier Gràcia, Miguel C. Muñoz-Lecanda, Narciso Román-Roy, "On some aspects of the geometry of differential equations in physics", math-ph/0402030
- Uri M. Ascher and Chen Greif, A First Course in Numerical Methods
- B. Grunbaum and G. C. Shephard, Tiling and Patterns
- Paul R. Halmos
- Finite Dimensional Vector Spaces
- An Introduction to Hilbert Space and the Theory of Spectral Multiplicity
- Measure Theory
- Naive Set Theory

- S. Helgason, "Radon-Fourier Transforms on Symmetric Spaces and Related Group Representations", Bulletin of the American Mathematical Society
**71**(1965): 757--763 - Einar Hille, Functional Analysis and Semi-Groups [I've read about half of this]
- Mark Kac, Selected Papers
- Gerald Kaiser, A Friendly Guide to Wavelets
- Yitzhak Katznelson, An Introduction to Harmonic Analysis
- Solomon Lefschetz, Algebraic Geometry
- Lipkin, Lie Groups for Pedestrians
- Sanders MacLane, Categories for the Working Mathematician
- James Munkres, Topology
- J. D. Murray, Mathematical Biology
- A. Neumaier, Introduction to Numerical Analysis
- Yves Nievergelt, Wavelets Made Easy
- Ivan Niven, Mathematics of Choice: Or, How to Count without Counting
- James Oxley, Matroid Theory
- Jonathan Partington, Interpolation, Identification and Sampling
- Vern I. Paulsen and Mrinal Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces
- Pedersen, Analysis Now
- Chris Preston, "Some notes on standard Borel and related spaces", arxiv:0809.3066
- Mark Ptekovsek, Herbert Wilf and Doron Zeilberger, A = B
- R. Tyrell Rockafellar, Convex Analysis
- Walter Rudin
- Functional Analysis
- Principles of Mathematical Analysis

- Gunther Schmidt, Relational Mathematics
- R. E. Showalter, Hilbert Space Methods for Partial Differential Equations
- Barry Simon, Convexity: An Analytic Viewpoint
- Michael J. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities
- Elias M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
- Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis
- Ian Stewart
- Galois Theory
- Lie Algebras

- Stroock, Probability Theory: An Analytic View
- Endre Sülli and David F. Meyers, Introduction to Numerical Analysis
- Klaus Truemper, Matroid Decomposition
- Warwick Tucker, Validated Numerics: A Short Introduction to Rigorous Computations
- Ulam, Analogies between Analogies
- R. F. C. Walters, Categories and Computer Science
- Robert J. Zimmer, Essential Results of Functional Analysis