Abstract
This chapter addresses the question of the ontology of mathematical entities, particularly geometrical figures (though I also touch on numbers), in Spinoza. I discuss the status of mathematical entities as beings of reason and mount a case against mathematical antirealism by linking it with acosmism. Despite the fact that geometrical figures per se are beings of reason, I argue for a realist interpretation of geometrical figures as the determinations of finite bodies. Advancing this argument requires me to examine Spinoza’s discussions of physical individuals in the Ethics and elsewhere. In addition to marshaling textual evidence for my interpretation, I address some questions regarding the property ontology of geometrical figures.
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Notes
- 1.
Spinoza targets specifically those who divide being into real being and being of reason as well as those who say that a being of reason is not a mere nothing. It is not entirely clear whom Spinoza has in mind, though the clearest candidates would appear to be those thinkers that Suárez characterizes in Metaphysical Disputation 54 as believing that beings of reason “somehow agree with real beings in the character of being” on the grounds that beings of reason, like real beings, are “said to be” (Suárez 1995, 60). For instance, since we say that someone is blind, then blindness, a being of reason, must have some manner of being. Spinoza explicitly criticizes this basis for dividing being into real being and being of reason, saying, “I do not wonder that Philosophers preoccupied with words, or grammar, should fall into such errors. For they judge the things from the words, not the words from the things” (CM 1.1/G 1:235). In the notes to his translation of Metaphysical Disputation 54, John P. Doyle cites Cajetan, Capreolus, Deca, Ferrara, and Soncinas as belonging to this category (Suárez 1995, 60, n.14). Since Suárez himself says that beings of reason do not belong to the genus of being, he would presumably be immune to Spinoza’s main criticism (Suárez 1995, 60–64). Cf. Curley 1:301, n. 2.
- 2.
Spinoza at one point says that beings of reason “can not in any way be classed as ideas” (CM 1.1/G 1:234), but this is misleading. In this context, Spinoza seems to be using “idea” in a restricted sense to mean a mode of thinking with a real object in nature. Elsewhere, Spinoza says that any mode of thinking is an idea (E2a3), and I take this to be his more consistent usage. Since beings of reason are modes of thinking, then they are ideas, albeit without real objects.
- 3.
Spinoza’s imaginatio includes experience by way of the five senses, as well as memory and our capacity for dreams and other fictive sensory experiences (including what we more narrowly understand as the purview of the imagination today). If we take “sensation” nowadays to refer narrowly to the deliverances of the five senses, then Spinoza’s imaginatio is clearly broader. But I think our more contemporary notion of “sensation” admits of a broader construal that encompasses memory and dreams insofar as the latter have a sensory component. On a broad construal of sensation, then, Spinoza’s imaginatio is a similar concept. Spinoza defines the imagination, or the first kind of knowledge, in E2p40s2/G 2:122, but he first explains image formation in E2p17s/G 2:105–6.
- 4.
In explaining his use of the term imago, Spinoza states, “to retain the customary words, the affections of the human Body whose ideas represent external bodies as present to us, we shall call images of things [rerum imagines]” (E2p17s/G 2:106).
- 5.
See E2p17dem2/G 2:105.
- 6.
Cf. Della Rocca 1996, 57–64. According to Della Rocca’s reading, a sensory idea is confused because it is literally of both the (proximate) cause of the bodily affection and the bodily affection itself (qua effect). As I have argued elsewhere (Homan 2014, 82), since the mind is the idea of its body alone, and since the cause of the bodily affection is external to the body, I do not think Della Rocca’s literal interpretation can be quite right.
- 7.
Cf. Melamed’s discussion of how the account of abstraction in E2p40s1 can be adapted to explain the formation of our ideas of numbers (Melamed 2000, 12–13).
- 8.
Cf. Descartes’ similar analysis of time in CSM 1:212.
- 9.
Frege 1980, 28. Unlike Frege, however, I do not see any evidence that Spinoza is committed to securing realism about numbers (by affirming a third realm, for instance). I return to this issue in the last section of this chapter.
- 10.
- 11.
Hobbes 2005, 139.
- 12.
Somewhat surprisingly, Letter 50 is not discussed in the antirealist literature from what I can tell. (I do not see references to it in Melamed (2000), Schliesser (2014), or Peterman (2015).) Far more emphasis is placed, in general, on Spinoza’s critique of time, number, and measure in Letter 12. Nevertheless, Schliesser (2014) deploys the remark, “Being finite is really, in part, a negation” (E1p8s1), as an epigraph to his article, and Melamed describes mathematics as a “science of non-beings” (2000, 20), suggesting a similar line of argument to the one outlined in this subsection.
- 13.
E1a4, the so-called causal axiom, has both normative, epistemic force (one must know the causes of things in order to have knowledge, as is the emphasis of the similar passage in TIE 92) and descriptive, metaphysical force (it simply is the case that all knowledge depends on knowledge of causes, and involves the latter). This leads to an interpretive difficulty that has exercised scholars inasmuch as it seems to imply that all knowledge is adequate. After all, to know something adequately is to know it per causam, but, as a matter of fact, all knowledge is knowledge per causam. But how can this be, since Spinoza recognizes inadequate knowledge, which he compares to conclusions without premises (E2p29s), ideas of effects without ideas of their causes? The fact that Spinoza invokes the causal axiom in arguing that the mind perceives external bodies insofar as external bodies cause the body to be affected in different ways (E2p16dem, c1) complicates matters further, since in this context Spinoza is discussing inadequate sensory perception. Wilson (1991) stresses E2p16 in critiquing Gueroult’s position that the axiom applies only to adequate knowledge (Gueroult 1968, 95–8). But I think E2p16 can be interpreted in a way that is consistent with Gueroult’s interpretation, and the general problem raised by the causal axiom can be allayed, if we recall what I said above (n. 1 in Chap. 1) in explanation of my decision to translate cognitio as “knowledge.” That is, in a sense all knowledge is adequate, since inadequate knowledge is privative, and to have inadequate knowledge is to constitute a part of a whole, adequate understanding, that is, God’s understanding, which provides the standard for what constitutes knowledge. As I read E2p16, when my body is affected by an external body, my mind has an idea only of my body qua affected. So, my mind lacks the idea of the bodily affection’s cause, but nevertheless involves an idea of the cause to the extent that my body bears the literal impress of the cause. The result of the idea of the affection absent its cause is a confused idea of an external body. (See my paper, Homan 2014, for further discussion of the mechanics of mental representation involved here, as well as Chap. 4.) All of this is, it seems to me, consistent with the notion of inadequate knowledge as privative, and thus does not require a special extension of the axiom to inadequate knowledge . Cf. Morrison 2015.
- 14.
Cf. Marshall 2008, 56–62. Marshall identifies two requirements for an adequate idea: (1) the idea must be wholly (not only partially) contained in the human mind; and (2) the idea must also include an adequate idea of the object’s cause. I think the brief account I have given of adequacy in light of Spinoza’s genetic definitions is consistent with Marshall’s. Genetic definitions are only genetic insofar as they include an adequate cause of their object. Moreover, I have argued, on the basis of that cause, all the properties of the thing can be deduced, which guarantees that the object is wholly contained in the idea, thereby satisfying Marshall’s first requirement. Cf. Descartes, CSM 2:155. Descartes denies that we can ever be sure that our idea of an object is complete in the sense of containing all the properties that are in the thing. On this basis, Descartes denies that we can ever have adequate knowledge. I discuss this discrepancy between Spinoza and Descartes on the subject of adequate knowledge in the concluding chapter.
- 15.
Cf . Gueroult 1968, 422. Gueroult distinguishes between imaginative and intellectual conceptions of figure, but concedes that even the intellectual conception involves elements of the imagination, and so the distinction does not hold: “However, as the geometrical essence can be conceived only by the delimitation of indeterminate extension by means of figure taken in its first sense, the two senses of the word figure tend to run together, and the result is an equivocation” (my translation).
- 16.
See Hegel 1995, 281.
- 17.
For other evidence that Spinoza believed in the existence of finite modes, see Melamed 2010, 89–91.
- 18.
- 19.
See n. 12.
- 20.
See CSM 2:44/AT VII:63; CSM 1:89/AT XI:26; CSM 1:217; AT VIIIA:33.
- 21.
CSMK: 119.
- 22.
CSM 1:61.
- 23.
CSM 1:212, first italics mine.
- 24.
Aristotle 1984, vol. 1, 331.
- 25.
Exactly how to understand mathematical objects and their relation to the sensible things from which they are abstracted in Aristotle are open questions. One issue is whether sensible things can instantiate the exact mathematical objects studied by mathematicians. If not, then an account of abstraction needs to be provided to explain the discrepancy. For such an account, which highlights the role of intelligible (as opposed to sensible) matter in mathematical thinking, see Mueller 1970. For a contrasting account, according to which mathematically exact properties are instantiated in sensible things (and can thus be abstracted, as it were, more directly from the latter), see Lear 1982.
- 26.
I am indebted here to Nolan’s discussion of Cartesian abstraction in terms of selective attention (1998, 167, 174).
- 27.
Nolan (1997) provides an anti-Platonist, conceptualist reading of Meditation Five’s immutable essences with which I am generally sympathetic. For a Platonist reading, see Kenny 1970. Schmaltz (1991) outlines a “middle way between Platonic realism and conceptualism” (170) according to which immutable essences exist in God.
- 28.
- 29.
An analogy from chemistry may be of some use. Chemical compounds, such as sodium chloride, are analogous to individuals, because in sodium chloride, the constituents, sodium and chlorine, bind together in a fixed, determinate relation. Mixtures, such as saltwater, by contrast, provide an analogue for what fails to constitute an individual because while salt and water may be physically intermingled, they do not form any chemical bond and, as a result, are much easier to separate than the constituents of a compound. While this distinction between chemical compound and mere mixture may illustrate the sort of thing that Spinoza may have been trying to define (despite its anachronism), it does little to clarify the terms of the definition itself. In particular, scholars have debated whether the ratio of motion and rest should be understood in numerical terms or not. For a numerical reading, see Lachterman 1978, 85–6. For a criticism of the numerical reading, see Lin 2005, 248–54. I do not take a position on this matter here. A numerical reading would, of course, have to be squared with what I said about numbers as beings of reason above.
- 30.
If Spinoza’s definition of an individual in E2p13d is a definition of a composite body (as it seems to be), then what individuates the bodies that make up the composite body? One place to look is Spinoza’s definition of body (“corpus”) in E2d1 as “a mode that in a certain and determinate way expresses God’s essence insofar as he is an extended thing” (G 2:84), but this is very general and raises precisely the questions about what makes something a “determinate” expression of God’s essence qua extended that one might have hoped Spinoza’s Physical Digression would clarify. Another place to look is immediately prior to the definition of individual bodies in the Physical Digression, where Spinoza describes the “simplest bodies” as “distinguished from one another only by motion and rest, speed and slowness” (E2p13a2″/G 2:99). It is far from clear, however, how motion and rest, speed and slowness, could distinguish bodies unless they were already individuated.
- 31.
In Letters 39 and 40, Spinoza argues for the superiority of circular (or spherical) lenses. (I discuss these letters in Chap. 5.) In Letter 32, Spinoza speaks of “size and shape” (magnitudinis, et figurae) (G 4:171a), along with motion and rest, in explaining the nature and composition of bodies. Finally, in E1p15s, Spinoza provides a kind of unofficial definition of body as follows: “by body we understand any quantity, with length, breadth, and depth, limited by some certain figure” (G 2:57). I call this a kind of unofficial definition because, as we saw above, Spinoza defines body officially in E2d1 as “a mode that in a certain and determinate way expresses God’s essence insofar as he is considered as an extended thing” (G 2:84).
- 32.
There is a debate about the extension of Spinoza’s notion of individuality, centering, in particular, around the question of whether a political state can be an individual. But even skeptics of the state qua individual, such as Steven Barbone, accept the whole of physical nature as an infinite individual. On this, see Barbone 2002.
- 33.
As John Grey pointed out to me, we might also derive this point from E1d2: “That thing is said to be finite in its own kind that can be limited by another of the same nature” (G 2:45). If we interpret this to mean that something can be limited by another of the same nature, if and only if it is finite, then it follows that if something is not finite, then it is not limited by something of the same nature. Thus, it cannot have any shape.
- 34.
In contemporary terminology, the question is whether Spinoza is a moderate nominalist or a realist. Extreme nominalism (the third major category) was ruled out earlier in the chapter. For an overview of the contemporary debates concerning universals, see Moreland 2001.
- 35.
Cf. Newlands 2017. Newlands highlights the ambivalence in Spinoza’s treatment of universals, as I have done, but interprets it as an inconsistency. By contrast, I have suggested that the ambivalence can be resolved if we distinguish between universals formed through the imagination, on the one hand, and universals (or quasi-universals) formed through reason, on the other. Spinoza rejects the former but embraces the latter. Newlands (2017, 84–5) acknowledges this interpretive move, but questions whether Spinoza is entitled to the distinction. I think Spinoza does have a fairly compelling argument for distinguishing his common notions (at least) from imaginative universals. I do not interpret common notions to be abstractions as Newlands does. I discuss the origins and adequacy of common notions in the next chapter. For a more sympathetic handling of Spinoza’s ambivalence about universals (i.e., which does not see it rooted in inconsistency), see Hübner 2015, 66–8. I discuss Hübner’s interpretation in Chap. 6.
- 36.
See Carriero 1995 for a useful discussion of this issue in relation to the Aristotelian notions of substance and accident. What I am speaking of as numerically different, but resembling, properties, Carriero (1995, 256–59) speaks of as particular or individual accidents. Like Carriero (and contra Bennett , whose position on the matter, along with Curley’s, Carriero criticizes) I see nothing about the notion of property itself that would determine one way or the other whether properties are particular or universal (or both). Cf. Bennett 1984, 94.
- 37.
Melamed 2000, 15–16.
- 38.
Melamed 2000, 16.
- 39.
CSM 2:262/AT VII:382.
- 40.
Palmerino finesses a similar distinction between the ontology of mathematical entities and epistemological constraints on our knowledge of them with respect to Galileo. Referencing a passage from the Assayer in which Galileo acknowledges the indefinability of infinitely complex irregular lines, Palmerino writes, “In Galileo’s eyes the problem is hence not that irregular lines (and physical accidents) are not mathematical, but rather that their mathematical structure is beyond the reach of our intellectual skills” (2016, 39).
- 41.
For an additional consideration in favor of interpreting modes as non-multiply exemplifiable particulars, see Melamed 2013, 58. Melamed presents the counterfactual possibility that there are two substances, A and B, sharing mode m. He argues that, if we assume a change in m, the change must come from either A or B. But if A, say, caused the change in m, then A would have caused a change in B (since m is also a mode of B). But this would violate Spinoza’s prohibition against causal interaction between substances. From this, Melamed concludes that modes are unrepeatable properties.
- 42.
In discussing Spinoza’s fourth proportional example, Matheron claims that proportions and ratios can be genetically conceived. It is unclear whether he thinks this applies to arithmetic operations in general. See Matheron 1986, 127–28.
- 43.
In the TIE, Spinoza says, somewhat surprisingly, that through the fourth mode of knowing, “we know that two and three are five, and that if two lines are parallel to a third line, they are also parallel to each other, etc.” (TIE 22/G 2:11). This is surprising since the TIE’s fourth mode of knowing is the most perfect form of knowledge, but the simple mathematical truths cited in this comment are beings of reason, and beings of reason do not supply the content of the most perfect form of knowledge, as Spinoza indicates on numerous occasions (TIE 75/G 2:28–9; TIE 95/G 2:35; TIE 99/G 2:36). Furthermore, the fourth mode of knowing “comprehends the adequate essence of the thing” (TIE 29/G 2:13), but a number of passages suggest that number can have nothing to do with the essence of anything (E1p8s2/G 2:50–1; Ep. 50/G 4:239b). As a result, I think the mathematical examples cited here should be dealt with in the same way as the fourth proportional example. They may represent the form of true ideas (grasped through the fourth mode of knowing, in this case), but do not provide the proper content of true ideas (or the proper content of intuitive knowledge, pace Sandler 2005, 84). It must be admitted that the comment is bedeviling, nevertheless.
- 44.
CSM 1:10–13.
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Homan, M. (2021). Realism and Antirealism About Mathematical Entities. In: Spinoza’s Epistemology through a Geometrical Lens. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-76739-6_3
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