Maths versus Philosophy: Picking teleological arguments apart

Christians often point to teleological arguments as evidence of God’s existence, based on what they perceive as signs of design and purpose in the universe. In essence, they see the chances of the universe being able to sustain intelligent life as so small that there must be design, and therefore a designer, underlying it. And there’s some truth in that – as far as we can tell, the precise conditions and physical laws of the universe do need to be in an extremely precise range. So the odds are massively in favour of a designer, right?

Well, not necessarily – this is an example of the Prosecutor’s Fallacy, as it attempts to draw a conclusion based on the calculated odds of a particular event occurring, not the conditional odds given the known facts. So what’s the difference? With apologies for gratuitous maths, I’ll try to explain with a simple example.

Let’s say you’ve just been screened for a fatal disease, and you’ve been told that the error rate of the screening is 1 in 100. (In reality, there are usually different probabilities for false positives and false negatives, but for the sake of the example, let’s say this error rate applies in either direction.)  So given that, if get a letter saying that your screening test indicated that you have the disease, you’d probably start dusting off your will. That’s a reasonable reaction, but it may be premature.

What you don’t know – but is vital to interpreting your test result – is what the odds are of anyone actually having this disease. I expect you’re saying to yourself, “What difference does it make? I know that the test’s right 99 times out of 100, and that’s all that matters.” And you’d be half right. The rate of error is very important to understanding the result, but it isn’t enough on its own.

Let’s say that you were screened because you’re part of a high-risk group with a 1 in 10 chance of having the disease. I’ll attempt to deal in whole numbers, rather than probabilities, in the hope that it’ll make things easier to understand. Remember, the test will give the wrong outcome for 1 in 100 people, so over the course of a million screenings you’d expect roughly these numbers:

In this situation, we know that the screening came out positive, so you’re one of 108,000 in that situation. 99,000 of those will prove to have the disease, meaning that there’s an 11 in 12 chance (91.67%) that you have it. OK, you’re saying, it’s not the 99% I instinctively came up with, but that small improvement in the odds doesn’t exactly offer much comfort. Fair enough. But what if this is routine screening of the general population, and only 1 in 10,000 will actually have the disease?

Now the chance of having the disease is only 99 in 10,098, which equates to 1 in 102, or just less than 1%. Despite a positive result in your screening, you would actually be overwhelmingly more likely not to have the disease – counterintuitive, but true nonetheless, and something that is often taken into account when deciding whether to run screening programmes. So the same error rate can mean very different things: as the basic odds of having a disease lengthen, so do the odds of anyone who tests positive actually having it.

This is directly applicable to the teleological argument: We believe we can calculate the odds of a suitable universe occurring by chance, but that’s only half the story. If we want to know the odds that it occurred by chance, given that we know we live in a suitable universe, we also need to know the relative probabilities of a designer/creator existing or not existing. As that’s the very question teleological arguments are meant to answer, it’s not very useful in drawing a conclusion.

Even if the odds of the universe coming about by chance are vanishingly small, it doesn’t tell us anything about the existence or otherwise of God. We have no way of putting any kind of range on metaphysical questions like God’s existence, so that probability may be much, much smaller than the chance probability of an inhabitable universe, or it may even be zero. Conversely, it may be quite high – the point is that we need to know in order to make sense of the parameters we can actually calculate.

So after all that, we’re back to square one.

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About Recovering Agnostic

I'm Christian by upbringing, agnostic by belief, cynical by temperament, broadly scientific in approach, and looking for answers. My main interest at the moment is in turning my current disengaged shrug into at least a working hypothesis.

6 responses to “Maths versus Philosophy: Picking teleological arguments apart”

  1. Uncle Tree says :

    Hello there! 🙂

    I’m not here to argue, and since I’ve been in your shoes, so to speak, I can easily understand your questions, your doubts, and your conclusions. I’ve also intensely enjoyed the ongoing debate back yonder a ways. Just so ya know. 😉

    I’m siding with the designer crew on this one (because of personal reasons – due to my experiences), but until we know whether or not The Universe is infinite, I’ll have to live with my belief, even if it acts as a thorn in my rational side. The infinite argument goes way back, so I’m somewhat surprised that the idea never comes up when this subject is up for debate. We just don’t know.

    Speculating should be fun, and we should try to imagine and guess for ourselves. Why not create solutions, and then do the math? There’s no such thing as an evil number. They are merely representations. We decide what’s what, and then we find out that Life is a game you can’t win, nor lose. Perhaps..

    🙂 Peace to you today, UT

    • Recovering Agnostic says :

      Sure. I concluded that there’s no way of drawing any conclusion about the origins of the universe on the back of teleological arguments, and I mean that – in either direction. This is just my response to those arguments being advanced as evidence for God, explaining why they don’t really stack up.

      That’s not to say there definitely isn’t a God at the root of it all (although I have my views on that), but the standard arguments don’t demonstrate it at all.

  2. unklee says :

    G’day RA, I hope you are expecting a response from me! : )

    I’m glad you have tried to use statistics here. Your maths is based on Bayes Theorem (the Wikipedia reference says this) and I have played around with Bayes Theorem a little myself (see this reference).

    I think your statistical analysis of the medical examples is correct (I worked it out using Bayes Theorem and got the same results) but I think your explanation missed the main point. Sure, in the second example, it is true that despite the positive result, it is still unlikely you’ll have the disease. But what you didn’t point out was that the positive screening did increase the probability that you have the disease, from 1 in 10,000 (the figure for the population as a whole) to about 1 in 100, an enormous increase in probability (a hundredfold increase).

    For this is what Bayes Theorem does. It takes a starting probability and modifies it in the light of new knowledge. This is important when we look at the universal fine tuning example, and unfortunately you have made a misleading statement. You say:

    “we want to know the odds that it occurred by chance, given that we know we live in a suitable universe, we also need to know the relative probabilities of a designer/creator existing or not existing”

    But there are two mistakes here, I think.

    1. “given that we know we live in a suitable universe” is not relevant. It is not a term in the Bayes equation, so it cannot affect the outcome of the calculation. And logic shows us this isn’t relevant. Imagine I enter a lottery and win it. The odds of my winning were (say) 100,000:1 against when I bought the ticket. But once I know I have won, the odds of my winning do not magically change, they remain the same – the only thing that has changed is that I now know I did in fact win, against the odds.

    2. “we also need to know the relative probabilities of a designer/creator existing or not existing. As that’s the very question teleological arguments are meant to answer, it’s not very useful in drawing a conclusion.”
    This too is mistaken. The information we need is an initial estimate of the probability of God existing, so we can use the information about fine-tuning to modify that probability (just as we modified the initial 1 in 10,000 to 1 in 100 with the additional information from the screening).

    So, let’s use Bayes Theorem to look at a few possibilities. We need three estimates – (A) the initial probability of God before we look at the information, (B) the probability that the universe would be well-designed if God did exist, and (C) the probability that the universe would be well-designed if God did not exist.

    If you are an agnostic, you would presumably assess A = 0.5. Who can say what B is, but let’s assume 1 in a million (surely if God exists, there is some reasonable chance he might create a finely tuned universe). C is a very large number according to the cosmologists (Lee Smolin estimates 1 in 10^229 for stars to exist, Roger Penrose estimates 1 in 10^10^123 for our low entropy universe), but let’s use a much smaller number, say 1 in 10^100. Put these numbers into Bayes equation and you find the answer (the probability that God exists) is effectively 1, or certain!

    We can vary these numbers quite a bit, but the probability that the universe is the way it is without God and by chance remains so high that the probability that God exists remains effectively 1. The only way you can make it work out differently is either to distort the initial probability of God’s existence and the probability he might create to an incredible degree, or to contradict the cosmologists.

    Now that isn’t the whole story, but it is enough for one comment. Your use of probability has “proved” exactly the opposite of what you intended. I encourage you again to read the references, study Bayes Theorem, do the maths and tell me what you think.

    Best wishes.

    • Recovering Agnostic says :

      “I’m glad you have tried to use statistics here” – Tried? Ooh, thank you so much! That’s not really a good start to an allegedly constructive discussion.

      But Bayes is powerless here – you can’t iteratively rerun the creation of the universe, and metaphysical issues don’t lend themselves to the creation of well-informed priors, so whether you’re an objectivist or subjectivist, there’s nothing to start from. Any use of Bayes’ Theorem is only as good as its priors, and when they’re junk like this, you’re just whistling in the wind. I love the idea of putting a probability of 0.5 on God, though. It’s like saying that you’ve got a 50-50 chance of winning the lottery – either you will or you won’t!

      Face it, any idea that you can meaningfully calculate the probability of God’s existence based on the observed characteristics of the universe is flawed from the start. You and many other apologists make a big thing about the calculated odds of the universe existing with qualities that make it suitable for life, but I’ll repeat, we know the universe has those qualities because we exist. If it didn’t, we wouldn’t be asking the question. So the important issue is the conditional probability of God’s existence, given that the universe is capable of supporting life.

      I’m tempted to just return the compliment and advise you to read up on the Prosecutor’s Fallacy, but I’ll flesh it out a bit for you. Some multiverse hypotheses suggest that mini universes keep popping up and either surviving and expanding or disappearing as not viable, and that the number of universes created and destroyed in this process is many, many orders of magnitude greater than the apparently huge numbers you mention in your calculations. I have no idea whether that’s correct or not – metaphysical issues don’t suddenly become quantifiable when they’re favourable to my argument – but for the sake of argument, and just to humour me, assume it is, and that the total number of possible universes is 10^10^10^7 (after Linde and Vanchurin here). Do you see how that would affect your calculations?

  3. unklee says :

    ““I’m glad you have tried to use statistics here” – Tried? Ooh, thank you so much! That’s not really a good start to an allegedly constructive discussion.”
    Sorry, it wasn’t intended as a put-down.

    “Face it, any idea that you can meaningfully calculate the probability of God’s existence based on the observed characteristics of the universe is flawed from the start. “
    Yes, but the calculation can reveal things nevertheless, as you have illustrated. The important point for me is not the initial probability (that is why I suggested several values for that), but the amount that the new information changes the initial setting. And on any numbers I have seen, that is a big difference when considering the design question (i.e. universal design increases the probability that God exists).

    “for the sake of argument, and just to humour me, assume it is, and that the total number of possible universes is 10^10^10^7 (after Linde and Vanchurin here). Do you see how that would affect your calculations?”
    I’ve read that the total number of universes in string theory is said to be 10^500, obviously a much smaller number than the one you quote. But I note that this article says 10^10^10^7 was rejected and the number became 10^10^16.

    Now Roger Penrose has estimated the probability of a low entropy universe like ours at 1 in 10^10^123 (I didn’t use this before because I didn’t need to), a bigger number than Linde and Vanchurin’s actual answer, though not as big as the number you quote but which they rejected.

    So of course these numbers affect the calculations. If your number was right, we have quite sufficient universes to do the trick. But since your number wasn’t what they said, there iaren’t sufficient universes, and my argument holds.

    My conclusion is that the maths approach is too speculative to be all that useful, but it was you who raised it. So I played along, and it remains true, on the best numbers we have, that the maths works against what you have said, and for what I believe. If you thought it worthwhile making a point, surely it is worthwhile for me to correct that point?

    Best wishes.

  4. bigstick1 says :

    I like your blog. However, I am not going to try to prove that a unknown phantom exists or doesn’t exist. Too me it really doesn’t matter as humans have never seen the phantom nor will they ever in their lifetime. However, religion is a bunch of bunk based on superstitution, contradictions, and the constant ever changing see this is what it meant.

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