Intro:
Greetings readers. I'd really like some help on this from the math gurus out there on xanga and/or Planet Atheism. Below is my latest response to atheist historian, Richard Carrier on his blog post, "Are We Doomed?" It's a bit of a tangent from the original post, but it relates to issues dealing with William Lane Craig's Kalam argument. But it's even a tangent from that, too.
I think I'm following along in concept just fine, but at some level Rick is appealing to his math proofs (which can be found in his latest comments in that link above) to show I'm breaking some rules and I'm not entirely sure what's going on there other than what I can infer from the things I'm pretty sure I do understand. My position is that infinity is infinity is infinity, and Rick's position is that one infinity can be a larger quantity than another and so relative proportions actually add up somehow in the ultimate sense depending on the thought experiment.
Some help from an outside party would be appreciated. Granted we've been over this before (link), but it never hurts to triple check.
Rick,
Thanks for responding. To recap, you had said:
But to give you an example of a counter-argument, if you are falling toward an infinite platform, and one out of every ten planks on that platform is missing, and the odds of hitting any one plank (missing or not) across that whole infinite expanse is equal, then it doesn't seem plausible to argue there is a 50% chance you will fall through the platform. Surely the odds are 1 in 10.
You said "surely" to appeal to my common sense, but common sense doesn't deal with the actuality, only the appearances. To make the sensibility "work" I had to assume you meant this:
If someone is falling from a finite height, there's only so much surface area of an infinite plane you are actually dealing with. And the density (and consistency) of the plank ratio matters. [...] You can put all the planks to one side of the infinite plane or do whatever, and it shouldn't effect the probability in any way shape or form. [...] As I see it, it is straight 50/50 and the idea that one infinite can be "more" than another is incoherent.
But, you had something more creative in mind:
Actually, you can just change the laws of physics any way you want to make the stipulated conditions hold, e.g. have it so people don't fall in a fixed direction but any direction at random that is below the parallel. [...] The point is not the incidentals of physics, but what happens to probabilities when you change ratios in an infinite series.
If we allow for this omni-fall, which is fine, my point is reinstated. There are two infinites on the infinite plane beneath you no matter how they are arranged and you can hit *any* plank space *anywhere* on it. Your chances on balance, since that's actually what we have to deal with despite appearances, is still 50/50.
One of the few things that has been proven (deductively, formally proven) for transfinites is that there are infinities that are objectively bigger than others. Cantor demonstrated this, and thus originated the concept of cardinality.
Cardinality may be the proper way to describe the 9/10 *arrangement* but cardinality does not magically make one infinity more or less than another infinity. Mathematicians suck at communication and "bigger" in this sense is not the bigger you need to mean anything relevant here.
However, producing a one-to-one correspondence is not by definition the same operation as producing a ratio, thus it is a conflation fallacy to convert the one into the other.
A 1:1 correspondence is a freaking ratio, by definition. How exactly is the "law of identity" a "fallacy of conflation"?
If in N for every x(f) there are ten ~x(f)'s, and you extract one item from N and it is a ~x(f), you can pull one x(f) out of N to pair with it. If you repeat the operation indefinitely, you will exhaust all the x(f)'s by the time you've exhausted only a tenth of the ~x(f)'s, but you cannot do the reverse, i.e. even after infinite operations, there will still be ~x(f)'s left over, but no x(f)'s.
So you think you can prove that you can somehow "run out" of an infinite quantity, one by one? I'm completely baffled.
It works the same for transfinites, as long as you use proper definitions and rules, which actually forbid tricks using "one-to-one" correspondence as a device to dink the quantities.
Okay, so there's some magical rule in math that says I'm "not allowed" to take 10 Easter eggs from each of an infinite number of Easter egg baskets and spread the eggs out so that there is only one egg in each basket? I'm not going to have nine eggs left over per basket because I'm going to have completely spread them out...to infinity. I could set 9 eggs beside every basket if I wanted to, but I don't have to because the arrangement of infinities is meaningless (though we may feel the need to articulate the specific arrangements). The end quantities are all that matter and none of the rearrangement stuff changes them. One infinity equals another infinity. What you call "dinking" (and "cheating") I call "demonstrating you are wrong."
Outro:
But who knows. Maybe I'm the one that is wrong. Rick has put a lot of effort into his response that is several comments long and unfortunately I absolutely disagree with him as much as he does me. I've gone back and forth with him on other issues in the past in ways we sharply disagree to no avail and I guess we're both quite stubborn. So I was hoping that someone out there might be able to shed some light on things and take us in a different direction. Or at least just show me how I'm wrong in a different way than Rick is attempting to.
Rick takes forever to respond to comments so there's plenty of time between cycles for me to reevaluate my response here.
Ben
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