09 Oct 06
What is the solution to Zeno's Paradox, and does it have any religious implications - particularly concerning the limits or limitless bounds of God's knowledge?
Or - is infinity real?
Are the set of natural numbers infinite?
Is God's knowledge infinite?
Just some question I've been thinking about and debating elsewhere. And I can usually count on a few interesting answers from RHP'ers and any subject with philosophical implications.
09 Oct 06
Originally posted by ColettiIs time continuous? If not, there is no paradox, if yes, then what's the problem with the calculus solution? I don't see how it relates with God. Can you explain? I'd think it would get the debate started faster if you explain where you are coming from with this relationship.
What is the solution to Zeno's Paradox, and does it have any religious implications - particularly concerning the limits or limitless bounds of God's knowledge?
Or - is infinity real?
Are the set of natural numbers infinite?
Is God's knowledge infinite?
Just some question I've been thinking about and debating elsewhere. And I can usually count on a few interesting answers from RHP'ers and any subject with philosophical implications.
Why should infinity unreal? Again, if time is continuous then infinity is necessarilly real as you can divide it infinitely.
The set of natural numbers is infinite by definition, as that set is an abstract concept.
09 Oct 06
Originally posted by NemesioActually I'm debating someone on the infinity of God's knowledge and I came across Zeno's Paradox.
I predict that Coletti will depart from the realm of logic within 4 of his substantial posts.
Any other takers?
I don't have a problem with saying God's knowledge is infinite, but the complaint was that infinite knowledge implied there was always something beyond what is known (or something like that). I didn't quite understand the complaint.
Anyhow, I'm trying to understand the solution to Zeno's Paradox in order to better understand the nature of infinity. I think if I can explain the solution the Zeno's, it might help me explain the nature of God's omniscience. (Maybe not. I'm still wrapping my mind around it.)
P.S. As long as I'm not typing complete gobbledygook - I haven't departed from the realm of logic. But 3 more posts might do it. 😉
09 Oct 06
If you're referring to this one:
Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
http://en.wikipedia.org/wiki/Zeno%27s_paradox#Achilles_and_the_tortoise
Then the infinite number of points that must be traversed is cancelled by the infinitely small size of each point.
09 Oct 06
Originally posted by AThousandYoungThat's the idea.
If you're referring to this one:
[i]Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
http://en.wikipedia.org/wiki/Zeno%27s_pa ...[text shortened]... umber of points that must be traversed is cancelled by the infinitely small size of each point.
I'm trying to make the solution as rigorous as I can.
I think the solution to Zeno's will have to do with the definition of motion and time.
Also, one can divide a line up into any number of sections, but the sum of the sections will always add up to the same amount. So even though there can be infinite divisions of a line - the sum of those individual sections will always add up to the total length of the original line. I think I need to take this fact (the sum of the lengths of an infinite number of lines can be finite) and combine it with the definition of motion (dist/time) to solve the paradox.
1/2 + 1/2 = 1
1/2 + 1/4 + 1/4 = 1
1/2 + 1/4 + 1/8 + 1/8 = 1
... and so on.
Any one know if I'm heading in the right direction to solving the paradox?
09 Oct 06
Originally posted by ColettiI was under the impression that there are several paradoxes attributed to Zeno. Are you talking about one of the plurality paradoxes?
What is the solution to Zeno's Paradox, and does it have any religious implications - particularly concerning the limits or limitless bounds of God's knowledge?
Or - is infinity real?
Are the set of natural numbers infinite?
Is God's knowledge infinite?
Just some question I've been thinking about and debating elsewhere. And I can usually count on a few interesting answers from RHP'ers and any subject with philosophical implications.
http://plato.stanford.edu/entries/paradox-zeno/
09 Oct 06
Originally posted by ColettiHere's one somewhat related, in terms of point of reference.
Actually I'm debating someone on the infinity of God's knowledge and I came across Zeno's Paradox.
I don't have a problem with saying God's knowledge is infinite, but the complaint was that infinite knowledge implied there was always something beyond what is known (or something like that). I didn't quite understand the complaint.
Anyhow, I'm trying ...[text shortened]... ook - I haven't departed from the realm of logic. But 3 more posts might do it. 😉
"The past is theory. It has no existence except in the records of the present." Dr. JA Wheeler.
10 Oct 06
Originally posted by ColettiI think you view the infinite as linear. Perhaps it is circular?
What is the solution to Zeno's Paradox, and does it have any religious implications - particularly concerning the limits or limitless bounds of God's knowledge?
Or - is infinity real?
Are the set of natural numbers infinite?
Is God's knowledge infinite?
Just some question I've been thinking about and debating elsewhere. And I can usually count on a few interesting answers from RHP'ers and any subject with philosophical implications.
10 Oct 06
Originally posted by ColettiI guess I can understand where the person is coming from. I believe if you replace the claim "God's knowledge is infinite" with "Each and every thing that can be known is known by God" that should take care of it. I think the problem is more semantic than substantive.
Actually I'm debating someone on the infinity of God's knowledge and I came across Zeno's Paradox.
I don't have a problem with saying God's knowledge is infinite, but the complaint was that infinite knowledge implied there was always something beyond what is known (or something like that). I didn't quite understand the complaint.
Then again, when it comes to discussing 'God' there's little that I find substantive. 🙂
10 Oct 06
Originally posted by ColettiPlease eplain which paradox you are talking about.
[b]What is the solution to Zeno's Paradox, and does it have any religious implications - particularly concerning the limits or limitless bounds of God's knowledge?
Or - is infinity real?
No, infinity is not real, it is a concept.(And yes the real numbers are infinitate)
Are the set of natural numbers infinite?
Yes the set of natural numbers is infinite. This follows from the definition of infinity and the definition of natural numbers[/b]
Is God's knowledge infinite?
I dont believe in God, and I dont believe that the writers of the Bible had a solid understanding of mathematics, so I find it unlikely that it is spelled out in the Bible.
However the real question is: Is knowledge countable? If not then there is a pretty good chance that my own knowledge is infinate! But this does not imply that I know everything. If Gods knowlege was infinate it would not imply he knows everything. Merely knowing every natural number would suffice without having to know every real number.
10 Oct 06
1 edit
Originally posted by twhiteheadGood point. I hadn't thought of that. It seems a rephrasing like the one I suggested is actually critical.
Please eplain which paradox you are talking about.
[b]Or - is infinity real?
No, infinity is not real, it is a concept.(And yes the real numbers are infinitate)
Are the set of natural numbers infinite?
Yes the set of natural numbers is infinite. This follows from the definition of infinity and the definition of natural numbers[/b]
ng. Merely knowing every natural number would suffice without having to know every real number.[/b]
10 Oct 06
Originally posted by twhiteheadThat is a good point. I was think about how, even when we move a fraction of an inch, we have covered infinite points. Most people think of infinity as something overwhelmingly large, like the fact that there are infinite natural numbers - an amount that is literally immeasurable. But even in the finite we find infinity. The number of real numbers between 1 and 10 are infinite, even though none is smaller than one or greater then 10.
... If Gods knowledge was infinite it would not imply he knows everything. Merely knowing every natural number would suffice without having to know every real number.
So knowledge could be infinite (theoretically) and still not be comprehensive. That's really interesting.
The Jews did not use the expression infinite in the Hebrew Old Testament (or use Calculus of indefinite integration). But Psa 147:5, the Hebrew "ayin mispar" is translated infinite in the New King James. The phrase means literally "beyond number". But this does not mean something that is without limit. The Bible uses the same phase in several passages as a hyperbole to describe things that are to numerous to count - but are clearly finite like the sands of the sea (Gen 41:49), the men and camels of Israel's army like locust (Jud 6:5). So I can only determine of Scripture support the premise that God's knowledge is without limit. I can't do it with the Hebrew phrase in Psa 147:5.
Anyhow, I've read others say the infinite knowledge is a kind of contradiction. I'm not sure. It is true that no human or man-made mind is able to hold infinite facts.
I can easily write a computer program that lists the natural numbers starting at one that could go on infinitely. There would be no logical stopping point withing the programs logic. And I could write a similar program that could take a given length and list different ways it could be divided up - which could go on forever. There is potential infinity there if not actual.
But is it a contradiction to propose an infinite divine mind (assuming the divine mind is part of an omniscient and omnipotent and eternal being)? I am looking at infinity as it is defined mathematically - as boundless or without limit. I think the point of conflict is in saying God is "all knowing" and has "infinite knowledge". Maybe the potential contradiction is between "all" and "infinity".
Just thinking out loud.
10 Oct 06
Originally posted by LemonJelloThere are indeed.
I was under the impression that there are several paradoxes attributed to Zeno. Are you talking about one of the plurality paradoxes?
http://plato.stanford.edu/entries/paradox-zeno/
http://mathworld.wolfram.com/ZenosParadoxes.html
http://en.wikipedia.org/wiki/Zeno's_paradoxes
But I think the phrase "Zeno's Paradox" usually refers to the race between Achilles and the tortoise. Most of them involve the same principle.
Rather, this much simpler paradox simply states that: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible".
[url]http://en.wikipedia.org/wiki/Zeno's_paradoxes#Issues_with_the_proposed_calculus-based_solution[/url]
The paradox, how is it possible to get past an infinity series of points if infinity by definition means "endless". One can not logically get to the end of an endless series of points, much less get past them.