Monday, August 15, 2011

God's Infinite Thoughts

In any defense of the kalam cosmological argument the absurdity of an actual infinite existing in reality is demonstrated. This is done by discussing how an infinite series cannot be traversed (e.g., the man who claims he has just finished counting all the negative numbers back to 0), or by showing the absurd results of instantiating just such an infinite (e.g., even though two planets orbit at speeds so that one is twice as fast as the other, if they have been orbiting for an infinite amount of time in the past, the number of their orbits is identical!). However, a possible objection is lodged by the atheist who claims a problem with consistency. “After all,” it is alleged, “the Christian God is supposed to be omniscient, is he not? Therefore, it can be said that God’s knowledge constitutes an actual infinite. Hence, either there cannot be a Christian God or there can be an actual infinite.”

This may not be as knock-down as it sounds. First, even if we grant the entire argument, all that follows is that we cannot use the argument against actual infinites as strongly as we had before. After all, if God’s knowledge were to be an actually infinite set of things, why would this make the planet illustration any less absurd? It wouldn’t. It would undercut our warrant for the argument; that is to say, we shouldn’t be as confident in it as we were before. But it by no means defeats the idea that actual infinites, as a part of the created/existing universe, are clearly absurd.

Second, the kalam’s truth does not depend upon the argument against actual infinites. Rather, well-known defenders of the cosmological argument also point to scientific models whereby the best models tell us the universe, or physical reality, had a definite beginning. This means that the kalam may yet still stand even if this objection does.

Finally, we may dispute the entire objection itself (and not merely its implications). God’s knowledge, as far as I can tell, is not propositional in nature. That is, propositions are conventions where we finite knowers take bits and pieces of truth to disseminate and discuss. God has no need of such a convention. In fact, the Scriptures teach that God is truth (John 14:6).

God’s knowledge is instead an undifferentiated whole. I have read William Lane Craig explaining the idea as analogous to one’s field of vision. Despite the fact that such a field of vision may be broken up into parts, it nonetheless remains an undivided whole ontologically. As any defender of the absurdity of actual infinites will tell you, simply because one makes smaller and smaller divisions of a whole to form an infinite does not an actual infinite make. The whole exists independently of and prior to the divisions made. So it is with God’s knowledge.

In case one is not yet convinced, consider also that I would challenge the notion that the events of the past, present, and eternal future constitute an actual infinite. The problem with actual infinites: they never end. That’s why one cannot build propositional truths, such as “Peter sings one praise to God at T1…Peter sings one praise to God at T2, etc.,” to a point where the set is both complete and infinite. If the set is complete, it is not infinite. If the set is not complete, then it is not an actual infinite, but a potential infinite. A potential infinite is one that can go on and on without limit, but because it is a set built one event at a time it nonetheless is never completed, and so is never actually infinite.

I hope these reasons are enough to cast serious doubt on the soundness of the objection. Any questions or comments? Leave them below!
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  1. A couple of thoughts:

    1. The number counting example misses actual infinites with its intuitive thrust, as do all such successive building examples. The intuition relies on starting. You're really saying you can't start counting and reach an end. So, I would say the argument really just implies that a potential infinite cannot become an actual infinite. When you say we could never get to 0, I would ask which number I should start with. What do you say then? The beginning wouldn't make sense. I don't think we can really wrap our heads around any adequate example.

    2. Regarding your planet example, one infinite set can be larger than another, or so I am told by people better at math.

    3. You may be right about the scientific models, but I don't think it's clear. The evidence available doesn't seem really useful for drawing confident conclusions.

    4. I would argue that the idea of a completed infinite set is really more like trying to make sense of a concept that is practically inaccessible to us. It reminds me of trying to recreate the globe on a 2D map. We're giving it a reasonably adequate try, but we could never quite get the representation right.

  2. By the way, I wrote something on this idea here if you or anyone else is interested in reading and/or commenting:

  3. Hi Dr. Mike (remember I said I would call you that!), long time no see! :)

    Remember that the number counting example makes no such assumption of starting; in fact I deliberately chose this example because of the fact it deals with negative numbers and completing a set. Even extending it infinitely into the "past" of numbers, it seems no less absurd precisely because there is no starting point. If you told me he started, at least that would make sense (though, as you point out, in that case it wouldn't be an actual infinite).

    As to the planets, I'm still not sure how that refutes the answer. If Planet A orbits at two times the speed of B, we have: A=2*B. Now let us suppose no starting point of actual orbiting, but of merely our observing. We observe one rotation of B every year. So after one year, we have: B=1*2=A, or A=2. After 100 years, we have B=100*2=A, or A=200, and so on. The disparity grows every year that passes, since time is a series of events one after the other. Given the idea of an infinite amount of time, however, we get this: B=∞*2=A, A=∞. Because the series of events is built one after the other, and because there is no limit to the number of moments (or years in our example) in either direction, then the number of orbits is also similarly unlimited. Yet we now have a contradiction from the premise that A's number of orbits is twice that of B's; in fact they are precisely the same!

    I agree that we shouldn't be too confident about any one specific model, but it's telling that all of the major proposed models at least implicitly and inferentially rely on an absolute beginning.

    I also agree the actual infinite is inaccessible, and I appreciate your analogy. :)

  4. Hi, Randy. I appreciate you remembering my fake honorific!

    I still don't get the force of the negative numbers counting example. I can't divorce the idea of counting from the idea of beginning to count. I think the intuitive force is based on that reasoning (and it works for people precisely because there is no beginning postulated).

    The math of infinite sets is not my specialty, but according to those who are (I'll provide a peer-reviewed paper if you'd like to see my source), two infinite sets are not necessarily equinumerous. So, the number of orbits would not be precisely the same. I have other problems with the analogy, but that is the main one.

  5. I like the idea of God's knowledge being an undifferentiated whole. I don't see why God's knowledge has to be infinite though. Surely there must be a finite number of pieces of knowledge? It's just that the number is ridiculously large and ever-increasing. We can't quantify it (obviously) so it's easier to represent the set as infinite.

  6. Your discussion about the the potential infinity of an eternal future reminds me of the discussion Wes Morriston had with Craig where he postulates that God commands that Wes and Bill praise God alternately for the rest of eternity. God, in his foreknowledge, has complete knowledge of the eternal future: there is nothing about these future events that makes God's knowledge incomplete.

    So, then, if one were to ask how many praises Wes and Craig will make throughout eternity, what should God reply? Clearly any determinate finite amount is inappropriate. Now, we have to decide between an indeterminate finite amount (potential infinity) or determinate infinite amount (actual/completed infinity). However, it looks like the potential infinity is out because it implies a perpetual deficiency in God's eternal foreknowledge.

    Perhaps worse, though, is if we put our concern in the language of omnitemporal truths (so as to bypass the issue of omniscience). If it is the case that X occurs at time T, then it is true at all times that X occurs at T. But if Wes and Bill alternate praises eternally, it is the case NOW that there WILL BE as many praises by Wes and Bill as there are natural numbers (even though there will be no time at which Wes and Bill *complete* their praises). The idea here is that the current omnitemporal truth produced by the conjunction of the individual truths for each future praise by Wes and Bill entails that the set of praises that WILL BE had is complete and infinite.

    What Morriston's example shows is that even though the series of praises by Wes and Bill, from a temporal perspective, may always be indefinitely finite (potentially infinite) at all times, there is still the logical truth that, at every time, there is an omnitemporal truth about the entire set of future praises which is not indeterminate in that way, and we can ask for a determinate answer to the question: "How many praises will be had?". The only appropriate determinate answer to this question is, "Infinitely many".


  7. Hi Mike. While I appreciate your concerns, I think it is this problem of not being able to conceive the negative numbers without the beginning in your case to show that we don't think it's really possible after all! :)

  8. Hi coltrane02, thanks for your thoughts! Indeed, I think this view of God's omniscience means his knowledge is not actually infinite, but potentially infinite only.

  9. Hi Jake, thanks for your comments and concerns! Briefly, I think the problem is solved when we consider two things: 1. For any infinite set, the set is never complete, else it is not actually infinite. 2. Therefore, if God knows a complete set, it by definition cannot be actually infinite. 3. The set of all future events is never complete. 4. God knows the set of all future events. 5. Therefore, what God knows is not an actual infinite.

    In defense of (4), consider tensed truths, like what time it is "now" or "I will leave five minutes from now." Even granting omnitemporal truths, if time holds tensed facts then there are constantly new "truths" becoming true, and then passing into falsehood. Since that is merely a logical consequence, I think it no more constitutes a deficiency in God's foreknowledge than God not being able to sin (another logical impossibility) constitutes a deficiency in his power.

    As to the answer to the normal set of praises, whby wouldn't God respond, "Potentially infinitely many"? Postulating that God must respond "actually infinitely many" seems just to be question-begging, especially since the question itself assumes an answer in a propositionally-based form (rather than an undifferentiated whole). Thanks for your thoughts!

  10. Randy, you said:

    "I think it is this problem of not being able to conceive the negative numbers without the beginning in your case to show that we don't think it's really possible after all!"

    If we are assigning possibility only to things I can conceive, then we're in trouble! In all seriousness, though, my point was not about the idea of an infinite past, it was about whether your the force of your analogy is due to a misconception. Saying that you can't count down is still invoking the idea of beginning to count without actually stating it. That's just how we think and that's why the analogy seems to have force. But that's simply a framing issue. If I said the same type of example but said you were counting for infinitely many moments, then we might say that the numbers counted by me were equal to the number of moments. Then, it's not so clear that it's an intuitive problem.

    In the end, though, we really aren't equipped to conceive of infinity. This doesn't make it impossible. It means we can't come up with an adequate representation and if we could we wouldn't understand it anyway.

    I don't know whether the past was infinite. But I do know that these examples, and every such example I've seen, suffers from vagueness. The examples are not very precise, rely too much on intuition about a difficult concept, are not clear on how they really show a 1:1 correspondence with what is claimed, and are not clear on why they have negative consequences for an infinite past.

  11. Hi Dr. Mike, I guess I just don't see it. :) What I mean by that is that the example just doesn't presuppose it. Now you're right that the reasoh I think it to be quite absurd is precisely because there is no beginning--hence, if it is going to be not-absurd, we'd require a beginning. But I by no means presuppose it in the example. We may say the man never started counting, and yet finished now. This is analogous to the number of moments, since this moment exists and is part of the infinite number of moments that preceeded it. Yet the claim is that actually infinite set can be infinite and yet end now. So it is with the man counting. The claim is that he can actually traverse the beginningless infinite set and finish at this present moment!

  12. Hey Randy. Thanks for the reply, though I don't think that the counterargument works. First, the nature of a "potential infinite" set is a finite but progressive set with the property of being actually infinite in its limit. So, properly speaking, an infinite set is complete, contra premise 1. Further, note what your first premise actually implies. Actual infinites are complete, a la Cantor. But, if we accept your first premise, it follows that actual infinites cannot be complete and, thus, actual infinites are *logically* impossible. This is rather surprising because the problem was supposed to lie in *metaphysics* not *logic*.

    Furthermore, premise 3 should be problematic, even on your view of the non-propositional divine foreknowledge. God's eternal non-propositional foreknowledge cannot be progressive in the sense necessary for a potential infinite. If some event will occur sometime in the eternal future, God's non-propositional foreknowledge already provides the grounds such that, if God so were to turn his attention to it, would have propositional knowledge that such an event occurs. And since the veracity of God's foreknowledge consists in the omnitemporal truths that such events occur in the future (regardless of whether they are represented propositionally by God or not), it still follows that there is an actually infinite set of events represented in God's foreknowledge.

    Further, it seems like the gist of my point about the omnitemporal truths was missed. I shifted to talk in that way to show that it's a problem that is broader than one's view of omniscience (and, hence, why it doesn't turn on one's particular view of omniscience). For instance, the argument I presented works equally well against a materialistic determinist. So, the charge of question-begging is a bit too quick. And I'm not quite sure I get what the relevance is of the tensed truths you're getting at. I don't think it's a problem that God doesn't timelessly know tensed truths, as that doesn't make any sense. But God should know, in some form or another, omnitemporal truths (or tenseless) truths about an eternal future. For every event that occurs in this eternal future, there is an omnitemporal fact that it occurs at such-and-such time. Taking the conjunction of these, we have a completed fact about the future which is true even at present. With that, we can easily set up the necessary bijection to establish that the set of future events is (actually) infinite.

    Though I still think there is a problem here, I definitely appreciate the always-courteous and thought-out replies.

  13. HI Jake! I guess I just don't see the problem with the first premise. It would mean the actual infinite, while working with certain fiats attached by mathematics, will nonetheless never be instantiated in reality--indeed, could never be!

    I do apologize that I seemed to miss your point with omnitemporality. I want to be sure I have you correctly: no matter tensed truths, it is always true that "On december 16, Randy breathes," and "On december 17, Randy breathes," so on and so forth. But with all the truths, why couldn't we add one more? Or ten more? Or infinitely many more? If we can, then the set considered previously can only be potentially infinite. Now, it occurs to me you may respond, "we postulated that God would have non-propositional knowledge of all the truths there are, so by definition none can be added." I agree. However, why is it that the particular number of truths in the first place could not be added to? If it cannot, why not? If it can, the previously postulated set was not actually infinite after all, but merely approaching infinity as a limit.

    I also want to stress that God's non-propositoional knowledge means he doesn't turn his attention to it propositionally; propositions are constructs of finite knowers such as ourselves.

    I have an interesting quote by Craig on just this subject here: "Morriston reiterates his intuition that the number of angelic praises that will be said in an endless series is actually infinite. But the only praises that are actual are the ones that are said, and they will always be finite in number." This article can be found here:

    I also very much appreciate our dialogue! Tell me, do you have or are you working towards a philosophy degree?

  14. Hey Randy,

    I completed my undergrad in math and philosophy in December, and I would like to eventually pursue philosophy in grad school. I'd like to get involved in some stuff on philosophy of mind or on decision theory, but we'll see how that goes. :)

    I guess my point about the first premise of your counterargument is that we have to be careful about how it's scoped. If it's a (narrowly) logical truth, then mathematics is riddled with contradictions. If it's supposed to be a metaphysical truth, then there doesn't appear any reason to accept it unless one is already convinced by the impossibility of real infinites, whereupon it follows trivially. If this is all you were after, then I guess I missed it.

    Regarding the omnitemporal truths, you correctly predicted one response. Your question, however, presses further in asking why one couldn't add some other truth to the set of omnitemporal truths anyway, perhaps by imagining some other possible world where Wes praises God twice the first time. To this, I would point out that being infinite doesn't imply being complete. Consider the set {...-3, -2, -1} U {T | 0<=T<=number of seconds since midnight 1/1/11}. This set is progressive just as potential infinites (progressive finites) are progressive, and yet always has an infinite cardinality.

    Thanks for the clarification about your view of God's non-propositional knowledge. I have some prima facie concerns about it, though my questions would turn this post into a novel. Are there any particularly good resources about it for me to get better acquainted with the particular view you espouse?

    Having read the article by Craig that you link to, I think he, by and large, missed the point. In effect he renders, "How many praises will be said?" in two ways: How many praises will have been said? (potentially infinitely many) and How many praises are there in the future temporal series? (none, given his particular view on A-theory). In addition to begging the question against certain alternate A-theories (shrinking block and moving spotlight, for example), these answers don't address the original question! To render the original question as Craig's first formulation misses the distinction I make in my original post and a similar distinction Dretske makes in "Counting to Infinity" (Analysis, 25, 1965). So, this rendering is not the question I'm asking (nor is it, I think, the question Morriston is asking). With regard to the second question, he argues that, in the temporal series, there are no future events because all future events exist merely in potentia. But we don't need them to exist in actua in order for us to be certain of the fact that each of those events in potentia will, in fact, occur, and that we're able to count all of them with the natural numbers and set up our nice 1-1 correspondence. Again, it looks like Craig doesn't address the question here. (I'll probably have to read his book on the tensed theory of time in order to really get why he privileges the past over the future when it comes to determining what is actual, so I won't raise it as an objection.)

    His earlier discussion in the article about Morriston assuming Platonism about propositions is quite interesting because the argument is couched in the omnitemporal truths that he himself accepts, at least in "Divine Foreknowledge and Newcomb's Paradox" (Philosophia, 17, 1987). Also interesting in that paper his that he holds that God knows all true propositions, and I don't think it makes sense to know a proposition non-propositionally. I'm also unconvinced that a nominalist would be inconsistent if he were to accept Morriston's argument, at least not without bringing in a substantive theory of time and truthmaking. If this is the case, then the proponent of the Kalam has even more assumptions to make (aside from mere A-theory) before the argument even gets started.

  15. Hi Jake, I would recomment "The Only Wise God," by WLC, as it contains both individual chapters dedicated to these subjects as well as references for other works on the subjects.

    As far as the A-theory goes, Craig only presupposes that objective becoming is a feature of time, and hence the future is not "real."

    After reading and re-reading, I think we'll have to agree to disagree. I actually agree that such an infinite series can be potentially enumerated, but I don't see how a potential infinite can be itself an actually infinite set. That's the view from an ontological status. From an epistemological status in relation to God, he does not have enumerated propositions. Craig does say that God knows all true propositions, but he also means this is to be taken as a manner of speaking. God knows all true propositions precisely because God knows all truth; the former is a logical construct of the ontological status of the latter. I think Craig's comment concerning nominalism and Platonism follows if and only if Morriston is committed, whether by entailment or by claim, to saying something like "there exist X number of events in the future," which Craig takes Morriston to be implicitly committed to. Also, Craig (and many kalam proponents) merely take the truth as correspondence theory, so that how we know something is true is something like: "X is true if and only if it is the case that X."

    I appreciate very much the correspondence between us, precisely because I am a philosophy amateur (never taken a formal credit hour in it); but that hopefully will change in the future. God bless :)

  16. Nice article.

    I'm surprised as many people think this as a formidable attack against actual infinity.

    If we look in the dictionary we'll see that the word 'infinite' is properly used both in the quantitative AND qualitative sense.

    Ergo, God can have infinite knowledge (omniscience) and not know an infinite amount of propositions. Which makes sense, since many theists argue against an actually infinite amount of anything -- but simply differentiating the term "infinite" from its quantitative and qualitative sense makes a world of difference. :) I enjoy your website, Randy. Lord bless you!


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