Mathematics and Poetry1

1. I thank Peter Atkinson, Nuccia Bencivenga, and an anonymous referee for their comments on a previous draft of this paper.

Abstract

Since Descartes, mathematics has been dominated by a reductionist tendency, whose success would seem to promise greater certainty: the fewer basic objects mathematics can be understood as dealing with, and the fewer principles one is forced to assume about these objects, the easier it will be to establish a secure foundation for it. But this tendency has had the effect of sharply limiting the expressive power of mathematics, in a way that is made especially apparent by its disappointing applications to the social sciences. We should move in the opposite direction: toward a mathematics that deals in general with constructed objects, and whose scope includes fictional, poetic characters as much as numbers and sets.

Notes

1. I thank Peter Atkinson, Nuccia Bencivenga, and an anonymous referee for their comments on a previous draft of this paper.

2. Kant, I. [1936, 1938] (1993) Opus Postumum, Trans. Förster, E. and Rosen, M. (Cambridge: Cambridge University Press), p. 139.

3. Readers who are interested in knowing more about the history of mathematics briefly sketched here could consult a standard reference book, such as Cajori, F. (1961) A History of Mathematics (New York: Macmillan).

4. This larger scope, hence also higher status, of geometry was one main reason why to prove something mathematically, for a long time, was to prove it “the way the geometers do it”: more geometrico.

5. One could get a long way with potential infinity, witness the notion of a limit. But the various definitions of irrational numbers offered in the 1870s all implied a reference to infinities that were actually given.

6. I am referring primarily to the logicist and the formalist projects. Of course many mathematicians and philosophers today would claim that mathematics has been given a foundation in set theory; but that is a very loose and weak sense of foundation, given how many competing set theories there are and how uncertain their metaphysical and epistemological status is—as compared, say, with Frege’s highly plausible (but inconsistent) logic of sets. Nothing in the present paper, however, depends on sharing this view of mine; in fact, I will immediately proceed to grant the hypothetical success of some such foundationalist project.

7. Note that it is no part of this view that mathematics is essentially concerned with quantity; this point will become relevant later. Note also that, though I am a Kantian, most of what I say below could be accepted by people who are fictionalists about mathematical entities, without necessarily sharing my Kantian commitments (see also the following note 9).

8. See Devlin, K. (1997) Goodbye, Descartes (New York: Wiley and Sons). I have also used Devlin, K. “Are mathematicians turning soft?”, MAA Online, April 1, 1996, and Devlin, K. (1996) “Soft mathematics: the mathematics of people”, mathforum.org.

9. It will soon become apparent that novels and films are as relevant to mathematics for me as what is conventionally understood as poetry; so the term “fiction” might be considered more appropriate for expressing my view. And, of course, that terminological choice would give my view a less controversial ring, since a number of authors have already claimed that mathematical objects are fictions. But note that I am primarily defending the converse (and much stronger) claim: that all fictions can be part of the kind of expanded mathematics we need—which requires reconceptualizing mathematics as well as fiction. So a more controversial terminology might be helpful in reminding us how much is at stake here, as well as letting us make better contact with the Vichian suggestions to be mentioned shortly. Ultimately, I do not care which word is used, as long as both “fiction” and “poetry” are meant to bring out the creativity which I take to be definitional of both (and which will make me say, later on, that there is poetry in a film or a novel insofar as their author is being creative).

10. Vico, G. [1744] (1948) The New Science of Giambattista Vico, Trans. Bergin, T. and Fisch, M. (Ithaca, NY: Cornell University Press).

11. It is fair to say that what follows is as controversial an interpretation of Vico as it is a controversial proposal about mathematics. I will be claiming that Vico’s “new science” was new insofar as it was a science that was also poetry, and not too many scholars would agree with that claim. But the present paper is not a contribution to the history of philosophy; so I am happy if readers take my reading of Vico as suggestive of where I think my main intellectual debts lie, rather than as assertive of what the correct interpretation of his thought is.

12. Vico, G. [1709] (1990) On the Study Methods of Our Time, Trans. Gianturco, E. (Ithaca, NY: Cornell University Press).

13. Vico, G. [1710, 1711, 1712] (1988) On the Most Ancient Wisdom of the Italians, Trans. Palmer, L. (Ithaca, NY: Cornell University Press), p. 181.

14. Vico, G. [1731] (1944) The Autobiography of Giambattista Vico, Trans. Fisch, M. and Bergin, T. (Ithaca, NY: Cornell University Press), pp. 137–38.

15. Note the emphasis in this passage on maintaining consistency with the earlier description of the characters, that is (in the language I used earlier) with the conceptual specifications that define those characters. This incidentally does not rule out a dialectical kind of consistency, in which a character might even turn into its opposite; on this topic see Bencivenga, E. (2000) Hegel’s Dialectical Logic (New York: Oxford University Press).

16. See the New Science pp. 21–22, 100.

17. Note in passing that Vico himself was also a (mediocre) poet.

18. Bartra, R. (2002) Blood, Ink, and Culture, Trans. Healey, M. (Durham: Duke University Press), p. 16.