Mathematics: original works

Dirichlet, Peter Gustav Lejeune, Vorlesungen über Zahlentheorie (edited by Richard Dedekind), 1837. Translated with introductory notes by John Stillwell as Lectures on Number Theory, AMS, 1999. Stillwell's introduction is available online.

Dedekind, Richard, Sur la théorie des nombres entiers algébrique, translated with introductory notes by John Stillwell as Lectures on the Algebraic Integers, Cambridge University, 1996.

Struik, Dirk, A Source Book in Mathematics: 1200-1800, Harvard University Press, 1969; in particular, papers by Fermat, Euler, Legendre, and Gauss.

Weddernburn, Maclagan, "A theorem on finite algebras," Transactions of the American Mathematical Society, 6:349-352, 1905. Pitt and CMU students should be able to get this online from JSTOR.

(Also landmark textbooks on algebraic number theory by Hasse, Weyl, ...)

(Interested students can also look into work by Euler, Gauss, Kummer, Kronecker, Hilbert, ...)

Mathematics: modern presentations

Goldman, Jay, The Queen of Mathematics: a historically motivated guide to number theory. A K Peters, 1998.

Stillwell, John, Mathematics and its History, second edition, Springer, 2001.

Conway, John, The Sensual (Quadratic) Form, Mathematical Association of America, 1997.

A number of introductory texts can be used to provide the requisite background in algebra and number theory. Here are some examples:

Nathanson, Melvyn, Elementary Methods in Number Theory, Spring, 2000

Niven, Ivan and H. S. Zuckerman, An Introduction to the Theory of Numbers, fourth edition, John Wiley and Sons, 1980.

Birkhoff, Garrett and Saunders MacLane, A Survey of Modern Algebra, third edition, MacMillan, 1965.

Grossman, Israel and Wilhelm Magnus, Groups and their Graphs, Random House, 1964.

Hill, Victor, Groups, Representations, and Characters, Hafner, 1975.

Ireland, Kenneth and Michael Rosen, A Classical Introduction to Modern Number Theory, second edition, Springer, 1990

Ian Stewart and David Tall, Algebraic Number Theory and Fermat's Last Theorem, third edition, AK Peters, 2002.

Harry Pollard and Harold Diamond, The Theory of Algebraic Numbers, 1998 Dover reprinting.

History of mathematics

Corry, Leo, Modern Algebra and the Rise of Mathematical Structures. Birkhäuser, 1996.

Edwards, Harold, Fermat's Last Theorem: a genetic introduction to algebraic number theory, Springer Verlag, 1977.

Ferreiros, Jose, Labyrinth of Thought: a history of set theory and its role in modern mathematics, Birkhäuser, 1999.

Klein, Felix, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, vols I and II originally published in 1926-1927, reissued by Chelsea; there is also a hard-to-find English translation, Development of Mathematics in the 19th Century.

Kolmogorov et al. (editors), History of 19th Century Mathematics. Birkhäuser, 1992.

Wussing, Hans, The Genesis of the Abstract Group Concept, MIT Press, 1984.

Gray, Jeremy, "Mathematicians as philosophers of mathematics," For the Learning of Mathematics, part I: 18(3):20-24, November 1998; part II: 19(1):28-31, March 1999.

Gray, Jeremy, "The nineteenth-century revolution in mathematical ontology," in Donald Gilles, editor, Revolutions in Mathematics, Oxford, 1996, pages 226-248.

Edwards, Harold, "Dedekind's invention of ideals," Bulletin of the London Mathematics Society, 15:8-17, 1983. Also in E. R. Phillips, editor, Studies in the History of Mathematics, MAA, 1988.

Edwards, Harold, "An appreciation of Kronecker," The Mathematical Intelligencer, 9:28-35, 1987.

Edwards, Harold, "Kronecker's views on the foundations of mathematics," in Rowe and McCleary, editors, The History of Modern Mathematics, Academic Press, 1989.

Edwards, Harold, "Mathematical Ideas, Ideals, and Ideology," The Mathematical Intelligencer, 14:6-19, 1992.

Kleiner, Israel, "From Numbers to Rings: the early history of ring theory," Elemente der Mathematik, 53:12-35, 1998.

Marion, Mathieu, "Kronecker's 'safe haven of real mathematics'," in Marion and Cohen, eds., Québec Studies in the Philosophy of Science I, Kluwer Academic Publishers, 1995, 189-215.

Stein, Howard, “Logos, Logic, Logistike: Some philosophical remarks on nineteenth-century transformation of mathematics.” In Aspray and Kitcher, editors, History and Philosophy of Modern Mathematics, University of Minnesota Press, 1988.

Philosophy and foundations: historical works

Ewald, William, editor, From Kant to Hilbert: A source book in the foundations of mathematics. Oxford, 1996.

van Heijenoort, Jean, editor, From Frege to Godel: A source book in mathematical logic, 1879-1931. Harvard, 1967.

Dedekind, Richard,"Was sind und was sollen die Zahlen" (translated, in Ewald)

Dedekind, Richard, "Stetigkeit und die Irrationale Zahlen" (translated, in Ewald)

Dedekind, Richard, "Letter to Keferstein" (translated, in van Heijenoort)

Kronecker, Leopold, "Über den Zahlbegriff" ("On the concept of number", translated, in Ewald)

Hilbert, David, Grundlagen der Geometrie,First edition, Teubner, 1899. Tenth edition translated by Leo Unger as Foundations of Geometry, Open Court, 1997

Hilbert, David, "Über den Zahlbegriff" ("On the concept of number")

Bourbaki, Nicholas, "The architecture of mathematics," in Ewald

Philosophy and foundations: contemporary

Benacerraf, Paul, and Hilary Putnam, editors, Philosophy of Mathematics: Selected readings, Cambridge University Press, second edition, 1983.

Shapiro, Stewart, Philosophy of Mathematics: structure and ontology, Oxford University Press, 1997.

Shapiro, Stewart, Thiking about Mathematics, Oxford University Press, 2000. See especially Chapter 10, "Structuralism."

Resnik, Michael, Mathematics as a Science of Patterns, Oxford University Press, 1997.

Hellman, Geoffrey, Mathematics without Numbers, Oxford University Press, 1989.

Benacerraf, Paul, "What numbers could not be," Philosophical Review 74:47-73, 1965. Reprinted in Benacerraf and Putnam.

Parsons, Charles, "The structuralist view of mathematical objects," Synthese 84:303-346, 1990.

McClarty, Colin, "Numbers can be just what they have to," Noûs 47:487-498, 1993.

In recent years, there have been a number of articles on structuralism in the journal Philosophia Mathematica, including a special issue, "Mathematical Structuralism," 4(2), May 1996. A few particular articles are indicated below.

Awodey, Steve, "Structure in mathematics and logic: a categorical perspective," Philosophia Mathematica 4:209-237, 1996.

Mac Lane, Saudners, "Structure in mathematics," Philosophia Mathematics 4:174-183, 1996

Keranen, Jukka, "The Identity Problem for Realist Structuralism," Philosophia Mathematica 9:308-30, 2001.