| Subject: Yes - I agree |
Author:
Duane
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Date Posted: 09/11/04 2:27am
In reply to:
Damoclese
's message, "I think" on 09/10/04 10:22am
Damocles:
Well, I think we've beaten this horse carcass into jello.
I agree with your assessment:
>Other questions
>spring to mind such as if you are in an infinity, why
>would you use an Earth method of counting the years?)
Because using our unidirectional understanding of time MAKES THIS STORY SEEM WEIRD.
>Clearly, one can see that this function is always
>going to leave behind more and more days that haven't
>been written about all the way to infinity, but what
>of the number of days he has to write? It's true that
>that particular number is infinite also, but there's a
>provision. He has to experience 365 days before he can
>write about a day. That means at any given point along
>the line, there are going to be more days left to
>write about as opposed to days already written.
Which is what us humans, who experience time in a directional fashion, would tend to think. Which, again, is why Bertrand Russell used this example.
>So now that we know what the points are doing, lets
>talk about them. What happens to the points as they
>approach infinity?
"approaching" infinity is impossible - infinity "is." The whole idea of "getting to" infinity by subsequent addition is doomed to fail (you can always add one more).
>Infinity isn't talking about the days and years as
>separate entities here; we're talking about given what
>we know to be true about the relationship, what's the
>amount of overall time do we have that this
>relationship holds steady? It continues to hold
>forever. As the line goes infinitely out, the points
>upon it hold fast to the rules.
This is intuitively true. And I don't disagree with you. However, this paradox is a product of the axioms of Cantor set theory.
>As with any other mathematical function that goes to
>infinity, it's assumed that infinity means "this
>relationship goes on for as long as you care to take
>it" It isn't that the days and years both eventually
>reach a quantity known as infinity, because that'd be
>breaking the function into bits.
Actually, I have to disagree with you on this point. When talking about mathematics, we use the phrase, "goes to infinity."
"Unfortunately", that's a byproduct of our reality-based minds. (insert sarcasm here) Things don't "go to" infinity - they are infinite, or they are not.
To finally put this thread to sleep, let me give you the history of this paradox:
The exemplar of mathematical systems is to be able to define a minimal set of axioms and inference rules that can generate a complex, robust, and useful system. Of all systems of mathematics, set theory is the new kid on the block. Geometry and Algebra have been around centuries longer. (In fact, boolean algebra is one of the only "complete" mathematical frameworks.)
Set theory was invented in the late 1800s, as an attempt to define all of mathematics and understand the universe using a VERY minimal set of axioms and inference rules. It seemed to work out pretty well, until Cantor and Russell started coming up with evidence that set theory, as it had been originally defined, was not internally consistent.
This was kind of like engineers torture-testing a product before trying to mass-market it. The problem with Cantor's theory was with one axiom:
"Any property can be used to define a set"
By "any property," he meant properties like, "red," "furry," "men that only have one leg," etc. Russell was the one who came up with the property that created a contradiction. The property was:
"The set containing sets that do not contain themselves."
The paradox arose from the following observation regarding this property:
If a set contains itself, then it does not contain itself.
If a set does not contain itself, then it contains itself.
Thus, Cantor set theory was shown to be internally inconsistent and abandoned for a much more complicated system describing sets. Still, since set theory was supposed to make sense of not only the world of mathematics, but our own universe, Bertrand Russell proffered the Tristam Shandy paradox as demonstration that set theory could not be used to describe the universe.
The key point to Tristam is this:
"Paradoxes and contradictions do not exist. If you find one within a system, one of your premises is wrong."
Tristam Shandy IS a paradox. And the incorrect premise is NOT, as Wade seems to think, that "an infinite past is impossible." The incorrect premise is that set theory can be used to understand our reality-based world.
If we throw out this assumption, then set theory remains a valid mathematical system. And that's it.
To say that "set theory disproves an infinite past" would be the same as saying, "The light microscope disproves the existence of atoms."
The fallacy is that a light microscope is only a tool to see things we normally couldn't. Atoms exist. We've seen them using "particle" microscopes (I say "particle microscopes" meaning particle accelerators, which are a type of microscope). Just because a limited tool cannot be used to elucidate everything does not mean that those things don't exist.
Duane
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