# General request for a book on mathematical history, for a VERY advanced reader.

I am aware that there are answered similar questions on here, however I am specifically after a text that would be engaging for a professor of mathematics, also Fellow of the Royal Society (FRS).

He is unwell and in the hospital, and I would like to get him something to pass the time. However anything aimed at undergraduate / postgraduate level is going to be far too patronising. Honestly, I'm not sure if there exists such a book, but if anyone has any recommendations, I would be extremely grateful.

• Not a single book encompassing "all" history of math... Some recent suggestions : Jeremy Gray, The Real and the Complex : A History of Analysis in the 19th Century, Springer (2015) and Thomas Sonar, The History of the Priority Dispute between Newton and Leibniz (2018, Birkhäuser). – Mauro ALLEGRANZA May 16 '18 at 15:55
• I am not so sure such a thing exists. However if you know his area of mathematics, you can find plenty of biographies of mathematicians is most any given field that detail their life and mathematics at very advanced levels. Perhaps that might be closer to what you wanted? – mathematics2x2life May 16 '18 at 16:00
• It would be helpful if you know the field of interest of this Fellow, then we can recommend a book that might be more or less of interest to them. – Asaf Karagila May 16 '18 at 18:03
• @Kevin: Unfortunately, I think that getting Erdős to help is impossible nowadays. – Asaf Karagila May 16 '18 at 20:22
• At first glance of your title I thought you were being grandiose, but then I read the question body, and I have to say, your thoughtfulness really warms my heart. I’m sure any book you deliver to him will lift his spirits, just because it shows you care. I wish you the best of luck $\ddot\smile$ – gen-z ready to perish May 17 '18 at 2:17

I suppose that a very good choice is A History of Mathematics, by Victor J. Katz.

• History is written by the Victors, so that would be a very appropriate choice! – Asaf Karagila May 16 '18 at 18:02
• An explanation of the cultural reference can be found here. – Mikhail Katz May 17 '18 at 10:27
• This was the textbook of an undergraduate course I took decades ago. So I'm not sure it passes the OP's ban on "anything aimed at the undergraduate/postgraduate level". – hmakholm left over Monica May 17 '18 at 12:17
• Exactly, @HenningMakholm. It's also a textbook used in undergraduate teacher education programs for elementary and secondary school teachers to-be. – Namaste May 17 '18 at 21:22

I have not read it myself, but I have heard excellent things about The Princeton Companion to Mathematics. It is not specifically a history book, but apparently has a decent amount of history in it, with many pages devoted to mini-biographies of mathematicians. It is written by, and primarily for, mathematicians.

• At 1000+ pages and over 2.5kg, you might want to consider buying an e-reader and an electronic copy for your sick friend. – Robin Whittleton May 18 '18 at 9:44
• This book has similar one for applied math as well: The Princeton Companion to Applied Mathematics. – Sil May 19 '18 at 10:35

In the comments, Kevin Long suggested

Hopefully this isn't getting too off topic, but I've heard that when Stan Ulam was in the hospital for encephalitis, Paul Erdos went to meet him when he was discharged and spent a few weeks at his house, plying him with math questions and playing chess with him. At the time, Ulam was afraid that the incident would have affected his mathematical ability, but Erdos helped him build his confidence back up. So if the professor in question is feeling suboptimal, some (small) math problems might be good.

Personally, when I had to stay for longer periods at the hospital, the most difficult part for me was to overcome the boredom. There aren't that many books I would enjoy reading for 8 hours a day, 7 days a week... from that perspective, math problems might make sense, because mathematicians like to spend sheer endless amounts of time on problems that they find engaging. In that spirit, I'd like to suggest The Art of Mathematics: Coffee Time in Memphis by Béla Bollobás. It contains many interesting problems, all with elegant solutions, some due to famous mathematicians such as Erdos.

Littlewoods’s Miscellany

• It is a classic.
• Littlewood does not write for the general public.
• There are wonderful anecdotes about the british academic life in the 1st half of the 20th Century.
• There is plenty of hard-core mathematical content.
• I might be wrong here, but I find it quite improbable that a FRS has not yet read Littlewood's Miscellany... – Klangen May 17 '18 at 10:19

not sure why you emphasize history... There are now translations of Fricke and Klein into English. Favorite of mine. The preface is pretty funny.

https://bookstore.ams.org/ctm-3/

Your ailing friend can try to find where the theorems are. A major one I have used with considerable enjoyment is on pages 507-508 of the German original, page 412 in the translation. I guess we know it is a theorem or lemma or the like as it is in italics. There is also one in the middle of the page where they admit everything, "obtain the following theorem:"

I bought a cheap reprint of the 1897 German original. Cheap because it was scanned online by a major library, but I wanted a paper version

The World of Mathematics by James R. Newman, 1956, 4 volumes. This is a collection of 133 essays written by the pantheon of mathematical thinkers: Descartes, Archimedes, Newton, Euler, Galileo, Bernoulli, Malthus, Laplace, Poincare, Mach, Einstein, Boole, Turing and dozens more.

Chapters average about 20 pages and need not be read in order, so the book is ideal to pick up for a brief diversion and then put aside for later.

For more detail, see the review by David E.H. Jones in Nature, 337 (February 2 1989), p. 420. (Link here.)

• I don’t know if Newman is advanced enough for an FRS, but I highly recommend it for kids in high school or advanced elementary. I remember rolling my eyes when my dad gave it me for Xmas - not because I wasn’t interested in math, but because it was so different from how math was taught in school - yet I recall stories from it all the times. – Krazy Glew May 17 '18 at 0:50
• ...yet I recall stories from Newman all the time [fixed spelling]. I have long thought that there should be classes taught using Newman and similar material - it might inspire mathematical interest in students more literary than formulaic. – Krazy Glew May 17 '18 at 0:56

A classic is the four-volume set The World of Mathematics, edited by James R. Newman (GoodReads, Amazon). The original 1956 hardcover edition has 2500 super-thin pages, but there also exists a more recent, and more affordable, paperback Dover edition. The collection contains essays by mathematicians including Von Neumann, Russell, Descartes, Galileo... There's bound to be material in there of interest even to an FRS.

Not a book on history, but if he hasn't read it already, Gödel, Escher, Bach is a good option.

• I wouldn't recommend it for advanced reader. Hofstadter is as patronising as one can be. – Crazy Yoghurt May 17 '18 at 8:41
• This book is so poor, that I don't understand where it gets its fame from. Probably from being a pompous bunch of intellectual trivialities that make the average reader feel smart. – Alex M. May 20 '18 at 13:53

I suggest

• Mathematical Thought from Ancient to Modern Times, Vol. 1&2, by Morris Kline
• Mathematics and Its History, by John Stillwell
• Mathematics and Its History is the book we used in our undergraduate math history course, and it is certainly aimed towards undergraduates – CulDeVu May 16 '18 at 18:43

Another really good read is Constance Ried's book Hilbert, which (as the title suggests) covers the history of one mathematician in depth, but also touches upon many other mathematically "significant figures" as it tells its tale.

An alternative to a history may be a serious introduction to an advanced field of mathematics that FRS has never touched. Mathematics is so broad now that no one can cover it all. A stay in a hospital is a chance not to recapitulate the old but to explore something genuinely new. That spirit of discovery can be more healing, and who knows what serendipity is possible.

Probably, any comprehensive book about history of mathematics is intended for undergraduates (almost all subjects are presented, but none of them is deeply discussed). So, here are two suggestions that tell very specific stories (about fairly sophisticated results):

Both authors are fields medalists and, in these books, they talk about their important discoveries.

• The book by Schwartz talks at length about how his family almost disappeared during WWII in France, and a lot of national and international politics starting in the 1930s up to the 1980s, from his initial trotskism up to his support for decolonization. His mathematical research, Bourbaki, the birth of distribution theory takes at most 1/5 of the book. There is also the french specific story about the math department of the Ecole Polytechnique. – ogerard Jan 21 '19 at 14:53

Stanislaw Ulam's autobiography Adventures of a Mathematician is a very uplifting personal history.

For an engaging read, you could try Cauchy, infinitesimals, and ghosts of departed quantifiers.

Here I'd like to propose four books which might at least partly met also more sophisticated demands. The first two cover the Hilbert problems, one is about the history of a specific mathematical subject and the last one is about one of the great masters.

Bair, J.; Błaszczyk, P.; Heinig, P.; Katz, M.; Schäfermeyer, J.; Sherry, D. "Klein vs Mehrtens: restoring the reputation of a great modern." Mat. Stud. 48 (2017), no. 2. See arxiv.

Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). We argue that Klein and Hilbert, both at Goettingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. Klein's Goettingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings.

Mehrtens' claim that [the future SS-Brigadefuehrer] "Theodor Vahlen ... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position.

• Of course, there's no mention of the relation between the user posting this paper and the authors of the paper. But it's important to say it's an engaging read. – Asaf Karagila May 17 '18 at 10:45

Since you haven't mentioned the professor's field of study, though I am sure you must know their specialized interests, I'd suggest you make sure they have access to the the current (top) journal(s) in their favorite field of study, which will help them stay abreast in the field.

If the professor's field of study is history in mathematics, or they simply stay finely attuned to the subject, e.g., may I suggest keeping the professor supplied with recent copies of the journal Historia Mathematica?

Also look into the most recent issue of BSHM Bulletin: Journal of the British Society of the History of Mathematics.

It is on one (two?!) specific topic, but still... maybe Jean Dieudonné's "A History of Algebraic and Differential Topology, 1900 - 1960" could be an interesting choice that fits OP needs (it is actually rather advanced).

History is also in the great original works, or in a book from the personal library of someone famous - ideally with margin side-notes in the latter case. Or all of these at once. Sometimes such books are obtainable today, for a reasonable amount of money. Not too long ago I saw a signed copy of a math book from the library of Fields Medalist Lars Ahlfors come up in a search engine. Since buying a book that would appeal to the professional interest of such a person as you describe is inherently difficult, you might do better in seeking something to be surely treasured.

I took History of Mathematics as an optional subject in my final year. There is one book really. "A brief history of mathematics" by Howard Eves. Super book, both for ordinary readers and academia. However, I do not know if it is still in print :D as I graduated in 1997. cheers!!

Carl Boyer's A History of Mathematics, might be an enjoyable read. I've read the 1991 revised edition, ed. Uta C. Metzbach (New York: John Wiley, 1991). It's about 700 pages. The first 260 pages are dedicated to mathematics before the Renaissance, starting with ancient Egypt and dwelling at length on the ancient Greeks. The author narrates the history of pre-modern mathematics in an engaging style. For a specialist in the field, the history might be interesting.

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

The mathematics in this book is pretty light, but nevertheless, the book presents an interesting case study in the ways that non-mathematical concerns affect the history of mathematics.

• The mathematics in it are light, indeed, but the history and cultural references within are not. I was going to post the same answer. – Alex M. May 20 '18 at 13:54

Jeremy Gray's "Plato's Ghost: The Modernist Transformation of Mathematics" tells the story of how mathematics became modern during several decades before and after 1900. Worth reading to understand how mathematics got to be so far from common intuition.

Amir Alexander's "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World" is a cultural fresco of the 17th century, the one when Europe became modern, the one that ends with the birth of real analysis as conceiver by Newton and Leibniz.

Not mathematics, but very close: Arthur Miller's (not that one!) "Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911)". A deeply intellectual book, not content with the just the physical theory but also presenting the general cultural context in pre-war Vienna.

• Hi Alex, in my opinion Jeremy Gray's book is a profoundly flawed one and I wouldn't recommend it to anyone. For one thing, he gobbles up hook, line, and sinker Mehrtens' crazy thesis about Felix Klein being leader of a countermodern camp allegedly opposed to the modern camp led by Hilbert; see this 2018 forthcoming publication in Mat.Stud. for details. – Mikhail Katz May 21 '18 at 15:06

Number Theory: An approach through history from Hammurapi to Legendre by Andre Weil is a classic. In it Weil covers some thirty-six centuries of arithmetical progress, with a close account of the founding fathers of modern number theory: Fermat, Euler, Lagrange and Legendre.

The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae edited by Catherine Goldstein, Norbert Schappacher, Joachim Schwermer. In this book eighteen authors - mathematicians, historians, philosophers - discuss the impact C.F. Gauss's Disquisitiones Arithmeticae (1801) has had on mathematics.

From Kant to Hilbert: A Source Book in the Foundations of Mathematics Volume I and Volume II by William Bragg Ewald. An historical overview starting from Kant's Critique of Pure Reason, widely taken to be the starting point of the modern mathematics, up to the end of the nineteenth century with Hilbert. Ewald's two-volumes contain translations of works by Bolzano, Cantor, Dedekind, Gauss, Hamilton, Kronecker, Riemann, Poincare, and Zermelo showing the links between algebra, geometry, number theory, analysis, logic, and set theory.

Geometry by Its History by Alexander Ostermann and Gerhard Wanner. This is an undergraduate book, but that's not the point: it's one of the most beautiful historical books on geometry out there.

The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore by Karen Hunger Parshall and David E. Rowe. In this book we see what the title suggests: Sylvester at John Hopkins, Moore at the University of Chicago, with Klein popping over to tour the mathematical scene.

The Mathematics of Plato's Academy: A New Reconstruction by David H. Fowler. In this book Fowler examines what we really know of the mathematics done in Plato's Academy, and before Euclid: not much! A brilliant book written by a classicist for fans of Euclid.

Get well soon.

Perhaps an older book if you can find it. (The chances are higher he hasnt read it...) So I recommend E T Bell The Development of Mathematics.

Two popular science books by Simon Singh:

Fermat's Last Theorem and The Code Book about cryptography and its history.

• These books are far more interesting before you know mathematics, and if you like history, then before you know the actual history. Simon Singh is a good writer, but he took some... artistic freedoms with the history of things. Which might be a bit rattling to a mathematics professor. – Asaf Karagila May 19 '18 at 0:30
• @Asaf Karagila: yes, I agree, Simon Singh is a good writer, and the books looks like well-suitable for recreational reading. – g.kov May 19 '18 at 1:25

Men of Math E.T Bell is a great book

I would suggest Mathematics Emerging by Jacqueline Stedall. While not a complete history, it does cover a rather important period from the late 16th century to the 20th century.

I believe that he may like Godël, Escher, Bach by Douglas Hofstadter, a brilliant read not so much about mathematical history as it is about discussions pertaining to knowledge and such metaphysical aspects of mathematics, and full of the most beautiful, sophisticated and interesting puzzles.

Hope you find a good book.

All the best!

Hermann Weyl wrote a 43 page essay on the life and work of David Hilbert that I think is quite nice.