Abstract
Descartes’s metaphysics posits a sharp distinction between two types of non-finitude, or unlimitedness: whereas God alone is infinite, numbers, space, and time are indefinite. The distinction has proven difficult to interpret in a way that abides by the textual evidence and conserves the theoretical roles that the distinction plays in Descartes’s philosophy—in particular, the important role it plays in the causal proof for God’s existence in the Meditations. After formulating the interpretive task, I criticize extant interpretations of the distinction. I then propose an alternative at whose core is the idea that whereas the indefinite is a structural, iterative notion, designating the absence of an upper bound, the infinite is an ontic notion, signifying being in general, or what is, without qualification.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
References to Descartes’s works cite the volume and page number in Descartes (1996) (abbreviated ‘AT’), followed by the volume and page number in Descartes (1985–1992), vols. 1 and 2 (abbreviated ‘CSM’), or by the page number in vol. 3 (abbreviated ‘CSMK’). I use ‘Meditations’ for Meditations on First Philosophy, and ‘Principles’ for Principles of Philosophy.
- 2.
In the Fourth Meditation, Descartes says of his own will that it is “so great that the idea of any greater faculty is beyond my grasp; so much so that it is above all in virtue of the will that I understand myself to bear in some way the image and likeness of God.” (AT 7.57/CSM 2.40) This has sometimes been read as a statement to the effect that the human will is infinite, perhaps contrary to the claim in the Principles that God alone is infinite. (See Naaman-Zauderer (2010, ch. 4) and Boehm (2014) for discussion.) Although I personally do not read Descartes as claiming that the human will is infinite, I will remain neutral on this issue here, for (I think) it can be properly addressed only after the infinite-indefinite distinction has been clarified.
- 3.
The labels “metaphysical” and “epistemological” are due to Wilson (1999), who speaks of metaphysical and epistemological “criteria”. Wilson does not identify a scope distinction (or criterion) as such, though it is implicit in her discussion.
- 4.
But cf. Descartes’s April 15th 1630 letter to Mersenne (AT 1.146-7/CSMK 23). Descartes’s dismissal of certain queries about infinity is not unusual in the period. See Mancosu (1996, esp. chs. 2 and 5) for discussion.
- 5.
- 6.
It has been claimed that the distinction plays a non-theoretical, political role as well, serving to fend off potential Church sanctions that Descartes would face were he to claim that the world is infinite (and consequently, perhaps, not geocentric). Such a role is suggested by Descartes’s June 6th, 1647 letter to Chanut : “I recollect that the Cardinal of Cusa and many other Doctors have supposed the world to be infinite without ever being censured by the Church … And my opinion is not so difficult to accept as theirs, because I do not say that the world is infinite, but only indefinite .” (AT 5.51/CSMK 319) Koyré (1957) famously took the distinction to play only this political role. But as shown by the theoretical roles discussed in the main text, this position is untenable. For a similar verdict, see Ariew (1987, §3.2). For further discussion of the political role of the distinction, see Vilmer (2011).
- 7.
See, e.g., Ariew (1987, 156): “Descartes’ indefinite is to be understood as a notion stemming from a defect of our understanding and not from the nature of things.” Cp. North (1983) and McGuire (1983). Wilson (1999) offers an interesting and subtle variant on the ignorance interpretation, which I discuss below, in note 11. Janiak (2015) seems to endorse the ignorance interpretation as well, though he acknowledges that it is difficult to reconcile with the existence of the metaphysical distinction , for reasons that will be discussed in the main text shortly.
- 8.
See also Descartes’s February 5th, 1649 letter to More : “God is the only thing I positively understand to be infinite. As to other things like the extension of the world and the number of parts into which matter is divisible, I confess I do not know whether they are absolutely infinite; I merely know that I know no end to them, and so, looking at them from my own point of view, I call them indefinite .” (AT 5.274/CSMK 374) Additional passages in this spirit are to be found in the French version of Principles 1.27 (AT 9B.37) and the Conversation with Burman (AT 5.154/CSMK 339ff), both of which are discussed by Ariew (1987).
- 9.
Proponents of the ignorance interpretation tend to embrace this implication, maintaining that Descartes was unjustified, mistaken, or confused in endorsing the metaphysical distinction . See, e.g., Ariew (1987) and Janiak (2015). See also Leibniz’s (1969, 139) remark that “the indefinite of Descartes is not in the thing but in the thinker”, and More’s verdict in his letter to Anne Conway from May 5th 1651: “For infinite and indefinite in Des Cartes sense, truly Madam, I can not easily absteine from being of your Ladiships opinion in that, that they come much to one” (quoted in Conway 1992, 486–9).
- 10.
See, e.g., the Sixth Meditation: “I have never judged that something could not be made by [God] except on the grounds that there would be a contradiction in my perceiving it distinctly.” (AT 7.71/CSM 2.50) Granted, Descartes famously holds that God is the creator of the eternal truths , and that he could have made necessary truths false, and contradictory claims possible (indeed, true), had he chosen to do so. However, as Wilson (1999, 116) observes in a similar dialectical context, it seems clear that Descartes intended for this doctrine to explain, rather than undermine, the necessity of eternal truths , or the impossibility of contradictions.
- 11.
In her interesting and subtle discussion of the epistemological distinction , Margaret Wilson appears to treat (G) as providing reason to reject the entailment from inconceivability to impossibility in Descartes. She writes (1999, 115): “Descartes’s view all along has been, I suggest, that there is something inconceivable to us in the idea that the world has limits , some conceptual barrier to positing limits to matter . Yet he seems to hold that this fact does not commit him to the view that the world lacks limits .” (Earlier in the discussion, she makes it clear that the evidence for the latter claim is (G); ibid., 114). Hence Wilson’s interpretation is ultimately a variant on the ignorance interpretation, where ignorance is replaced with inconceivability. However, as suggested in the main text, and as we will see below, (G) can be sensibly understood without appeal to the ignorance interpretation, or to Wilson’s variant on it.
- 12.
This might be thought to resemble Spinoza’s distinction between what is unlimited in its kind , and what is absolutely infinite, or infinite in all kinds. See Ethics, part I, definition 6 (Spinoza 1985).
- 13.
- 14.
The respects interpretation coheres with a certain natural translation of two key elements in the Principles passages (quoted in Sect. 3.1), the expressions omni ex parte and alique ex parte (recall F and G). Omni and alique are often translated using the quantifiers “all” and “some”, while parte is translated using the term “respects ”. It therefore becomes natural to interpret this element as claiming that the infinite does not have limits “in any respect ” [omni ex parte]. However, this translation is not obligatory; instead, omni ex parte can be translated as “completely” or “absolutely”, where such completeness need not be understood as holding “in any respect ”. This option will be exploited by the alternative interpretation of the metaphysical distinction developed in Sect. 3.6.
- 15.
Cp. Wilson (1999, 115). And contrast Spinoza’s Ethics, part I, proposition 15, scholium 2.
- 16.
See, e.g., Principles I.23 (AT 8A.13/CSM 1.201): “[T]he nature of body includes divisibility along with extension in space, and since being divisible is an imperfection , it is certain that God is not a body.”
- 17.
At the very least, it is not clear that such a difference justifies the attention it receives in Descartes’s publications and correspondence.
- 18.
The latter is the distinction’s third theoretical role, foreshadowed in Sect. 3.1.
- 19.
As should be clear, this property is distinct from, and is not implied by, iterative unlimitedness : even if an entity is such that for any one of its parts, a greater one exists, this does not entail that the entity itself is greater than any of its parts. The latter would follow only if the entity as a whole has a measure, and hence is capable of being compared, with respect to measure, to other entities.
- 20.
The labels ‘iterative’ and ‘quantitative’ are from my Schechtman (forthcoming), where I discuss these two kinds of unlimitedness (or two kinds of infinity, as they are called there) in greater detail. I prefer these labels to Nolan and Nelson’s . They use “actual” (or “complete” and “perfect”) for the infinite, and “potential ” (or “incomplete” and “imperfect”) for the indefinite . In the Aristotelian tradition, something is potentially infinite only if it is finite, though it is possible for it to become greater without limit. Yet as Nolan and Nelson themselves observe, the collection of natural numbers is not finite: for every natural number there is an actual, and not a merely possible, successor. Similarly, as discussed in Sect. 3.3, Descartes thinks that each region of extension is exceeded by an actual, and not merely possible, greater region. For these reasons, Descartes’s indefinite cannot be understood as Aristotelian potential infinity—contra, e.g., Curley (1978, 224), Nolan and Nelson (2006), and Janiak (2015).
- 21.
I say ‘merely’ because otherwise the cardinality reading would imply that God is indefinite —the wrong result. Hereafter, I elide ‘merely’ for ease of exposition.
- 22.
Elsewhere in their article Nolan and Nelson seem to want to deny this, but their denial rests on two assumptions, both of which are problematic. The first is that the indefinite is equivalent to potential infinity; I explained why this equivalence fails in note 20. The second assumption is that the idea of “that which is subject to augmentation and potentially infinite ” cannot be the cause of the idea of “that which is complete and actually infinite” (1996, 110). But the process they describe in the initial quotation in the main text suggests a direct route from the former to the latter.
- 23.
Descartes’s position bears interesting affinities to a traditional view of God in medieval philosophy and theology. This view conceives God as identical to or as possessing unlimited “Being ” or “being itself” [ipsum esse], and earthly creatures as possessing qualified, limited being derived from God, by whom they were created and on whom they depend. See, e.g., Augustine (1991, 7.10.16).
- 24.
See especially AT 7.46/CSM 2.31.
- 25.
This claim is central to the argument for God’s existence in the Third Meditation. I will explain how my proposed interpretation handles the crucial assumption in this argument below. Throughout I use ‘being’ and ‘reality’ interchangeably.
- 26.
See the Third Replies (AT 7.185/CSM 2.130). I discuss Descartes’s treatment of the relevant dependence relations in Schechtman (2016).
- 27.
Recall note 14, where I noted that omni ex parte can be translated as “completely” or “absolutely”.
- 28.
In this way, all of (C), (F), and (G) can be sensibly interpreted without appealing to our ignorance .
- 29.
This serves to elucidate (A), (D), and (E).
- 30.
This upholds (B).
- 31.
I have received helpful suggestions from audience members at a conference on infinity in early modern philosophy at the Van Leer Institute in Jerusalem and at the 2017 Atlantic Canada Seminar in Early Modern Philosophy at Dalhousie University. I am also grateful to Ohad Nachtomy for helpful comments, and in particular to John Bengson for extensive input, both critical and constructive.
References
Ariew, R. (1987). The infinite in Descartes’ conversation with Burman. Archiv für Geschichte der Philosophie, 69(2), 140–163.
Augustine (1991). Confessions (C. Henry, Trans.). Oxford: Oxford University Press.
Beyssade, J. M. (1979). La philosophie première de Descartes. Paris: Flammarion.
Beyssade, J. M. (1992). The idea of God and the proofs of his existence. In J. Cottingham (Ed.), The Cambridge companion to Descartes (pp. 174–199). Cambridge: Cambridge University Press.
Boehm, O. (2014). Freedom and the Cogito. British Journal for the History of Philosophy, 22(4), 704–724.
Conway, A. (1992). In M. Nicolson & S. Hutton (Eds.), The Conway letters: The correspondence of Anne, Viscountess Conway, Henry more, and their friends (pp. 1642–1684). Oxford: Oxford University Press.
Curley, E. (1978). Descartes against the skeptics. Cambridge: Harvard University Press.
Janiak, A. (2015). Mathematics and infinity in Descartes and Newton. In V. De Risi (Ed.), Mathematizing space (pp. 209–230). Basel: Birkhäuser.
Kendrick, N. (1998). Uniqueness in Descartes’ ‘infinite’ and ‘indefinite’. History of Philosophy Quarterly, 15(1), 23–36.
Koyré, A. (1957). From the closed world to the infinite Universe. Baltimore: Johns Hopkins University Press.
Leibniz, G. W. (1969). In L. E. Loemker (Ed.), Philosophical papers and letters. Dordrecht: Reidel.
Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. New York: Oxford University Press.
McGuire, J. E. (1983). Space, geometrical extension, and infinity: Newton and Descartes on extension. In W. R. Shea (Ed.), Nature mathematized (pp. 69–112). Reidel: Dordrecht.
Naaman-Zauderer, N. (2010). Descartes’ deontological turn. Cambridge: Cambridge University Press.
Nolan, L., & Nelson, A. (2006). Proofs for the existence of God. In S. Gaukroger (Ed.), The blackwell guide to Descartes’ meditations (pp. 104–121). Oxford: Blackwell.
North, J. (1983). Finite and otherwise. Aristotle and some seventeenth century views. In W. R. Shea (Ed.), Nature mathematized (pp. 113–148). Reidel: Dordrecht.
Schechtman, A. (2014). Descartes’s argument for the existence of the idea of an infinite being. Journal of the History of Philosophy, 52(3), 487–517.
Schechtman, A. (2016). Substance and independence in Descartes. Philosophical Review, 125(2), 155–204.
Schechtman, A. (forthcoming). Three infinities in early modern philosophy. Mind.
Spinoza, B. 1985. The collected works of Spinoza. Vol. 1 (Edwin Curley, Ed. and Trans.). Princeton: Princeton University Press
Vilmer, J. B. J. (2008). La veritable nature de l’indéfini cartésien. Revue de Métaphysique et de Morale, 60, 503–515.
Vilmer, J. B. J. (2011). L’indéfini cartésien entre politique et langage. Revue philosophique de Louvain, 109(3), 443–460.
Wilson, M. (1999). Can I be the cause of my idea of the world? (Descartes on the infinite and the indefinite). In M. Wilson (Ed.), Ideas and mechanism: Essays in early modern philosophy (pp. 108–125). Princeton: Princeton University Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Schechtman, A. (2018). The Ontic and the Iterative: Descartes on the Infinite and the Indefinite. In: Nachtomy, O., Winegar, R. (eds) Infinity in Early Modern Philosophy. The New Synthese Historical Library, vol 76. Springer, Cham. https://doi.org/10.1007/978-3-319-94556-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-94556-9_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94555-2
Online ISBN: 978-3-319-94556-9
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)