| Subject: Finally.... Tristam meets his doom.... (bleah) |
Author:
Duane
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Date Posted: 08/31/04 5:56am
Wade:
Ugh. I've been on-and-off thinking about this, and I think I've finally located a deficiency in the argument. The deficiency is mathematical in nature, and I'll get to it, but first I wanted to bring something up:
I've read most of your and Damocles's dialogue about this, and I've noticed that some of it focuses on "absurdity."
>Wade said:
>Is it? You be the judge. The essence of the argument is >that an infinite past implies an absurdity and therefore >cannot be correct.
You of all people should recognize that "absurdity" does not equal disproof. "Absurd" means that you can't understand it. If you claim something to be "absurd," and then say "because I declare this to be absurd, I have disproven it!" you are arguing from the standpoint of proof/disproof by personal incredulity. Just because the claims of quantum mechanics seem absurd to my mom doesn't mean she's disproven Einstein, Heisenberg and Planck!
In doing some background reading, I came across the original statement of the paradox - Bertrand Russell was just trying to illustrate interesting properties of infinite sets. These other people then grepped his paradox, and wove this tangled web of crap around it, claiming they'd "disproven" an infinite past just because they can't seem to understand elementary set theory. (W. L. Craig, P. L. Huby, etc. - I'm sure you know of these people, Wade - they're the ones that wrote your Tristam Shandy argument).
So enough of this. On to the math.
=======================================================
== The Math Part ======================================
=======================================================
>Wade Said:
>Because the above is a deductive argument, it can fail in >only one of two ways. Either the argument is invalid (i.e. >the conclusion does not logically follow from the premises) >or at least one of the premises is incorrect.
The reason why Wade's paradox is incorrect is that his first premise is false. It's not egregiously false, just subtly wrong. Let's look at it.
> 1. There is a one-to-one correspondence between years >passed and days passed.
False. Here's why.
Functions are simply a mapping of the elements of one set to the elements of another. "One-to-one correspondence" is a specific class of function.
First, let's be clear about what is meant by "one to one correspondence." 1-1 correspondence is formally referred to as "bijection." It means that every element in set A maps to a single element in set B, that no two elements in A map to the same element in B, and that every element in B is "mapped to" by an element of A.
The important part is that every element in B is mapped to, that there are no elements in B that are NOT mapped to. Hmm.. Let's look at Tristam's situation formally, and we'll see that this isn't the case.
D: set of all days, d0, d1, d2, ..., dn | D = N
Y: collection of sets, y0, y1, y2, ..., yn |
yi = {d(0+i), d(1+i), ..., d(365+i)} &&
all di are members of D
So this reads, in English:
D contains an infinite number of distinct elements.
Y contains an infinite number of distinct elements, each containing 365 distinct elements of D
Let's try to find a function that either maps D->Y or Y->D in a "one-to-one correspondence" relationship.
F(yi) -> di
OK, this clearly doesn't work. For every yi, we have 364 elements of D that are not mapped to.
G(di) -> yi
And this, too - for every yi, there are 364 elements of D that are not mapped to.
So what, then, was Wade trying to say?
Like I said, the falsity of his first premise isn't horrible, it's rather subtle. Here's why - There exists a class of functions called "one-to-one functions", or "injection" functions. (1-1 correspondence vs 1-1 function) where each member of set A maps to a distinct member of set B, and no two members of set A map to the same member of set B, but there can be members of set B that are "left over." - That is, no member of set A maps to them.
Clearly, the relationship of days to years, assuming infinite time, is a one-to-one FUNCTION, not a one-to-one correspondence.
The "paradox" comes from this fact:
2 different infinite sets are numerically equivalent - they contain the same number of elements (infinity!).
BUT!!! There may exist elements in one set that do not correspond to elements in the second set.
Huh? How is that possible? They have the same number of elements, but one has "more" than the other? The short (and correct) answer is: YES
This is an axiomatic property of infinite sets. It arises from an interesting property of infinity - that infinity = infinity. But that doesn't neccessarily prove that there exists a one-to-one correspondence function from one to the other.
Here's a simple example -
Let N be the set of natural numbers (which is infinite)
Let E be the set of even natural numbers. E is constructed using the following function:
E = {H(n) -> 2n | n is a member of N}
If N is infinite, then E is also infinite. And infinity equals infinity, so they have the same "number" of elements (they have the same "cardinality")
But the function
I:E->N where I(e) = e
is clearly NOT a one-to-one correspondence - there exist members of N that are not mapped to by a member of E. (there are functions that map E to N with one-to-one correspondence, but there is NO function that is the equivalent of the statement:
"There is a one-to-one correspondence of untransformed members of E to members of N"
which is what Wade is trying to say.
So Wade's first premise is wrong. Here's the correct statement:
1) There exists a one-to-one function that maps the elements of Y to the elements of D
2) There exist elements of D that are not mapped to by any element of Y
3) Therefore, there exist "days" that Tristam has not written about, and so he will never finish.
So let's, in light of the above, look at more of what Wade says.
>Wade Said:
>One could deny the existence of the one-to-one >correspondence between years passed and days passed (the >first premise), but that too is ridiculous since we can >easily prove otherwise given an infinite number of years >and an infinite number of days (for each day that has >passed there exists a different year that has passed, and >vice versa).
Clearly, we've seen that this is not the case. There exist, even assuming an infinite number of elements in D, elements of D that do not correspond to an element of Y.
Essentially, Wade's argument is, "That's ridiculous! That can't be the case!"
Wade, you could define your own system of mathematics where the Tristam Shandy "paradox" would truly be a paradox. But it's not a paradox in Cantor set theory, nor is it a paradox in Peano arithmetic (if you want a proof of that, just ask).
But here's the real ass-kicker: if you defined such a system, it would be internally inconsistent, and therefore less "correct" than any of the other systems you attempted (and failed) to disprove.
>Wade Said:
>Can you think of a way to resolve it? I cannot. Let me know >what you think.
In answer to your questions Wade, "Yes." "Read the above."
If anyone wants clarification of any of this, please post or email me. Any arguments are welcome, but please, please, please, do us all a favor and do some background reading on set theory. Math is created by man, and it's axiomatically defined - you have to know those axioms before you can try to do anything, and Wade, you don't know the axioms.
If it sounds like I'm saying, "You have to play by the rules we humans have defined about math," then you understand me perfectly. That's the only way we can possibly accomplish anything productive in mathematics and phiolsophy, since they are not grounded in objective reality.
Duane
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