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Descartes on the limited usefulness of mathematics

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Abstract

Descartes held that practicing mathematics was important for developing the mental faculties necessary for science and a virtuous life. Otherwise, he maintained that the proper uses of mathematics were extremely limited. This article discusses his reasons which include a theory of education, the metaphysics of matter, and a psychologistic theory of deductive reasoning. It is argued that these reasons cohere with his system of philosophy.

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Notes

  1. ‘Philosophy’ will be used in its current sense to include metaphysical, epistemological, and ethical topics. Similarly, ‘science’ will denote the treatment of specific natural phenomena and topics in fundamental physics such as laws of nature.

  2. I take the concept of anti-mathematicism from Schliesser Schliesser which regards it as an important eighteenth century development. This article argues that some central anti-mathematical ideas are found earlier in Descartes.

  3. Descartes’ works are cited from Adam and Tannery (1982–91) as translated in Cottingham et al. (1984–91). They are abbreviated by AT followed by the roman numeral of the Adam and Tannery volume, the page number, then followed by a semi-colon and the Cottingham, et al. volume number and page number.

  4. I shall use the term ‘mathematics’ generally to refer to geometry, algebra, applied mathematics, or all of these at once depending on the context. How Descartes understood the relationship between geometry and algebra is a difficult and controversial matter. See Bos (2001), Domski (2015), Grosholz (1991). Liu (2017) reviews the literature and argues that for Descartes geometry and algebra have a common subject matter, but that geometry is conceptually prior to algebra. I shall adopt her conclusion in what follows.

  5. A good guide to the intricate issues of dating and interpretation is provided in Schuster (2013, Chapters 5–7).

  6. In this passage, Descartes goes on to say that imagination and sensation also depend on intellect as well as on the body. On the connection between intellect and imagination and the complications it involves see e.g., Sepper (1996) and Nolan (2005).

  7. “This discipline [mathesis universalis] should contain the primary rudiments of human reason and extend to the discovery of truths in any field whatever” (Rule 4, AT X 374; CSM 1 17).

  8. More on this in Sect. 4 below.

  9. In the Rules, Descartes did not bother to distinguish intellect from will as he did later. In his mature writings, virtue is connected with the exercise of the will. A good statement of this doctrine is in The Passions of the Soul in the article introducing the important passion of “generosity.” (AT XI 445-46; CSM 1 384)

  10. This has been documented and discussed in rich detail in Jones (2006, Chapters 1, 2). For the connection between mathematical exercise and intellectual development, also see Lachterman (1989, Chapter 3).

  11. Later in the Principles, Descartes explains his purpose thus: “Finally in the Geometry, I aimed to demonstrate that I had discovered several things which had hitherto been unknown, and thus to promote the belief that many more things may yet be discovered, in order to stimulate everyone to undertake the search for truth” (AT IX-2 187; CSM 115–16). It would seem he regarded his advances in geometry primarily as advertisements for the more important applications of the method.

  12. Daniel Garber, in his influential (1992, Chapter 3), argues that Descartes’ views on method fall into three sharply distinguished phases and that the Rules express the first of these stages. He focuses, however, on the example of the anaclastic and not on the “finest” example discussed above. If the thesis of this article is correct, Descartes’ thinking shows more continuity than many interpreters have allowed. Other scholars stressing the ways in which much of the philosophy of the Rules is never abandoned include Marion (1981, 1992), David Lachterman (1989, Chapter 3), and Recker (1993).

  13. For interpretations of the Geometry itself, again see the works cited in footnote 3 and also Mancosu (1996, Chapter 3).

  14. Clarke (1992) is representative of this reading.

  15. On this point also see Garber (2002).

  16. The term “Galilean paradigm” is taken from the important article, (Garber 2000).

  17. A very detailed analysis of the relevant texts is provided in Schuster (2013, Chapters 3, 8, and Appendix 1). See also Gaukroger (1995, Chapter 3).

  18. Descartes makes complaints of this kind while criticizing Galileo in an 11 October 1638 letter to Mersenne. (AT II 380-400; CSMK 124-28). Again, see (Garber 2000) for further discussion.

  19. A full argument for this and a discussion of some of its consequences has been provided by Alice Sowaal (2004; 2005). The concept and terminology of “mutual transfer” are from Garber (1992, Chapter 6). Descartes does often refer to bodies at rest (e.g. when discussing collisions in the Principles), but he means only that they are considered at rest with respect to some reference point (VIII-1 55–57; 1 234–36). For other features of Cartesian extension relevant to the possibility of quantitative physics see Lennon (2007) and Smith and Nelson (2010).

  20. For a discussion of Newton’s criticism of Descartes on quantity of matter and density see Janiak (2008, pp. 102–18).

  21. Although Descartes does not explicitly encumber his exposition with this final consequence of matter as extension, one can scarcely believe that it escaped him. For additional discussion of the “micro-chaos” that pervades Cartesian extension see Nelson (1995). In some ways this strongly anticipates Leibniz. A careful treatment of Leibniz on the metaphysical physics of the plenum is Crockett (2009). The endless division of Cartesian matter in the case of fluids flowing through pipes is discussed in M. Wilson (forthcoming).

  22. For examples of Descartes displaying a disregard for numerical exactitude, see (AT 1 100 and AT VIII-B 70; CSM 1 245). For an account of why Descartes rejected the Galilean paradigm different from the one presented here see Gaukroger (1980).

  23. See Gueroult (1968, Chapter XVII) on how the human being constitutes an exception to Descartes’ rejection of appeals to divine ends.

  24. Making use of old dictionaries, Clarke (1977) has shown that Descartes’ usage was in accord with contemporary standards.

  25. There is no evidence that Descartes ever modified his views on the theory of logical deduction. As we shall see, it coheres with his account of analysis and synthesis in the Second Set of Replies to the Meditations. There is also strong external evidence that Descartes’ account in the Rules had lasting impact. His account is adopted by such highly influential figures as the authors of the extremely important Port-Royal Logic. See the introductory material in Buroker (1996) and especially the Fourth Part on method. For Locke’s connection with the Rules, see Aaron (1971, Ch. 7) and for Spinoza’s see Nelson (2015).

  26. (AT X 368; CSM 1 14). This characterization of intuition corresponds perfectly with what Descartes will later refer to as clear and distinct perception, “I call a perception ‘clear’ when it is present and accessible to the attentive mind... I call a perception ‘distinct’ if, as well as being clear, it is so sharply separated from all other perceptions that it contains within itself only what is clear” (Principles, AT VIII-1 22; CSM 1 207-8).

  27. For an interpretation that bottoms out in intuitions of linkage, see Gaukroger (1989, p. 51).

  28. Descartes insists that extension is only distinct by reason from extended body and therefore nothing but body in, for example, the Rules (AT X 443-46; CSM 1 59-61), and in the Principles (AT VIII-1 30-31, CSM 1 215).

  29. For a more detailed account of this interpretation see Rogers and Nelson (2015).

  30. I am using the term ‘principles’ to refer to natures that are perfectly simple and, therefore, not capable of being deduced from any others. Descartes later settles on the term as well (e.g. in the Principles (IX-1 2-3; CSM 1 179).

  31. See Gaukroger (1980) for example.

  32. For more detail on this point, see Rogers and Nelson (2015).

  33. The main problems concerning hypotheses in Descartes are clearly formulated in (Garber 1992, Chapters 5 and 6), which also reviews some of the literature on these topics. Garber presents a more pessimistic evaluation of the cogency of Descartes’ claims about knowledge.

  34. Descartes’ scientific explanations recall Stephen Toulmin’s account of reductions to explanatory ideals (Toulmin 1963). Ideals such as general equilibrium, rational choice, perfect information, etc. are still useful in the social sciences. This is developed in Nelson (2001).

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Acknowledgements

I benefited from discussions of these ideas with audiences at Ghent University, UNC at Chapel Hill, and the University of Texas, Austin. I have also received helpful comments from anonymous reviewers for Synthese.

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    Alan Nelson

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Nelson, A. Descartes on the limited usefulness of mathematics. Synthese 196, 3483–3504 (2019). https://doi.org/10.1007/s11229-017-1328-9

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