a list compiled by Alex Kasman (College of Charleston)
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Three Socratic dialogues by the Hungarian mathematician Alfréd Rényi that address mathematical topics such as Platonism and the differences between pure and applied math.
A Socratic dialogue is not fiction in the usual sense of the word. In particular, it generally lacks a plot and a setting. However, it does have characters who "speak" the words put into their mouths by the author. Moreover, to be effective, a Socratic dialogue must get the reader to recognize the different participants of the discussion as having different knowledge, viewpoints, and desires. It is with this in mind that I have listed works such as Douglas Hofstadter's dialogues in Gödel, Escher, Bach in this database. Using such characters to convey some deep ideas about mathematics is achieved quite well by Rényi in these dialogues that feature famous historical figures. I especially admire the first dialogue, in which Socrates himself addresses what math actually is and whether new math is created or discovered. This is a difficult question, and Rényi resists the urge to oversimplify it, offering good reasons for either viewpoint before reaching a conclusion concerning the "unexpected effectiveness of mathematics" (i.e., why math is useful in reality even if it is a creation of our minds). He actually delivered this dialogue as a speech to a 1963 joint meeting of the American Physical Society and the Canadian Association of Physicists in Edmonton. It was published as Canad. Math. Bull., vol. 7, no. 3, July 1964. It appears to be available for free from the website of Cambridge University Press. In the second dialogue, the author uses Archimedes to address the relationship between pure math and applied math. In fact, Archimedes is a very good choice of guide for this particular question. (See also The SandReckoner.) Finally, Galileo himself is involved in a dialogue addressing his famous claim that the book of nature is written in the language of mathematics. Fortunately, it is possible to find free copies of Dialógusok a matematikáról online. It can be read in the original Hungarian here and a scanned copy of the 1967 English translation is available through Archive.org. I offer thanks to Paul Nevai who brought these very nice dialogues to my attention. 
More information about this work can be found at archive.org. 
(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.) 

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Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in nonfictional math books let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.
(Maintained by Alex Kasman, College of Charleston)