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This is a fairly unambigious question but it hasn't been asked before so I thought I would ask it myself:

Which old books do the modern masters recommend?

There are old books where the mathematical fields explored there have been so thoroughly plowed by later mathematicians that it would be extremely naive and foolish to think that anything of value can still be salvaged from them. Those books are no longer read for the purposes of inspiring research in that area, but are curiosities which are mainly consulted for historical purposes. These are not the old books I am talking about.

The type of old book I refer to is one which still has treasures hidden inside it waiting to be explored. Such is for example Gauss's Disquisitiones Arithmeticae, which Manjul Bhargava claimed inspired his work on higher composition laws, for which he won the 2014 Fields medal.

This is why we need the opinion of the modern masters in the field as to which books are worth consulting today, because only a master in any given field (with his experience of the literature etc) can point us to the fruitful works in that field.

If you list a book, please include the quote of the master who recommended it.

Here is my attempt at the first two:

Fields medallist Alan Baker recommends Gauss's Disquisitiones Arithmeticae in his book A Comprehensive Course of Number Theory: "The theory of numbers has a long and distinguished history, and indeed the concepts and problems relating to the field have been instrumental in the foundation of a large part of mathematics. It is very much to be hoped that our exposition will serve to stimulate the reader to delve into the rich literature associated with the subject and thereby to discover some of the deep and beautiful theories that have been created as a result of numerous researches over the centuries. By way of introduction, there is a short account of the Disquisitiones Arithmeticae of Gauss, and, to begin with, the reader can scarcely do better than to consult this famous work."

Andre Weil recommends Euler's Introductio in Analysin Infinitorum for today's Precalculus students, as quoted by J D Blanton in the preface to his translation of that book: "... our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than of the available modern textbooks."

I feel this question will be found useful by many people who are looking to follow Abel's advice in a sensible and efficient manner, and I hope this question is clear-cut enough that it doesn't get voted for closure.

Edit: Thanks to Bye-World for bringing up the question of who qualifies as an old master. My response is that any great dead mathematician should qualify as an old master, so Grothendieck is an old master for instance.

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    $\begingroup$ How old is old? Who qualifies as a modern master? For instance I know Terence Tao called Vladimir Arnold's 1966 (which is nearly 50 years old at this point) "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits" a beautiful paper on his blog. $\endgroup$ – user137731 Jan 6 '16 at 19:41
  • $\begingroup$ @Bye_World: We could call any great dead mathematician an old master I guess, so VI Arnold's paper would qualify. $\endgroup$ – user45220 Jan 6 '16 at 20:06
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    $\begingroup$ But there are also great living mathematicians like Serre, I don't think this should be limited to dead people. $\endgroup$ – Brandon Thomas Van Over Jan 6 '16 at 20:26
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    $\begingroup$ @ElChapo: No - good idea! $\endgroup$ – user45220 Jan 10 '16 at 13:38
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    $\begingroup$ I've greatly enjoyed researching for this question.... I have gotten to become a lot more familiar with the history of mathematics!! Out of curiosity though, why did you pose this question? The fact a bounty was set implies a strong desire for an answer, although you mention no personal need within your post. Is this for education, for personal interest, or merely to pose a question you feel that the community would benefit from? Regardless, excellent question! $\endgroup$ – Brevan Ellefsen Jan 16 '16 at 4:25
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In Experimental and computational mathematics: Selected writings Jonathan Borwein states the following about The psychology of invention in the mathematical field by Jacques Hadamard

[The psychology of invention in the mathematical field is] a book that still rewards close inspection

[Hadamard] was perhaps the greatest mathematician to think deeply and seriously about cognition in mathematics

Borwein also gives an analysis of A Mathematician's Apology. Borwein does state that some of the idea's expressed throughout the book are outdated, but nevertheless quotes Hardy's view on mathematical beauty multiple time in his paper.

The Apology is a spirited defense of beauty over utility: ‘‘Beauty is the first test. There is no permanent place in the world for ugly mathematics.’’

In a publication of Journal of Recreational Mathematics Charles Ashbacher wrote the following about A Course of Pure Mathematics by G. H. Hardy

"Although the sequence of the presentation of the fundamentals of mathematics has changed over the last century, the substance has not. There is no greater evidence of this fact than this classic work by Hardy, which could be used without alteration or additional explanation as a text in modern college mathematics courses... The mathematical influence of G. H. Hardy over mathematical education was and remains strong, as can be seen by reading this masterpiece."

The user Nathan provides the following quote by Herman Weyl in an article for the Mathematical Review concerning G. Polya's How to Solve It (added by permission):

This Elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected.

E. T. Bell also praised How to Solve It, having written in Mathematical Monthly that

Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.

Continuing the trend of Polya's works, Mathematics and Plausible Reasoning is key. In The Mathematics Teacher Bruce E. Meserve defends the book as

...a forceful argument for the teaching of intelligent guessing as well as proving. . . . There are also very readable and enjoyable discussions of such concepts as the isoperimetric problem and 'chance, the ever-present rival of conjecture.'

$$---------------------------------------$$ Naoki Saito, Professor of Mathematics Department of UC Davis, has published a list of his recommended books. Unfortunately, the author does not provide a quote for each (in order to save space in the already extremely long list). However, I have condensed a number of my personal favorites he recommends that fit your criteria (except for quotes). I have placed an asterisk by books that lean more towards mathematical physics, as I feel these should be included but should be noted to not be pure mathematics.

E. T. Bell: Men of Mathematics

Max Born: Principles of Optics*

R. Courant & David Hilbert: Methods of Mathematical Physics*

F. R. Gantmacher & Mark Krein: Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems

R. P. Feynman: Lectures on Physics*

F. R. Gantmacher: The Theory of Matrices

P. R. Garabedian: Partial Differential Equations

G. H. Hardy: A Course of Pure Mathematics

G. H. Hardy: Divergent Series

G. H. Hardy, J. E. Littlewood, & G. Pólya: Inequalities

G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers

H. Helmholtz: On the Sensations of Tone*

H. Helmholtz: Treatise on Physiological Optics*

T. Kato: Perturbation Theory for Linear Operators

O. Kellogg: Foundations of Potential Theory

C. Lanczos: Applied Analysis

C. Lanczos: Linear Differential Operators

C. Lanczos: Discourse on Fourier Series

P. M. Morse & H. Feshbach: Methods of Theoretical Physics*

G. Pólya: Mathematics and Plausible Reasoning

J. W. S. Rayleigh: The Theory of Sound*

W. Rudin: Real & Complex Analysis

V. I. Smirnov: A Course in Higher Mathematics

E. C. Titchmarsh: The Theory of Functions

G. N. Watson: A Treatise on the Theory of Bessel Functions

E. T. Whittaker & G. N. Watson: A Course of Modern Analysis

K. Yosida: Functional Analysis

A. Zygmund: Trigonometric Series

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  • $\begingroup$ This list actually contains several physics books. Bell's book (1st in the list) contains only biographies of mathematicians. Although it is quite old now, I enjoyed reading it as I was a student. If you are interested in physicists biographies, I warmly recommend Subtle is the Lord, by Abraham Pais, about Einstein's life and science. The book contains many details concerning Einstein's research. It is certainly of interest for mathematicians too (and the author himself is physicist). $\endgroup$ – Tom-Tom Jan 14 '16 at 19:38
  • $\begingroup$ @Tom-Tom interesting recommendation! The list does mention a number of works that are either physics or biographies, but I find them all interesting reads nevertheless for a mathematician; I feel that having some knowledge of applications is critical. I do preface that I include asterisks next to writings that are heavily about physics though, as I was unsure whether mathematical physics is in the spirit of the question $\endgroup$ – Brevan Ellefsen Jan 14 '16 at 19:42
  • $\begingroup$ Might want to look at your first sentence again. Is Prof. Saito the one at UC Davis? $\endgroup$ – epimorphic Jan 15 '16 at 2:45
  • $\begingroup$ @epimorphic ...maybe.... (oops. Hey, who doesn't love the Mathematics Department of Mathematics? It's like "ATM Machine" XD) $\endgroup$ – Brevan Ellefsen Jan 15 '16 at 2:58
  • $\begingroup$ @epimorphic thanks for that. Keen eye :) $\endgroup$ – Brevan Ellefsen Jan 15 '16 at 3:00
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I would like to add another mathematics book to Breven Ellefsens list of books above. Its also a great guide for grad students or undergrad college kids struggling with math in general because it goes over general problem solving techniques along with examples from a variety of math topics such as geometry, calculus, proofs both direct and indirect.

G. Polya: How to Solve It

Herman Weyl in an article for the Mathematical Review, had this to say about the book:

"This Elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected."

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  • $\begingroup$ A great book... I would highly recommend it! I absolutely would have recommended it if I had a quote like that! Thank you for sharing this! $\endgroup$ – Brevan Ellefsen Jan 14 '16 at 22:43
  • $\begingroup$ I probably should say that answers in this format generally get removed (using an answer to replace a comment). I would either edit this to be a full answer (remove the stuff about not being able to comment) or I can include this reference in my answer with a disclaimer that the quote is due to you. $\endgroup$ – Brevan Ellefsen Jan 14 '16 at 22:56
  • $\begingroup$ While I have edited out this, you @Brevan Ellesfen are free to use this reference if you wish. I just want to help contribute. :) $\endgroup$ – Nathan Jan 14 '16 at 23:06
  • $\begingroup$ I completely understand! I simply recommended you remove that part about the comments to avoid people downvoting, so you can get more rep and get commenting faster :) just one of those weird things you figure out after a while :P $\endgroup$ – Brevan Ellefsen Jan 15 '16 at 0:29
  • $\begingroup$ You should also include his books "Mathematics and Plausible Reasoning" volumes 1 and 2. $\endgroup$ – Brandon Thomas Van Over Jan 15 '16 at 2:05
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Ian Stewart, both an influential mathematician and mathematics popularizer, recommends Isaac Newton's Principia.

The Mathematical Principles of Natural Philosophy by Isaac Newton

There ought to be a great classic in this top 10, and there is none greater. I've put it last because it's not popularisation in the strict sense. However, it slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics. Using nothing more complicated than Euclid's geometry, Newton developed his laws of motion and gravity, applying them to the motion of the planets and strange wobbles in the position of the Moon. He famously said that he "stood on the shoulders of giants", and so he did, but this book set the scientific world alight. As John Maynard Keyes wrote, Newton was a transitional figure of immense stature: "the last of the magicians … the last wonderchild to whom the Magi could do sincere and appropriate homage." No mathematical book has had more impact. Source: Ten Books recommended by Ian Stewart about Mathematics

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    $\begingroup$ One of my friend once said to me, "If you want to know what ugly mathematics really is, take a look at Newton's Principia and if you want to know what mathematics really is, take a look at Russell and Whitehead's Principia." Ironic, isn't it? $\endgroup$ – user 170039 Mar 11 '16 at 13:57

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