In Book I of the Ethics, it is Spinoza's intention to explain reality, i.e. everything there is, as a part of one vast interconnected system, which may be variously called substance, God, or Nature. That is to say, that nothing may be adequately understood except as it is understood to be a part of this entire system. After explaining Spinoza's conception of substance, attribute, modification, and, most importantly, his misconception of infinity, as well as his argument that there can be only one substance, I will argue that in order to remain consistent with these conceptions, Spinoza must deny that there really are any finite things, and must instead maintain that the commonly held belief that there are finite things is a result of trying to understand reality by means of the inferior imagination, rather than understanding things by means of the intellect. Fortunately for Spinoza, I will further argue, he is in no way ontologically committed to the existence of finite things, as his observations about finite modes are couched in hypothetical terms. Spinoza, of course, must then pay a steep price for consistency, namely the implausibility of his entire enterprise.
Spinoza defines substance as "that which is in itself and can be conceived through itself" (def. III). Spinoza's use of the word 'can' is somewhat misleading here, for it turns out (proposition II) that not only can substance be conceived thorough itself, it must be conceived through itself. This definition of substance is quite at odds with more traditional conceptions of substance as found in Aristotle, Descartes, and Leibniz, for whom substances are those things of which any property may be correctly predicated. So different is Spinoza's conception of substance, it is a legitimate question whether Spinoza is talking about the same thing by his use of 'substance' that Aristotle and Descartes are talking about by use of the word 'substance'. According to the traditional conception of substance, since it is true that Fido is a dog, it is also true that Fido is a substance. This is not so according to Spinoza. For to understand a thing, i.e. adequately conceive of it, we must understand its cause (axiom III). Thus to understand Fido, we must understand the cause of Fido, at least part of which is Fido's parents. Fido cannot be conceived through himself, so Fido cannot be a substance.
What, then, is Fido, given that he is not a substance? To answer this, Spinoza defines another type of being: a modification, or mode, of substance, which he defines as "that which is in another thing, through which also it is [can be, must be] conceived." Since an adequate conception of Fido involves a conception of another thing, viz. Fido's parents, Fido cannot be a substance. And by axiom I, which states that everything is either in itself (i.e. can/must be conceived through itself) or in another (i.e. can/must be conceived through another), Fido must be in another, i.e. another thing through which Fido must be conceived. That is to say, Fido is a modification of substance.
Spinoza's notions of being in something and being conceived through something, are, if not entirely synonymous, at least necessarily coextensive. What Spinoza means both by being "in" something and being "conceived through" something can best be illustrated by an analogy which he made in a letter to Henry Oldenburg. Imagine, Spinoza asks Oldenburg, that there is a parasitic worm living in the bloodstream, which can see each of the particles of lymph and chyle which make up the bloodstream. This worm conceives of each of the particles of the blood as a whole, particular thing, and fails to see the relationships which hold between the particles in blood and which determine the motions of each particle. This worm is in the same position as those who think that Fido is a substance, and do not understand that relationship which holds between Fido and his parents, and with all the other dogs which are Fido's ancestors. The worm will fail to understand adequately any individual particle of blood, for it cannot see the whole of which each individual is a part, i.e. is in. If something is in another thing, i.e. is causally determined by the system as a whole, it cannot be understood by itself, i.e. conceived through itself. The contrapositive of this claim is that if anything can be conceived through itself, it must not be in another thing, i.e. part of a greater system. And if it is not a part of some greater system, i.e. in something else, there is no way to understand it, i.e. conceive of it, except through itself.
A third important term in Spinoza's system is 'attribute', which he defines as "that which the intellect conceives of a substance as constituting its essence." He mentions only two attributes, thought and extension, although he maintains in definition VI that God has an infinite number of attributes. A substance may have more than one attribute, i.e. there may be more than one way that the intellect may adequately understand that substance. An attribute might be likened to the twentieth century idea of a conceptual scheme, 1 to which the truth of sentences or propositions are relative, and outside of which the notions of truth and meaning make no sense. This comes out clearly in proposition X, in which he proves that "each attribute of a substance must be conceived through itself," and in the note to proposition X: "It is very far from being absurd, therefore, to ascribe to one substance a number of attributes, since nothing in Nature is clearer than that every substance must be conceived under some attribute. . . ." Under the attribute of thought, God will be seen to be an infinite thinking thing, and under the attribute of extension, God will be seen to be an infinitely extended thing (proposition XIV, corollary 2). For Descartes, this would be a complete contradiction, as he maintains that whatever is thinking is necessarily non-extended, and that which is extended is necessarily non-thinking. But for Spinoza there is no contradiction, for saying that God is an infinite thinking thing and that God is an infinitely extended thing are merely two different ways of saying the same thing. Thus Spinoza manages to avoid the mind-body problem which plagued the Cartesians, without postulating some sort of mysterious interaction between extended, corporeal substance and non-extended, mental substance, or resorting to some sort of parallelism thesis, according to which there is no causal interaction between mind and body, merely two systems which act in parallel, like two synchronized watches.
A fourth important notion in Spinoza's system, the one which must ultimately cost him either the consistency of his system or its plausibility, is his notion of infinity. Spinoza holds three things true of infinity that, on our contemporary conception of infinity, derived from Cantor, are mistaken. First, Spinoza holds that if something is infinite "in its own kind," there can be nothing greater of that kind. He builds this right into his definition of finitude (def. II): "That thing is called finite in its own kind which can be limited by another thing of its own nature. For example, a body is called finite because we always conceive of another which is greater." Our contemporary Cantorian conception infinity allows, in fact requires, that for every infinity, there is a greater infinity. For example, the number of integers, although infinite, is less than the infinite number of real numbers.
The second thing Spinoza holds of infinity is that an infinite quantity cannot be divided into parts. He asserts this to avoid certain "paradoxes" of infinity. One of the puzzles (note to proposition XV) runs thus: suppose we have something which is of infinite quantity, such as the set of all integers, and that we divide it into two parts, the set of all natural numbers and the set of all negative integers. Either these parts are finite, or they are infinite. If they are each finite, then a infinite quantity is composed of two finite quantities, which is absurd. If they are both infinite, then there is an infinite quantity which is twice as great as another infinite quantity, which is also absurd. 2 Having the benefit of Cantor's insight, we now see that there is nothing absurd in this second option: an infinite quantity (e.g., aleph0, the cardinality of the set of natural numbers) plus an equal infinity yields the very same infinite quantity. For Spinoza, however, this seemed to be a paradox.
A second puzzle (also in the note to proposition XV), related to the first, runs as follows. Suppose we have an body of infinite extension. We may measure it in feet, in which case the body will be an infinite number of feet long. We may also measure it in inches, in which case the body is an infinite number of inches long. But it seems to follow that there is one infinite quantity (the number of inches) that is twelve times as great as another infinite quantity (the number of feet). Today this is seen as unproblematic, but this seemed paradoxical to Spinoza and his contemporaries. These puzzles drove some of Spinoza's contemporaries to deny that there could even be an infinitely extended body, and thus to deny that extension is an attribute of God, for it seemed necessary that any extended body could be measured, in inches, feet, or miles. These seeming paradoxes even drove some philosophers to deny that there was even such a thing as infinity. But neither of these ways out of the seeming paradoxes is open to Spinoza. He cannot take the first way out, as he wishes to attribute extension to God (which, of course, must be infinite extension, for otherwise by definition II there would something greater than God) and thus avoid the mind-body interaction problem. And the second way out would be even worse for Spinoza, for the concept of infinity figures in his very definition of God, and in proposition VIII he purports to prove that substance is necessarily infinite. If there were nothing infinite, there could be no substance, on Spinoza's view, which would be the height of absurdity, for then there could be nothing at all, since everything is either a substance or a modification of substance.
Spinoza's way out of these seeming paradoxes is to deny that something which is infinite has any parts whatsoever: ". . . Indeed it is not less absurd to suppose that [infinite] corporeal substance is composed of bodies or parts than to suppose that a body is composed of surfaces, surfaces of lines, and that lines, finally are composed of points" (note to proposition XV); "No attribute of substance [e.g. extension] can be conceived from which it follows that substance can be divided" (proposition XII); "Substance absolutely infinite is indivisible" (proposition XIII). As we shall see, this way in which he gets out of having to deny infinite quantities will lead him into trouble about explaining the existence of finite things.
The third incorrect assumption which Spinoza makes about infinity is that infinity is identical with totality. On Spinoza's view, if I had an infinite number of sheep in my back yard, then there could not be any sheep outside my backyard. This view is, in all likelihood, prompted by the same sort of seeming paradox which motivated his view that infinity quantities do not have parts. For if there could be something of a given kind outside of an infinite set of objects of that kind, there would be one infinite quantity greater than another. The infinite number of sheep in my back yard would be less than the number of the sheep in my back yard plus all the sheep not in my backyard. This, once again, is not seen as a paradox at all today, in light of Cantor's insights. This equating of infinity with totality is crucial to Spinoza for his proof in proposition XIV that there cannot be more than one substance, but it will lead him ultimately to face the problem of having to deny that there are any finite things, lest his system be inconsistent.
Given the geometrical style of Spinoza's system, it will be best to go through the steps which lead Spinoza into his quandary about finite things, although, of course, not in as much detail as Spinoza presents. Propositions I through VII constitute Spinoza's version of the cosmological argument that substance necessarily exists, i.e. that every substance is cause of itself, and that its existence involves essence (these are synonymous by definition I). Proposition VIII proves that every substance is necessarily infinite, according to his definition of infinity (def. II), according to which if something is infinite if and only if it cannot be limited by something of its own kind, i.e. if there could not be something greater of that kind. Proposition IX is Spinoza's version of the ontological argument, used to prove that not only is there a substance which necessarily exists (proposition VII), but that God, which he defines (def. VI) as a substance consisting of infinite attributes, exists. The argument used to prove this proposition is, of course, heir to all the problems of Anselm's original ontological argument. For just as Gaunilo "proved" that the greatest conceivable island exists in reality, using reasoning precisely parallel to the reasoning Anselm had used to "prove" that the greatest conceivable being, i.e. God, exists in reality, one could produce an argument precisely parallel to Spinoza's argument in proposition IX, purporting to show that there is a substance with exactly one attribute, or exactly two attributes, etc., rather than a substance with an infinite number of attributes.
Proposition XIV is Spinoza's proof that God is the only substance, in which he first uses the mistaken principle that infinity is identical with totality: ". . . God is a being absolutely infinite, of whom no attribute can be denied which expresses the essence of substance. . . ." God is, by definition, a being with infinite attributes, which he has already "proved" to exist in proposition IX, so, by conflating infinity and totality, God is a substance having all attributes there are. Since in proposition V Spinoza proved that there could not be two distinct substances possessing the same attribute (for if there were, they could be conceived through one another, contrary to definition III), and God has all attributes, it follows that God must be the only substance. Proposition XV states that "Whatever is, is in God, and nothing can be conceived without God." The argument for this proposition shows clearly what I above explained via Spinoza's worm-in-the-bloodstream analogy, that Spinoza regards 'being in' and 'being conceived through' as, if not synonymous, at least necessarily coextensive, for throughout the argument the relations being in and being conceived through are used in conjunction.
With proposition XVI, we begin to see the beginnings of Spinoza's problem about explaining finite modes:3 "From the necessity of the divine nature infinite numbers of things in infinite ways (that is to say, all things which can be conceived of by the divine intellect) must follow" [italics added]. Here we see again Spinoza's characteristic conflation of an infinite number of things with all things. Also in this proof of this proposition, Spinoza departs from the strict geometrical method, for the only previous definition, axiom, or proposition, to which he appeals is definition VI, his definition of God as a substance having an infinite number of attributes. (It is these attributes, it seems, that the "infinite number of ways" follow from.) Instead of appeal to previous definitions, axioms, or propositions, Spinoza appeals to a principle that from a definition of a thing a number of properties follow. Thus, he reasons, since God is defined as absolutely infinite, an infinite number of things (i.e. all things) must follow. It is important to note that nowhere in the Ethics does Spinoza make any distinction between logical implicature and causal implicature. 4 It seems that this relation of following from, which holds between God and all things ("conceived of by the divine intellect"), includes both logical and causal implicature.
And with propositions XXI and XXII, the problem of finite modes becomes clear. Proposition XXI states that "all things which follow from the absolute nature of any attribute of God must forever exist, and must be infinite, that is to say, through that same attribute they are eternal and infinite." The proof of this proposition is unquestionably the most convoluted proof in all of book I of the Ethics, and suffice it to say that I do not understand it fully. The 'eternal' part of the proposition, if 'eternal' is understood as necessary existence (def. VIII), may be understood as a straightforward theorem of modal logic: That which is entailed by a necessary truth is itself necessary. The proof that only infinite things can follow from an attribute of God, on the other hand, has no straightforward translation, as far as I know, into any system of modern logic. Proposition XXII is so similar to proposition XXI that one why Spinoza did not merely present it as a corollary: "Whatever follows from any attribute of God, insofar as it is modified by a modification which through the same attribute exists necessarily and infinitely, must also exist necessarily and infinitely." Unfortunately, the demonstration of proposition XXII is of no help in understanding the demonstration of proposition XXI, for we are merely referred back to the previous demonstration: "This proposition is demonstrated in the same manner as the preceding proposition."
Despite my inability to give a coherent account of the demonstrations of propositions XXI and XXII, it should now be clear what problem Spinoza has with finite modes. Since according to propositions XXI and XXII, all things which follow from an attribute of God are infinite, we may conclude that finite modes do not follow from an attribute of God. But according to proposition XVI, all things conceived of by the divine intellect follow from "the necessity of the divine nature." So either finite things are not conceived of by the divine intellect, which would seem to be an unacceptable limitation on the divine intellect, or finite things somehow follow from the necessity of the divine nature without following from any attribute of God.
The second horn of the dilemma, however, seems equally untenable, for everything must be conceived under some attribute or other (note to proposition X). How could it be possible that a finite thing be conceived through an attribute if it does not follow from that attribute, for by axiom IV we know that the knowledge of an effect (in this case, the finite thing) depends upon knowledge of the cause, i.e. what it follows from (given that Spinoza makes no distinction between logical and causal implicature)? Furthermore, it would seem difficult to draw this distinction between the necessity of the divine nature and the attributes of God, maintaining that finite things follow from the latter and not the former. For it would seem possible to construct an argument parallel to that of proposition XVI, showing that from an infinite attribute infinite numbers of things (i.e. all things which fall under that attribute) must follow. Since finite modes must fall under some attribute, finite modes must then follow from that attribute. Equally, it would seem that a proof parallel to that of XXI and XXII could be constructed, showing that anything which follows from God, or "the necessity of the divine nature," being an infinite and eternal substance, must itself be infinite and eternal.
My suggestion will be that Spinoza's only consistent way out of this problem will be to grasp the first horn of the dilemma, and that he may legitimately do this without placing any limits on the divine intellect. But first, we should look at proposition XXVIII, the only place where Spinoza deals directly with finite modes.
In proposition XXVIII, Spinoza shows that a finite mode must follow from another finite mode, which in turn must follow from another finite mode, and so on, ad infinitum. This is so because finite modes cannot, by propositions XXI and XXII, follow from any infinite mode. This infinite causal chain might itself be considered to be an infinite mode (after all, it's infinite, and it's not a substance, so it must be a mode), but then from this infinite chain of causes none of its finite constituents can follow. Furthermore, this infinite chain cannot even be considered to have these finite causes as parts, for Spinoza holds that nothing infinite has parts (note to proposition XV). And since substance is prior to its modifications (proposition I), including its finite modifications, we are left without any explanation for the existence of finite modifications. Spinoza's only consistent way out of this problem, it seems, is to deny that there really are any finite modifications. Thus he may take hold of the first horn of the dilemma without placing any limits on the divine intellect, for if there are no finite modes, it is not a problem that the divine intellect cannot conceive of them. Indeed, if there are no finite modes, the divine intellect had better not conceive of them, lest it be mistaken. Rather than explaining finite modes, Spinoza must explain them away.
Denying that there are any finite modes does not, I believe, result in any inconsistency for Spinoza's system, at least insofar as book I of the Ethics is concerned. For as I have said, the only proposition which directly deals with finite modes in book I is proposition XXVIII, and I think that this proposition could be consistently regarded as a hypothetical one, concerning only how finite modes would have to be produced if there really were any such things. 5 Spinoza is no more ontologically committed to the existence of finite things in proposition XXVIII than I am committed to the existence of unicorns by my holding true the sentence 'All unicorns are white'. Given the non-existence of unicorns, this sentence is true vacuously. I might even hold true a stronger sentence, the counterfactual claim 'If there were any unicorns, they would be white', without committing myself to the existence of unicorns. 6
If, however, Spinoza chose to deny that there are any finite modes (he does not, of course, do this explicitly), he would then owe us an account of why it is that just about everyone thinks that there are finite things. We do not have to look far for an answer. In the note to proposition XV, he must do something quite similar, viz. explain why it is we think that all quantities, even infinite quantities, can be divided:
If, nevertheless, anyone should ask why there is a natural tendency to consider quantity as capable of division, I reply that quantity is conceived by us in two different ways: either abstractly and superficially, that is to say, as we imagine it, or else as substance, in which way it is conceived by the intellect. If, therefore, we regard quantity (as we do very often and easily) as it exists in the imagination, we find it to be finite, divisible, and composed of parts; but if we regard it as it exists in the intellect, and conceive it in so far as it is substance, which is very difficult, then . . . we find it to be infinite, one, and indivisible. [italics added]
It is my suggestion that Spinoza might well explain our mistaken belief that there are finite modes in the same way, as being what we believe when these seemingly finite things are conceived of abstractly and superficially by the imagination. When we conceive of things truly, by means of the intellect, we will see that there is nothing finite, only infinite substance with infinite attributes, and infinite modes following from those attributes, all of which is in the one substance: God or Nature.
But how can such a claim be plausibly made? For surely it is a truism, which philosophers may deny only at their own peril, that there are finite things: Fido, Spinoza, and the Empire State Building. Ultimately, I believe, Spinoza must choose between inconsistency and implausibility. He may explain the existence of finite things only by being inconsistent, yet he may explain them away only by being implausible.
2. Spinoza fails to notice a third option, viz. that one part is finite and the other infinite. But this would be no better an option in Spinoza's view, since then one infinity (the infinite part) would still be smaller than another infinity (the infinite part plus the finite part).
3. Finite things must be modes, for, by proposition VIII, all substances are infinite.
4. It is possible that Spinoza regards logical implicature and causal implicature to be one and the same thing, the former being conceived under the attribute of thought and the latter under the attribute of extension.
5. Not that I think Spinoza would be happy with this interpretation of proposition XXVIII, just that he had better hold that his musings regarding finite modes are hypothetical, if he wishes to remain consistent.
6. If, as Kripke has suggested, it is necessarily true that unicorns do not exist, then this counter-factual claim is also vacuously true.
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