Math I Ought to Learn

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. Functional analysis (for real, not just the rudiments needed for probability and Markov processes).

Paul Alexandroff, Elementary Concepts of Topology [With an introduction by Hilbert!]
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 Russian professor would tell you. Now available in a cheap, one-volume Dover edition.]
V. I. Arnol'd, Ordinary Differential Equations
J. L. Berggren, Episodes in the Mathematics of Medieval Islam
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.]
G. H. Hardy, A Mathematician's Apology
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
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
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]
Neil Gershenfeld, The Nature of Mathematical Modeling
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]
John McCleary, A First Course in Topology: Invariance and Dimension [Mini-review]
W. V. O. Quine, Mathematical Logic [Rashly reviewed]
David Ruelle
- "Conversations on Mathematics with a Visitor from Outer Space"
- Chance and Chaos
- The Mathematician's Brain [Mini-review]
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[Long review, with some history and explication of differential geometry]
Michael Spivak, Calculus on Manifolds
Ian Stewart and David Tall, Complex Analysis, the Hitchhiker's Guide to the Plane
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]
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

Courant, with 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 [The blurb for this book (1964) is so remarkable I can't resist quoting it at length.
The mathematicizing of science has led to revolutionary changes in our civilization, especially in recent years. In addition, new approaches to the study of many aspects of human knowledge have been made possible by what scholars have learned through their struggle to undersand the nature of mathematics. Thus a person living in our time can hardly be characterized as educated unless he has some understanding of the fundamental concepts of mathematical science, their meaning, and their proper utilization.
Yet, hardly any expositions treating these subjects are available to the inquisitive reader who possesses limited mathematical background. Some excellent books and articles upon the foundations of mathematics have been written for the professional mathematician or logician, but the sophistication of their approach is such that only a few members of the reading public can understand them. The concern of most popular books on mathematics, on the other hand, is merely with interesting problems and anecdotes that appear upon the fringe of mathematical knowledge; little attention is given to mathematics as a systematic discipline.
This little treatise, therefore, written for the nonmathematician, is concerned with mathematical discourses, which are central to all mathematical study. Unless a person has some familiarity with the concept of mathematical discourse, he is not prepared to understand even the rudiments of modern mathematics and its significance. An understanding of the subject of this book, in other words, is eesential to mathematical literacy.
]
Hugo Steinhaus, Mathematical Snapshots
Ian Stewart
- Nature's Numbers: the Unreal Reality of Mathematical Imagination
- The Problems of Mathematics

Andrew Aberdein, "The Uses of Argument in Mathematics", math.HO/0504090
Archimedes, Works
Jody Azzouni, Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences [Blurb]
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 ("Edited and with a pref. by Karl Pearson; newly edited and with an introd. by James R. Newman; pref. by Bertrand Russell")
E. J. Dijksterhuis, Archimedes
Charles Coulston Gillispie, Pierre-Simon Laplace, 1749--1827: A Life in Exact Science
Luke Hodgkin, A History of Mathematics: From Mesopotamia to Modernity [Blurb]
Jens Hoyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin
Penelope Maddy, Naturalism in Mathematics
Reviel Netz, The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations [blurb]
Polya
- Patterns of Plausible Inference
- How to Solve It
Constance Reid
- Courant and Hilbert
- Julia
- A Long Way from Euclid [blurb]
William P. Thurston, "On proof and progress in mathematics", arxiv:math.HO/9404236 [Comments by Jordan Ellenberg]
V. S. Varadarajan, Algebra in Ancient and Modern Times

Walter Appel, Mathematics for Physics and Physicists [blurb]
Axler, Linear Algebra Done Right
Richard Beals, Analysis: An Introduction
Adam Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction [Blurb]
Victor Bryant, Yet Another Introduction to Analysis [Blurb]
Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms
J. Scott Carter, How Surfaces Intersect in Space: An Introduction to Topology
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 looks like a very nice book.)]
Jürgen Jost
- Partial Differential Equations
- Postmodern Analysis
- Riemannian Geometry and Geometric Analysis
Tristan Needham Visual Complex Analysis

Artin, Modern Algebra
Keith Ball, "An Elementary Introduction to Modern Convex Geometry"
Vasile Berinde, Iterative Approximation of Fixed Points
Birkhoff, Lattice Theory
Birkhoff and MacLane, A Survey of Modern Algebra
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 [on-line]
Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick, Analysis, Manifolds and Physics
Paul M. Cohn, Universal Algebra [Anyone have a copy they'd be willing to sell?]
Thierry Coquand and Henri Lombardi, "A logical approach to abstract algebra", Mathematics Structures in Computer Science 16 (2006): 885--900
Courant
- Introduction to Calculus and Analysis
- and David Hilbert, Methods of Mathematical Physics
B. A. Davey and H. A. Priestly, Introduction to Lattices and Order
Reinhard Diestel, Graph Theory [online]
Timothy Gowers (ed.), The Princeton Companion to Mathematics
Xavier Gr´cia, Miguel C. Munoz-Lecanda, Narciso Roman-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
- An Introduction to Hilbert Space and the Theory of Spectral Multiplicity
- Measure Theory
- Naive Set Theory
Mark Kac, Selected Papers
Gerald Kaiser, A Friendly Guide to Wavelets
Yitzhak Katznelson, An Introduction to Harmonic Analysis [Blurb]
Solomon Lefschetz, Algebraic Geometry
Lipkin, Lie Groups for Pedestrians
Sanders MacLane, Categories for the Working Mathematician
James Munkres, Topology
J. D. Murray, Mathematical Biology
Yves Nievergelt, Wavelets Made Easy
Ivan Niven, Mathematics of Choice: Or, How to Count without Counting
Jonathan Partington, Interpolation, Identification and Sampling
Pedersen, Analysis Now
Mark Ptekovsek, Herbert Wilf and Doron Zeilberger, A = B [online]
Frigyes Riesz and Béla Sz.-Nagy, Functional Analysis
Walter Rudin
- Functional Analysis
- Principles of Mathematical Analysis
R. E. Showalter, Hilbert Space Methods for Partial Differential Equations [Free online]
Michael J. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities
Ian Stewart
- Galois Theory
- Lie Algebras
Stroock, Probability Theory: An Analytic View
Klaus Truemper, Matroid Decomposition [online]
R. F. C. Walters, Categories and Computer Science
Robert J. Zimmer, Essential Results of Functional Analysis

Notebooks

Math I Ought to Learn