PvI and an Unobservable Infinite Being
July 4, 2010 — 21:56

Author: Andrew Moon  Category: Concept of God Existence of God  Comments: 15

In The Problem of Evil, PvI presents a very interesting argument in the following dialogue between an atheist and a theist:
Atheist.If the stars in the sky were suddenly rearranged so as to spell out ‘I am who am’, I’d believe in the existence of God then, all right…
Theist.… even the ‘I am who am’ story wouldn’t make the existence of God evident to a sufficiently determined skeptic–for even the (apparent) rearrangement of the stars could be the work of a lesser being than God. We can imagine no sign that would have to be the work of a necessary, omnipresent, omnipotent being. Any sign one might imagine could be ascribed to a contingent, locally present being whose powers, though vastly greater than ours, were finite. I should expect that someone like you would say that if two hypotheses explain the data equally well, and if they are alike but for the fact that one of them postulates an unobservable infinite being and the other an unobservable finite being, one should always prefer the latter hypothesis, since it does the same explanatory work as the former, but is, literally, infinitely weaker.
Atheist… You argument has convinced me of something you didn’t forsee: that you theists have invented a being whose existence no one could possibly rationally believe in, since the hypothesis that he exists is necessarily infinitely stronger than other hypotheses that would explain any possible observations equally well… (pp. 141-142)
There’s a lot of interest in here (e.g., the non sequitur that Atheist makes in that last sentence), but I’m interested in this claim:

“if two hypotheses explain the data equally well, and if they are alike but for the fact that one of them postulates an unobservable infinite being and the other an unobservable finite being, one should always prefer the latter hypothesis, since it does the same explanatory work as the former, but is, literally, infinitely weaker.”

Unless I missed it, PvI never addresses this claim in the book. Do you guys think it’s true? I wonder what arguments there are for or against it. (If people are interested in commenting on other parts of the quote, please go on right ahead.)

Comments:
  • gwern

    > I wonder what arguments there are for or against it.
    Sounds like one of those horrible little arguments about ‘what do we actually mean by Occam’s razor?’
    You could easily argue both ways. Infinity is bigger than any finite number, so clearly is more complex. On the other hand, if we prefer something like Kolmogorov complexity, then infinity can easily be simpler than our finite number – ‘x=x+1’ is a very short and simple program, while computing (the number given by Ackermann’s function of a googol) might be quite a long program.

    July 5, 2010 — 0:37
  • Andrew Moon

    gwern,
    Thanks for the feedback.
    Hmm, I don’t know if I understood your last sentence, probably because I don’t know what Kolmogorov complexity is, and I also don’t know what Ackermann’s function of a googol is.
    However, is complexity what we’re concerned with? It’s not obvious to me that a being who, for example, knows all truths is more complex than a being who knows a trillion truths. Actually, if both beings are immaterial objects, they might be equally simple (or complex).
    Yet, in PvI’s terminology, the positing of a being who knows a trillion truths seems to be weaker than a being who knows all truths. Still not sure what to think.

    July 5, 2010 — 12:20
  • christian

    andrew,
    what is an “unobservable infinite being”?
    the statement you quote appears to involve an equivocation on ‘infinite’. On the one hand, it’s supposed to modify ‘being’. on the other, it’s supposed to modify the strength of a hypothesis, as when it says “infinitely weaker”. it’s not clear that these claims are even related.
    in any case, the argument can be strengthened. the activities of a finite or an infinite being are not the only candidates for explaining what we observe. we can make the options much more fine-grained.

    July 5, 2010 — 13:51
  • Normally, we say that a hypothesis H1 is stronger than a hypothesis H2 iff H1 entails H2. The hypotheses H1 “There is an infinite unobservable being” and H2 “There is a finite unobservable being” are not related in this way, and hence in the standard sense of “stronger”, H1 isn’t stronger than H2.
    However, H1 is stronger than H2* “There is a finite or infinite unobservable being”. Indeed, unless H1 has probability 1, H2* will have probability greater than H1. But that does not mean we shouldn’t accept H1: maybe we should accept both H1 and H2*.
    Moreover, any particular finite unobservable being may be more complex than an absolutely infinite one. For an infinite being has all positive attributes maxed out, and that’s a particularly simple way to be. But a finite being has different positive attributes at different levels between 0 and infinity. So any particular finite unobservable being hypothesis is apt to have greater complexity than an absolutely infinite being hypothesis.

    July 5, 2010 — 13:58
  • Mike Almeida

    Andrew,
    Let’s suppose what is meant by ‘infinite being’ can be spelled out in terms of the properties the being instantiates. To keep it simple, focus on what appears to be the property most relevant to “star-graphy”, and that’s omnipotence. But suppose it’s true that a less-than-omnipotent, but sufficiently powerful being would also explain the star writing: say, a being whose power is at least P. Here are two hypotheses:
    H1: A being whose power is at least P caused the star writing.
    H2: An omnipotent being caused the start writing.
    Certainly hypothesis (1) is the weaker hypothesis, and so more confirmed by the observation of the star writing. That’s evidence for the existence of the less-than-omnipotent being. But you’ve also got some evidence for the infinite being.

    July 5, 2010 — 15:33
  • “Certainly hypothesis (1) is the weaker hypothesis, and so more confirmed by the observation of the star writing”
    It does not follow from the mere fact that H1 is a weaker hypothesis that it is more confirmed by the observation of the star writing.
    Consider:
    H2*: H2 or there is no star writing.
    Then, H2* is weaker than H2, but it is not true that H2* is more confirmed by the star writing than H2 is. Not only is it the case that P(H2*|star writing)=P(H2|star writing), but depending on one’s probability assignments, one could have P(H2*|star writing)<P(H2*) and P(H2|star writing)>P(H2), so H2* is incrementally disconfirmed by star writing, while H2 is incrementally confirmed by it.

    July 5, 2010 — 19:40
  • Andrew Moon

    Hello Christian,
    When PvI refers to an infinite being, he is referring to God, and I think that the ‘infinite’ modifies ‘being’ and is just pointing to some of God’s qualities, e.g., omnipresence and omnipotence. He explains these qualities in the second chapter of the book.
    I think that it’s supposed to follow that the postulation of a finite being would be an infinitely weaker explanation than the postulation of an infinite being. That’s how I understand the passage. I’m not sure if it’s true.
    You’re right that an infinite being and finite being are not the only candidates for explanations for what we observe. I am, however, concerned with the specific quote I highlighted toward the end of the opening post, which is comparing the strength of explanations which appeal to an infinite being versus explanations which appeal to a finite being.

    July 5, 2010 — 23:47
  • Andrew Moon

    Alex and Mike,
    Alex wrote: “Moreover, any particular finite unobservable being may be more complex than an absolutely infinite one. For an infinite being has all positive attributes maxed out, and that’s a particularly simple way to be. But a finite being has different positive attributes at different levels between 0 and infinity. So any particular finite unobservable being hypothesis is apt to have greater complexity than an absolutely infinite being hypothesis.”
    Yeah, I was thinking of this. Didn’t Swinburne say something like this? I think that this would be the way to go to object to the position defended in that dialogue.
    Mike, if Alex is right about this, then it may not be that your H1 is the weaker hypothesis. However, I am still inclined to think that H1 is the weaker hypothesis. I am not sure why.

    July 5, 2010 — 23:52
  • Brian

    A quick comment:
    the fact that we have reasons for suspecting a being capable of star writing exists already (infinite unobservable via ontological, cosmological, and teleological arguments defended throughout the history of philosophy) make it such that the conditions for Occam’s razor do not obtain. It is only certitus paribus that the more complex proposition is cut down in favor of a simpler one, and we ought to be prima facie more inclined to believe in an infinite unobservable than a finite unobservable who meddles in our sky in our own language for no discriminable reason.

    July 6, 2010 — 7:28
  • Mike Almeida

    “H1: A being whose power is at least P caused the star writing.
    H2: An omnipotent being caused the start writing.”
    If the evidence is that there is star writing (and not merely “there is the appearance of star writing”), then H1 is no doubt better confirmed that H2. H1 simply states that something powerful enough to produce the star writing caused the star writing. The probability of H1 on the evidence has to approximate 1. Obviously, H2 does not have probability 1 (or anything close to 1) on the evidence that there is star writing. So, H1 is better confirmed than H2. I have no idea how anyone could arrive at the conclusion that H1 is about as confirmed as H2.

    July 6, 2010 — 7:32
  • Ted Poston

    Andrew,
    Jeremy Gwiazda, a recent CUNY grad, has been working on simplicity and the infinite in Swinburne. Here’s his dissertation title: “Probability, Simplicity, and Infinity: A Critique of Richard Swinburne’s Argument for Theism.” He also has a few papers in Religious Studies on this as well.
    Re Swinburne (and following Alex’s comments): Swinburne thinks having a property without any finite limit is one of the simplest ways to have a property (the other being 1, whatever that means). I have the intuition that other ways of having a property–e.g., limited power–call out for some explanation. Why is it that Jones can bench 175lbs? Limited property exemplification gives rise to contrasts in ways that infinite properties don’t.
    Re Mike: It’s interesting that you set up H1 with the ‘at least’ clause. I’m inclined to agree with you on empirical confirmation but disagree on what the balance of evidence supports once on takes into account the priors.

    July 6, 2010 — 8:01
  • Mike Almeida

    Mike: It’s interesting that you set up H1 with the ‘at least’ clause. I’m inclined to agree with you on empirical confirmation but disagree on what the balance of evidence supports once on takes into account the priors.
    Ted,
    Of course, there is no non-tendentious set of priors. This is why Rowe (in other contexts) just sets them at .5 to invite everyone to the party. Alston once told me, on the other hand, that this invites only the agnostics to the party. But, to your point on priors, there’s something like a spectrum of problems of old evidence. As the priors for theism go up, the very same amount and quality of evidence E disconfirms theism less and less until it reaches zero (when theism is certain). So you’re right that the relative confirmation might not amount to much, if the priors for theism are sufficiently high. Wykstra makes this point against Rowe in his noseeum paper and draws the deep moral: the more committed you are to theism, the less otherwise seriously disconfirming evidence will matter to you.

    July 6, 2010 — 9:09
  • Enigman

    A nice analogy might be with our instinctive postulation of infinite Euclidean space to account for our observations of the external world, since space is not a directly observable being. A weaker hypothesis would be to postulate some space that is locally at least approximately Euclidean and big enough to account for all our actual observations. But that is not as simple (not so intuitive), so it is not obvious that before the Einsteinian revolution it was even preferable (was more confirmed). Furthermore, any hypothesis of some particular finite part of Euclidean space big enough to account for all our actual observations would not even be weaker, e.g. it would imply the existence of a particular boundary to space.

    July 6, 2010 — 9:27
  • ” As the priors for theism go up, the very same amount and quality of evidence E disconfirms theism less and less until it reaches zero ”
    That depends on how you measure the amount of disconfirmation / confirmation. On my intuitive measure, a move from 0.999 to 0.998 is quite a significant move–it doubles the probability of atheism. For instance (and this isn’t a serious proposal, but it captures my intuitions), you might measure the amount of disconfirmation in a move from probability P1 to a smaller probability P2 as: P1/P2+(1-P2)/(1-P1). (This is undefined if P1=1, which seems the right way to think about that case.) In this measure, it need not be true that the same amount and quality of evidence disconfirms theism less and less.

    July 6, 2010 — 13:02
  • Mike Almeida

    That depends on how you measure the amount of disconfirmation / confirmation
    Obviously. I’m measuring it the standard way as the difference between prior and posterior probability. There are all sorts of alternatives, if one disprefers that typical measurement.

    July 6, 2010 — 13:09