Planck limits

Enrico,

It seems to me that here you make another mistake as Zeno did in assumeing an infinite series must sum to infinity. You now assume that an infinite computation must be performed to perform an infinite calculation. Here is an example of an infinite computation done in a finite amount of time:

Multiply the infinite number 0.1111.... by 2

Hmmm. I bet the answer is 0.2222.... I wonder how I did that without spending the rest of eternity at my calculator.

Enrico: Frankly, although your position seems unyielding opposed to mine, I like your spirit and determinedness to defend your viewpoint. Let's see if we can keep digging until we unearth the real roadblock here.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>I can give you a good reason for my rejection. ... You may start dividing 1 meter in half, and then the remaining in half, and so on. Then, keep adding them together, not in your mind but with your calculator. Are you ever going to finish? The answer is no.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

You are asking me to add, one term at a time, the series 1/2 + 1/4 + 1/8 + 1/16 + ... = Summation (1/2^n) for n = 1 to infinity. But notice that n can serve two purposes at once: It can be the power in the power series. And it can be a simple counter of terms in the series (n = 1, 2, 3, 4, ..., infinity).

Now if you asked me how many terms are in the series for n, I would immediately answer "infinity", and that would be the right answer. But you would ask me to prove that by counting them on my calculator one at a time. Why would I do that? It is madness. The answer is already known, and can be proved rigorously by induction. And the calculator can never verify that answer no matter how high I count. Who can prove that the number of positive integers is infinite by counting them?!

I repeat yet once again this essential distinction: All integers are finite. Yet the total number of integers is infinite. The former are real and tangible. The latter is a pure mathematical or logical concept without real existence. In general, all real forms are finite, and the dimensions we use to measure or count them are infinite. Yet the latter, while not existing outside our minds, has meaning to us and is useful in proving things and in describing them.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Then, your answer that a finite series converges is only in mathematics which you claim in the first place to have nothing to do with reality in the sense of the infinite, but them you invoke mathematics of infinity to provide a proof of your argument because you cannot demontrate a proof by experiment.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

This is very confused. Infinite series can have a finite sum. One can use a 1-to-1 correspondence between an infinite series and infinite steps toward a goal, and the finite sum or the finite goal they are equivalent to. Another example is a finite line segment, which also has a 1-to-1 correspondence with the distance to a goal. The line segment and the distance to a goal can both be divided into an infinite number of infinitesimal points or parts. But that makes them no less finite than they were before the division.

The "proof by experiment" is far inferior to the proof by induction, which is in turn inferior to the proof by calculus (taking limits as quantities approach zero or infinity). However, we can prove by experiment that the infinite series [Summation (1/2^n) for n = 1 to infinity] = 1. We can show by trial and error that it is impossible for the sum to exceed 1, no matter how many terms are taken. And we can likewise prove that any number we specify below 1 will eventually be exceeded by the sum. Therefore, the sum is exactly 1, based on trial-and-error experiment.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>But you cannot and will never be able to prove that experimentally.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

That is not true. The proof I just outlined is considered on sound footing. The inductive and calculus proofs are also available.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>even in mathematics you will never get there unless you drop the final infinitesimal and consider it the smallest possible element, a delta or epsilon, which makes your continous space idea a contradiction in terms, even in mathematics.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

This makes no sense to me. Are you denying the mathematics of infinities too? What about calculus (limits)?

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>those proposing naive convergent series abstractions must live in a well known contradiction their argument presents.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

I see no "well-known" contradiction. The field of math contains the math of infinities as an integral component. You seem to be denying the rigor of that discipline.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>I still expect you to invoke the "convergent series" argument as a solution to Zeno's paradox. But think about it: by invoking that argument have you realized a solution for the cause of motion?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

No, of course not. That was never the purpose of Zeno's paradoxes, even for Zeno. In MM, I used the reality of motion and Zeno's paradoxes to eliminate all possibilities for the composition of space, time, and scale except the one that led to no paradox: infinite divisibility.

So my conclusion is about the nature of dimensions, not about the nature of motion. In MM, motion arises because there exists no framework or universal standard of rest, so absolute non-motion has no meaning. Only relative motion has meaning. And it is as omni-present as is substance at every scale because neither substance nor motion can ever come into or pass out of existence. But that is an argument (about the nature of motion) that is completely unrelated to Zeno's paradoxes and the nature of dimensions.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>If you say yes, Newton will jump out of his grave and kiss you! If you say no, you will be at least an honest man. If you say that the cause of motion is known, then I say we have been wasting our time here.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

I apparently get a very low score from you. I said "no", but for the wrong reason, so I get no kiss from Newton, I get no credit for being an honest man, and I have apparently been wasting your time here.

Nevertheless (as Galileo said first), "it moves!" -|Tom|-