Computability, Ad Hominem, Faith and Reason

Consider the function:

Π(n) = { the position of the first occurrence of n consecutive zeros in the decimal expansion of π if it exists, 0 otherwise }

For example, "0" first occurs at position 32, so Π(1) = 32; the first few values are: 32, 307, 601, 13,390, 17,534... (according to the Pi-Searcher). This function is "computable" in a sense, for a trivial reason: for any given n, it is possible to output the correct integer (because it's possible to output any integer). But this function doesn't really teach us anything — it doesn't demonstrate a necessary relationship between the input and output values.

Now consider ad hominem, one of the most popular logical fallacies. In essence, it works by citing the origin of a belief as evidence against the belief. Sometimes it's just delicate slander, but other times the attack is on the argument itself. If I say "I'm alive because Andy Warhol walks among us.", you might respond "That's ridiculous, there is no relation between those two claims — and furthermore, Andy Warhol has most certainly kicked the can..." It's fine to argue with the soundness of my argument, but the problem lies in the conclusion of ad hominem: "...therefore, you are not alive." (a negation of my conclusion). This might sound silly, but it happens constantly. This would be as ludicrous as claiming that 32, 307, 601... could not possibly be the first three values for Π(n) simply because I haven't provided an algorithm associating those values with the function definition.

This all comes back to the question of where faith is justified. Imagine I've just given you 100 arguments for the existence of God. If you go through each one and explain why the argument is unsound, you haven't disproven God's existence (that would be a case of denying the antecedent — which Popper brings up when discussing falsifiability). Unfortunately, too many atheistic arguments are based on this fallacy. For example, Bertrand Russell employs one of the most subtle forms of ad hominem:

Religion is based, I think, primarily and mainly upon fear. It is partly the terror of the unknown and partly... the wish to feel that you have a kind of elder brother who will stand by you in all your troubles and disputes. Fear is the basis of the whole thing -- fear of the mysterious, fear of defeat, fear of death...

For another example, see this previous post.

In short, there's some interesting symmetries between computability, ad hominem, denying the antecedent, falsifiability and questions about the justification of beliefs... maybe it's just me, but I think symmetries help me see more clearly.