11 Apr 13
Originally posted by sonshipFunny how you failed to read my whole post.
Thankyou. That is why I suspect the cosmologist are correct who maintain that time had a beginning. I suspect that a beginning to the universe and time is the better scientific theory.
thanks
But at least you now admit that all you have is suspicions, and your previous claims that you had an argument based on philosophy or mathematics have turned out to be incorrect.
11 Apr 13
4 edits
Originally posted by wolfgang591. In Zeno's Paradox Achilles does catch the tortoise. (He must!!!)
Of course t=2 is reached!!!
We are talking about a mathematical model (t = 2-1/n) which considers the amount of steps in an amount of time. Time will progress! We can calculate the number of steps for any amount of time [b]except t=2.
That does not mean that t=2 is never reached! It means that n is undefined at t=2.[/b]
Your argument here is with Zeno of Elea, not me—
From The Cambridge Dictionary of Philosophy: “Zeno argued that Achilles can never catch up with the tortoise no matter how fast he runs and no matter how long the race goes on.” (p.987)
And from The Stanford Encyclopedia of Philosophy, at http://plato.stanford.edu/entries/paradox-zeno/#AchTor:
“And so on to infinity: every time that Achilles reaches the place where the tortoise was, the tortoise has had enough time to get a little bit further, and so Achilles has another run to make, and so Achilles has in infinite number of finite catch-ups to do before he can catch the tortoise, and so, Zeno concludes, he never catches the tortoise.”
This essay goes on to note that: “But what the paradox in this form brings out most vividly is the problem of completing a series of actions that has no final member—in this case the infinite series of catch-ups before Achilles reaches the tortoise. But just what is the problem? Perhaps the following: Achilles' run to the point at which he should reach the tortoise can, it seems, be completely decomposed into the series of catch-ups, none of which take him to the tortoise. Therefore, nowhere in his run does he reach the tortoise after all. But if this is what Zeno had in mind it won't do. Of course Achilles doesn't reach the tortoise at any point of the sequence, for every run in the sequence occurs before we expect Achilles to reach it! Thinking in terms of the points that Achilles must reach in his run, 1m does not occur in the sequence 0.9m, 0.99m, 0.999m, … , so of course he never catches the tortoise during that sequence of runs! The series of catch-ups does not after all completely decompose the run: the final point—at which Achilles does catch the tortoise—must be added to it. So is there any puzzle? Arguably yes.
“Achilles run passes through the sequence of points 0.9m, 0.99m, 0.999m, … , 1m. But does such a strange sequence—comprised of an infinity of members followed by one more—make sense mathematically? If not then our mathematical description of the run cannot be correct, but then what is? Fortunately the theory of transfinites pioneered by Cantor assures us that such a series is perfectly respectable. It was realized that the order properties of infinite series are much more elaborate than those of finite series. Any way of arranging the numbers 1, 2 and 3 gives a series in the same pattern, for instance, but there are many distinct ways to order the natural numbers: 1, 2, 3, … for instance. Or … , 3, 2, 1. Or … , 4, 2, 1, 3, 5, … . Or 2, 3, 4, … , 1, which is just the same kind of series as the positions Achilles must run through. Thus the theory of the transfinites treats not just ‘cardinal’ numbers—which depend only on how many things there are—but also ‘ordinal’ numbers which depend further on how the things are arranged. Since the ordinals are standardly taken to be mathematically legitimate numbers, and since the series of points Achilles must pass has an ordinal number, we shall take it that the series is mathematically legitimate. (Again, see ‘Supertasks’ below for another kind of problem that might arise for Achilles’.)”
From: http://www.mathcs.org/analysis/reals/history/zeno.html:
“Today, armed with the tools of converging series and Cantor's theories on infinite sets, these paradoxes can be explained to some satisfaction. However, even today the debate continues on the validity of both the paradoxes and the rationalizations.”
That does not mean that t=2 is never reached!
The function in JS357’s paradox is of the form y = f(x) = 2-1/X, where X stands for an infinite sequence of positive integers: x=1,2,4,8, . . .. y=2 is undefined: x never “equals infinity” (of course infinity is not a number, it is the sequence that is infinite); x is always a positive number in an infinite sequence. The common language would be that lim(y)=2 as x “approaches infinity”. The function can be graphed, which will show that y never “reaches” the limit, or crosses it—for that function. That function will never yield a value for y>2.
The problem, as indicated by the Stanford article cited above, is with the function. I do not think that Achilles can actually run forever without catching up or that anyone can count forever, or actually count all the positive integers (and have never so implied). Nevertheless, that is the function that describes JS357’s case (substituting t for y, and n for x).
To “decompose” the function in the manner suggested by the Stanford article would seem to mean that one just—stops, having counted some finite number of positive integers.
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I’m happy to be corrected. The number of exclamation points, however, carries no weight. If you’re frustrated, just point me to some reference material, and I’ll struggle along.
11 Apr 13
2 edits
Originally posted by vistesdThe thing about Achilles, is that although he must go through an infinite sequence, each entry in the sequence takes a shorter amount of time. So the total time used remains finite, so there never is a question of him running forever as in taking a very long or infinite amount of time. The difficulty is with him going through an infinite sequence and apparently getting to the end then continuing beyond it. But it is our minds, and not the mathematics that has the problem.
I do not think that Achilles can actually run forever without catching up ...
Mathematics has no problem taking two infinities and matching them up pairwise and showing for example that there are the same number of elements in both, or in some cases different numbers. So one could presumably create a set with exactly infinity plus one elements.
On the topic of infinities:
I recently saw a claim that the number pi, being non-repeating and thus having infinite variation, contains a digital version of every picture you ever took, or book that ever existed, or web page on the internet etc. Having given it some thought though, I believe this is in fact incorrect. One can have a non repeating decimal that does not contain all possible finite sequences. Once could for example create an irrational number that does not contain the digit 9.
11 Apr 13
1 edit
Originally posted by twhiteheadSo one could presumably create a set with exactly infinity plus one elements.
The thing about Achilles, is that although he must go through an infinite sequence, each entry in the sequence takes a shorter amount of time. So the total time used remains finite, so there never is a question of him running forever as in taking a very long or infinite amount of time. The difficulty is with him going through an infinite sequence and appa ...[text shortened]... sequences. Once could for example create an irrational number that does not contain the digit 9.
Thank you. Am I correct, or not, in saying that the f(x) that I described for JS357's paradox is fairly straightforward (I am trying to recall limits and calculus)—but that that is a trap in terms of resolving the paradox, which requires moving to the mathematics of infinite sets?
11 Apr 13
Originally posted by vistesdI am not entirely certain what your question is, but the integer 2, is not a member of the sequence generated by the function y = f(x) = 2-(1/2)^i, where i is the set of positive integers (essentially the same function as yours)
Am I correct, or not, in saying that the f(x) that I described for JS357's paradox is fairly straightforward (I am trying to recall limits and calculus)—but that that is a trap in terms of resolving the paradox, which requires moving to the mathematics of infinite sets?
Another cool fact is that there is an irrational number between every member of the above set and 2. So although the set gets infinitely close to 2, it is always separated from it by another number. So we have an infinitely small interval that contains a number.
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11 Apr 13
Originally posted by twhiteheadNo. You admitted that the problem of an infinitely past time universe is indeed an insurmountable mathematical and philosphical dead end.
Funny how you failed to read my whole post.
But at least you now admit that all you have is suspicions, and your previous claims that you had an argument based on philosophy or mathematics have turned out to be incorrect.
11 Apr 13
Originally posted by sonshipNo, I did not. Go back and read the whole post. You quoted only half my sentence. This is why I accuse you of thinking yourself infallible, rather than admit you were wrong, you quote me out of context and lie about my conclusion.
No. You admitted that the problem of an infinitely past time universe is indeed an insurmountable mathematical and philosphical dead end.
12 Apr 13
2 edits
Originally posted by wolfgang59No. I was just pointing out that your statement—“In Zeno's Paradox Achilles does catch the tortoise. (He must!!!)”—is incorrect. In Zeno’s Paradox, Achilles never catches the tortoise; otherwise, there would be no paradox.
Do you honestly believe that Zeno thought Achilles would not catch the tortoise?.
12 Apr 13
4 edits
Originally posted by wolfgang59t and n are undefined at 2 seconds. But the function never touches the asymptote at 2 seconds, which is why for that function t is never equal to or greater than 2. So the “JS357 paradox” cannot be resolved by appeal to that function. t=2 is the limit of the curve as it “approaches infinity”.
APOLOGIES
My previous math does not make sense.
The function is
t = 2-1/x where x=2^n where n= 0 to inf
the conclusion is the same though
x = 1/(2-t) and therefore x and n are undefined at 2 seconds
Perhaps, though, your point was, in part, that that function does not define the “mathematical space”, and that one cannot conclude, from the confines of that function, that there is nowhere in the “mathematical space” where t>2? As it were, “on the other side of the t=2 asymptote?
12 Apr 13
Originally posted by vistesdThe paradox is that Zeno's argument "proves" that Achilles does not catch the tortoise however Achilles obviously does.
No. I was just pointing out that your statement—“In Zeno's Paradox Achilles does catch the tortoise. (He must!!!)”—is incorrect. [b]In Zeno’s Paradox, Achilles never catches the tortoise; otherwise, there would be no paradox.[/b]
12 Apr 13
1 edit
Originally posted by vistesdHow can t be undefined at 2 seconds?
t and n are undefined at 2 seconds. But the function never touches the asymptote at 2 seconds, which is why for that function t is never equal to or greater than 2. So the “JS357 paradox” cannot be resolved by appeal to that function. t=2 is the limit of the curve as it “approaches infinity”.
Perhaps, though, your point was, in par ...[text shortened]... ere in the “mathematical space” where t>2? As it were, “on the other side of the t=2 asymptote?
t=2
The problem is that you are looking at t as a variable dependent on n (the number of iterations) but that is nonsensical (but tempting due to the way
the problem has been set). Time, as we know, progresses regardless of what activity we do (including counting ) so t=2 is reached (after 2 seconds!)
What we cannot resolve is the value of n at 2 seconds.
Any value of t other than 2 will give a value for n.
The limit of n as t approaches 2 is infinite.
13 Apr 13
4 edits
Originally posted by wolfgang59You’re right that I have been treating t as the dependent variable. That is why I substituted y=f(x): to get away from thinking about physical time, and just focus on the function itself first, and what values for y (or t) it would yield. Treating it that (seemingly straightforward) way, the function f(x) never yields y=2. So, treating it that way, the function never "decomposes" to resolve the paradox (or the absurdity). That is why I said that the problem is with [my treatment of] the function.
How can t be undefined at 2 seconds?
t=2
The problem is that you are looking at t as a variable dependent on n (the number of iterations) but that is nonsensical (but tempting due to the way
the problem has been set). Time, as we know, progresses regardless of what activity we do (including counting ) so t=2 is reached (after 2 seconds!)
What we ...[text shortened]... alue of t other than 2 will give a value for n.
The limit of n as t approaches 2 is infinite.
You’re saying, if I understand you, that I was looking at it the wrong way ‘round . . .