The Best Textbooks on Every Subject

lukeprog

811 The Best Textbooks on Every Subject

16th Jan 2011

9 min read

811

For years, my self-education was stupid and wasteful. I learned by consuming blog posts, Wikipedia articles, classic texts, podcast episodes, popular books, video lectures, peer-reviewed papers, Teaching Company courses, and Cliff's Notes. How inefficient!

I've since discovered that textbooks are usually the quickest and best way to learn new material. That's what they are designed to be, after all. Less Wrong has often recommended the "read textbooks!" method. Make progress by accumulation, not random walks.

But textbooks vary widely in quality. I was forced to read some awful textbooks in college. The ones on American history and sociology were memorably bad, in my case. Other textbooks are exciting, accurate, fair, well-paced, and immediately useful.

What if we could compile a list of the best textbooks on every subject? That would be extremely useful.

Let's do it.

There have been other pages of recommended reading on Less Wrong before (and elsewhere), but this post is unique. Here are the rules:

Post the title of your favorite textbook on a given subject.
You must have read at least two other textbooks on that same subject.
You must briefly name the other books you've read on the subject and explain why you think your chosen textbook is superior to them.

Rules #2 and #3 are to protect against recommending a bad book that only seems impressive because it's the only book you've read on the subject. Once, a popular author on Less Wrong recommended Bertrand Russell's A History of Western Philosophy to me, but when I noted that it was more polemical and inaccurate than the other major histories of philosophy, he admitted he hadn't really done much other reading in the field, and only liked the book because it was exciting.

I'll start the list with three of my own recommendations...

Subject: History of Western Philosophy

Recommendation: The Great Conversation, 6th edition, by Norman Melchert

Reason: The most popular history of western philosophy is Bertrand Russell's A History of Western Philosophy, which is exciting but also polemical and inaccurate. More accurate but dry and dull is Frederick Copelston's 11-volume A History of Philosophy. Anthony Kenny's recent 4-volume history, collected into one book as A New History of Western Philosophy, is both exciting and accurate, but perhaps too long (1000 pages) and technical for a first read on the history of philosophy. Melchert's textbook, The Great Conversation, is accurate but also the easiest to read, and has the clearest explanations of the important positions and debates, though of course it has its weaknesses (it spends too many pages on ancient Greek mythology but barely mentions Gottlob Frege, the father of analytic philosophy and of the philosophy of language). Melchert's history is also the only one to seriously cover the dominant mode of Anglophone philosophy done today: naturalism (what Melchert calls "physical realism"). Be sure to get the 6th edition, which has major improvements over the 5th edition.

Subject: Cognitive Science

Recommendation: Cognitive Science, by Jose Luis Bermudez

Reason: Jose Luis Bermudez's Cognitive Science: An Introduction to the Science of Mind does an excellent job setting the historical and conceptual context for cognitive science, and draws fairly from all the fields involved in this heavily interdisciplinary science. Bermudez does a good job of making himself invisible, and the explanations here are some of the clearest available. In contrast, Paul Thagard's Mind: Introduction to Cognitive Science skips the context and jumps right into a systematic comparison (by explanatory merit) of the leading theories of mental representation: logic, rules, concepts, analogies, images, and neural networks. The book is only 270 pages long, and is also more idiosyncratic than Bermudez's; for example, Thagard refers to the dominant paradigm in cognitive science as the "computational-representational understanding of mind," which as far as I can tell is used only by him and people drawing from his book. In truth, the term refers to a set of competing theories, for example the computational theory and the representational theory. While not the best place to start, Thagard's book is a decent follow-up to Bermudez's text. Better, though, is Kolak et. al.'s Cognitive Science: An Introduction to Mind and Brain. It contains more information than Bermudez's book, but I prefer Bermudez's flow, organization and content selection. Really, though, both Bermudez and Kolak offer excellent introductions to the field, and Thagard offers a more systematic and narrow investigation that is worth reading after Bermudez and Kolak.

Subject: Introductory Logic for Philosophy

Recommendation: Meaning and Argument by Ernest Lepore

Reason: For years, the standard textbook on logic was Copi's Introduction to Logic, a comprehensive textbook that has chapters on language, definitions, fallacies, deduction, induction, syllogistic logic, symbolic logic, inference, and probability. It spends too much time on methods that are rarely used today, for example Mill's methods of inductive inference. Amazingly, the chapter on probability does not mention Bayes (as of the 11th edition, anyway). Better is the current standard in classrooms: Patrick Hurley's A Concise Introduction to Logic. It has a table at the front of the book that tells you which sections to read depending on whether you want (1) a traditional logic course, (2) a critical reasoning course, or (3) a course on modern formal logic. The single chapter on induction and probability moves too quickly, but is excellent for its length. Peter Smith's An Introduction to Formal Logic instead focuses tightly on the usual methods used by today's philosophers: propositional logic and predicate logic. My favorite in this less comprehensive mode, however, is Ernest Lepore's Meaning and Argument, because it (a) is highly efficient, and (b) focuses not so much on the manipulation of symbols in a formal system but on the arguably trickier matter of translating English sentences into symbols in a formal system in the first place.

I would love to read recommendations from experienced readers on the following subjects: physics, chemistry, biology, psychology, sociology, probability theory, economics, statistics, calculus, decision theory, cognitive biases, artificial intelligence, neuroscience, molecular biochemistry, medicine, epistemology, philosophy of science, meta-ethics, and much more.

Please, post your own recommendations! And, follow the rules.

Recommendations so far (that follow the rules; this list updated 02-25-2017):

On history of western philosophy, lukeprog recommends Melchert's The Great Conversation over Russell's A History of Western Philosophy, Copelston's History of Philosophy, and Kenney's A New History of Western Philosophy.
On cognitive science, lukeprog recommends Bermudez's Cognitive Science over Thagard's Mind: Introduction to Cognitive Science and Kolak's Cognitive Science.
On introductory logic for philosophy, lukeprog recommends Lepore's Meaning and Argument over Copi's Introduction to Logic, Hurley's A Concise Introduction to Logic, and Smith's An Introduction to Formal Logic.
On economics, michaba03m recommends Mankiw's Macroeconomics over Varian's Intermediate Microeconomics and Katz & Rosen's Macroeconomics.
On economics, realitygrill recommends McAfee's Introduction to Economic Analysis over Mankiw's Principles of Microeconomics and Case & Fair's Principles of Macroeconomics.
On representation theory, SarahC recommends Sternberg's Group Theory and Physics over Lang's Algebra, Weyl's The Theory of Groups and Quantum Mechanics, and Fulton & Harris' Representation Theory: A First Course.
On statistics, madhadron recommends Kiefer's Introduction to Statistical Inference over Hogg & Craig's Introduction to Mathematical Statistics, Casella & Berger's Statistical Inference, and others.
On advanced Bayesian statistics, Cyan recommends Gelman's Bayesian Data Analysis over Jaynes' Probability Theory: The Logic of Science and Bernardo's Bayesian Theory.
On basic Bayesian statistics, jsalvatier recommends Skilling & Sivia's Data Analysis: A Bayesian Tutorial over Gelman's Bayesian Data Analysis, Bolstad's Bayesian Statistics, and Robert's The Bayesian Choice.
On real analysis, paper-machine recommends Bartle's A Modern Theory of Integration over Rudin's Real and Complex Analysis and Royden's Real Analysis.
On non-relativistic quantum mechanics, wbcurry recommends Sakurai & Napolitano's Modern Quantum Mechanics over Messiah's Quantum Mechanics, Cohen-Tannoudji's Quantum Mechanics, and Greiner's Quantum Mechanics: An Introduction.
On music theory, komponisto recommends Westergaard's An Introduction to Tonal Theory over Piston's Harmony, Aldwell and Schachter's Harmony and Voice Leading, and Kotska and Payne's Tonal Harmony.
On business, joshkaufman recommends Kaufman's The Personal MBA: Master the Art of Business over Bevelin's Seeking Wisdom and Munger's Poor Charlie's Alamanack.
On machine learning, alexflint recommends Bishop's Pattern Recognition and Machine Learning over Russell & Norvig's Artificial Intelligence: A Modern Approach and Thrun et. al.'s Probabilistic Robotics.
On algorithms, gjm recommends Cormen et. al.'s Introduction to Algorithms over Knuth's The Art of Computer Programming and Sedgwick's Algorithms.
On electrodynamics, Alex_Altair recommends Griffiths' Introduction to Electrodynamics over Jackson's Electrodynamics and Feynman's Lectures on Physics.
On electrodynamics, madhadron recommends Purcell's Electricity and Magnetism over Griffith's Introduction to Electrodynamics, Feynman's Lectures on Physics, and others.
On systems theory, Davidmanheim recommends Meadows' Thinking in Systems: A Primer over Senge's The Fifth Discipline: The Art & Practice of The Learning Organization and Kim's Introduction to Systems Thinking.
On self-help, lukeprog recommends Weiten, Dunn, and Hammer's Psychology Applied to Modern Life over Santrock's Human Adjustment and Tucker-Ladd's Psychological Self-Help.
On probability theory, SarahC recommends Feller's An Introduction to Probability Theory + Vol. 2 over Ross' A First Course in Probability and Koralov & Sinai's Theory of Probability and Random Processes.
On probability theory, madhadron recommends Grimmett & Stirzaker's Probability and Random Processes over Feller's Introduction to Probability Theory and Its Applications and Nelson's Radically Elementary Probability Theory.
On topology, jsteinhardt recommends Munkres' Topology over Armstrong's Topology and Massey's Algebraic Topology.
On linguistics, etymologik recommends O'Grady et al.'s Contemporary Linguistics over Hayes et al.'s Linguistics: An Introduction to Linguistic Theory and Carnie's Syntax: A Generative Introduction.
On meta-ethics, lukeprog recommends Miller's An Introduction to Contemporary Metaethics over Jacobs' The Dimensions of Moral Theory and Smith's Ethics and the A Priori.
On decision-making & biases, badger recommends Bazerman & Moore's Judgment in Managerial Decision Making over Hastie & Dawes' Rational Choice in an Uncertain World, Gilboa's Making Better Decisions, and others.
On neuroscience, kjmiller recommends Bear et al's Neuroscience: Exploring the Brain over Purves et al's Neuroscience and Kandel et al's Principles of Neural Science.
On World War II, Peacewise recommends Weinberg's A World at Arms over Churchill's The Second World War and Day's The Politics of War.
On elliptic curves, magfrump recommends Koblitz' Introduction to Elliptic Curves and Modular Forms over Silverman's Arithmetic of Elliptic Curves and Cassel's Lectures on Elliptic Curves.
On improvisation, Arepo recommends Salinsky & Frances-White's The Improv Handbook over Johnstone's Impro, Johnston's The Improvisation Game, and others.
On thermodynamics, madhadron recommends Hatsopoulos & Keenan's Principles of General Thermodynamics over Fermi's Thermodynamics, Sommerfeld's Thermodynamics and Statistical Mechanics, and others.
On statistical mechanics, madhadron recommends Landau & Lifshitz' Statistical Physics, Volume 5 over Sethna's Entropy, Order Parameters, and Complexity and Reichl's A Modern Course in Statistical Physics.
On criminal justice, strange recommends Fuller's Criminal Justice: Mainstream and Crosscurrents over Neubauer & Fradella's America's Courts and the Criminal Justice System and Albanese' Criminal Justice.
On organic chemistry, rhodium recommends Clayden et al's Organic Chemistry over McMurry's Organic Chemistry and Smith's Organic Chemistry.
On special relativity, iDante recommends Taylor & Wheeler's Spacetime Physics over Harris' Modern Physics, French's Special Relativity, and others.
On abstract algebra, Bundle_Gerbe recommends Dummit & Foote's Abstract Algebra over Lang's Algebra and others.
On decision theory, lukeprog recommends Peterson's An Introduction to Decision Theory over Resnik's Choices and Luce & Raiffa's Games and Decisions.
On calculus, orthonormal recommends Spivak's Calculus over Thomas' Calculus and Stewart's Calculus.
On analysis in Rⁿ, orthonormal recommends Strichartz's The Way of Analysis over Rudin's Principles of Mathematical Analysis and Kolmogorov & Fomin's Introduction to Real Analysis.
On real analysis and measure theory, orthonormal recommends Stein & Shakarchi's Measure Theory, Integration, and Hilbert Spaces over Royden's Real Analysis and Rudin's Real and Complex Analysis.
On partial differential equations, orthonormal recommends Strauss' Partial Differential Equations over Evans' Partial Differential Equations and Hormander's Analysis of Partial Differential Operators.
On introductory real analysis, SatvikBeri recommends Pugh's Real Mathematical Analysis over Lang's Real and Functional Analysis and Rudin's Principles of Mathematical Analysis.
On commutative algebra, SatvikBeri recommends MacDonald's Introduction to Commutative Algebra over Lang's Algebra and Eisenbud's Commutative Algebra With a View Towards Algebraic Geometry.
On animal behavior, Natha recommends Alcock's Animal Behavior, 6th edition over Dugatkin's Principles of Animal Behavior and newer editions of the Alcock textbook.
On calculus, Epictetus recommends Courant's Differential and Integral Calculus over Stewart's Calculus and Kline's Calculus.
On linear algebra, Epictetus recommends Shilov's Linear Algebra over Lay's Linear Algebra and its Appications and Axler's Linear Algebra Done Right.
On numerical methods, Epictetus recommends Press et al.'s Numerical Recipes over Bulirsch & Stoer's Introduction to Numerical Analysis, Atkinson's An Introduction to Numerical Analysis, and Hamming's Numerical Methods of Scientists and Engineers.
On ordinary differential equations, Epictetus recommends Arnold's Ordinary Differential Equations over Coddington's An Introduction to Ordinary Differential Equations and Enenbaum & Pollard's Ordinary Differential Equations.
On abstract algebra, Epictetus recommends Jacobson's Basic Algebra over Bourbaki's Algebra, Lang's Algebra, and Hungerford's Algebra.
On elementary real analysis, Epictetus recommends Rudin's Principles of Mathematical Analysis over Ross' Elementary Analysis, Lang's Undergraduate Analysis, and Hardy's A Course of Pure Mathematics.

If there are no recommendations for the subject you want to learn, you can start by checking the Alibris textbooks category for your subject, and sort by 'Top-selling.' But you'll have to do more research than that. Check which textbooks are asked for in the syllabi of classes on your subject at leading universities. Search Google for recommendations and reviews.

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811 The Best Textbooks on Every Subject

by lukeprog

16th Jan 2011

9 min read

418

811

What if we could compile a list of the best textbooks on every subject? That would be extremely useful.

Let's do it.

There have been other pages of recommended reading on Less Wrong before (and elsewhere), but this post is unique. Here are the rules:

Post the title of your favorite textbook on a given subject.
You must have read at least two other textbooks on that same subject.
You must briefly name the other books you've read on the subject and explain why you think your chosen textbook is superior to them.

I'll start the list with three of my own recommendations...

Subject: History of Western Philosophy

Recommendation: The Great Conversation, 6th edition, by Norman Melchert

Subject: Cognitive Science

Recommendation: Cognitive Science, by Jose Luis Bermudez

Subject: Introductory Logic for Philosophy

Recommendation: Meaning and Argument by Ernest Lepore

Please, post your own recommendations! And, follow the rules.

Recommendations so far (that follow the rules; this list updated 02-25-2017):

On history of western philosophy, lukeprog recommends Melchert's The Great Conversation over Russell's A History of Western Philosophy, Copelston's History of Philosophy, and Kenney's A New History of Western Philosophy.
On cognitive science, lukeprog recommends Bermudez's Cognitive Science over Thagard's Mind: Introduction to Cognitive Science and Kolak's Cognitive Science.
On introductory logic for philosophy, lukeprog recommends Lepore's Meaning and Argument over Copi's Introduction to Logic, Hurley's A Concise Introduction to Logic, and Smith's An Introduction to Formal Logic.
On economics, michaba03m recommends Mankiw's Macroeconomics over Varian's Intermediate Microeconomics and Katz & Rosen's Macroeconomics.
On economics, realitygrill recommends McAfee's Introduction to Economic Analysis over Mankiw's Principles of Microeconomics and Case & Fair's Principles of Macroeconomics.
On representation theory, SarahC recommends Sternberg's Group Theory and Physics over Lang's Algebra, Weyl's The Theory of Groups and Quantum Mechanics, and Fulton & Harris' Representation Theory: A First Course.
On statistics, madhadron recommends Kiefer's Introduction to Statistical Inference over Hogg & Craig's Introduction to Mathematical Statistics, Casella & Berger's Statistical Inference, and others.
On advanced Bayesian statistics, Cyan recommends Gelman's Bayesian Data Analysis over Jaynes' Probability Theory: The Logic of Science and Bernardo's Bayesian Theory.
On basic Bayesian statistics, jsalvatier recommends Skilling & Sivia's Data Analysis: A Bayesian Tutorial over Gelman's Bayesian Data Analysis, Bolstad's Bayesian Statistics, and Robert's The Bayesian Choice.
On real analysis, paper-machine recommends Bartle's A Modern Theory of Integration over Rudin's Real and Complex Analysis and Royden's Real Analysis.
On non-relativistic quantum mechanics, wbcurry recommends Sakurai & Napolitano's Modern Quantum Mechanics over Messiah's Quantum Mechanics, Cohen-Tannoudji's Quantum Mechanics, and Greiner's Quantum Mechanics: An Introduction.
On music theory, komponisto recommends Westergaard's An Introduction to Tonal Theory over Piston's Harmony, Aldwell and Schachter's Harmony and Voice Leading, and Kotska and Payne's Tonal Harmony.
On business, joshkaufman recommends Kaufman's The Personal MBA: Master the Art of Business over Bevelin's Seeking Wisdom and Munger's Poor Charlie's Alamanack.
On machine learning, alexflint recommends Bishop's Pattern Recognition and Machine Learning over Russell & Norvig's Artificial Intelligence: A Modern Approach and Thrun et. al.'s Probabilistic Robotics.
On algorithms, gjm recommends Cormen et. al.'s Introduction to Algorithms over Knuth's The Art of Computer Programming and Sedgwick's Algorithms.
On electrodynamics, Alex_Altair recommends Griffiths' Introduction to Electrodynamics over Jackson's Electrodynamics and Feynman's Lectures on Physics.
On electrodynamics, madhadron recommends Purcell's Electricity and Magnetism over Griffith's Introduction to Electrodynamics, Feynman's Lectures on Physics, and others.
On systems theory, Davidmanheim recommends Meadows' Thinking in Systems: A Primer over Senge's The Fifth Discipline: The Art & Practice of The Learning Organization and Kim's Introduction to Systems Thinking.
On self-help, lukeprog recommends Weiten, Dunn, and Hammer's Psychology Applied to Modern Life over Santrock's Human Adjustment and Tucker-Ladd's Psychological Self-Help.
On probability theory, SarahC recommends Feller's An Introduction to Probability Theory + Vol. 2 over Ross' A First Course in Probability and Koralov & Sinai's Theory of Probability and Random Processes.
On probability theory, madhadron recommends Grimmett & Stirzaker's Probability and Random Processes over Feller's Introduction to Probability Theory and Its Applications and Nelson's Radically Elementary Probability Theory.
On topology, jsteinhardt recommends Munkres' Topology over Armstrong's Topology and Massey's Algebraic Topology.
On linguistics, etymologik recommends O'Grady et al.'s Contemporary Linguistics over Hayes et al.'s Linguistics: An Introduction to Linguistic Theory and Carnie's Syntax: A Generative Introduction.
On meta-ethics, lukeprog recommends Miller's An Introduction to Contemporary Metaethics over Jacobs' The Dimensions of Moral Theory and Smith's Ethics and the A Priori.
On decision-making & biases, badger recommends Bazerman & Moore's Judgment in Managerial Decision Making over Hastie & Dawes' Rational Choice in an Uncertain World, Gilboa's Making Better Decisions, and others.
On neuroscience, kjmiller recommends Bear et al's Neuroscience: Exploring the Brain over Purves et al's Neuroscience and Kandel et al's Principles of Neural Science.
On World War II, Peacewise recommends Weinberg's A World at Arms over Churchill's The Second World War and Day's The Politics of War.
On elliptic curves, magfrump recommends Koblitz' Introduction to Elliptic Curves and Modular Forms over Silverman's Arithmetic of Elliptic Curves and Cassel's Lectures on Elliptic Curves.
On improvisation, Arepo recommends Salinsky & Frances-White's The Improv Handbook over Johnstone's Impro, Johnston's The Improvisation Game, and others.
On thermodynamics, madhadron recommends Hatsopoulos & Keenan's Principles of General Thermodynamics over Fermi's Thermodynamics, Sommerfeld's Thermodynamics and Statistical Mechanics, and others.
On statistical mechanics, madhadron recommends Landau & Lifshitz' Statistical Physics, Volume 5 over Sethna's Entropy, Order Parameters, and Complexity and Reichl's A Modern Course in Statistical Physics.
On criminal justice, strange recommends Fuller's Criminal Justice: Mainstream and Crosscurrents over Neubauer & Fradella's America's Courts and the Criminal Justice System and Albanese' Criminal Justice.
On organic chemistry, rhodium recommends Clayden et al's Organic Chemistry over McMurry's Organic Chemistry and Smith's Organic Chemistry.
On special relativity, iDante recommends Taylor & Wheeler's Spacetime Physics over Harris' Modern Physics, French's Special Relativity, and others.
On abstract algebra, Bundle_Gerbe recommends Dummit & Foote's Abstract Algebra over Lang's Algebra and others.
On decision theory, lukeprog recommends Peterson's An Introduction to Decision Theory over Resnik's Choices and Luce & Raiffa's Games and Decisions.
On calculus, orthonormal recommends Spivak's Calculus over Thomas' Calculus and Stewart's Calculus.
On analysis in Rⁿ, orthonormal recommends Strichartz's The Way of Analysis over Rudin's Principles of Mathematical Analysis and Kolmogorov & Fomin's Introduction to Real Analysis.
On real analysis and measure theory, orthonormal recommends Stein & Shakarchi's Measure Theory, Integration, and Hilbert Spaces over Royden's Real Analysis and Rudin's Real and Complex Analysis.
On partial differential equations, orthonormal recommends Strauss' Partial Differential Equations over Evans' Partial Differential Equations and Hormander's Analysis of Partial Differential Operators.
On introductory real analysis, SatvikBeri recommends Pugh's Real Mathematical Analysis over Lang's Real and Functional Analysis and Rudin's Principles of Mathematical Analysis.
On commutative algebra, SatvikBeri recommends MacDonald's Introduction to Commutative Algebra over Lang's Algebra and Eisenbud's Commutative Algebra With a View Towards Algebraic Geometry.
On animal behavior, Natha recommends Alcock's Animal Behavior, 6th edition over Dugatkin's Principles of Animal Behavior and newer editions of the Alcock textbook.
On calculus, Epictetus recommends Courant's Differential and Integral Calculus over Stewart's Calculus and Kline's Calculus.
On linear algebra, Epictetus recommends Shilov's Linear Algebra over Lay's Linear Algebra and its Appications and Axler's Linear Algebra Done Right.
On numerical methods, Epictetus recommends Press et al.'s Numerical Recipes over Bulirsch & Stoer's Introduction to Numerical Analysis, Atkinson's An Introduction to Numerical Analysis, and Hamming's Numerical Methods of Scientists and Engineers.
On ordinary differential equations, Epictetus recommends Arnold's Ordinary Differential Equations over Coddington's An Introduction to Ordinary Differential Equations and Enenbaum & Pollard's Ordinary Differential Equations.
On abstract algebra, Epictetus recommends Jacobson's Basic Algebra over Bourbaki's Algebra, Lang's Algebra, and Hungerford's Algebra.
On elementary real analysis, Epictetus recommends Rudin's Principles of Mathematical Analysis over Ross' Elementary Analysis, Lang's Undergraduate Analysis, and Hardy's A Course of Pure Mathematics.

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Epictetus11y200

I suppose I can think up a few tomes of eldritch lore that I have found useful (college math specifically):

Calculus:

Recommendation: Differential and Integral Calculus

Author: Richard Courant

Contenders:

Stewart, Calculus: Early Transcendentals: This is a fairly standard textbook for freshman calculus. Mediocre overall.

Morris Kline, Calculus: An Intuitive and Physical Approach: Great book. As advertised, focuses on building intuition. Provides a lot of examples that aren't the usual contrived "applications". This would work well as a companion piece to the recommended text.

Courant, Differential and Integral Calculus (two volumes): One of the few math textbooks that manages to properly explain and motivate things and be rigorous at the same time. You'll find loads of actual applications. There are plenty of side topics for the curious as well as appendices that expand on certain theoretical points. It's quite rigorous, so a companion text might be useful for some readers. There's an updated version edited by Fritz John (Introduction to Calculus and Analysis), but I am unfamiliar with it.

Linear Algebra:

Recommended Text: Linear Algebra

Author: Georgi Shilov

Contenders:

David Lay, Linear Algebra and its Applications: Used this in my undergraduate class. Okay introduction that covers the usual topics.

Sheldon Axler, Linear Algebra Done Right: Ambitious title. The book develops linear algebra in a clean, elegant, and determinant-free way (avoiding determinants is the "done right" bit, though they are introduced in the last chapter). It does prove to be a drawback, as determinants are a useful tool if not abused. This book is also a bit abstract and is intended for students who have already studied linear algebra.

Georgi Shilov, Linear Algebra: No-nonsense Russian textbook. Explanations are clear and everything is done with full rigor. This is the book I used when I wanted to understand linear algebra and it delivered.

Horn and Johnson, Matrix Analysis: I'm putting this in for completion purposes. It's a truly stellar book that will teach you almost everything you wanted to know about matrices. The only reason I don't have this as the recommendation is that it's rather advanced and ill-suited for someone new to the subject.

Numerical Methods

Recommendation: Numerical Recipes: The Art of Scientific Computing

Author: Press, Teukolsky, Vetterling, Flannery

Contenders:

Bulirsch and Stoer, Introduction to Numerical Analysis: German rigor. Thorough and thoroughly terse, this is one of those good textbooks that only a sadist would recommend to a beginner.

Kendall Atkinson, An Introduction to Numerical Analysis: Rigorous treatment of numerical analysis. It covers the main topics and is far more accessible than the text by Bulirsch and Stoer.

Press, Teukolsky, Vetterling, Flannery, Numerical Recipes: The Art of Scientific Computing: Covers just about every numerical method outside of PDE solvers (though this is touched on). Provides source code implementing just about all the methods covered and includes plenty of tips and guidelines for choosing the appropriate method and implementing it. THE book for people with a practical bent. I would recommend using the text by Atkinson or Bulirsch and Stoer to brush up on the theory, however.

Richard Hamming, Numerical Methods for Scientists and Engineers: How can I fail to mention a book written by a master of the craft? This book is probably the best at communicating the "feel" of numerical analysis. Hamming begins with an essay on the principles of numerical analysis and the presentations in the rest of the book go beyond the formulas. I docked points for its age and more limited scope.

Ordinary Differential Equations

Recommended: Ordinary Differential Equations

Author: Vladimir Arnold

Contenders:

Coddington, An Introduction to Ordinary Differential Equations: Solid intro from the author of one of the texts in the field. Definite theoretical bent that doesn't really touch on applications.

Tenenbaum and Pollard, Ordinary Differential Equations: This book manages to be both elementary and comprehensive. Extremely well-written and divides the material into a series of manageable "Lessons". Covers lots and lots of techniques that you might not find elsewhere and gives plenty of applications.

Vladimir Arnold, Ordinary Differential Equations: Great text with a strong geometric bent. The language of flows and phase spaces is introduced early on, which becomes relevant as the book ends with a treatment of differential equations on manifolds. Explanations are clear and Arnold avoids a lot of the pedantry that would otherwise preclude this kind of treatment (although it requires more out of the reader). It's probably the best book I've seen for intuition on the subject and that's why I recommend it. Use Tenenbaum and Pollard as a companion if you want to see more solution methods.

Abstract Algebra:

Note: I am mainly familiar with graduate texts, so be warned that these books are not beginner-friendly.

Recommended: Basic Algebra

Author: Nathan Jacobson

Contenders:

Bourbaki, Algebra: The French Bourbaki tradition in all its glory. Shamelessly general and unmotivated, this is not for the faint of heart. The drawback is its age, as there is no treatment of category theory.

Lang, Algebra: Lang was once a member of the aforementioned Bourbaki. In usual Serge Lang style, this is a tough, rigorous book that has no qualms with doing things in full generality. The language of category theory is introduced early and heavily utilized. Great for the budding algebraist.

Hungerford, Algebra: Less comprehensive, but more accessible than Lang's book. It's a good choice for someone who wants to learn the subject without having to grapple with Lang.

Jacobson, Basic Algebra (2 volumes): Note that the "Basic" in the title means "so easy, a first-year grad student can understand it". Mathematicians are a strange folk, but I digress. It's comprehensive, well-organized, and explains things clearly. I'd recommend it as being easier than Bourbaki and Lang yet more comprehensive and a better reference than Hungerford.

Elementary Real Analysis:

"Elementary" here means that it doesn't emphasize Lebesgue integration or functional analysis

Recommended: Principles of Mathematical Analysis

Author: Walter Rudin

Contenders:

Rudin, Principles of Mathematical Analysis: Infamously terse. Rudin likes to do things in the greatest generality and the proofs tend to be slick (i.e. rely on clever arguments that don't really clarify the thing being proved). It's thorough, it's rigorous, and the exercises tend to be difficult. You won't find any straightforward definition-pushing here. If you had a rigorous calculus course (like Courant's book), you should be fine.

Kenneth Ross, Elementary Analysis: The Theory of Calculus: I'd put this book as a gap-filler. It doesn't go into topology and is rather straightforward. If you learned the "cookbook" approach to calculus, you'll probably benefit from this book. If your calculus class was rigorous, I'd skip it.

Serge Lang, Undergraduate Analysis: It's a Serge Lang book. Contrary to the title, I don't think I'd recommend it for undergraduates.

G.H. Hardy, A Course of Pure Mathematics: Classic text. Hardy was a first-rate mathematician and it shows. The downside is that the book is over 100 years old and there are a few relevant topics that came out in the intervening years.

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Richard_Kennaway6y20

Here's another. I learnt point-set topology from Bourbaki, borrowing the books from the public library.

1Oxide6y

Why not? I used it and thought it was wonderful.

0pinyaka11y

I purchased Shilov's Linear Algebra and put it on my bookshelf. When I actually needed to use it to refresh myself on how to get eigenvalues and eigenvectors I found all the references to preceding sections and choppy lemma->proof style writing to be very difficult to parse. This might be great if you actually work your way through the book, but I didn't find it useful as a refresher text. Instead, I found Gilbert Strang's Introduction to Linear Algebra to be more useful. It's not as thorough as Shilov's text, but seems to cover topics fairly thoroughly and each section seems to be relatively self contained so that if there's a section that covers what you want to refresh your self on, it'll be relatively self contained.

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