Abstract
This chapter is devoted, in part, to developing an example of Spinozan science in practice, and exhibiting the role of geometry therein. In this regard, I offer a reading of Spinoza’s epistolary writings on optics and his treatment of a question of optimal lens shape. I also address a further objection to my realist interpretation of geometrical figures stemming from Letter 12, as well as the difficulty raised by the incompleteness of Spinoza’s thinking about physics for any interpretation of Spinozan science.
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Notes
- 1.
For further discussion in support of this reading of the infinite modes as encompassing the finite modes (such that the latter are the parts of the former), see Melamed 2013 130–32.
- 2.
Geometry was classically defined as the science of (continuous) magnitudes, but I think it is clear that this is insofar as magnitudes can be determined in various ways (e.g., as points, lines). Cf. Hobbes’ statement of the subject matter of geometry in De Corpore (with his particular emphasis on motion):
first we are to observe what effect a body moved produceth, when we consider nothing in it besides its motion; and we see presently that this makes a line, or length; next, what the motion of a long body produces, which we find to be superficies; and so forwards, till we see what the effects of simple motion are; and then, in like manner, we are to observe what proceeds from the addition, multiplication, subtraction, and division, of these motions, and what effects, what figures, and what properties, they produce; from which kind of contemplation sprung that part of philosophy which is called geometry. (Hobbes 2005, 71)
- 3.
See Petry 1985, 97. Petry describes Huygens’ admiration for Spinoza’s practical skills in optics and doubts about Spinoza’s capacity to contribute to optical theory. Cf. Von Duuglas-Ittu (2008c), who questions Petry’s evidence on this score. See also Vermij (2013, 66), whose assessment is more in line with Petry’s.
- 4.
On Hudde’s influence on Spinoza’s optical thought and practice, see Vermij (2013, 79–81).
- 5.
Von Duuglas-Ittu (2008b) provides an insightful, in-depth reading of these letters with which the points I make below are largely sympathetic. Especially helpful is his speculative reconstruction of the questions from Jelles—in letters now lost—that prompted Spinoza’s responses, as well as his discussion of the contemporaneous optical writings on which Jelles and Spinoza probably drew.
- 6.
See DM, 117–19.
- 7.
For further analysis of Descartes’ discussion of the imperfections of natural vision and means of enhancement in the Optics, see Ribe 1997.
- 8.
Gabbey 1996, 153.
- 9.
DM, 160–1.
- 10.
Von Duuglas-Ittu (2008a) also rejects Gabbey’s interpretation.
- 11.
Vermij and Atzema 1995, 106.
- 12.
See Vermij and Atzema 1995, 113.
- 13.
See Vermij and Atzema 1995, 113–14. The relevant section of Hudde’s text reads, “The crucial thing is that we cause the preponderance of parallel rays to be refracted by the glass through which they pass in such a way that they tend afterward to one and the same point. But this point can be considered either mathematically or mechanically [aut mathematice aut mechanice]. And although it is certain that circular figures have neither the power nor property (as do elliptical or hyperbolic figures and many others more composite) to refract parallel rays in such a way that afterward they tend to one mathematical point, nonetheless a preponderance of them can be turned toward the same place, such that that space in which they all converge can be considered a mechanical point. Now, I call a point mechanical either when it is not divisible in mechanical terms or when its parts are not worth consideration” (my translation).
- 14.
See also CSM 1: 288: “I do not recognize any difference between artefacts and natural bodies except that the operations of artefacts are for the most part performed by mechanisms which are large enough to be easily perceivable by the senses…so it is no less natural for a clock constructed with this or that set of wheels to tell the time than it is for a tree which grew from this or that seed to produce the appropriate fruit.” I am grateful to John Grey for pointing me to this passage. It is well known that Spinoza goes even further than Descartes in subsuming everything under the same laws of nature, inasmuch as he includes therein the workings of the human mind (E3pref).
- 15.
See n. 32 in Chap. 1.
- 16.
DM, 87–91.
- 17.
See DM, 91.
- 18.
For further discussion of Descartes’ account of vision in the Optics, see Osler 2008, 134–36.
- 19.
For discussion of the history of the term “a priori” and Spinoza’s usage of it therein, see Miller 2004, 556–66.
- 20.
I have in mind Spinoza’s offer to “suspend judgment” until Leibniz has had a chance to explain further his own arguments in favor of a lens shape other than the circle (G 4:233b).
- 21.
G 2:59.
- 22.
See Peterman 2015, 6–7.
- 23.
On the question of the identity of Spinoza’s targets in these texts, Wolfson writes, “It is safe to say that whomsoever in particular and directly Spinoza may have had in mind when assailing his opponents for denying the infinity of corporeal substance, it is ultimately the views and arguments advanced by Aristotle that he is contending with” (1934, 265).
- 24.
Cf. Spinoza’s critique of Zeno in the DPP:
He supposes, first, that bodies can be conceived to move so quickly that they cannot move more quickly, and second, that time is composed of moments, just as others have conceived that quantity is composed of indivisible points.
Both assumptions are false. (DPP2p8s, emphasis added)
- 25.
Bennett 1984, 89. See also, more generally, the “field metaphysic” interpretation that Bennett gives of substance and its modes (1984, 88–106). Bennett’s field metaphysic interpretation of extension has proven controversial insofar as it attributes (divisible) spatial extent or dimensionality to God qua extended substance. One problematic implication of this view is that by analogy with God’s having spatial extent (qua extended), he also has temporal extent (Bennett 1984, 206). Schmaltz criticizes Bennett on this score, drawing a contrapositive lesson from the space-time analogy: since God does not have temporal extent (but is eternal), God also does not have spatial extent (Schmaltz 1999, 199). I agree with Schmaltz that God is eternal, but I do not follow him in concluding that God is extended only in an eminent (not formal) sense (Schmaltz 1999, 188). Schmaltz’s main motivation for considering God to be extended only eminently appears to be his concern that viewing God as actually (or formally) extended (i.e., as having “spatial extent”) would make God divisible, but God qua substance is not divisible. Schmaltz’s arguments raise metaphysical issues that it is not possible to delve into here. (I touch on some of these issues in the next chapter, though only cursorily.) I will just say that what is most important in my view is Spinoza’s distinction between intellectual and imaginative conceptions of substance qua extended. When we conceive extended substance using the imagination (i.e., inadequately), we conceive it as divisible, having parts, enduring, and so on. When we conceive it, by contrast, via the intellect (i.e., adequately), we conceive it as indivisible, eternal, and so on, but also as extended. In this light, I agree with Bennett’s reading that God is spatially extended and with Schmaltz’s point that God is eternal (not enduring). This suggests perhaps that the space-time analogy breaks down on an intellectual view. I will return to the eternity/duration distinction in the next chapter.
- 26.
Cf. Melamed 2000, 15: “For when we prove a certain mathematical property of a body, we are completely uninterested in the way this body ‘flows from eternity,’ or even whether a body, which has such proportions, is instantiated in the extended world at all” (2000, 15). Here Melamed appears to invoke the Aristotelian notion that mathematicians study natural mathematical properties, but not qua naturally instantiated. In other words, mathematicians study objects abstracted from material instantiation (and the way bodies flow from eternity). Granting this to be the case, it still does not preclude the subsequent application of mathematical conclusions to the properties of natural bodies, at which stage instantiation would be of interest.
- 27.
Wilson 1996, 115.
- 28.
For a discussion of how Spinozan mechanistic science can be reconciled with Letter 12 with which I am generally sympathetic, see Lecrivain 1986, 20–23.
- 29.
My claim that intellectual conceptions of bodies are figural raises a question about the relation between conceiving the figural properties of a body and conceiving a body’s essence. This is a difficult question that I take up at the end of the next chapter.
- 30.
A number of commentators have interpreted Spinoza’s criticism of Cartesian extension in Letters 81 and 83, as I have done, as a criticism of its intrinsic inertness, and the attendant need to invoke an external (transcendent) cause to bring about motion (and, thus, individuation). See, for instance, Lachterman 1978, 101–3; Klever 1988, 165–71; and Viljanen 2011, 76. Peterman (2012) argues that Spinoza’s objection to Cartesian extension is more fundamental than the common reading would have it, applying even on the assumption that motion is imposed by God. Leibniz argues in De Ispa Natura (1989, 505) that (extrinsic) motion is unable to bring about the individuation of a uniform plenum (as he understands Cartesian extension to be), on the grounds that if there is no way to tell bodies apart prior to motion, there will be no way to tell them apart after the transposition of parts. (See Garber 1992, 179–81, for discussion of Leibniz’s critique.) Spinoza may well have had something similar in mind, objecting, thus, to Descartes’ physics not just because it requires God to transcend nature, but that it fails to account for individuation in any case. On either construal, Spinoza’s objection is against the externality of motion to matter.
- 31.
See Iltis (1971) for discussion of the vis viva controversy sparked by Leibniz’s criticisms of Descartes’ principle of the conservation of the quantity of motion. On the other hand, Leibniz is reported to have said, after having met with Spinoza in November 1676 (and after Spinoza’s letters to Tschirnhaus), that Spinoza did not see the problems with Descartes’ laws of motion. (See Gabbey 1996, n. 55.) For further discussion of Spinoza’s remarks about Descartes’ laws in Ep. 32, see Gabbey 1996, 165–66.
- 32.
For an interpretation that answers this question affirmatively, see Peterman 2012. Peterman argues that Spinoza’s critique of Descartes’ physics encompasses the latter’s conception of extension itself in terms of spatial dimensionality; extension is spatial, for Spinoza, only as it is imagined, not as it is in itself. (This argument is developed in greater depth in Peterman 2015.) The “commensurability response” outlined above serves, in part, to respond to this mathematical antirealist line of argument. For another response to Peterman, see Schmaltz 2020, 232–37.
References
Bennett, Jonathan. 1984. A Study of Spinoza’s Ethics. Indianapolis, IN: Hackett.
Gabbey, Alan. 1996. “Spinoza’s natural science and methodology,” in The Cambridge Companion to Spinoza, edited by Don Garrett, 142–191. Cambridge, UK: Cambridge University Press.
Garber, Daniel. 1992. Descartes’ Metaphysical Physics. Chicago, IL: University of Chicago Press.
Hobbes, Thomas. 2005. The English Works of Thomas Hobbes of Malmesbury. Vol. 1. Translated by William Molesworth. London: John Bonn, 1839. Replica edition, Elbiron Classics.
Iltis, Carolyn. 1971. “Leibniz and the Vis Viva Controversy.” Isis 62: 21–35.
Klever, W.N.A. 1988. “Moles in Motu: Principles of Spinoza’s Physics.” Studia Spinozana 4: 165–95.
Lachterman, David R. 1978. “The Physics of Spinoza’s Ethics.” In Spinoza: New Perspectives, edited by by Robert W. Shahan and J. I. Biro, 71–112. Norman: University of Oklahoma Press.
Lecrivain, André. 1986. “Spinoza and Cartesian Mechanics,” in Spinoza and the Sciences, edited by Marjorie Grene and Debra Nails, 15–60. Dordrecht: D. Reidel.
Leibniz, Gottfried. 1989. Philosophical Papers and Letters. Edited and translated by Leroy E. Loemker. Dordrecht: Kluwer Academic Publishers.
Melamed, Yitzhak Y. 2000. “On the Exact Science of Nonbeings: Spinoza’s View of Mathematics.” Iyyun, The Jerusalem Philosophical Quarterly 49: 3–22.
———. 2013. Spinoza’s Metaphysics: Substance and Thought. Oxford, UK: Oxford University Press.
Miller, Jon. 2004. “Spinoza and the a priori.” Canadian Journal of Philosophy 34(4): 555–590.
Osler, Margaret J. 2008. “Descartes’ Optics: Light, the Eye, and Visual Perception.” In A Companion to Descartes, edited by Janet Broughton and John Carriero, 124–141. Malden, MA: Blackwell Publishing.
Peterman, Alison. 2012. “Spinoza on the ‘Principles of Natural Things’.” The Leibniz Review 22: 37–65.
———. 2015. “Spinoza on Extension.” Philosopher’s Imprint 15: 1–23.
Petry, M.J. (ed.) 1985. Spinoza’s Algebraic Calculation of the Rainbow and Calculation of Chances. Dordrecht: Martinus Nijhoff Publishers.
Ribe, Neil M. 1997. “Cartesian Optics and the Mastery of Nature.” Isis 88(1): 42–61.
Schmaltz, Tad M. 1999. “Spinoza on the Vacuum.” Archiv für Geschichte der Philosophie 81(2): 174–205.
———. 2020. The Metaphysicsof the Material World: Suárez, Descartes, Spinoza. New York, NY: Oxford University Press.
Vermij, Rienk, and Atzema, Eisso. 1995. “Specilla circularia: an Unknown Work by Johannes Hudde.” Studia Leibnitiana 27: 104–121.
Vermij, Rienk. 2013. “Instruments and the Making of a Philosopher. Spinoza’s Career in Optics.” Intellectual History Review 23: 65–81.
Viljanen, Valtteri. 2011. Spinoza’s Geometry of Power. Cambridge, UK: Cambridge University Press.
Von Duuglas-Ittu, Kevin. 2008a. “Spinoza’s Blunder and the Spherical Lens.” Frames/sing (blog). Posted June 21, 2008. Available at: https://kvond.wordpress.com/2008/06/21/spinozas-blunder-and-the-spherical-lens/
———. 2008b. “Deciphering Spinoza’s Optical Letters.” Frames/sing (blog). Posted August 17, 2008. Available at: https://kvond.wordpress.com/2008/08/17/deciphering-spinozas-optical-letters/
———. 2008c. “Spinoza: Not As Abused as Is Said.” Frames/sing (blog). Posted September 14, 2008. Available at: https://kvond.wordpress.com/2008/09/14/spinoza-not-as-abused-as-is-said/
Wilson, Margaret D. 1996. “Spinoza’s theory of knowledge.” In The Cambridge Companion to Spinoza, edited by Don Garrett, 89–141. Cambridge, UK: Cambridge University Press.
Wolfson, Harry Austryn. 1934. The Philosophy of Spinoza: Unfolding the Latent Process of His Reasoning. Volume 1. Cambridge, MA: Harvard University Press.
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Homan, M. (2021). Geometry and Spinozan Science. In: Spinoza’s Epistemology through a Geometrical Lens. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-76739-6_5
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