Continuing my exciting following of the God Delusion, chapter 3 concerns common arguments for god’s existence and rebuttals. I quite liked this chapter, although anyone with an interest in the subject will already have heard the main arguments listed here.

The most interesting of these arguments are the ontological proofs, which are clever mathematical/logical (I’ll treat these two words as interchangeable in this entry) arguments which proceed as follows:

1) Imagine a super perfect who cannot be any more super-perfect

2) Suppose said god does not exist

3) Said god would be even more perfect if he did exist

(3) contradicts (1), meaning (2) lead to a contradiction so it must be false. Therefore god must exist.

Dawkins ridicules this a bit, which I find pretty weak, because he can’t actually find a flaw in it. If you study maths at all you’ll see proofs like this all the time, it’s a perfectly valid proof and technique, although Dawkins seems to dismiss it as immature. Even so, this proof isn’t awfully convincing. I think the problem with this particular one, which makes it distinct from ones we are happy to see as valid, is that we are granting a mathematical proof the power to tell us things about our world. This is not what they are supposed to do. They are supposed to tell us things about a mathematical system. Many people think that maths is somehow woven into the fabric of the universe, but I think this is just a fairly naive Platonist interpretation.

The result (god does exist) is actually contradicted again by the fact that being ‘perfect’ in every way is impossible. One cannot be perfectly evil and perfectly good, and if you’re not both, you’re not perfect. Nobody claims god is perfectly evil (of course), but they do claim he is omniscient and omnipotent, and these two properties seem necessary for perfection. These two properties also imply a contradiction: if god is omniscient he already knows the course of the universe which means he cannot later change it, which means he is not omnipotent (as mentioned in the book). As does perfectly merciful and perfectly just: just implies punishing everyone as they deserve, merciful implies being a bit soft and not (as mentioned on Wikipedia). The fact that our system apparently allows two contradictory theorems shows that there’s something a bit wrong. [Or actually that’s being a bit generous. Since overall perfection is contradictory and the above argument depends upon it, the argument is vacuous anyway. Although you can loosen your definition of perfection]

I think the actual fallacy in these arguments is in the word ‘imagine’. If you **imagine** a god who does not exist then **you’re** not imagining the best god possible. I can imagine an elephant that flies, but that doesn’t bring it into existence, not even if I set up an argument analogous to the above where its nonexistence is contradictory … imagine a super perfect elephant, suppose elephant lacks the ability to fly, uh oh then it’s not perfect so it must be able to fly, now suppose the elephant doesn’t exist, uh oh then it’s not perfect so it must exist. So where’s my flying elephant? Just for geographical clarity I can define that ‘being in my room’ is necessary for perfection. And yet still no flying elephant. An exercise to the over-enthusiastic reader is to investigate what happens when you want to introduce a second god who’s better than the first.

And this basically highlights why this stuff works in maths and not in real life: because maths is pretty much imaginary. We can imagine something, whether it’s the number 14, the square root of minus one, a circle or a universal Turing machine, as long as we can define it in words or symbols, it ‘exists’ as much as it needs to for it to be a valid, usable mathematical entity. That doesn’t mean it really physically exists or that it needs to; you can argue that ’14’ exists by giving me 14 apples if you want^{1} but you will find it more difficult to show that (-1)^{1/2} exists because it’s a nonsensical operation on anything we can show physically (yet we can handle it just fine in maths), and you would definitely have trouble physically showing a number greater than the number of atoms in the universe^{2} really exists, but it would still be perfectly valid to use it. My flying elephant is imaginary so it exists and is usable as far as the proof is concerned, but the proof is not able to reach out and effect a flying elephant into the real world.

Therefore a proof like this can assert the existence of something and be entirely correct logically, but have no link to real life, hence you have to be a bit sceptical when someone uses supposedly infallible reasoning to ‘prove’ some less than obvious statement about the universe without worrying about empirical evidence; it’s a clear sign they have a naive view of maths/logic.

I believe this also addresses Gödel’s stronger and more rigorous ontological proof, but it’s a bit cryptic and I haven’t sat down and gone through it.

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1. I’d say this was rather missing the point, but we define natural numbers in terms of sets so maybe not. Although technically numbers are defined in terms the empty set so maybe it is. Actually this was a really big problem: what is a number? we have all these things and we never said what they actually were. How can our maths be truly rigorous if we’ve not addressed what these things are? The solution is bizarre, clever, and thoroughly brilliant, we define them in terms of NOTHINGS:

0 = {} [the brackets denote a set, and its contents are separated by commas. In this case, there is nothing between the brackets because the set is empty and we call it the empty set]

n+1 = n **U** {n} [the joining together of n and the set containing n,

e.g. {0, 1} **U** {2, 3} = {0, 1, 2, 3} ]

so,

0 = {}

1 = {} **U** {{}} = {{}} = {0}

2 = {{}} **U** {{{}}} = {{}, {{}}} = {0, 1}

3 = {{}, {{}}} **U** { {{}, {{}}} } = {{}, {{}}, {{}, {{}}}} = {0, 1, 2}

and so on. Obviously it’s easier if you start substituting in the numbers rather than keeping the indecipherable lists of brackets, but I thought it was interesting to highlight that each natural number is defined by clever arrangements of nothing. This should convince you that maths is something quite separate from real life.

2. Assume this is finite. Even if it’s not, it is sufficient that the statement would be correct if the number was finite.

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