a simple mathematical proof that god most likely does not exist

A SIMPLE MATHEMATICAL PROOF THAT GOD MOST LIKELY DOES NOT EXIST

John [email protected]

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ABSTRACT

Given the basic theistic assumptions of God’s omnipotence, probability theory, and Bayes’ Theorem, we can demonstrate that theism is almost certainly infinitely improbable.

INTRODUCTION• P(h|e&k) = P(e|h&k) x P(h|k) / P(e|k)

• The probability of the hypothesis (theism) (and background knowledge), given the evidence (our universe, and background knowledge), is equal to the prior (a priori) probability of the hypothesis of theism (and background knowledge), multiplied by the probability that this evidence (universe) would be observed (given theism and background knowledge), divided by the prior probability of the evidence itself (and background knowledge).

INTRODUCTION

• Disclaimer:

• I am aware that P(e) can be expanded. since h v ¬h:

• P(e|k) = P(h|k)P(e|h&k) + P(¬h|k)P(e|¬h&k)

• Let’s omit the derivation of this, see Hacking p70

MUTUAL SUPPORT

• If h, the hypothesis of theism, renders e (our universe) more likely, it follows that e (the existence of a life-capable universe) supports the hypothesis of h (theism).

• This is prima facie plausible, and indeed, it is more or less what Swinburne (2004) argues, in brief.

MUTUAL SUPPORT

• P(e) x P(h|e)/P(h) = P(e) x P(e|h)/P(e) [Multiply throughout by P(e)]

• P(e) x P(h|e) / P(h) = P(e|h) [Result 2]

MUTUAL SUPPORT

• P(e|h) > P(e) [Given]

• P(h|e) > P(h) [Calculated]

• Therefore P(e|h) > P(e) ⊃ P(h|e) > P(h)

MUTUAL SUPPORT• In short, this means that where the probability of the

likelihood of this universe goes, so goes the probability of theism.

• Since, according to theists, this universe is highly likely to exist, given God, it follows that this good universe supports the truth of theism, or stands as evidence for theism.

• The probability of P(e|h) is close to or very similar to the probability of P(h|e).

MUTUAL SUPPORT

• Proof:

• If we chart this with the following values obtained by substituting actual figures into Bayes’ Theorem, using P(h) = 0.5 and P(e) = 0.55 for both graphs, and varying only P(e|h), we get the following graph.

MUTUAL SUPPORTTable 1. With P(h) and P(e) neutral to P(e|h) P(h|e)

0 00.05 0.045

0.1 0.0910.15 0.136

0.2 0.1820.25 0.227

0.3 0.2730.35 0.318

0.4 0.3640.45 0.409

0.5 0.455

P(e|h) P(h|e)0.55 0.5

0.6 0.5450.65 0.591

0.7 0.6360.75 0.682

0.8 0.7270.85 0.773

0.9 0.8180.95 0.864

MUTUAL SUPPORT

MUTUAL SUPPORT

• Claim T: If we assume that we agree on the prior probability of h and e, then it follows that the posterior probabilities not only are directly proportional, but they track each others’ values closely, as shown above.

THE REFUTATION OF THEISM

• 1. God is omnipotent [By definition, he can create anything logicallypossible]

• 2. ∴ ne = ∞ [1: ∃ an infinite number of possible creatable universes]

• 3. ∴ P(e|h) = 1/∞ [From frequentism, (2)]

THE REFUTATION OF THEISM• Frequentism: that probability is a measure of frequency or how

often a datum turns up in data. An infinite range means that any item in the infinite range has a frequency of 1/∞. This means that our universe, given theism, is infinitely improbable (2).

• The cosmological argument for the existence of God claims precisely the opposite (Swinburne, 1991, p119, p135), that a benevolent God would likely create an ordered universe such as ours (Swinburne, 1991, p144, p299).

THE REFUTATION OF THEISM• However, this argument has a crucial flaw, in that it does

not give solid reasons why God should have selected precisely this universe out of an infinite number of slightly different universes. Divine Arbitrariness.

• At most, the cosmological argument can claim that God will select a “best type” of universe from a range of “best types”, above a minimum level of goodness. (Swinburne, 2001, p79; 2004, p102, pp131-2).


• There are, Stenger tells us, at least 10^500 combinations of fundamental forces that could lead to habitable universes (Stenger, 2011, p23).

• Theists themselves say that our universe is vastly improbable.


• 6. P(h|e) ≈ 1/∞ [From T, and from (3, 5)]

• 7. ∴ P(h|e) ≈ 0

• This means that theism is infinitely improbable; its posterior probability is infinitely close to zero.

OBJECTIONS• 1. Deny Step 6. P(e|h) does not equal P(h|e), so

P(h|e) might be much greater than P(e|h).

• Response (a): P(h|e) and P(e|h) track each other’s values; they are proportional and close in value (claim T). This is because if P(h|e) is highly probable, say 0.9, i.e. if God likely exists, then it is likely that God would make just this universe, i.e. P(e|h) will be close to 0.9 too. A theist must accept this point. And the mathematics is proof.

OBJECTIONS• But what if P(e) depends on the evidence,

and therefore cannot be held constant? TPR says it does: P(e) = P(h)P(e|h) + P(¬h)P(e|¬h)

• Response (b): Ok, we can graph that too. We’ll see that at the low end of the scale, for P(e|h) ≈1/∞, that P(h|e) is still infinitessimal, and δ is maximum 0.33.

OBJECTIONS

Green = P(h|e) ; Blue = P(e|h)

note:End-points

touching at low end

δ

OBJECTIONS• Suppose we argue that P(h) and P(¬h) might both be low, and P(¬h) is

lower than P(h).

• Response (c): Firstly, what alternatives are you proposing to theism and materialism? Because claiming that P(h) and P(¬h) may both be low entails that the probability space is filled by other possible hypotheses. Polytheism? Pantheism? Swinburne (p147) rejects these.

• Response (d): P(e) depends on h and/or ¬h, and P(h) + P(¬h) = 1.0 due to sum of probabilities. Meaning if P(e|h) is low, and P(h|e) is low due to mutual support, it follows that P(¬h) must be high — in fact, exactly (1-P(h)), in our case 0.999…

OBJECTIONS• 2. Deny Step 2. God might have a much smaller than

infinite set of universes to choose from.

• Response: yes; he’d not choose vastly evil universes. However, that means we have at most a probability of 1/2 x ∞ which is still infinitessimal.

• Even if we go with Stenger’s value of 10^500 possibly actualisable life-capable universes, it still results in an infinitessimal probability for theism. See the following graphs.

OBJECTIONS• For 2 universes, P is 0.5

• For 128 universes, P is 0.0078

• 512: 0.002

• 10^6 : 0.000001

• 10^500 : 0.000…500 x … 1

Table 2No. of possible creatable universes N

P(e) = 1/N P(h) ≈ P(e)

1 1.00002 0.50004 0.25008 0.1250

16 0.062532 0.031364 0.0156

128 0.0078256 0.0039512 0.0020

OBJECTIONS

At the top of the graph,theism is true and our universe is certain.

However, it drops off rapidly as we gain extra universes.

As you can see, our universeis almost impossible after 150 universes.

OBJECTIONS• 3. But because there are an almost infinite number of universes under

Multiple Universe Hypothesis, materialism is also improbable.

• Response: The probability of our specific universe being found in the range of all parallel universes is low, however: (a) The Anthropic Principle entails that we would exist in our universe (b) If there is a large range of actual universes, their being parallel and large in number does not entail that our universe is unlikely. In fact, quite the opposite; it guarantees that our universe would exist somewhere within the parallel universe ensemble. In the theistic case, the universes are only possible, and there is only one universe actualised by God, and the probability that he would actualise just that one universe, is extremely low.

A SIMPLE MATHEMATICAL PROOF THAT GOD MOST LIKELY DOES NOT EXIST

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a simple mathematical proof that god most likely does not exist

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