Can Two Equal Infinity?
The Attributes of God in Spinoza
Alexander R. Pruss
March 30, 2002
Spinoza’s God is a being with infinite attributes, each of which expresses infinite essence. Does this mean that God has infinitely many attributes, each of which expresses infinite essence, or does God simply have attributes, each of which is infinite and expresses infinite essence? Spinoza’s argumentation in Letter 9 and the Scholium to Prop. I.10 clearly indicates that it is not just each individual attribute that is infinite, but there are in some sense infinitely many of them. This would seem to settle the issue, except that Spinoza never talks of more than two attributes, Extension and Thought, and this prompts Bennett to say that in fact Spinoza’s God has only two attributes. At first sight this reading seems absurd, and incompatible with, say, the argument in Letter 9 and the Scholium to Prop. I.10 which can be naturally read as saying that God has infinite attributes because how real something is depends on how many attributes it has, and so a maximally real being will have infinitely many attributes. After all, how can two be infinite? Bennett responds that Spinoza sometimes uses infinity for a totality and that anyway all that Spinoza is saying is that God has all possible attributes, and if only two are possible, then so it is. The weakness of this interpretation is that if matters are thus, why doesn’t Spinoza say so?
I will offer something between an interpretation of Spinoza’s system and an examination of its consequences which is compatible with Thought and Extension being the only attributes of God and also compatible with them not being the only attributes of God. However, the interpretation I will offer will be one that Spinoza cannot state on paid of self-defeat. In dogmatic summary: For aught that we human beings can know, Thought and Extension are the only attributes of God. But one can argue that if Spinoza’s system is correct, then neither the claim that there are only two attributes nor the claim that there are more than two can be entertained by human beings. And if this is so, then unless Spinoza wished to whistle that which cannot be said or to employ Kierkegaardian or Wittgensteinian indirection, or unless he wished to repudiate his own system, he could not say some of these things which follow from his system. Nonetheless, it cannot be denied that Spinoza thought that he thought there were more than two attributes of God, but I will argue that for Spinoza’s sake this had better be a mere obiter dictum.
The traditional pre-modern notion of the infinite is that something is infinite if and only if it is not bounded by or comprehended in something greater—the infinite exceeds without being exceeded. Thus, St. Augustine was able to write that the natural numbers are finite in that they are contained in the mind of God. In this sense of infinity, it is impossible for there to be a greater infinity. Cantor’s theorem that any set can be embedded in a strictly larger set then can be taken as showing that in this sense of infinity, there are no infinite sets. We will come back shortly to why this is a claim Spinoza would actually welcome. Let us call this traditional notion of the infinite “unboundedness”.
Spinoza himself uses unboundedness as a notion of the infinite in the Scholium to Prop. I.13:
The indivisibility of substance can be more easily understood merely from the fact that the nature of substance can be conceived only as infinite, and that a part of substance can mean only finite substance, which involves an obvious contradiction (Pr.8).
This argument presumes that if x is a proper part of y, then x is finite, and this assumption only makes sense on the unboundedness view of infinity—it is, for instance, clearly false on the modern mathematical view: the aggregate (i.e., mereological sum) of integers is a proper part of the aggregate of rational numbers, but both are infinite in the mathematical sense.
In Letter 12, Spinoza distinguishes three sorts of infinity. Some things are “infinite by their own nature.” Something infinite in this sense cannot be broken down into parts. This kind of infinity is only conceived by the intellect, not the imagination. I will call this “absolute infinity.” The second sort of infinity derives from the first, when the imagination is brought into play. In the second case, there can indeed be an infinitude of parts, but the parts are imaginary: they are projected by our imagination onto reality. We can call this “mathematical infinity”, an infinite number of parts. Finally, a third kind of infinity is also a species of indefiniteness. Spinoza’s example is not particularly clear, but it seems to deal with a case where an explicit number cannot be given for a quantity, but it is not excluded that a lower and upper bound can be given. We can call this “indefiniteness.”
The three notions of infinity are not obviously all mutually exclusive. Conceivably, something could be both indefinite and absolutely infinite. For Spinoza, something that is absolutely infinite cannot, after all, have its infinitude expressed mathematically, and hence is indefinite, though no upper bound can be given in this case.
Infinity in the sense of unboundedness cannot coincide with the mathematical infinity, since mathematical infinities certainly can be contained in greater wholes: the aggregate of even integers is a part of the aggregate of integers. Should it be objected—though Spinoza would not want to object given the argument in the Scholium to Prop. I.13—that this is not good enough since the integers can in fact be put in one-to-one correspondence with the even integers, we can now pull out Cantor’s theorem to show that one can come up with a set that cannot be put in one-to-one correspondence with the even integers but in which we can contain the even integers.
Nor does infinity in the sense of unboundedness coincide with indefiniteness, since indefiniteness is compatible with the existence of a maximum and unboundedness is not. If then we are to relate infinity in the sense of unboundedness with one of the three sorts of infinity in Letter 12, it has to be by identifying it with absolute infinity. This fits well with the fact that Prop. I.13 is clearly talking both of unboundedness and of a real infinity conceivable in the intellect, and hence of absolute infinity.
Spinoza then need not fear that modern mathematical developments would undercut his notion of infinity. On the contrary, the Cantorian arguments only show that the mathematical infinity is not a truly absolute infinity, and hence should be welcome by Spinoza.
Now that we have three notions of infinity, which one of them enters into the infinitude of attributes of God? Since the notion of God is a notion of the intellect and not of the imagination, it cannot be the mathematical notion. This leaves indefiniteness and absolute infinity. Indefiniteness is compatible with there being an upper bound, and that does not sit that well with the considerations of Letter 9 and the Scholium to Prop. I.10. Moreover, one would expect absolute infinity to be what is primarily at issue when we are talking about the nature of God considered in himself. At the same time, this is compatible with indefiniteness. On this interpretation, there are both absolutely infinitely and indefinitely many attributes of God.
In Letter 64, Spinoza argues that we can only know two of God’s attributes. Our mind is the idea of body, though additionally we have the idea of this idea. We can only know the ideates or objects of our ideas, and hence we can only know God under the attributes of Extension and Thought. Observe that the argument, if sound, proves more than a claim about what we can know, but also about what we can think: we can only think of God under the attributes of Extension and Thought. If there were any other attributes, they would be literally unthinkable, though Spinoza says more moderately that we could not attain to their knowledge. Bennett notes that Spinoza’s very discussion of why it is that we can know only two attributes suggests strongly that he thinks there are more.
Even more strongly, in Letter 56 Spinoza talks of his “ignorance of very many attributes” of God in the course of arguing that this ignorance does not bar him from knowing some attributes. One might try to make out that Spinoza is merely conceding this ignorance for the sake of the argument, but Spinoza’s explicit affirmation that he does not understand even “the greater part” of the attributes makes this reading doubtful.
Moreover, Spinoza’s notes to his Short Treatise say:
After the preceding reflections on Nature we have not yet been able to find in it more than two attributes that belong to this all-perfect being. And these give us nothing by which we can satisfy ourselves that these would be the only ones of which this perfect being would consist. On the contrary, we find in ourselves something which openly indicates to us not only that there are more, but also that there are infinite perfect attributes which must pertain to this perfect being before it can be called perfect.
And where does this Idea of perfection come from? It cannot come from these two, for two gives only two, not infinitely many. From where, then? Certainly not from me, for then I would have had to be able to give what I did not have. From where else, then, than from the infinite attributes themselves, which tell us that they are, though they so far do not tell us what they are. For only of two do we know what they are.
If this view is indeed retained by Spinoza in his more mature years, then Spinoza appears to think that there are “more” than two attributes of God. Or can one perhaps finesse one’s way out of this by denying that Spinoza asserts that there are in fact more than two attributes of God, and making him claim that rather than there being only two, there are infinitely many attributes of God? We will explore whether this option makes any sense, but in any case it will not take care of Letter 56.
On textual grounds, it is clear that the historical Spinoza did hold that there were more than two attributes of God. However, neither does anything in Spinoza’s metaphysical system appear to hang on this, nor do Spinoza’s arguments ever establish this. The ontological arguments that Spinoza use to argue for the existence of God nowhere need the assumption that there are more than two attributes in God, though it may need the unboundedness of the attributes, considered individually and collectively. Indeed, as Leibniz and Plantinga have shown, ontological arguments require only one truly controversial premise: that the essentially perfect being which is such as to have necessary existence is logically possible. Nor was Spinoza ignorant of this: he explicitly says that the only way out of his argument is to claim that the concept of God is self-contradictory (second proof of Prop. I.11). The possibility requires that all possible attributes be mutually compatible, but this, if anything, is harder to support the more attributes there are.
Nor does Spinoza’s central argument for there being only on substance depend on there being more than two attributes. This argument depends on three assumptions. First, that God exists; second, that God has all attributes; and, third, that no two distinct substances can have an attribute in common (Prop. I.2). The first assumption, as we have seen, is independent of there being more than two attributes. The second, too, will be satisfied as long as God’s having “infinite attributes” entails his having all attributes. And the argument for the third does not depend on any considerations of the number of attributes. Rather, one can (controversially) reconstruct Spinoza’s argument as follows: If substances A and B share an attribute X, then to fully understand B, one needs to understanding X. But since X is an attribute of A as well, then an understanding of X also constitutes an at least partial understanding of A. Thus, to fully understand B one must employ an act of understanding that constitutes an at least partial understanding of A. But if B is a substance, then by Def. I.3 the only way this can happen is if in fact A and B are identical. Nothing in this argument presupposes any claim about the number of attributes.
The affirmations of there being more than two attributes of God can, apparently, thus be detached from Spinoza’s system and handled as mere obiter dicta that Spinoza may well have taken himself to have good reason to hold. After all, prima facie, it would seem rather unlikely that the absolutely infinite being, should it have more than one attribute, should have exactly two. One might reasonably suppose that the absolutely infinite being is likely to have either exactly one attribute or for the number of attributes to exceed any finite number. Since Spinoza’s God has at least two attributes, the former option is impossible, and the latter is thus likely. Moreover, Spinoza argues that he cannot know about attributes other than Thought and Extension. But would it not be particularly surprising if what aspects of an infinite being one can know about should neatly coincide with what there is? This argument would be particularly convincing psychologically to Spinoza insofar as the Judeo-Christian notion of an infinite God about whom only a little can be known should still have a hold on him.
An argument that Spinoza’s view that there are more than two attributes is a mere obiter dictum will only really be convincing if in addition to showing that the claim is not essential to Spinoza’s philosophy one can show that Spinoza’s philosophy is better off without it. I will argue that this is so, but at the same time I will argue that neither should Spinoza’s philosophy affirm, as Bennett might wish, that there are in fact only two attributes. In fact, I will argue that Spinoza cannot talk of how many attributes there are beyond saying that Thought and Extension are distinct attributes. Prima facie reason in favor of this is the general idea that number is supposed to be a creature of the imagination on Spinoza’s view, and yet imagination presumably cannot be applied to the attributes of God as such. But when one says or denies that there are more than two attributes in Spinoza’s God, is one not speaking of number?
There is in fact a serious problem with Spinoza even entertaining the thought that there are more than two attributes of God. The ideate of such a thought would be neither anything having to do with Thought nor anything having to do with Extension. And, thus, it is unclear that the idea of God having more than two attributes would even be thinkable in Spinoza’s epistemology. In the passage from the Short Treatise, Spinoza says that the notion of there being infinitely many attributes seems to come from something other than Thought and Extension; indeed, it seems to come from “the infinite attributes themselves”. But if this is understood as saying that the notion comes from the attributes beyond Thought and Extension, then this is indeed most problematic in light of the epistemology of Letter 64.
Thus, it would be problematic for Spinoza to claim that there are more than two attributes of God, as this claim appears to be unintelligible on his view of mind. At the same time, it would be shortchanging the texts to suppose that Spinoza does not want to affirm and show in the Ethics more than just that there are two attributes of God. On Bennett’s interpretation, Spinoza is saying that God has all possible attributes. This is prima facie compatible with, but not entailed by, God having only two attributes.
Observe now that just as thinking that God has more than two attributes is problematic from a Spinozistic viewpoint, so is thinking that God has only two attributes. For that God has only two attributes, Thought and Extension, entails that every attribute of God is one of the two. To think this would require that one think of the totality of all of God’s attributes as a totality of God’s attributes and to say that each member of this totality is either Thought or Extension. But such a totalization may be found problematic by Spinoza in light of each attribute being itself infinite. For the totality then would seem to be a whole composed of infinite parts, and that cannot be by the Scholium to Prop. I.13. Moreover, the totality would be an absolute infinity and yet composed of parts—the attributes—contrary to the claim that only a mathematical infinity can be composed of parts.
There are also considerations against the thinkability of the claim that there are only two attributes of God that go beyond Spinoza’s accounts of infinity. The claim that Thought and Extension are only two attributes of God can be rephrased negatively: there are no attributes of God other than Thought and Extension. But if not only Spinoza’s metaphysical results are Parmenidean, but Spinoza’s more basic sympathies are also Parmenidean, then such a negative existential claim could well be problematic for Spinoza: it would be speaking of what is not.
Spinoza’s ideate-based theory of knowledge commits him to the truthmaker principle, the principle that each true proposition is true in virtue of the existence of a truthmaker, a positive worldly item in virtue of which the proposition is true—that there are horses is made true by the horses of the world and that Socrates is sitting when true is made true by the sitting Socrates or by Socrates’ sitting. But the truthmaker principle is simply a restatement of the Parmenidean claim that what is not is unspeakable and unthinkable. It is well known that the truthmaker principle makes negative existential claims problematic when these claims cannot be grounded in some clever way in a positive reality.
It is not obvious in what positive reality Spinoza could ground the claim that there are no attributes but Thought and Extension. It is true that Spinoza is willing to make the negative existential claim that there are no substances other than God. But this claim’s truth is grounded in the positive reality of God’s infinitude and the conceptual analysis thereof. Like Descartes who thought that the notion of the infinite is prior to that of the finite, Spinoza thought that absolute infinity was a positive concept. Otherwise, the remark that it is absurd to think that the notion of an absolutely infinite Being is self-contradictory (Proof 2 of Prop. I.11) is quite implausible—it is surely the positivity of the constituent concepts of the ens realissimum that gives to this claim whatever plausibility it might have. Likewise, the argument for the infinitude of attributes in Letter 9 and the Scholium to Prop. I.10 is explicitly based on the idea that there is more reality in an absolutely infinite being. Now in general the fact that there is more reality in an absolutely infinite being does not imply that absolute infinity is a positive concept, but given an identity between the order and connection of things and the order and connection of ideas, the implication does indeed follow.
Now even though the proof of Prop. I.14 is by reduction ad absurdum and the statement is negative—“There can be, or be conceived, no other substance than God”—the proof and statement of Prop. I.14 can easily be rephrased in purely positive terms: The infinitude of God’s attributes implies that every attribute is had by God, and hence if something is a substance then it has an attribute in common with God, and hence is God. All this is merely a restatement of the idea, thought by Spinoza to be positive, that God has an absolute infinity of attributes.
Now, one might suppose in principle that there is a similar purely positive argument showing that any attribute is either Thought or Extension, so that the claim that Thought and Extension are the only two attributes could be grounded in such an argument. But the prospects for such an argument are slim given the disjunctive conclusion. Such an argument would presumably proceed by a conceptual analysis of the notion of an attribute, showing that whatever is an attribute is either Thought or Extension. The only way this could plausibly work would be if the notions of Thought and Extension were somehow built-in to the very notion of an attribute. But there is no reason to think they are thus built-in to the very notion of an attribute. When Spinoza shows that Thought or Extension are attributes in Props. II.1–2, he does not proceed by pulling, say, Thought out of the notion of an attribute. On the contrary, Spinoza’s arguments clearly presuppose on a posteriori grounds that there are thinking and extende things. Indeed, given Spinoza’s rationalism, were the notions of Thought and Extension built-in to the very notion of an attribute, then they would not be attributes. For then there would be a more general notion, Having Attributes, from which they would be derivable, and presumably then we would say not that it is Having Attributes that is the real attribute of substance, not Thought or Extension, as Having Attributes would express the essence of the substance, and Thought and Extension would be mere modes of the substance under the description of Having Attributes. No longer would we have two parallel worlds, the world of Thought and the world of Extension, but a single world hierarchically unified under Having Attributes.
In fact, this argument suggests that Spinoza might well be a nominalist about attributes in the sense that he needs to hold that there are attributes but there is no such universal as Having Attributes. Now, we know nominalism about the concept of a horse, the view that there are horses but no Horseness, spells trouble for making claims of the form “Necessarily, every horse is F”, since it is difficult to see what the truthmaker of such a claim is: one would like to say that the truthmaker is Horseness’s involving Fness, but the nominalist bars the move. Of course, the problem is not insoluble, but it is a problem nonetheless, and suggests that a claim that, necessarily, every attribute is either Thought or Extension will be semantically problematic if we are to be nominalists about attributes.
We see then that both the claims that there are only two attributes and the claim that there are more than two attributes are semantically problematic. The expressibility of the claim that there are two attributes is problematic given Spinoza’s truthmaker- or ideate-based semantics. And the expressibility of the claim that there are more than two is problematic given Spinoza’s analysis of our ideas all being either about ideas or extended things.
But we want to say: “Surely there are either two attributes or not two attributes, tertium non datur; and if there are not two, there are either fewer than two, which Spinoza denies, or more than two, again tertium non datur.” The impulse to say this is one that Spinoza must reject.
But why doesn’t Spinoza say the things I said above. There is a good philosophical reason for him not to say these things, since when spoken in the first person about one’s own views they are self-defeating. To say that it is meaningless to claim, e.g., that there are only two attributes is either to speak meaninglessly or to speak falsely. For if we say that claiming that p is meaningless, then either we have made ourselves understood as to what proposition it is the claiming of which is meaningless or we have not made ourselves understood. If we have made ourselves understood, then we know what proposition it is the claiming of which is meaningless, and since we know what proposition it is, claiming that proposition is not meaningless. On the other hand, if we have not made ourselves understood, then we spoke meaninglessly. Nor will it do to say that when we said that the claim that p is meaningless we spoke merely about our language. For our language is a language only insofar as its sentences have meanings. If this is correct, then the only way to make such claims of meaninglessness is as the Kierkegaard of the Postscript or the Wittgenstein of the Tractatus: by withdrawing them as soon as the reader has, as it were (and much is hidden under the “as it were”), understood that she understands “them” not. But we would not expect a rationalist like Spinoza to take such an indirect and roundabout approach. It is much more in character for him simply not speak whereof, if his theory is correct, one cannot speak.
But Spinoza is wrong. The claim that if it makes sense to talk of attributes then there are either more than two attributes or two or fewer than two is more plausible than the conjunction of Spinoza’s assumptions. But it is not the purpose of this paper to criticize Spinoza but to explore the issues Spinozistically.
A problem remains. The attributes are infinite, both individually and in some way collectively. But what can it mean to say that they are collectively infinite? Bennett proposes that it simply means that God has all attributes. But what is the truthmaker of the claim that God has all attributes, i.e., what is the ideate of the collective infinity of attributes?
Truthmaker theorists approach the problem of universal propositions in several ways. Consider the proposition p, which I shall for the sake of the argument assume to be contingently true, that all existing animals are carbon-based. It is tempting to say that the truthmaker is the worldly item x1’s being an animal and being carbon-based, x2’s being non-carbon based but a non-animal, x3’s being an animal and being carbon-based, ... where we thus go through the list of all objects in the universe. But this is not enough for the truthmaker, since this worldly item could exist without all animals being carbon-based. What is needed in the truthmaker is a totalizing clause that makes it true that x1, x2, x3, … are all the items in the cosmos.
Because of this, some truthmaker theorists say that the truthmaker of a universal proposition like p is the cosmos as a whole. However, for this it is necessary that we consider the cosmos not merely as a mereological sum of its objects, but as a genuine totality of all existing objects. For the cosmos considered as a mereological sum of its objects is not enough to make true the appropriate universally quantified claims: one also needs the consideration that this mereological sum is the sum of all there is.
What, then, can make true the claim that God has all the attributes? Earlier I said that this was the infinity of God’s attributes, with infinity considered as something positive, as in Descartes. This infinity of attributes must be more than a mereological sum of the attributes, as the above discussion of universal propositions shows. Rather, it must be a totality, an allness. This explains the pains to which Spinoza goes to distinguish absolute infinity from mathematical infinity, since a mereological sum of infinitely many items is mathematically but not absolutely infinite.
There are two aspects of the absolute infinite, plurality and wholeness. What is doing the important work here is not so much the plurality but the wholeness: the fact that the infinity is a totality of attributes, all the attributes seen as taken together. It seems, thus, that the concept of infinity is supposed to be a positive concept of allness, so that having an infinity of attributes is a positive feature of reality that can serve as a truthmaker for the claim that God has all attributes. There is a lot more, then, to the attributes than twoness. If the attributes were merely two, then they would not be sufficient to ground the claim that God has all attributes.
This may seem mysterious. What is this note of allness that gets added on to the mereological sum of the attributes to produce the infinity of attributes? But actually the answer may be quite simple. It is a necessary ontological fact, according to Spinoza, that all the attributes are attributes of the only possible substance, God. The truthmaker of the claim that God has all attributes then is God. There is nothing else. The claim that there is anything else would be a claim that could not possibly have a truthmaker. In fact, Spinoza’s system provides a neat solution to the truthmaker problem for all universal propositions: all of them have as their truthmaker reality-as-a-whole, i.e., the one necessarily existing substance.
If no solution to the truthmaker problem for universally quantified propositions were available to the non-monist, then this would constitute an argument for monism on the basis of the truthmaker principle. Basically, the argument goes, if there is to be a truthmaker for a claim like “All life-forms are carbon based”, this truthmaker will presumably be the universe-as-a-totality—call this concrete item U. Now, the truthmaker for a proposition p must be sufficient to guarantee the truth of p. But the universe can only be a totality if it is one thing and moreover if U is of such sort that there could not be another thing of this sort—otherwise, the existence of U would not guarantee that all life-forms are carbon based since U could co-exist with another collection of physical objects (perhaps causally separated from U) and were that to be the case then it would be true that U exists but not all life-forms are carbon based if the other collection contained a non-carbon-based life-form. Therefore, the universe-as-a-totality must be one thing without the possibility of anything else of the same sort being added on to it, i.e., co-existing with it. Parmenides would be happy.
Of course, the non-monist in reaction to this may give up on or weaken the truthmaker principle. Or she might try to opt for a different truthmaker for universal propositions, e.g., the totality of a non-pantheistic necessarily existent God’s creative activity considered as a unity either through being represented as such in God’s mind or through its being the effect of the simple action of a God that satisfies the doctrine of divine simplicity.
In fact, the Spinozistic solution to the truthmaker problem gives us a sketch of another version of the ontological argument that Spinoza could have used. Consider the notion of a perfection. Whenever we make universally quantified claims about perfections, such as “Every perfection is distinct from being a rotten tomato”, then our claim needs to have some truthmaker, T. This truthmaker has to comprise the totality of all perfections in such a way that T’s existence would suffice to make it true that every perfection is distinct from being a rotten tomato. Now, there really are only two plausible ways we could see an object T as comprising all perfections: either by actually having the perfections within it in some way or by including the thoughts of all perceptions. If the latter, then there is still the question of what makes it be the case that T is thinking about all perfections: what guarantees that T involves all possible thoughts of perfections. It would seem that unless T is the sort of thing that is the only possible producer of perfections, there can be no guarantee that all the perfections it thinks of are all the perfections. But if T is the only possible producer of perfections, then we’ve shown the existence of a God who at least has all perfections, eminently, as Descartes would say. On the other hand, if all perfections are to be actually found within T, then we have found a totality, one thing, which has all perfections within it, formally, as Descartes would say. Q.E.D.
 Indeed, some scholars give up on the attempt to reconstruct this argument except in the special case where there is only one attribute had by each substance. See Bennett (??ref).
 The line of reasoning here is inspired, though not endorsed by, Scruton (??ref).
 This particularly well comes out in Gödel’s version of the ontological argument (??ref).