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Mathematics and Infinity in Descartes and Newton

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Mathematizing Space

Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))


The concept of the infinite has often been regarded as inherently problematic in mathematics and in philosophy. The idea that the universe itself might be infinite has been the subject of intense debate not only on mathematical and philosophical grounds, but for theological and political reasons as well. When Copernicus and his followers challenged the old Aristotelian and Ptolemaic conceptions of the world’s finiteness, if not its boundedness, the idea of an infinite, if not merely unbounded, world seemed more attractive. Indeed, the infinity of space has been called the “fundamental principle of the new ontology” (Koyré in From the Closed World to the Infinite Universe. Johns Hopkins University Press, Baltimore, 1957, p. 126). Influential scholarship in the first half of the twentieth century helped to solidify the idea that it was specifically in the seventeenth century that astronomers and natural philosophers fully embraced the infinity of the universe. As Kuhn writes in his Copernican Revolution (1957, p. 289): “From Bruno ’s death in 1600 to the publication of Descartes ’s Principles of Philosophy in 1644, no Copernican of any prominence appears to have espoused the infinite universe, at least in public. After Descartes , however, no Copernican seems to have opposed the conception.” That same year saw the publication of Alexandre Koyré ’s sweeping volume about the scientific revolution, From the Closed World to the Infinite Universe. The decision to describe and conceive of the world as infinite might be seen as a crucial, if not decisive, aspect of the overthrow of Scholasticism. As Kuhn and Koyré knew, one finds a particularly invigorating expression of this historical-philosophical interpretation in an earlier article by Marjorie Nicholson (Studies in Philology 25:356–374, 1929, p. 370).

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  1. 1.

    The next section of Principia Philosophiae presents a clearer view: we should reserve the term ‘infinite’ for God, since we positively recognize that God has no limits, which apparently is not true in the cases quoted above.

  2. 2.

    Descartes seems to have thought that the idea of the world as indefinite was original with him, but there were Scholastic philosophers who held similar, if not identical, views; others, including some Jesuits, criticized the notion. See Ariew (1999, pp. 165–171).

  3. 3.

    This is my way of approaching the material; in the Physica, Aristotle himself contends that “belief in the existence of the infinite” comes mainly from five considerations, including quantities dealt with in mathematics, the nature of time, the division of magnitudes, and so on (203b 15–24).

  4. 4.

    In the Physica, he writes of Zeno (233a 22–31; transl. Aristotle 1908-, as modified by Barnes 1984): “Hence Zeno ’s argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two ways in which length and time and generally anything continuous are called infinite: they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility; for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.” Perhaps Aristotle is suggesting here that although a person walking a hundred meters across the college quad cannot traverse an infinite number of things in the sense of things that are quantitatively infinite—which we can read as an actual infinity—she can traverse an infinite number of things in the sense of things that are infinitely divisible—which we can read as a potential infinity. Just as the hundred meters that the person walks is infinitely divisible into smaller and smaller segments, from meters to centimeters to millimeters and so on, the time that it takes her to cross the quad is also infinitely divisible into smaller and smaller moments, from minutes to seconds to milliseconds and so on.

  5. 5.

    In 1671, Leibniz argued that Descartes ’s distinction between the infinite and the indefinite is merely epistemological; he may have been the first to endorse an epistemic reading of Descartes ’s view. For his part, Leibniz certainly endorsed the idea that features of reality involve actual infinities, at least in his late work: in the monadology, it seems clear that Leibniz thinks that the world, or perhaps even an individual object like a chair, contains an actual infinity of monads. See Moore (1990, p. 79).

  6. 6.

    Thanks to Henry Mendell for making this point.

  7. 7.

    Perhaps the distinction between the metaphysical and the epistemic readings is not clear: either way, we still have only finite and infinite items in the ontology; according to one reading, the metaphysical, we can say that things like the material world are finite but potentially infinite, and therefore without limits in a certain sense—which would distinguish them from ordinary finite things like tables, which do have limits—and according to the epistemic reading, we would say that each item in our ontology is finite or infinite and by “indefinite” we would simply be signaling the fact that we do not know whether certain things, like the material world, are finite or infinite. But this might be compatible with the claim that the material world could be potentially infinite, i.e., finite but without limits. It could be. Of course, there would still be one distinction between the two readings: on the metaphysical reading, we would know that the material world is indefinite, by which we could mean, potentially infinite; and on the epistemic reading, we would not know whether the material world is finite—whether potentially infinite, and therefore without limits, or just plain finite, and therefore with limits—or infinite.

  8. 8.

    As Broughton highlights (2002, pp. 151–53), one might also find the third meditation proof unpersuasive because it seems to rely on an obscure, or at least not fully clarified, conception of what the representation of an infinite being involves. That representation is connected with a cluster of ideas, including the notion that there is more “reality” in an infinite being than in a finite one, that require clarification beyond what Descartes provides.

  9. 9.

    In a letter of 5 May 1651, More agrees with Anne Conway ’s claim that there isn’t any clear distinction in Descartes between the infinite and the indefinite (this is connected with an interpretation of section 21 of part two of Principia Philosophiae). Gabbey (1977, pp. 589–90).

  10. 10.

    It is possible that for More , who was a more or less standard Anglican, the notion of God’s substantial omnipresence had no special connection with the Trinity; but for Newton , who was obviously a heretical Anglican, one who rejected the Anglican view of the Trinity, it may also be possible that the doctrine of divine omnipresence was in fact connected with his view of the Trinity. See Snobelen (2005) and (2006). For his part, Clarke defended the view that God is substantially omnipresent in the twelfth section of his third letter to Leibniz , quoting from the General Scholium passage reproduced above; see Koyré (1957, p. 248).

  11. 11.

    Presumably, on any reasonable view of the dating of the former, it must have been written before the first edition of Principia mathematica, which was ready in 1686—see Ruffner (2012).

  12. 12.

    In his Trinity notebook, Newton may have made use of some of Wallis ’s techniques from Arithmetica infinitorum (Newton 1983, pp. 106–107, including footnote 168), and he made two pages of annotations from Wallis in that text (Newton 19671981, vol. 1, pp. 89–90). As Whiteside indicates, in another pocket book from 1664–1665, Newton made detailed entries concerning Wallis : Newton (19671981, vol. 1, pp. 91–141). In the Que[a]estiones quedam Philosoph[i]cae, Newton noted: “one infinite extension may be greater than another,” a key point from Wallis that Newton would describe to Bentley nearly 30 years later (Newton 19671981, vol. 1, p. 89). Whiteside notes, intriguingly, that Newton may have read Hobbes ’s attack on Wallis first, and then proceeded to read Wallis for himself (Newton 19671981, vol. 1, p. 89 note 1). Newton also retained a copy of Wallis ’s Opera mathematica in his personal library (Harrison 1978). David Rabouin points out that by roughly 1680, Newton had decided that he no longer needed the techniques of Wallis . This is an important point, but it’s compatible with the fact that Newton still regarded Wallis as indicating how we can avoid various kinds of paradoxes when thinking about infinitesimals, infinite divisibility, and infinity more generally, as his correspondence with Bentley a decade later indicates.

  13. 13.

    For discussions of Wallis ’s work, see Guicciardini (2009, pp. 140–47), which places it in the context of understanding Newton ’s work in mathematics; and, Stedall (2010), which places it within the history of mathematics more broadly.

  14. 14.

    This is uncontroversial, although it leaves open Wilson ’s intriguing question (1986, pp. 354–355): can a finite substance like me be the cause of my idea of the world, which is merely indefinite?

  15. 15.

    See also Descartes ’s letter to Mersenne of May 1630 (AT 1: 152), although that is obviously from a much earlier period in his career.

  16. 16.

    Descartes writes: “The only principles that I admit—or require—in physics are those of geometry and abstract mathematics; they explain all natural phenomena, and enable us to provide quite certain demonstrations concerning them” (Principia Philosophiae, Part two, § 64; AT 8: 78–79).

  17. 17.

    As Ted McGuire writes in a recent paper: “If Newton ’s theology of divine existence grounds the actuality of infinite space, geometry underwrites his claim to understand its infinite nature. Clearly, the depth of Newton ’s dialogue with Descartes must be appreciated if we are adequately to understand his path to this conception” (McGuire 2007, p. 125). I agree.


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For very helpful conversations that substantially altered my argument in this paper, I would like to thank Vincenzo De Risi and David Sanford. Many thanks to the audience at the Max-Planck-Institut für Wissenschaftsgeschichte in Berlin for their help on an earlier version of this paper—the audience included Lorraine Daston, Dan Garber, Graciela De Pierris, Michael Friedman, Jeremy Gray, Gary Hatfield, Doug Jesseph, Clarissa Lee, Brandon Look, Henry Mendell, David Rabouin. All translations are my own unless otherwise noted.

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Janiak, A. (2015). Mathematics and Infinity in Descartes and Newton. In: De Risi, V. (eds) Mathematizing Space. Trends in the History of Science. Birkhäuser, Cham.

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