Sunday, May 24, 2009

Aquinas and Malebranche on Mathematical Infinites and God

Thomas Aquinas (Summa Contra Gentiles 1.43):

When our intellect understands something, it extends to infinity. A sign of this is that when any finite quantity is given, our intellect can think a greater one. But this order of our intellect to an infinite would be in vain were there not some infinite intelligible. And so it is necessary that some infinite intelligible exists, which must be the greatest of things. And this we call God.


Nicolas Malebranche (Search after Truth, Elucidation Ten, LO 614):

The mind clearly sees that the number which when multiplied by itself produces 5, or any of the numbers between 4 and 9, 9 and 16, 16 and 25, and so on, is a magnitude, a proportion, a fraction whose terms have more numbers than could stretch from one of the earth's poles to the other. The mind sees clearly that this proportion is such that only God could comprehend it, and that it cannot be expressed exactly, because to do so, a fraction both of whose terms were infinite would be required. I could relate man such examples demonstrating not only that the mind of man is limited but also that the Reason he consults is infinite. For, in short, the mind clearly sees the infinite in this Sovereign Reason, although he does not comprehend it. In a word, the Reason man consults must be infinite because it cannot be exhausted, and because it always has an answer for whatever is asked of it....No creature is infinite; infinite reason, therefore, is not a creature.


Even though both of these arguments begin with our ability to think about the mathematical infinite and conclude to the existence of God, they are radically different kinds of argument. Malebranche's argument is, we might say, an argument that thinks in terms of formal causality: the mathematical infinites we can reason about are thought to be mathematical infinites in God Himself. When a geometer works up a proof in geometry, Malebranche thinks he is studying God -- in particular, he is studying the archetype of space as it is found in God, or, to put it in other terms, he is studying the limits that divine Reason places on God's power to create space. The mathematician literally and directly studies God, albeit in a very limited way; mathematical reason is divine Reason.

St. Thomas's argument, on the other hand, is purely in terms of final causality. The mathematical infinite with which the mathematician is concerned is not the divine infinite in any way, shape, or form, and the reason he consults is not the divine Reason. But the infinite is still an issue requiring explanation: mathematical reason would be, as we might say, massive overkill if there were no actual infinite intelligible to which the human mind is somehow suited. And thus the intellect's ability to think of the infinites in mathematics is a sign of the fact that it is disposed to know God.

A major difference between the two is that St. Thomas the Aristotelian has no problem with the idea that created substances may be in some real way infinite. And, indeed, the Aristotelian account of the human intellect requires that human beings be infinite in a certain respect. Actually, we can probably be much stronger: on Aquinas's view, everything is infinite in some sense. Some of these ways of being infinite are not particularly interesting for the purposes discussed here, but the relative infinity of the human intellect means that there is no fundamental problem with mathematical infinites being drawn from sense experience and being contemplated in our own minds. But Malebranche does not allow for this possibility; indeed, he thinks it manifestly absurd. On his view we are obviously and completely finite creatures with completely finite intellects having access only to completely finite sensible experience; and thus there is no way we could even recognize mathematical infinites as infinite unless our minds already had access to something that actually was infinite. Every attempt to explain how we have an idea of the infinite either collapses, so that we are forced to regard mathematical discussions of the infinite as utterly unintelligible, or relies on a pre-existing infinite idea, either openly or by smuggling it in through the back door. You can't get the idea of the infinite from sensible things, you can't get it from yourself, and thus you must be consulting some actual infinite accessible to the mind.

So they are actually very different arguments, despite some similarities. In a sense one might say that they are as far apart as Aristotle and Plato.

15 comments:

  1. Brandon,

    In the penultimate paragraph you state, "You can't get the idea of the infinite from sensible things, you can't get it from yourself, and thus you must be consulting some actual infinite accessible to the mind."

    But some have argued that you can get the idea of infinity from yourself. From '36 arguments for the existence of God' Rebecca Goldstein makes the following argument:

    "There are certain computational procedures governed by what logicians
    call recursive rules. A recursive rule is one that refers to itself,
    and hence it can be applied to its own output ad infinitum. For example,
    we can define a natural number recursively: 1 is a natural number, and if
    you add 1 to a natural number, the result is a natural number. We can
    apply this rule an indefinite number of times and thereby generate an
    infinite series of natural numbers. Recursive rules allow a finite system
    (a set of rules, a computer, a brain) to reason about an infinity of objects,
    refuting Premise 3."


    (Source: Argument 29 at http://www.randomhouse.com/pantheon/authors/goldstein/36%20Arguments.pdf)

    If thoughts are biochemical reactions and the brain is like a cellular computer that implements recursive rules, then it seems like a finite system can arrive at the idea of infinity.


    How would you meet this objection?

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  2. By the way, the same exact point is made here in :

    "As noted above, recursive processes and structures can in principle extend without limit, but are limited in practice. Nevertheless recursion does give rise to the concept of infinity, itself perhaps limited to the human imagination. After all, only humans have acquired the ability to count indefinitely, and to understand the nature of infinite series, whereas other species can at best merely estimate quantity, and are accurate only up to some small finite number."

    (Source: http://press.princeton.edu/chapters/s9424.pdf)

    So what do you say to this 'recursion'-objection? If this objection can be met then I believe your conclusion follows ("you must be consulting some actual infinite accessible to the mind").

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  3. branemrys1:27 PM

    This is a very old objection; it fails by sliding from 'indefinite' to 'infinite'; as Malebranche argues at considerable length, the two are not the same.

    In the particular example of the objection you note, it's also the case that talking about recursive rules as if they were automatically infinite is another problem, since it is easy to identify cases of recursion that are not infinite -- e.g., "Ancestors of ancestors are also ancestors", which is recursive but lasts only as long as there are actually ancestors for the rule to apply to. In order to identify rules as infinitely recursive (or to have a coherent conception of the infinite application of any rule, recursive or not), we have to already have an independent criterion for identifying the infinite, which means that we cannot be getting the idea of the infinite from recursive rules.

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  4. Are you saying recursion may possibly account for potential infinity but not actual infinity?

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  5. branemrys10:27 PM

    No; you seem to be making the mistake of assuming again that anything indefinite is infinite, which no Cartesian like Malebranche would grant without argument. (Descartes explicitly argues that it is wrong in several places, and as I note, Malebranche argues that it is wrong at some length.) Recursion rules of themselves don't necessarily imply the infinity of anything; that your ancestor's ancestor is your ancestor, for instance, although an obvious recursion rule, implies neither that you have infinite ancestors nor does it even imply that your lineage could be arbitrarily large -- for all it actually tells us, it might be impossible for ancestry to go beyond some unknown but very, very large number. The Malebranchean point would be that any kind of rule application, recursive or not, couldn't involve any infinity at all without a prior independent idea of infinity to make it possible for us to know that the application of the rule could be infinite (whether we are dealing potential or actual infinity) rather than an immensely large finite.

    The problem with the arguments is that they assume that we somehow magically know that recursive rules are infinite just by looking at them -- it's never explained how it is that we know that they are infinite at all. But, of course, Malebranche would say that even an idiot can derive an infinite if he assumes a recognized infinite as a starting point -- the question at hand, however, is how he would recognize the infinity of the starting point in the first place. Thus it's not enough to assume that recursion rules are infinite: the problem requires that you identify how recursion rules come to be known as capable of going on forever, and do so without first presupposing that the infinite is possible (since recognizing that it is possible for something to be infinite requires that one already have the idea of the infinite).

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  6. "One of the basic problems with the objections is that they assume that we somehow magically know that recursive rules are infinite just by looking at them -- it's never explained how it is that we know that they are infinite at all."

    But the objection is attempting to do just that. The objection is saying that recursion is somehow hardwired into the brain and because recursive rules are able to represent the infinite in a finite form, your neural recursion enables you to recognize things as finite or infinite. And this is why you're able to magically know that recursive rules are infinite just by looking at them when you encounter them in a formal mathematical context.

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  7. "Recursion rules of themselves don't necessarily imply the infinity of anything"


    So what? It doesn't matter if recursive rules necessarily imply infinity. It simply needs to be *possible* for them to do so. The right recursive neural circuitry set up in your head can grant you access to the infinite; no God is necessary.

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  8. branemrys4:21 AM

    It simply needs to be *possible* for them to do so.

    This makes an obvious modal error: recursion rules are formal and abstract entities. Thus it is only possible for them to imply infinity when they necessarily do so.

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  9. branemrys4:38 AM

    What the objection is "attempting" to do is not relevant to the point; it is how the objection actually does it without begging the question that matters. And again, it cannot be the mere fact of "recursion" that does the trick, because it is easy to identify merely finite recursions. In addition, there are infinitely many possible recursion rules, so which ones are "hardwired" into the brain? However, if we assumed that some recursion rules or other were the answer, it would still not suffice for the recursion rule to be "hardwired" into the brain, whatever precisely it means for a recursion rule to be hardwired. Since it is possible to use a recursion rule without recognizing that it is infinite, to get the infinity the recognition that these recursion rules are infinite would also have to be "hardwired", and it would have to be "hardwired" in such a way that the idea of the infinite could be applied independently of any specific recursion rule. This is logically equivalent to saying that we must have an idea of the infinite that is independent of any specific recursion rule in order to distinguish any infinite recursions from finite recursions, which concedes the primary point of the argument.

    The rest would just depend on the precise details of how the idea of the infinite is "hardwired". As specifically noted in the post, there are two distinct paths one can take given the problem: the Platonist path, of which Malebranche is a major example, and the Aristotelian path, of which Aquinas would be an example. This talk of "hardwiring" is vague and handwaving enough that it isn't able to rule out the Aristotelian version; it just doesn't seem to fit comfortably with the Platonist version. What you do not seem to be doing is suggesting anything that actually eliminates the problem in the first place.

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  10. "Now lets consider the following three propositions which might be used to justify premise 3 in the above argument:"



    There's a type in the above quote. I mean premise 4, not premise 3.

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  11. branemrys7:24 PM

    Not a problem, but it's not 'my' argument. It's a historical, classical argument that has two forms, of which you are arguing against the Platonistic or Malebranchean version.

    The objection you note here is in fact an old objection; it was the objection raised by Arnauld in his dispute with Malebranche. Unfortunately, the problem is that no one doubts that finite systems in some broad sense can represent the infinite; I can put a little sidewise 8 on a piece of paper and Lo! I have represented the infinite in a tiny figure on a piece of paper. But the representation not being what it represents, it can only be a representation of the infinite if one presupposes the idea of the infinite. For what, precisely, about a representation that is itself finite, in a medium or vehicle of representation that is itself finite, makes it a representation of the infinite rather than of an indefinite finite?

    How does the objection confuse indefinite with
    the infinite?


    If you go back and read the original versions you posted, you will see that they make use of the indefinite to try to 'get' the infinite -- very explicitly in the first one, in fact.

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  12. "If you go back and read the original versions you posted, you will see that they make use of the indefinite to try to 'get' the infinite -- very explicitly in the first one, in fact."

    That would be the case if the brain had to somehow *arrive* at the idea of infinity after an indefinite number of recursive computations. But as I stated in my previous comment this is an incorrect understanding of the objection. The recursive rule hardwired into the brain *just is* a representation of infinity; no indefinite amount of computation is necessary. Hence there is no jump from indefinite to infinite.

    "For what, precisely, about a representation that is itself finite, in a medium or vehicle of representation that is itself finite, makes it a representation of the infinite rather than of an indefinite finite?"

    I thought I told you the naturalist response to this question ad nauseam in my previous comment. You come to know the infinite because the postulated recursive neural architecture inside your head enables you to comprehend infinite recursions when you encounter them in a formal mathematical context. If you don't think the naturalist answer is plausible then please point out what is lacking.

    You still have not given a clear answer to the following question: If I can represent the infinite via some recursive neural architecture in the brain, why do I have to appeal to anything that is actually infinite? Your appeal to the actual infinite is unnecessary and needs justifying.

    What precisely is your solution anyways? I still don't know. Is it divine illumination? Innate ideas? Platonic reminiscence? Vision in God? How exactly do these fare any better than naturalism?

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  13. branemrys8:53 PM

    That would be the case if the brain had to somehow *arrive* at the idea of infinity after an indefinite number of recursive computations

    That's not what the original objection you quoted involved; if you are reworking the objection to eliminate the obvious problem, that's excellent, but it's important to be clear that you are putting forward a different objection with only loose similarities. Otherwise I have no way to have any idea which you objection you are talking about at any given moment.

    You come to know the infinite because the postulated recursive neural architecture inside your head enables you to comprehend infinite recursions when you encounter them in a formal mathematical context.

    But this was not the question. The question was, again: "what, precisely, about a representation that is itself finite, in a medium or vehicle of representation that is itself finite, makes it a representation of the infinite rather than of an indefinite finite?" The above is not an answer to this question at all.

    If I can represent the infinite via some recursive neural architecture in the brain, why do I have to appeal to anything that is actually infinite? Your appeal to the actual infinite is unnecessary and needs justifying.

    First of all, as I specifically have already told you, it's not "my" appeal, so stop pretending that it is. I am doing you the courtesy of explaining the actual historical argument as it actually historically was developed; this is pointless if you are going to repeatedly attack me because you can't bother to exercise the basic rational diligence of distinguishing between me and someone who lived three hundred years ago. As to your question, this was specifically answered in the previous comment when I referred to the actual historical dispute in which the kind of position you are suggesting first arose (three hundred years ago, in case you are getting confused again) and summarized, very briefly, a dispute between Arnauld and Malebranche that turned on precisely the point you are raising. If you are not going to pay attention to what I am saying in the first place, then you have no business harassing me about it.

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  14. So do you have a response to my updating of the objection?

    By the way I'm not even a naturalist; I'm simply playing devil's advocate to try and get objections to the argument I'm presenting. I apologize if I come off as harassing and confused. I'm a concerned theist who is a little in flux about how to work out this "recursion" wrinkle. I have always been thoroughly convinced by Descartes Trademark argument (mostly in part by your defense of it on Ed Feser's blog) but unfortunately since I've come across the "recursion" objection I've lost a lot of confidence in it because I now see how it is possible to naturalize the most important idea contained in the idea of God: The infinite. If a finite set of recursive rules programmed in the brain can represent the infinite AND the brain doesn't need to *arrive* at this idea then the naturalist may plausibly claim this neural architecture came about via some adaptive Darwinian selection process, thereby rendering appeals to an actual infinite superfluous. Moreover, if the concept of infinity can be naturalized then the idea of God can be naturalized too because the idea of God is really just an application of infinity to positive attributes (i.e. power + infinity --> omnipotence; knowledge + infinity --> omniscience; goodness + infinity --> omnibenvolence).

    Perhaps infinity can be saved if one distinguishes between potential and actual infinity and then claiming recursion can only account for potential infinity; not actual infinity.
    So an updated version could look like ...

    1. The idea of actual infinity exists.
    2. Ideas must come from somewhere.
    3. Either it comes from something actually infinite or not.
    4. But it cannot be derived from anything either potentially infinite or finite.
    5. God is actually infinite.
    6. God exists and gave us this idea.

    Now 4 eliminates the possibility of recursion since recursion can only account for potential infinities
    (or can it?). I don't know. I'm just thinking out loud.

    Also, I'm not at all familiar with the Malebranche-Arnauld dispute (which is why it seems like I've been ignoring what you've been saying). Can you recommend some resources please?

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  15. branemrys11:54 PM

    An adequate answer would require closely reading the entire discussion again, since I had been operating under the assumption that you were just adding bells and whistles to the original objections you quoted as being the problem. At a rough glance, though, I don't see how it does much more than posit, on the basis of what evidence is unclear, that we have the idea (i.e., representation) of the infinite built in. This does not appear to be a real naturalistic solution; if merely positing whatever was needed to solve a problem were naturalism, anything could be naturalism. (Contrast this with naturalistic finitism, for instance, which happens to be the most elegantly structured solution and directly solves the entire problem while not positing any magic at all; the only reason it's not more popular is that it ends up cutting out a great many things that even many naturalists think shouldn't be cut out.) Perhaps I'm not understanding what you mean by 'naturalizing' here; all that seems to have been proposed (again, on a rough glance, since a more serious response would require going through the entire discussion again) is that we are able to think about the infinite because we can just assume (on grounds that are not really very clear) that we have brain architecture that solves all the problems people have raised about thinking about the infinite, and (also on grounds that are not very clear) that this brain architecture is the kind that could be delivered by natural selection. Just-so stories are not naturalism; they are fake naturalism. Natural selection stories are not freebies, but require very detailed evidential construction to be of any value at all. Whether or not the proposal is merely a just-so story is impossible to determine without knowing the details.

    It's also (and relatedly) unclear what work recursion is doing in the actual explanation; since our brains are not themselves actually infinite recursions of anything, everything has to be done through the notion of representation, so the whole 'solution' seems to amount to nothing more than saying that "our brains just evolved so that they naturally represent the infinite, so there". But the whole question is about the conditions for representing the infinite so that it can be recognized as infinite. How recursion somehow gets one to a genuine account of our recognizing things as infinite is not very clear. Again, that's without going back and trying to piece together everything from the entire discussion so far.

    I don't think, for the purposes of these kinds of arguments, that the distinction of potential vs. actual infinites makes any difference whatsoever. (Malebranche, at least, would certainly deny that it makes any difference at all.)

    The Malebranche-Arnauld dispute is scattered over a very large number of books; it is not something I would generally recommend for beginners. But reading the first few chapters of Malebranche's Dialogues on Metaphysics and Arnauld's extended discussion in the Book of True and False Ideas would lay out some of the basic points relevant to this particular topic. Not, alas, all of the basic points; Malebranche's response to Arnauld's criticism, and Arnauld's response to Malebranche, etc., are all still only in the French.

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