If this wasn't clear: responses would be much more helpful than up/down votes.
How do you deal with Munchhausen trilemma? It used to not bother me much, and I think my (axiomatic-argument based) reasoning was along the lines of "sure, the axioms might be wrong, but look at all the cool things that come out of them." The more time passes, though, the more concerned I become. So, how do you deal?
Fine, Eliezer, as someone who would really like to think/believe that there's Ultimate Truth (not based in perception) to be found, I'll bite.
I don't think you are steelmanning post-modernists in your post. Suppose I am a member of a cult X -- we believe that we can leap off of Everest and fly/not die. You and I watch my fellow cult-member jump off a cliff. You see him smash himself dead. I am so deluded ("deluded") that all I see is my friend soaring in the sky. You, within your system, evaluate me as crazy. I might think the same of you.
You might think that the example is overblown and this doesn't actually happen, but I've had discussions (mostly religious) in which other people and I would look at the same set of facts and see radically, radically different things. I'm sure you've been in such situations too. It's just that I don't find it comforting to dismiss such people as 'crazy/flawed/etc.' when they can easily do the same to me in their minds/groups, putting us in equivalent positions -- the other person is wrong within our own system of reference (which each side declares to be 'true' in describing reality) and doesn't understand it.
I think this ties in with http://lesswrong.com/lw/rn/no_universally_compelling_arguments/ .
Now, I'm not trying to be ridiculous or troll. I really, really want to think that there's one truth and that rationality -- and not some other method -- is the way to get to it. But at the very fundamental level (see http://lesswrong.com/lw/s0/where_recursive_justification_hits_bottom/ ), it seems like a choice between picking from various axioms.
I wish the arguments you presented here convinced me, I really do. But they haven't, and I have no way of knowing that I'm not in some matrix-simulated world where everything is, really, based on how my perception was programmed. How does this work for you -- do you just start off with assumption that there is truth, and go from there? At some fundamental level, don't you believe that your perception just.. works and describes reality 'correctly,' after adjusting for all the biases? Please convince me to pick this route, I'd rather take it, instead of waiting for a philosopher of perfect emptiness to present a way to view the world without any assumptions.
(I understand that 'everything is relative to my perception' gets you pretty much nowhere in reality. It's just that I don't have a way to perfectly counter that, and it bothers me. And if I did find all of your arguments persuasive, I would be concerned if that's just an artifact of how my brain is wired [crudely speaking] -- while some other person can read a religious text and, similarly, find it compelling/non-contradictory/'makes-sense-ey' so that the axioms this person would use wouldn't require explanation [because of that other person's nature/nurture]).
If I slipped somewhere myself, please steelman my argument in responding!
Oops, misread that as sum(1/(2n))[1:infinity] (which it wasn't), my bad.
Hate to nitpick myself, but 1/2+1/4+1/8+... diverges (e.g., by the harmonic series test). Sum 1/n^2 = 1/4 + 1/9 + ... = (pi^2)/6 is a more fitting example.
An interesting question, in this context, is what it would mean for infinitely many possibilities to exist in a "finite space about any point that can be reached at sub-speed of light times." Would it be possible under the assumption of a discrete universe (a universe decomposable no further than the smallest, indivisible pieces)? This is an issue we don't have to worry about in dealing with the infinite sums of numbers that converge to a finite number.