F Rosa Rubicondior: Infinitely Impossible Gods

Saturday 9 June 2012

Infinitely Impossible Gods

Thinking more on how William Lane Craig was again caught misleading his audience, this time with a simple mathematical error, as exposed by DoctorFreed here and about which I wrote a couple of days ago (William Lane Craig's Cock-up) I began to think about how else probability theory can be used either for or against any idea. Imagine my delight when I realised William Lane Craig's blunder had lead me to understand better why his (or any other) 'eternal' god can't possibly have done what he claims it to have done.

We'll stick to the idea of a god here but of course we could be dealing with any other absurd notion for which there is no definitive evidence so we are having to try to get round it the way religious apologists need to, and for which apologetics was invented in the first place.

Now, one of the standard escape clauses apologists use when you ask them to subject their favourite god to the same tests they demand science passes, like explaining where it came from, what it's made of, who created it and whatever and wherever that came from in the first place; how gods can come from nothing when they insist that nothing can - that sort of thing - is to invoke infinity.

They simply assert that their favourite god has always existed so they don't need to explain its origins. This neatly absolves them of having to apply the same standards they demand of science and, if you fall for it, allows them to get away with a much lower standard of proof whilst you try to meet their impossibly high, and usually shifting, standard.

But, let's apply probability theory to this claim, in particular the probability that this hypothetical god could even decide to do something, let alone actually do it.

I won't bore you with an explanation of how probability works as there are several good on-line articles which do that better than I could. For example, Probability Theory and Basic Probability Theory.

Basically, to work out the probability of anything happening at any one moment in time, the maths is very simple: you simply divide 1 by the number of moments of time. Which means we need to know how many 'moments' of time there have been. We can use this to, for example, work out the 'half-life' of a radioactive isotope. We know the probability of an individual atom decaying in any one moment, so we can work out how long it will be before half of them have decayed. This if the half-life. It's an emergent property derived from chaotic mass action and an example of order emerging from chaos, but that's another story, and maybe another blog.

Let's define a 'moment' as the smallest unit of time which can exist - Planck Time - because that really is the only sensible unit we can use. All our other units, like seconds, hours, years, etc all depend ultimately on how fast earth rotates on it's axis and how long it takes to orbit the sun so they aren't really universal measures. We could take something like the vibration frequency of certain atoms, like atomic clocks do, but we might just as well use Plank Time, the result is the same.

Planck Time, by the way, is 10-43 seconds. In other words, there are 10-43 Plank Time units in ever second, so they are very small, but not zero. This means there have been a very large but finite number of Planck units of time since the start of the known universe.

Incidentally, to calculate the probability of the Big Bang occurring when it did is very simple. Since time also started then, at the moment of the Big Bang, there had only been 1 Planck unit of time, so it's probability was 1 (1/1), i.e. certainty. Of course, quite quickly there were lots more Planck units of time and the probability of another Big Bang happening in this universe rapidly became very small and is always getting smaller (less likely), which could explain why there has only been one so far.

In fact, thinking about it, since the universe was already 1 Planck unit old the instant it came into existence it could never get back to the initial conditions to happen again.

So, what about this hypothetical eternal god existing for eternity? Well, for that god, there will have been an infinite number of Plack units of time and there will be an infinite number more (which means it gets two infinities, presumably, one stretching into the past and one stretching into the future).

How do we calculate the probability of this god doing anything, or even deciding to do anything at any one of these infinite number of moments in time? Simple, again, we divide 1 by the number of moments, in this case, infinity (or two infinities).

So, the probability of an eternal god doing anything, is : \[ \frac{1}{\infty} \] (or, if that's all Greek to you, 1 divided by infinity) which equals zero, in other words, it's impossible.

Oops! So, rather than the invocation of an eternal god being the escape clause and easy cop-out religious apologists were hoping for, they've blundered into a mathematical claim that it's impossible for their god to have ever done anything, and always will be.

Nice one.

To escape from that, apologists now need to claim eternity started at some point in the past and will end at some point in the future. That will mean they need to explain what caused their god and how it came from nothing; something they've declared to be impossible.

As the infinity symbol shows, when you start going round it, you get nowhere and have to keep starting over.

As Thomas Dixon said in Science And Religion: A Very Short Intoduction, you have to pity the poor theologians. ("Science and Religion. A Very Short Introduction", by Thomas Dixon IBSN 978-0-19-929551-7)


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  1. Rosa,

    I enjoy collecting logical arguments that expose the fallacy of supernatural claims. But, it seems to me that there is a flaw in this argument of 1/infinity. Briefly, this leads to the conclusion that NO events can occur in a universe of infinite time. Another implication is that in a finite-time universe of say 100 Giga years events are half as likely to occur as compared to one of 50 Giga years. But what meaning can be assigned to such statements. After all events DO occur at one time or another.
    Am I missing something ?

    1. >Briefly, this leads to the conclusion that NO events can occur in a universe of infinite time. <

      Absolutely. Ergo, the universe does not exist in infinite time.

      >Another implication is that in a finite-time universe of say 100 Giga years events are half as likely to occur as compared to one of 50 Giga years. <

      No. At the moment of it's creation, the universe had existed for only 1 Planck Time, hence the probability of it occurring then was 1 (certainty).

      > After all events DO occur at one time or another.<

      Indeed. Ergo time isn't infinite.

      >Am I missing something ?<

      See above.

      Your approach should be to dispute the maths, not the consequences of it. There is no obligation on reality to be amenable to human intuition.

  2. Very well, I will attempt to dispute the math.
    (Reference: http://mathforum.org/library/drmath/view/66800.html)

    Estimating probability with 1/infinity is true only if a uniformly random probability distribution is assumed. But what if the physics governing an event are not random at all ?
    Imagine an infinite-time universe (both past & future) where physical constants fluctuate cyclically or randomly. An event might be triggered when the the gravitational constant reaches a threshold for example. Then the probability at some Planck times is non-zero.
    I agree that anything existing for infinite time without a beginning makes NO sense.
    But the 1/infinity is not convincing enough; unless my above scenario is wrong.

    I could be completely wrong: I also found a cosmology paper that predicts a finite end to
    time for the very reason that an infinite-time universe would make it impossible to calculate probabilities.

    1. >Estimating probability with 1/infinity is true only if a uniformly random probability distribution is assumed. <

      In the absence of any evidence to the contrary, why assume the probability distribution of a god deciding to create a universe would be non-random? What non-random event would trigger that decision?

  3. Pretending for a moment that there IS a god, maybe he was once a 'one-of-us' in another previous universe. He's having his turn.

    Likewise, we all go on to become gods, Oh, only the good ones of us.
    thats a lotta gods, and a lotta universes...


    All gotta fit in somewhere, or do we each get our own little corner of the Great Nothingness in which to build our individual universes?

    Can I have one with 7 dimensions please?

  4. I still haven't received a sensible answer when I've asked believers of an infinite god if that god could have a first memory...

    1. I love the way theists think invoking infinity gets them out of a tight spot, only to find it bites them in the bum. :-)

  5. 1/∞ does not equal 0, because that implies ∞ x 0 = 1. Anything x 0 = 0 even infinity. Admittedly 1/∞ is infinitely small but not zero nonetheless.

    1. The distinction is so small and subtle as to make no practical difference.


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