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From Kant to Hilbert, Volume I

William Ewald
Oxford University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

From Kant to Hilbert was originally published in 1996 as a much-too-expensive two-volume set in hard covers. The fact that it is now available in two paperback volumes at a much lower price is great news for anyone interested in reading and/or teaching from original sources.

This is a very rich selection of material indeed. The first volume opens with a long section devoted to George Berkeley’s philosophy of mathematics, including both the obvious (The Analyst, matched with selections from Newton’s Principia Mathematica) and the non-obvious (selections from Alciphron and other writings). This is followed by material from Colin MacLaurin and Jean D’Alembert on the foundations of the calculus, then selections from Kant, Lambert, Bolzano, Gauss, etc. It’s quite an impressive list, and goes far beyond the standard list of sources on foundational issues. I particularly like having material from Cayley, Clifford, and Sylvester. The last section in the first volume is named for Charles Sanders Pierce, but it also includes a selection Benjamin Pierce’s famous “Linear Associative Algebra.”

The second volume includes material from Riemann, Helmholtz, Dedekind, Cantor, Kronecker, Klein, Poincaré, Borel, Baire, Hilbert, Brouwer, Zermelo, Hardy, and Bourbaki. Having material from Helmholtz is particularly welcome, as he is not often given his due as a philosopher of mathematics. The middle portion is more standard, but here and there one notices something really nice. Overall, the selection of material is very intelligent, and most of the texts are definitely worth reading. This is Very Good Stuff, a book that you should make sure your library owns and which is a wonderful source of texts for a course in the history and philosophy of modern mathematics or for reading seminar for upper-level undergraduates.

Fernando Q. Gouvêa ( is the editor of MAA Online. He teaches both “History of Mathematics” and “Number Theory”, among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

1. George Berkeley (1965-1753)
2. Colin MacLaurin (1698-1746)
3. Jen LeRond D'Alembert (1717-1783)
4. Immanuel Kant (1724-1804)
5. Johann Heinrich Lambert (1728-1777)
6. Bernard Bolzano (1781-1848)
7. Carl Friedrich Gauss (1777-1855)
8. Duncan Gregroy (1813-1844)
9. Augustus De Morgan (1806-1871)
10. William Rowan Hamilton (1805-1865)
11. George Boole (1815-1864)
12. James Joseph Sylvester (1814-1897)
13. William Kingdon Clifford (1845-1879)
14. Arthur Cayley (1821-1895)
15. Charles Sanders Peirce (1839-1914)