Math I Ought to Learn
Last update: 21 Jul 2026 14:15First version: 11 December 2000 (if not before)
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 (for conditional independence).
- See also:
- Basis Selection in Function Decomposition
- Calculus of Variations and Optimal Control Theory
- Category Theory
- Cellular Automata
- Computation, Automata, Languages
- Dynamics
- Economics
- Ergodic Theory
- Gödel's Theorem
- Graph Limits and Infinite Exchangeable Arrays
- Graph Theory
- Information Theory
- "Math Methods"
- Mathematical Logic
- Numerical Methods
- Optimization
- Physics
- Probability Theory
- Operator Semigroups
- Random Matrix Theory
- Statistics
- 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 Soviet professors would tell you. Now available in a one-volume Dover edition.]
- Neil Gershenfeld, The Nature of Mathematical Modeling
- The Princeton Companion to Mathematics (ed. Timothy Gowers, June Barrow-Green and Imre Leader)
- The Princeton Companion to Applied Mathematics (ed. Nicholas J. Higham, Mark R. Dennis, Paul Glendinning, Paul A. Martin, Fadil Santosa and Jared Tanner)
- 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
- 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
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics
- Paul R. Halmos, "What Does the Spectral Theorem Say?", American Mathematical Monthly 70 (1963): 241--247 [JSTOR]
- Jürgen Jost, Postmodern Analysis
- 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. --- An appreciation of the book by Maria Popova]
- Probability and Related Topics in Physical Sciences
- Statistical Independence in Probability, Analysis and Number Theory
- Integration in Function Spaces and Some of Its Applications
- Selected Papers
- 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 successfully is seeing real analysis for the first time.]
- John McCleary, A First Course in Topology: Invariance and Dimension
- 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]
- Manya Raman-Sundstrom, "A pedagogical history of compactness", arxiv:1006.4131
- Frigyes Riesz and Béla Sz.-Nagy, Functional Analysis
- Bertrand Russell, Introduction to Mathematical Philosophy
- Bernard F. Schutz, Geometrical Methods of Mathematical Physics
- 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 exactly recommended:
- 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 in 1991 or 1992, with my own two hands I turned the pages.]
- Loren R. Graham and Jean-Michel Kantor, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity [Unlike Gerdes, these authors are serious scholars, but this is not a good book, for reasons I have tried to explain in my review]
- To finish, one of these decades:
- Paul M. Cohn, Universal Algebra
- 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.)]
- Paul R. Halmos, Measure Theory [This helped me to write one of my favorite papers...]
- Einar Hille, Functional Analysis and Semi-Groups
- T. W. Korner, Fourier Analysis
- R. Tyrell Rockafellar, Convex Analysis
- Stanislaw Ulam, Analogies between Analogies
- To read, popular and miscellanea:
- Richard Courant and Herbert Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods [2nd edition, revised by Ian Stewart]
- Philip Davis
- Thomas Gray, Philosopher Cat
- Thomas Gray in Copenhagen: In Which the Philosopher Cat Meets the Ghost of Hans Christian Andersen
- The Thread: a Mathematical Yarn
- A. T. Fomenko, Mathematical Impressions [Appreciation by Maria Popova. Unrelated (?): Oh, Anatoly Timofeevich, no!]
- Caroll V. Newsom, Mathematical Discourses: The Heart of Mathematical Science
- Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics
- 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
- 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
- Reviel Netz
- Reviel Netz and William Noel, The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity's Greatest Scientist
- Kim Plofker, Mathematics in India
- George Polya, Mathematics and Plausible Reasoning
- Constance Reid, 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
- Anna Sverdlik, Perfect Mathematics for Imperfect Minds: How Our Emotions and Bodies are Vital for Abstract Thought
- Fenner Stanley Tanswell, Mathematical Rigour and Informal Proof
- V. S. Varadarajan, Algebra in Ancient and Modern Times
- Benjamin Wardhaugh, Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World
- To read, the lives of the mathematicians:
- Tobias Dantzig, Henri Poincaré: Critic of Crisis
- E. J. Dijksterhuis, Archimedes
- Angelo Guerraggio and Giovanni Paoloni, Vito Volterra
- Paul R. Halmos, I Want to be a Mathematician: An Automathography
- Constance Reid
- Courant
- Hilbert
- Julia
- Stanislaw Ulam, Adventures of a Mathematician
- Ferdinand Verhulst, Henri Poincaré: Impatient Genius
- To read, pedagogical (see also math methods) (also there's some overlap with the next category...):
- Sheldon Axler, Linear Algebra Done Right
- 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
- Larry Gonick
- The Cartoon Guide to Algebra
- The Cartoon Guide to Calculus
- The Cartoon Guide to Geometry
- Jürgen Jost
- Partial Differential Equations
- 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
- Ian Bradley and Ronald L. Meek, Matrices and Society: Matrix Algebra and Its Applications in the Social Sciences
- Tristan Needham
- Walter Rudin
- Functional Analysis
- Principles of Mathematical Analysis
- Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis
- To read, technical / my learning stuff:
- Ravi P. Agarwal, Maria Meehan, and Donal O'Regan, Fixed Point Theory and Applications
- Artin, Modern Algebra
- Keith Ball, "An Elementary Introduction to Modern Convex Geometry", pp. 1--58 in Silvio Levy (ed.), Flavors of Geometry
- Alexander Barvinok, A Course in Convexity
- Marcel Berger, Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry [Review by Danny Yee]
- Vasile Berinde, Iterative Approximation of Fixed Points
- Garrett Birkhoff, Lattice Theory
- Garrett Birkhoff and Saunders Mac Lane, 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
- Yvonne Choquet-Bruhat, Cecile DeWitt-Morette and Margaret Dillard-Bleick, Analysis, Manifolds and Physics
- 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
- Xavier Gràcia, Miguel C. Muñoz-Lecanda, Narciso Román-Roy, "On some aspects of the geometry of differential equations in physics", math-ph/0402030
- B. Grunbaum and G. C. Shephard, Tiling and Patterns
- Paul R. Halmos
- G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities
- S. Helgason, "Radon-Fourier Transforms on Symmetric Spaces and Related Group Representations", Bulletin of the American Mathematical Society 71 (1965): 757--763
- Gerald Kaiser, A Friendly Guide to Wavelets
- Yitzhak Katznelson, An Introduction to Harmonic Analysis
- Solomon Lefschetz, Algebraic Geometry
- Harry J. Lipkin, Lie Groups for Pedestrians [Possibly-legit PDF]
- Jiri Matousek, Geometric Discrepancy: An Illustrated Guide
- James Munkres, Topology
- J. D. Murray, Mathematical Biology
- 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
- Gerd K. 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
- 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
- Ian Stewart
- Lloyd N. Trefethen and Mark Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators
- Klaus Truemper, Matroid Decomposition
- Herbert S. Wilf, Generatingfunctionology
- Robert J. Zimmer, Essential Results of Functional Analysis