Spinoza's God has Parts: The Mereological Reading of the Substance-Mode Relation

2.1. Why ‘indivisibility’ cannot mean ‘without parts’

Spinoza uses the notion of divisibility to make a distinction between two forms of quantity in a passage (henceforth ‘the passage on quantity’) that we can find in Letter 12:

But if you ask why we are so inclined, by a natural impulse, to divide extended substance, I reply that we conceive quantity in two ways: either abstractly, or superficially, as we have it in the imagination with the aid of the senses; or as substance, which is done by the intellect alone. So if we attend to quantity as it is in the imagination, which is what we do most often and most easily, we find it to be divisible, finite, composed of parts, and one of many. But if we attend to it as it is in the intellect, and perceive the thing as it is in itself, which is very difficult, then we find it to be infinite, indivisible and unique. (Ep. 12, G IV.56/C I.202–203)

This passage is repeated almost verbatim in E1P15S (G II.59/C I.423–424). Spinoza here describes divisibility as pertaining to the abstract notion of quantity (henceforth ‘abstract quantity’), whereas non-abstract quantity (henceforth ‘real quantity’) is indivisible. But does ‘indivisible’ mean a complete absence of parts? If so, the nasty implication is that real quantity would completely lack mereological complexity, thereby confining all such complexity to abstract quantity—a problematic view given Spinoza’s frequent use of parthood throughout his philosophy. This is why most scholars¹⁴ understand indivisible, real quantity as something that only applies to substance. This would allow us to interpret ‘indivisibility’ as the absence of parts without thereby denying the mereological structure of modes. However, there are quite a few reasons why real quantity cannot be limited to substance:

First, the passage itself does not oppose the two types of quantity in terms of the substance-mode distinction. Instead, Spinoza says that he opposes quantity as it is understood ‘by the intellect’ to quantity as it is understood ‘in the imagination’. Therefore, Spinoza’s defended notion of quantity applies to all things insofar as they are rightly understood by the intellect, that is, as Spinoza writes, insofar as we ‘perceive the thing as it is in itself’. In short, all real things, insofar as they are known by the intellect and insofar ‘as they are in themselves’ are characterized by the indivisibility of this real quantity.

Second, Spinoza’s characterization of real quantity clearly applies to modes, as shown in the rest of Letter 12. He critiques the common understanding of particular ‘things’ (i.e., modes) through abstract quantity, arguing that those who view things this way are ‘ignorant of the true nature of things’ (G IV.59/C I.204). Modes, he explains, ‘can never be rightly understood’ when interpreted using ‘aids of the imagination’, such as notions of number and measure—examples of abstract quantity (G IV.57/C I.203). In sum, it is not only the correct conception of substance that is at stake in Spinoza’s reconception of quantity, but also the correct conception of modes.

Third, if the second kind of quantity (i.e., real quantity) is understood to only apply to substance, the quantitative nature of modes becomes a problem. There are three options: option one is that modes have no quantitative nature. This conflicts with Spinoza’s references to their (in)finitude and mereological structure throughout the Ethics. Option two is that the first kind of quantity (i.e., abstract quantity) pertains to modes. However, this implies that modes, or at least their finitude and their mereological structure, are nothing but abstractions pertaining to the imagination, which ultimately leads to an ‘acosmist’ interpretation of Spinoza.¹⁵ The third option is that modes possess a quantitative nature distinct from the two listed by Spinoza. This means that there are actually three types of quantity: abstract quantity (pertaining to the imagination), real quantity (pertaining to substance), and another form of real quantity (pertaining to modes). Tad Schmaltz (2021b: 234–37) defends such a reading. However, there appears to be no clear textual basis for a third kind of quantity, and Spinoza explicitly distinguishes only two types in both the Ethics and Letter 12.

Fourth, the restriction of mereological structure to modes, by reserving the indivisibility of real quantity for substance, does not solve the paradoxes of infinite quantity because modes too (at least some of them) are said to be infinite (see E1P21–23). In E1P15S, Spinoza responds to those who have used the paradoxes that result from dividing infinite quantity as an argument for the incorporeality of God. But if Spinoza’s response would only consist of saying that substance does not have parts, then his reply would be insufficient because he clearly accepts that modes too can be infinite. Schmaltz thereby needs to add to the distinction between the non-mereological nature of substance and the mereological nature of modes a second distinction between the mereology of finite modes and the mereology of infinite modes (Schmaltz 2021b: 258).¹⁶ But, again, no such distinction can be found in the texts. Furthermore, when one attributes a different mereological structure to infinite modes, one also attributes a different quantitative nature. Such an interpretation thus ultimately leads to four kinds of quantity: one pertaining to abstractions, one pertaining to substance, one pertaining to infinite modes, and one pertaining to finite modes. However, again, given Spinoza’s repeated statement that there are only two kinds of quantity, postulating extra kinds of quantity is problematic. Therefore, I believe that the ‘indivisibility’ of real quantity applies to modes just as much as to substance.

To be fair, the passage on quantity does state that when we conceive quantity through the intellect, we conceive it ‘as substance’. This has led many to conclude that Spinoza’s reconception of quantity pertains only to substance. However, conceiving something ‘as substance’ here just means conceiving it through substance. This is indicated by the broader context of Letter 12, where quantity ‘as substance’ is contrasted with ‘Quantity abstracted from Substance’ (IV.56/C I.203). The latter is the abstract conception of quantity in which we ‘separate the Affections of Substance from Substance itself’ and ‘from the way they flow from eternity’ (G IV.57/C I.203). Thus, understanding things ‘as substance’ through the notion of real quantity should be seen as the opposite of ‘separating them from substance’ when conceived through the abstract notion of quantity. With the correct concept of quantity, things are modes of substance rather than independent entities. Consequently, the distinction between the two types of quantity does not reflect a substance-mode divide but rather the contrast between understanding things as modes of substance and misconceiving them as separate from it.

In conclusion, in the context of the passage on quantity, we simply cannot understand ‘indivisible’ as ‘without parts’ without stripping modes of their mereological complexity. For if we then take indivisible quantity to pertain to modes, all modes lack parts, but if we take divisible quantity to pertain to modes, the quantitative and mereological nature of modes falls under abstract quantity. I believe there is only one way out of this conundrum: as Letter 12 says, real, indivisible quantity also applies to modes. But given the clear and widely accepted mereological complexity of modes, the ‘indivisibility’ of real quantity must somehow be reconcilable with such complexity. As we will see now, indivisibility does not entail the complete absence of parts but only the absence of separable parts. In other words, something is ‘indivisible’ insofar as its parts are inseparable. And this applies to modes just as much as to substance.

2.2. Indivisibility as the inseparability of parts

Both in Letter 12 and in E1P15S, the passage on quantity is meant to defend the corporeality of God against those who ‘try to show that corporeal substance is unworthy of divine nature, and cannot pertain to it’ (G II.58/C I.422). Critics of divine corporeality contend that the quantitative properties of extension are incompatible with God’s nature. In Letter 12 and E1P15S, Spinoza reconceives quantity so that God can be quantitative and corporeal. Divisibility is one of the key features of the traditional notion of quantity that needs to be reconceived in order for God to be quantitative and corporeal. Opponents of divine corporeality assume that what is corporeal and quantitative is also divisible. Spinoza counters this by arguing that correctly understood, quantity—and thus corporeality—does not imply divisibility. However, upon examining his arguments, it is clear that he does not reject the general idea that quantity has parts. Instead, he challenges a specific conception of mereology.

In E1P15S, Spinoza argues that objections to God’s corporeality are ‘founded only on the supposition that corporeal substance is composed of parts’ (G II.58/C I.422; my emphasis). Throughout the scholium, he emphasizes that the idea of extension being ‘put together’ (conflare) or ‘composed’ (componere) of parts is problematic.¹⁷ But what exactly does Spinoza mean by ‘composition’?

Spinoza clarifies this concept in the Short Treatise: ‘A thing composed (tezamen gezet) of different parts must be such that each singular part can be conceived and understood without the others’ (KV I ii §19; G I.25/C I.71). In other words, in composition, the parts are independent from the whole (i.e., the other parts), while the whole depends on the parts from which it is constructed. This meaning of ‘composition’ is confirmed in Letter 35: ‘For component parts must be prior in nature and knowledge to what is composed of them whole’ (Ep. 35, G IV.181/C II.27). Spinoza’s specific wording in E1P15S indicates that in order to defend the corporeality of God, he rejects the idea that quantity has the structure of composition. In summary, when Spinoza asserts that God is indivisible, he does not deny the existence of parts, but rather the existence of independent parts that are prior to their whole.

It is also evident from Letter 12 that ‘indivisible’ has the narrower meaning of ‘without independent parts’. In the passage on quantity, where Spinoza argues that the real quantity of God is ‘indivisible’, this argument is directly preceded by a passage clarifying what he specifically opposes:

Hence they talk utter nonsense, not to say madness, who hold that Extended Substance is put together of parts, or bodies, really distinct from one another. This is just the same as if someone should try, merely by adding and accumulating many circles, to put together a square or a triangle or something else completely different in its essence. So that whole array of arguments by which Philosophers ordinarily labour to show that Extended Substance is finite falls of its own weight. For they all suppose that corporeal Substance is composed of parts. (Ep. 12, G IV.55–56/C I.202; my emphasis)

Once again, it is clear that ‘divisibility’, which cannot be attributed to God, refers to a bottom-up mereological structure where the whole is ‘put together of parts’. In this type of structure, the parts are independent from one another and ‘really distinct from one another’. Spinoza here applies Descartes’ theory of distinctions to mereology. For Descartes, a real distinction is a distinction between substances.¹⁸ In other words, as Spinoza writes, elements are ‘really distinct’ when ‘each can be conceived, and consequently can exist, without the aid of the other’ (CM II v, G I.257/C I.323). Therefore, the rejection of really distinct parts (see also TIE §87) amounts to a rejection of the structure of composition in which parts are prior and independent (i.e., substantial). In sum, Spinoza uses the term ‘divisible’ in the narrow meaning of ‘divisible into really distinct parts’, that is, ‘divisible into separable parts’. In that sense, and only in that sense, God is ‘indivisible’, that is, his parts cannot be separated (i.e., divided).¹⁹

The fact that divisibility does not refer to the existence of parts, but more specifically to the existence of separable or ‘really distinct’ parts is also clear from the already mentioned vacuum argument:

For if corporeal substance could be so divided that its parts were really distinct, why, then, could one part not be annihilated, the rest remaining connected with one another as before? And why must they all be so fitted together that there is no vacuum? Truly, of things which are really distinct from one another, one can be, and remain in its condition, without the other. Since, therefore, there is no vacuum in nature (a subject I discuss elsewhere), but all its parts must so concur that there is no vacuum, it follows also that they cannot be really distinguished, i.e., that corporeal substance, insofar as it is a substance, cannot be divided. (E1P15S, G II.59/C I.423; my emphasis)

Separable parts would make possible the absurdity of a vacuum, i.e., the existence of an emptiness within the extended world. Therefore, again, it is clear that Spinoza specifically rejects the existence of these independent and separable parts.²⁰

However, Spinoza endorses another kind of mereology in the passage just cited. He says that parts ‘so concur’ that they cannot be separated. In an early letter, Spinoza says that the parts of matter are so intertwined that ‘if one part of matter were annihilated, the whole of Extension would also vanish at the same time’ (Ep. 4; G IV.14/C I.172). Likewise, in Letter 32, Spinoza defines parts in terms of mutual coherence and agreement: ‘I consider things as parts of some whole to the extent that the nature of the one adapts itself to that of the other so that they agree with one another as far as possible’ (G IV.170a/C II.18; see also Ep. 30, G IV.166/C II.14). The same idea is expressed in the Ethics:

It is impossible that a man should not be a part of Nature, and that he should be able to undergo no changes except those which can be understood through his own nature alone, and of which he is the adequate cause. (E4P4, G II.212/C I.548)

But parthood does not only imply causal dependence on other parts. It also implies a logical or conceptual dependence. Throughout the Ethics, Spinoza notes that we ‘are a part of Nature which cannot be conceived through itself, without the others’ (E4P2; see also E3P3S; E4P4; E4App, G II.266/C I.588). But this logical and ontological dependence on other parts is ultimately a dependence on the whole of which they are a part.

Spinoza’s mereological model can be illustrated with the example of a liver as part of a human body. The liver is not a compositional part because, as Spinoza explains, composition implies the part’s priority over the whole. The liver, however, cannot be what it is, it cannot have its defining function, without the blood, the stomach, and other organs. Thus, while the body mereologically contains its organs, it is not composed of them. I believe Spinoza has such an organicist model in mind when he emphasises the dependence and inseparability of parts.

In summary (see Table 1), Spinoza rejects compositional mereology, where parts are prior to the whole, and instead proposes a mereology in which the whole is logically and ontologically prior to its parts.²¹ Thus, when Spinoza describes God as indivisible, he does not mean that God lacks parts. Rather, he emphasizes that ‘each part of the whole of corporeal substance pertains to the whole substance, and can neither be nor be conceived without the rest of the substance’ (Ep. 32, C II.20). God’s indivisibility thereby reflects the inseparability and interdependence of his parts.

Lastly, the fact that Spinoza’s notion of indivisibility is opposed to a compositional mereological structure, rather than to mereological structure in general, can be inferred from the manner in which Spinoza proves the indivisibility of God in E1P12D, E1P13D, and Letter 35 (G IV.181–2/C II.27).²² These passages reiterate different versions of the same dilemma: if God is divided, the substantial nature is either retained or not. In the first case, substances would result from the decomposition of another substance, meaning ‘so many substances will be able to be formed from one’ (E1P12D). This contradicts the causal isolation (E1P6) and the uniqueness of substance (E1P14). In the second case (i.e., substance is divided into non-substantial parts), substance ‘would lose the nature of substance, and would cease to be’ (E1P12D). This, of course, contradicts the indestructibility of substance (E1P7).

However, the absurdity of both options in this dilemma relies on the compositional model of mereology.²³ The first option is absurd because it assumes that a whole can be decomposed into independent or substantial parts. Likewise, the second option is absurd because it presumes that a whole is composed of parts and thus depends on those parts for its existence. In summary, these absurdities stem from the notion of substance as a composite whole, and it is this specific idea that Spinoza rejects when he proves God’s indivisibility in these passages.²⁴

In conclusion, the passages in Section 1, where Spinoza attributes parts to God, are perfectly consistent with God’s indivisibility. Indivisibility only opposes a mereological structure of separable parts. When parts are seen as interdependent and secondary to the whole, God can be both indivisible and mereologically complex.

2.3. Modal division

A likely objection to my reading of the indivisibility of corporeal substance is that it contradicts Spinoza’s idea that in matter ‘parts are distinguished only modally, but not really’ (E1P15S, G II.59/C I.424). This is often read as a confirmation of the idea that mereological complexity is limited to modes. Schmaltz, for example, understands this mereology of modal distinction as referring to the mereological structure of modal wholes (2021b: chap. 7).²⁵ Conversely, the rejection of real (i.e., substantial) distinction is read as the rejection of mereological structure of substance. The modal/real qualification of a distinction is thus understood to refer to the thing that is divided. Modally distinct parts are thereby taken to be parts of modes; and really distinct parts are then parts of substance.

But Descartes’ theory of distinctions, invoked by Spinoza, tells us something different: a modal distinction is a distinction that holds between modes rather than within a mode.²⁶ So when the parts of matter are modally distinct, this does not mean that the parts are part of a modal whole but that the parts are themselves modes. In short, the modal/real qualification refers to the status of the things that are distinguished from each other: modally distinct parts are modes; really distinct parts are substances. So, in light of Descartes’ theory of distinctions, Spinoza’s claim that there is only modal distinction in matter is a further confirmation of the inseparability (i.e., the non-substantiality) of parts and of the fact that the parts of matter are secondary to their whole.

Although the Cartesian theory of distinction cannot be invoked to prove Schmaltz’s claim that modally distinct parts are parts of a modal whole, neither does it contradict his view. One could concede to my point that modally distinct parts are modes and still argue that such modal parts must compose a modal whole, which would mean that Spinoza’s modally distinct parts are still not parts of God.²⁷ However, in addition to all of the systematic and textual arguments for my idea that modes are parts of God, there is one passage from Metaphysical Thoughts (CM) that shows that Spinoza understands ‘modally distinct parts’ in a manner that contradicts Schmaltz’ reading. Of course, the CM does not represent Spinoza’s own ideas; but it does often clarify the meaning of his terminology—in this case, the meaning that he gives to the Cartesian theory of distinction.

The passage is entitled ‘That God is a most simple Being’ (CM I v, G I.258/C I.324). The simplicity of God is here proven by considering two manners in which God could be composite—for, as Spinoza writes, ‘Whatever is not composed in these first two ways should be called simple’ (G I.258/C I.324). The preceding passage derives these two options for being composed from the Cartesian theory of distinctions: Option one is that God’s complexity involves parts that are ‘really distinct’, that is, ‘substances by whose coalition and union God is composed’ (G I.258/C I.324). It is clear from this that Spinoza, following Descartes, does not understand really distinct parts as ‘parts of a substantial whole’ but rather as ‘substantial parts’. In other words, as I have been arguing, the substantiality of the distinction clearly refers to the substantiality of the parts and not to the substantiality of the whole. Option two is that God’s complexity involves parts that are ‘modally distinct’, which means that there is ‘in God […] a composition from different modes’ (G I.258/C I.324). One could object here that Spinoza’s wording does not make clear that (as I argue) God is the one that is composed of those modally distinct parts. In other words, it might seem that Schmaltz’s idea that modally distinct parts are parts of a modal whole is still not really disproven. But of course, the whole point of the passage is to invalidate the two options for God to be composite. The CM thus treats composition by modally distinct parts as a potential form of God’s complexity, implying that such modes are indeed considered parts of the divine whole. If Schmaltz is right and modally distinct parts are parts of a modal whole, such a form of composition would be wholly irrelevant here. As Schmaltz himself argues (2021b: 251–52), the existence of modal wholes in God leaves his simplicity intact. Therefore, the fact that composition by modally distinct parts is considered a possible threat to God’s simplicity in the CM proves that these parts would be parts of God. Conversely, if ‘really distinct parts’ would mean ‘parts of substance’, then this would be the only possible meaning of God’s complexity, and Spinoza would not have to consider and disprove two options for the complexity of God.

However, the Ethics contains one passage that undeniably supports Schmaltz’s reading. Near the end of E1P15S, Spinoza offers the example of water, which is divided ‘insofar as it is water, but not insofar as it is corporeal substance’ (E1P15S, G II.59/C I.424). I believe this passage is a remnant of the KV. As noted in Section 1, the KV explores three different views on mereology: one passage argues that there are no parts in nature; several others argue that God is a whole composed of his creatures; and still others argue that only modes can have parts. One of the passages in this last group explicitly states that ‘division never occurs in substance’ and then gives the same water example. I therefore believe that the water example in the Ethics is a residue of one of Spinoza’s earlier views. Consequently, the inconsistency concerning mereology present in the KV has not completely disappeared in the Ethics. Nevertheless, as I have argued, the Ethics is still largely consistent with the mereological view of God found in the KV and Letter 32.

3.1. Which infinite can and which infinite cannot have parts

In E1P15S, Spinoza lists a number of paradoxes that result from dividing infinity:

If corporeal substance is infinite, they say, let us conceive it to be divided in two parts. Each part will be either finite or infinite. If the former, then an infinite is composed of two finite parts, which is absurd. If the latter [NS: i.e., if each part is infinite], then there is one infinite twice as large as another, which is also absurd. Again, if an infinite quantity is measured by parts [each] equal to a foot, it will consist of infinitely many such parts, as it will also, if it is measured by parts [each] equal to an inch. And therefore, one infinite number will be twelve times greater than another [NS: which is no less absurd]. (E1P15S, G II.57/C I.442)

As we have seen, E1P15S is meant to defend the corporeality of God against its deniers. One of the arguments invoked against the corporeality of God is based on these paradoxes of infinity. As Spinoza says, ‘since these absurdities follow—so they [his adversaries] think—from the fact that an infinite quantity is supposed, they infer that corporeal substance must be finite, and consequently cannot pertain to God’s essence’ (E1P15S, G II.58/C I.422). Spinoza argues, however, that these paradoxes do not result from the idea of corporeal substance being infinite, but from a flawed conception of quantity: ‘So from the absurdities which follow from that they can infer only that infinite quantity is not measurable, and that it is not composed of finite parts’ (E1P15S, G II.58/C I.423). Recognizing the true quantitative nature of corporeal substance reveals it can be both infinite and divine.

But what is the error in our conception of quantity that leads to the paradoxes of infinity? Spinoza writes that the absurdities result from conceiving infinite quantity to be ‘measurable’ and ‘composed of parts’. Now, in accordance with the common idea that God does not have parts, most commentators have assumed that the error here simply lies in attributing parts to infinite quantity and thus to corporeal substance. However, I believe that things are not so simple. For, as I will try to show here, Spinoza believes that infinity can have parts.

In Letter 12, Spinoza says that we should distinguish different concepts of infinity so that we can understand ‘what kind of Infinite [henceforth ‘Infinity¹’] cannot be divided into parts, or cannot have any parts, and, what kind of Infinite [henceforth ‘Infinity²’] can, without contradictions’ (G IV.53; my translation). The original Latin here reads ‘quale Infinitum in nullas partes dividi, seu nullas partes habere potest; quale vero contra, idque sine contradictio’. Curley translates this as ‘what kind of Infinite cannot be divided into any parts, or cannot have any parts, and what kind of Infinite can, on the other hand, be divided into parts without contradiction’ (C I.201). But the Latin passage does not specify whether Infinity² is opposed to Infinity¹ in terms of its capacity to be divided into parts or in terms of its mere capacity to have parts.

The traditional reading is that Infinity² is divisible, which means that it corresponds to the abstract notion of quantity in the passage on quantity (see Section 2.1). Infinity² is thus taken to be an abstract misconception of infinity (e.g., Nachtomy 2011: 948; Nachtomy 2018: 149; Schmaltz 2021b: 251). However, E1P15S contradicts this interpretation. As I said, Spinoza there argues that the paradoxes of infinity do not disprove the corporeality of God. He says that these paradoxes do not result from the idea of corporeal substance being infinite, but merely from a flawed conception of quantity. This is then followed by the passage on quantity, distinguishing abstract from real quantity. As shown above, the passage on quantity clearly describes abstract quantity as ‘composed of parts’. And in the subsequent paragraph of Letter 12, ‘measure’ is given as a key example of abstract quantity. In other words, there is no denying that when Spinoza writes that the paradoxes show that we cannot conceive infinite quantity to be ‘measurable’ and ‘composed of parts’, he argues that the paradoxes follow from conceiving infinity through the abstract notion of quantity. Therefore, when Spinoza says that Infinity² is capable of having parts ‘without contradiction’, this cannot refer to an abstract conception of infinity because, as he argues in E1P15S, this abstract notion of quantity precisely causes paradoxes when applied to infinity. This also means that, contrary to Curley’s translation, Infinity² cannot be opposed to Infinity¹ in terms of its divisibility. The opposition to which Spinoza refers must be between the inability to have parts of Infinity¹ and the ability to have parts (without contradiction) of Infinity².

In sum, given the fact that Infinity² cannot be the abstract notion of infinity, it must be the notion of infinity that Spinoza defends (i.e., infinity conceived in terms of the real notion of quantity). In fact, as we will see, the rest of Letter 12 confirms that he understands infinity to have parts ‘without contradiction’.

Letter 12’s idea of an infinite that allows parts likely draws from Hasdai Crescas’ defence of actual infinity. We know that this author was on Spinoza’s mind when he wrote the letter because near the end of the text, he explicitly refers to Crescas’ cosmological proof of God (G IV.61–62/C I.205). One of Crescas’ main philosophical aims was the vindication of actual infinity. His strategy for this was to resolve the paradoxes of infinity, on the basis of which Jewish and Arabic Aristotelians have refuted actual infinity. Like Crescas, Spinoza opposes this traditional Aristotelian rejection of actual infinity. In the letter, Spinoza rejects those who, because ‘they were ignorant of the true nature of things, denied an actual Infinite’ (G IV.59/204). And like Crescas, his defence of actual infinity relies on resolving the paradoxes of infinity, which he refers to as ‘great crowd of difficulties’ (G IV.53/C I.201) that make up ‘the problem of the Infinite’ (G IV.53/C I.201; G IV.61/C I.205). As shown, the central element in these paradoxes is the contradictions that result from partitioning infinity. Therefore, Infinity², which can have parts ‘without contradiction’, belongs to Spinoza’s attempt to solve the paradoxes of infinity by reconceiving infinite quantity.

3.2. A quantity is not infinite because of the multiplicity of its parts

A detailed analysis of the rejected ‘abstract’ notion of (infinite) quantity in Letter 12 can provide additional evidence for the fact that Spinoza’s defended notion of infinity allows parts.

Spinoza identifies three examples of abstract quantity: number, measure, and time. Number is the foundational concept from which the others derive. Measure expresses continuous quantity in terms of numbered parts, or units of measurement, while time measures duration. Thus, both measure and time reduce quantities to numerical terms (CM I i, G I.234/C I.300). In summary, number represents the most basic form of abstract quantity. Consequently, the abstract notion of infinity, criticized in Letter 12, is fundamentally a numerical conception.³⁰

Spinoza invokes the judgement of ‘mathematicians’, without giving names, who have perceived ‘clearly and distinctly’ the inaptitude of number to grasp infinity:

For not only have they discovered many things which cannot be explained by any Number—which makes quite plain the inability of numbers to determine all things—they also know many things which cannot be equated with any number, but exceed any number that can be given. Still they do not infer that such things exceed every number because of the multiplicity of their parts, but because the nature of the things cannot admit any number without a manifest contradiction. (Ep. 12, G IV.59/C I.204)

Infinity cannot be expressed by any number. As Spinoza states, infinity ‘exceed[s] every number’. This is obvious. However, the numerical conception of infinity targeted by Spinoza also includes the idea that infinity ‘exceed[s] every number because of the multiplicity of its parts’. This resembles what we now call a ‘countable’ or ‘denumerable’ infinity: a countable series of elements exceeding any finite number. Although no single number can express it, this multiplicity is still numerical because it shares the countable structure of numbers. Here, infinite quantity differs from finite quantity not in structure but in being endless. This corresponds to Spinoza’s description of infinity as ‘what is called infinite because it has no limits’ (Ep. 12, G IV.53/C I.201). But, as Spinoza argues, such a conception remains vulnerable to paradoxes, or ‘manifest contradiction’.

In the paragraph immediately following this statement, Spinoza illustrates this ‘manifest contradiction’ through his example of the non-concentric circles:

For example [see Figure 1], all the inequalities of the space between two circles, A and B, and all the variations which the matter moving in it must undergo, exceed every number. That is not inferred from the excessive size of the intervening space. For however small a portion of it we take, the inequalities of this small portion will still exceed every number. Nor is it inferred because, as happens in other cases, we do not know its maximum and minimum. For we know both in this example of ours: AB is the maximum and CD is the minimum. Instead it is inferred simply from the fact that the nature of the space between two non-concentric circles does not admit anything of the kind. So if anyone should wish to determine all those inequalities by some definite number, he will, at the same time, have to bring it about that a circle is not a circle. (Ep. 12, G IV.59–60/C I.204)³¹

Figure 1

Geometrical drawing in Ep. 12.

This example clearly aims to demonstrate that, as Spinoza asserts, the infinite inequalities between the two circles are ‘not inferred from the excessive size of the intervening space’. In other words, this falsifies the definition of infinity in terms of the absence of limits. However, at first sight, the example does not discredit the idea of infinity as a denumerable set of elements. In other words, the limited space between the two circles still seems to be infinite ‘because of the multiplicity of its parts’. Both Leibniz (A VI iii.281/LC 113) and Tschirnhaus make this objection. The latter addresses this issue in his correspondence with Spinoza:

And in the example of the two circles which you use there, you do not seem to show what you had undertaken to show. For there you show only that they do not infer this from the excessive size of the intervening space, and that they do not infer it from the fact ‘that we do not know its maximum and minimum’. But you do not demonstrate as you wanted to, that they do not infer it from the multiplicity of the parts. (Ep. 80, G IV.331/C II.484)

Spinoza replies:

What I said in my Letter concerning the Infinite, that [Mathematicians] do not infer the infinity of the parts from their multiplicity, is evident from the fact that if they inferred it from their multiplicity, we could not conceive a greater multiplicity of parts, but their multiplicity would have to be greater than any given multiplicity, which is false. For in the whole space between two circles having different centres we conceive twice as great a multiplicity of parts as in half of the same space. Nevertheless, the number of parts, both in the half and in the whole space, is greater than every assignable number. (Ep. 81, G IV.332–333/C II.484–485)

The idea that a quantity is infinite due to the multiplicity of its parts assumes that this multiplicity is endless—without limits. In this conception, infinite quantity differs from finite quantity not by structure but by being an unlimited, unexceedable countable series. However, this creates a contradiction: take half the space between two circles. This space contains an infinite quantity of inequalities and is considered infinite because its multiplicity is unexceedable. Yet, as this quantity is only part of the complete space between the circles, it is exceeded by the multiplicity contained in the full space. Hence the contradiction.

According to Spinoza, the source of this issue lies in the fact that, in this conception of infinity, quantity is still defined by the multiplicity of its parts. He does not reject the idea that infinite quantities have an innumerable multitude of parts. However—and this is crucial—according to Spinoza, the infinitude of a quantity does not result from this multiplicity. This is what he claims ‘the mathematicians’ know:

They also know many things which cannot be equated with any number, but exceed every number that can be given. Still they do not infer that such things exceed every number because of the multiplicity of their parts, but because the nature of the thing cannot admit number without manifest contradictions. (Ep. 12, G IV.59/C I.204; my emphasis)³²

What Spinoza says here is that infinite quantities should be understood to be infinite by their nature and not by the multiplicity of their parts. Although their infinite nature does entail an innumerable amount of parts, quantity is not defined by that. It is not infinite because of that.³³

Let us connect this to E1P15S and Spinoza’s rejection of composition. Addressing the paradoxes of infinity, Spinoza writes: ‘from the absurdities which follow from that they can infer only that infinite quantity is not measurable, and that it is not composed of finite parts’ (E1P15S, G II.58/C I.423). As we have seen, measure, a derivative of the numerical conception of quantity, involves dividing continuous quantity into units of measurement and numbering them. Therefore, Spinoza’s claim that infinite quantity is not measurable implies that it is not numerical. Number, as the most fundamental form of abstract quantity, is ‘composed of parts’. This compositional structure—a quantity defined by a multiplicity of parts—is precisely what Spinoza rejects for infinity. The paradoxes of infinity arise not from conceiving infinite quantity to have parts, but from assuming it has a numerical, measurable structure, which is infinite ‘because of the multiplicity of its parts’. This rejection aligns with Spinoza’s broader critique of divisibility and bottom-up mereology. Parts are neither logically nor ontologically prior to their whole. Infinite quantities are not defined or explained by their multiplicity of parts because they are not composed of them. In this way, E1P15S is perfectly consistent with the theory of infinity in Letter 12.

One might object that the idea that infinity has parts, even if it is not composed or defined by them, does not completely invalidate the paradox of infinity. Let us look again at the paradox in its most simple form:

If corporeal substance is infinite, they say, let us conceive it to be divided in two parts. Each part will be either finite or infinite. If the former, then an infinite is composed of two finite parts, which is absurd. If the latter [NS: i.e., if each part is infinite], then there is one infinite twice as large as another, which is also absurd. (E1P15S, G II.57/C I.442)

As discussed earlier, Spinoza argues that this paradox results from conceiving infinity through the abstract notion of quantity. According to my reading, Spinoza thereby rejects the idea that infinity is divisible or composed of parts, but not the idea that infinity has parts. A large part of the paradox is nullified when infinity is not divisible or composed. However, one could argue that the mere idea of infinity having parts still causes some absurdity. Take the example of the circles: even if infinity is not divisible into parts nor composed of then, the mere fact that the infinity of inequalities in half of the space between the two circles is part of the infinity of inequalities in the complete space still means that ‘there is one infinite twice as large as another, which is also absurd’ (E1P15S, G II.57/C I.442). Therefore, one could wonder, does the paradox of infinity not also invalidate my reading of Letter 12 as a defence of an infinite with parts? But this part of the paradox assumes that unequal infinities are absurd. Spinoza, however, rejects this assumption. This is something that he probably learned from Crescas’s response to al-Tabrizi’s proof for the impossibility of actual infinity. The proof involves two infinite lines. One line is infinite in both directions; the other line is infinite in one direction and limited in the other. This is meant to show that the idea of (corporeal) actual infinity leads to the absurdity of one infinite being greater than another. But Crescas argues that it is only impossible for one infinite to be greater than another when this quantity is taken to be measurable:

The impossibility of one infinite being greater than another is true only with respect to measurability, that is to say, when we use the term greater in the sense of being greater by a certain measure, and that indeed is impossible because an infinite is immeasurable.³⁴

Crescas then continues to argue that although the lines cannot be said to be greater or smaller in terms of measurability, the first infinite line does ‘extend beyond’ the other infinite line.

Spinoza’s response to the paradoxes of infinity thereby resembles the solution proposed by Crescas and others.³⁵ Spinoza identifies the paradoxes as arising from applying measure to infinite quantity: ‘So from the absurdities which follow from that they can infer only that infinite quantity is not measurable’ (E1P15S, G II.58/C I.423). Measure, as shown above, is a form of abstract quantity. But by abandoning the idea of measure and adopting the correct conception of quantity, Spinoza argues that ‘one infinite can be conceived to be greater than another Infinite’ (Ep. 12, G IV.53/C I.201). This is reaffirmed at the letter’s conclusion, where Spinoza explains, ‘if things cannot be equated by a number [i.e., when they are infinite], it does not follow that they must be equal’ (G IV.61/C I.205). The fact that one infinite is larger than another, if the first mereologically contains the latter, is thereby no longer an absurdity. Therefore, infinity, rightly conceived, can have parts ‘without any contradiction’.

In sum, when Spinoza writes that God has parts, this is not inconsistent with the infinity of God. Infinity is only inconsistent with a countable, bottom-up mereology that we find in numbers. Therefore, the interpretation that Spinoza’s God has parts, supported by the textual evidence presented in Section 1, cannot be refuted by the argument that mereological complexity contradicts God’s infinity.

4.1. The perfect form of passibility

In his response to the Cartesian imperfection argument, Spinoza argues that the passibility implied by extension, even if we assume it to be divisible, does not pose a problem to the idea of a corporeal God:

I do not know why [matter] would be unworthy of the divine nature. For (by P14) apart from God there can be no substance by which [the divine nature] would be acted on. All things, I say, are in God, and all things that happen, happen only through the laws of God’s infinite nature and follow (as I shall show) from the necessity of his essence. So it cannot be said in any way that God is acted on by another, or that extended substance is unworthy of the divine nature, even if it is supposed to be divisible, so long as it is granted to be eternal and infinite. (E1P15S, G II.60/C I.424)

The passibility implied by corporeality does not pose a threat to God’s imperfection because there is no outside actor to act on God. As Spinoza explains in the Short Treatise, so long as God is not acted on by an external actor, passibility is not an imperfection:

[B]eing acted on, when the agent and the one acted on are different, is a palpable imperfection. For the one acted on must necessarily depend on what, outside him, has produced this state of being acted on. In God, who is perfect, this cannot happen.

Further, one can never say of such an agent, which acts in himself, that he has the imperfection of being acted on, because he is not acted on by another; similarly, the intellect, as the Philosophers also say, is a cause of its concepts. But since it is an immanent cause, who would dare say that it is imperfect when it is acted on by itself. (KV I ii §§23–24, G I.26/C I.72)

In other words, since God is a causa immanens, his passibility does not amount to imperfection. Being acted on only constitutes imperfection if it is a form of dependence, that is, if the actor is an external thing.³⁸

These two passages together suggest that, for Spinoza, the corporeality of God involves a form of passibility that is not an imperfection because there is nothing outside God. Now the true cause of this passibility must be God’s mereological structure, for this has been the target of the Cartesian imperfection argument. Therefore, Spinoza seems to argue that although the mereological complexity of corporeal substance entails passibility, it is a perfect form of passibility (i.e., being acted on by itself). So, the mereological structure of God constitutes an action of God on himself. This is more explicitly confirmed in a passage already cited in Section 1, which again uses the intellect as an example:

For example, the intellect is the cause of its concepts; that is why I called the intellect a cause (insofar as, or in the respect that its concepts depend on it); and on the other hand, I call it a whole, because it consists of its concepts. Similarly, God is, in relation to his effects or creatures, no other than an immanent cause and also a whole, because of the second consideration [i.e., because he consists of his creatures]. (KV I 1^st dialogue §12, G I.30/C I.76)

The concepts of the intellect are its mereological parts (see also E2P11C; E2P15). However, this mereological complexity attests to the activity of the intellect; for the intellect causes the concepts it consists of. The passibility entailed by its mereological complexity is thus a perfect form of passibility, that is, being acted on by itself. And as Spinoza writes here, it is precisely in this sense that God is the whole of its creatures. In sum, God acts on himself, and causes himself, by mereologically complexifying himself.

As I wrote in the Introduction, I believe that the mereological reading of the substance-mode relation thereby offers a crucial element for making sense of Spinoza’s theory of self-causation. Self-causation is not mere conceptual self-containment. It is not being conceived through itself. Nor is it mere formal causation. Instead, it is efficient causation (Lærke 2013: 68–70; Melamed 2021: 121–23). God produces himself. But this production process seems incomprehensible if it is not identical to immanent causation, that is, the production of modes. As E1P25S says, God is the cause of himself ‘in the same sense’ that he is the cause of all things. Therefore, as Lærke writes, ‘the divine cause causing effects is in fact just the divine cause causing (modifications of) itself’ (2013: 71). However, I believe that one element is missing to make this case: for God to produce himself through the production of modes, he must consist of these modes. If God would be a bare substratum, underlying its modes, God would not be able to produce himself by producing modes.³⁹ There is much more to be said on this topic, but I will save this for another text.

4.1. How mereological complexity attests to power and activity

In the traditional bottom-up model, parts are independent of each other and prior to the whole. The whole is then dependent on the parts. But in Spinoza’s reversed mereological model, parts are dependent on each other, that is, they depend on the whole, and, as Spinoza argues in Letter 32, the whole now enjoys independence.⁴⁰

God not only enjoys independence by being a whole, his mereological structure is also an expression of his activity and power. Again, the key is to grasp that parts are not compositional elements. Instead, Spinoza understands them as variations or modifications of the whole.⁴¹ In E1P15S, Spinoza writes that ‘parts are distinguished in it [matter] only insofar as we conceive matter to be affected in different ways’ (G II.60/C I.424). In Letter 32, he writes that ‘every body, insofar as it exists modified in a definite way, must be considered as a part of the whole universe’ (G IV.173a/C II.19; my emphasis). In other words, it is by being affections or modifications⁴² of nature that things are parts of nature.

Many passages support the equivalence of parts and modes. Spinoza often equates ‘being part of (the intellect or power or love of) God’ with ‘being God not insofar as he is infinite, but insofar as he can be explained through the human mind (or the human essence)’ (see E2P11C; E4P4D; E5P36; E5P36D). But ‘being God not insofar as he is infinite but insofar as he is explained through X’ is also one of Spinoza’s standard ways of describing X as a mode of God. In sum, there seems to be a conceptual equivalence between ‘being a part of Y’ and ‘being a mode of Y’. A very straightforward example is this: Spinoza both writes that ‘the essence of man is constituted by certain modifications of God’s attributes [i.e., his essence]’ (E2P10C; my emphasis) and that ‘man’s power, […] insofar as it is explained through his actual essence, is part of God or Nature’s infinite power, that is (by IP34), of its essence’ (E4P4D; my emphasis). This, again, suggests that in Spinoza’s conceptual apparatus ‘being part of God’s essence’ is equivalent to ‘being a modification of God’s essence (or attribute)’.⁴³

In short, instead of understanding partition as dissection, it should be understood as a form of internal differentiation or modification. To explain this with an image, for Spinoza mereological complexity does not have the structure of a wall built from bricks but of an embryogenic cell differentiating itself through self-complexification. As such, mereological complexity attests to a form of power.

As parts are understood as modifications or internal differentiations of the whole, the mereological complexity of nature expresses God’s modifiability. But God’s power is his power to create modes, that is, his power to modify himself. In Letter 32, Spinoza writes that because ‘[nature] is absolutely infinite, the variations of its parts which can follow from this infinite power must be infinite’ (C II.20).⁴⁴ In other words, the endless mereological complexity of nature expresses the infinite power of God.⁴⁵ And each quantum of nature participates in this divine infinity by being itself infinitely modifiable and having infinite mereological complexity.⁴⁶

As Spinoza explains right after the passage on the two non-concentric circles in Letter 12, this correspondence between mereological complexity and power is why the finite mereological complexity of nature would contradict God’s power:

Similarly, to return to our theme, if someone should wish to determine all the motions of matter there have been up to now by reducing them and their Duration to a definite number and time, he will certainly be striving for nothing but depriving corporeal Substance (which we cannot conceive except as existing) of its Affections and bringing it about that it does not have the nature which it has. (G IV.60/C I.204)

As the example of the circles indicates (see note 31), Spinoza here follows Cartesian tradition and understands the complexity of matter in terms of motions in the material plenum. If we would limit those motions partitioning matter to a definite (i.e., finite) number, we would limit the modifiability of nature. But, again, since God’s power is his power to create modes, limiting the modifiability of nature amounts to limiting the power of God.

So, while mereological complexity, in Spinoza’s theory, does in fact attest to a form of passibility—namely, undergoing the action of complexification—it constitutes a ‘being acted on by itself’, which is a form of perfection and power rather than imperfection and passivity. Therefore, God’s perfection does not forbid him from having parts.