Infinity

In an interesting post "Paradoxicon" discusses Aquinas and Scotus on God's infinity. He gives the following passage from Aquinas' Summa theologiae Q.7 a.1:

We must consider therefore that a thing is called infinite because it is not finite. Now matter is in a way made finite by form, and the form by matter. Matter indeed is made finite by form, inasmuch as matter, before it receives its form, is in potentiality to many forms; but on receiving a form, it is terminated by that one. Again, form is made finite by matter, inasmuch as form, considered in itself, is common to many; but when received in matter, the form is determined to this one particular thing. Now matter is perfected by the form by which it is made finite; therefore infinite as attributed to matter, has the nature of something imperfect; for it is as it were formless matter. On the other hand, form is not made perfect by matter, but rather is contracted by matter; and hence the infinite, regarded on the part of the form not determined by matter, has the nature of something perfect. Now being is the most formal of all things, as appears from what is shown above (4, 1, Objection 3). Since therefore the divine being is not a being received in anything, but He is His own subsistent being as was shown above (Question 3, Article 4), it is clear that God Himself is infinite and perfect.”

Then the following from Scotus' De primo principio c.4:

Finally some argue to the proposed conclusion from the absence of any intrinsic cause. Since form is limited by matter, any form incapable of being in matter therefore is infinite. I do not think this argument is any good, because [its proponents] admit an angel is immaterial, but not infinite. And existence will never limit [its] essence, since they hold that it is posterior to essence. Now the intrinsic degree of perfection that any entity has is not just a vicarious possession. Furthermore, it is a fallacy of the consequent to argue that just because form is limited with reference to matter, therefore without such reference it is unlimited. [This is like arguing] one body is limited with reference to another, hence where there is no such reference a body will be infinite; hence the outermost heaven will be infinite. This is the fallacy of the Physics, Bk. III. Just as a body is first limited in itself, so too with form. Form is first limited in itself (because there is just this sort of nature among things) before it is limited by matter since the latter limitation presupposes but does not cause the first. An essence is first finite by nature, and hence is unable to be made finite by existence; hence it is not subsequently limited by existence.

Then he seems to get slightly confused. He writes:

Scotus isn’t talking about the hypothetical fallacy of affirming the consequent when he mentions the “fallacy of the consequent” (which Aristotle never directly addressed ). He means rather a violation of the rules of conversion regarding the relation of implication or (logical) dependence, described by Aristotle in the Sophistical Refutations:
"The refutation which depends upon the consequent arises because people suppose that the relation of consequence is convertible. For whenever, suppose A is, B necessarily is, they then suppose also that if B is, A necessarily is. This is also the source of the deceptions that attend opinions based on sense-perception. For people often suppose bile to be honey because honey is attended by a yellow colour: also, since after rain the ground is wet in consequence, we suppose that if the ground is wet, it has been raining; whereas that does not necessarily follow."

Paradoxicon seems to be unaware of how the scholastics used the phrase "affirming the consequent", which does not just mean "asserting a fallacious consequent, i.e., one that doesn’t follow (formally) from the premises" - in other words, a non sequitur. Rather the way Scotus argues here is regularly called "fallacy of the consequent" in mediaeval logic textbooks, for instance in Peter of Spain's Summulae logicales VII.150 et seq.

So Scotus accuses Aquinas' argument for God's infinity of "affirming the consequent" because it argues to God's infinity from his immateriality in the same way that in Aristotle's example someone argues to its having been raining because the ground is wet. Scotus accuses Aquinas of implicitly making the following argument:

A. Everything immaterial is infinite
B. God is immaterial
C. Therefore God is infinite

And getting A this way:

if something is material it is finite
Therefore if something is not material it is not finite

which is clearly fallacious. I'm not sure why Paradoxicon says that this isn't the fallacy of the consequent, though. Peter of Spain gives a precisely parallel argument to illustrate the fallacy of the consequent:

If something is a man it is an animal
Therefore if something is not a man it is not an animal

In any case, not even Aquinas believes (A), rather he believes

A1 Some immaterial thing is infinite,

as well as

A2 Some immaterial thing is finite

because angels are immaterial but not infinite. But you can't derive (C) from (B) and either (A1) or (A2).

Aquinas doesn't simply make that argument, of course, because in addition to God's immateriality he adduces the fact that God is "ipsum esse" and not received in any limiting or restricting principle. So he would say that the angels are not infinite, while God is, because angels have restricted esse while God does not. Scotus points out, though, that even according to Aquinas the essence of a creature is not restricted by esse, but rather esse is restricted by essence. Therefore the limitation of essence has to be derived from somewhere other than matter or esse, which Aquinas does not do, ergo etc.

* * *

In a comment to one of Faber's recent posts Paradoxicon, following upon the post I've just been talking about, asks "What is your take on Scotus's idea of infinity as a (positive) intrinsic mode vs. Aquinas's conception of infinity as a mere lack of (accidental?) limits? While it is easy to see that Aquinas's arguments for divine infinity in the Summa are formally invalid (as Scotus notes), it is not so easy to understand how Scotus's teaching is different from Aquinas's, regarding the concept of infinity itself. Any thoughts?"

Aquinas frequently gives me the impression that practically all of metaphysics boils down to immateriality and essence/existence. He uses these two concepts so often that in my opinion it's an unhealthy preoccupation, metaphysically speaking, because it frequently causes difficulties which could be otherwise avoided. This bit on proving God's infinity is a good example. It's not entirely fair to act as though this is all he has to say on the matter - the parallel chapter in the Contra gentiles for instance is much longer - but from Scotus' point of view the S.T. argument is pretty useless.

The first thing I would say about Scotus' approach would be to point out that in the Primo principio, a little before the quote that Paradoxicon gave, Scotus establishes God's infinity in a totally different way, without saying anything about intrinsic modes either. Scotus proves that God knows everything knowable earlier on, after showing that God is a perfect intellect. But the number of knowables is infinite, since the number of possibles is infinite and God knows everything of which he can be the cause, i.e. everything other than himself. But if God has actual simultaneous knowledge of infinites, his intellect itself must be infinite, and his intellect is identical with his nature, therefore he is infinite by nature.

So rather than proving merely a lack of a delimiting principle, as St Thomas does, Scotus proves the presence of a positive infinite in God. He has other approaches too, but this one has nothing to do with either immateriality or the essence/existence distinction. Rather only an intensively infinite and eternal being could have actual intuitive knowledge of, say, the entire series of integers at once. Contrast this with St Thomas' discussion of how God knows infinites in S.T. I q. 14 a. 11-12. First he says that God knows singulars, because he knows both the universal (immaterial!) form and the matter that he creates which he can apply the universal forms to. (What about the angels again?!) And since form can be applied to matter in an infinite number of ways, God knows infinites. This is quite different from Scotus, for whom form is singular through itself, not through matter, and so infinitely knowable not because it can be absorbed by matter in an indeterminate number of instances, but because there are an infinite number of distinct individual formal essences. The knowledge of infinites in God is not therefore merely virtual, insofar as God can produce his effects in an infinitely varied way, but actual, insofar as the number of possible producibles as distinctly present to the mind of God as an actually infinite series. I suspect then that for Aquinas God knows all the numbers because there's no multitude of things he cannot produce; whereas for Scotus God has a clear formally distinct intuitive conception of each and every integer all at once. So in S.T. I q 25 a.3 Thomas proves that God has infinite power simply because he has an infinite nature which is not delimited by anything, which is identical with his power; whereas as in the P.P Scotus shows that God's power is such that he could produce at one time an infinity of things, "if only they were able to exist simultaneously" (the inability of finite things to aggregate to infinity is a defect of the things themselves, not their cause), and so his power is intensively infinite.

I note also that in P.P. right after talking about infinity Scotus says "From infinity every type of simplicity is inferred." It is precisely the actual distinct intellection of an infinity of knowables that lets us infer that God must be absolutely simple. If God were not simple, then the distinct number of things he knows would have to aggregate, which could never happen to infinity. An infinite number of knowables cannot be grasped successively or even by instantaneous "grouping" or "collecting" or establishing a "set", but only in a single simple act which grasps each at once in its distinction from all the others. With the formal distinction divine simplicity looks really different for Scotus than it does for Aquinas.

This is pretty brief and there are a lot of deep waters here. Please nobody take this as anything but the merest first stab at the question. It's all I have time for now.