Zeno revisited

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</i>
<br />If a series never ends, there can be no sum either.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I gave an example of an infinite series with a finite sum: [1/2 + 1/4 + 1/8 + 1/16 + ...] = 1. This is the infinite series: Summation (1/2^n) from n = 1 to infinity.

I also explained how we know the sum is correct. There are mathematical ways to prove this, but the short story can be revealed by trial and error: The sum of these terms will always eventually exceed any real number less than 1 as n grows larger. But the sum can never exceed 1 no matter how large n gets.

Another way to see this is a reverse one-to-one correspondence. Consider Zeno's paradox that someone attempting to cross a street must first reach the half-way point (the first term in our series), then the point half-the-remaining-distance to the other side (the second term in our series), then ... Each series term corresponds to another half-the-remaining distance point along the way when crossing the street. There are an unlimited (i.e., infinite) number of such points and of corresponding terms in our series. Yet the distance across the street is finite, represented by the number 1 for the series sum.

OK, I've offered some further explanations for the position I stated. But you haven't yet provided an explanation why you declared "there can be no sum". The summation of infinite series is a well-established branch of mathematics. I have on my shelves a whole book devoted to nothing else: "Summation of series" by L.B.W. Jolley, Dover, 1961. I recommend a library visit, which will have many less-technical books on this than the one I mentioned. I highly recommend another Dover book, Gamov's "One, Two, Three...Infinity", which I read and appreciated in high school. (The internet cannot yet compete with books and classrooms for serious learning in most areas, although we can see the day coming when it probably will.)

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">That would be like trying to calculate the life expectancy of our universe.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I don't see the connection, or why you mentioned this at all. If it is still relevant, please explain. -|Tom|-

Its easy enough to show logically that there can be no smallest distance in space or smallest size of physical objects. Given this and the fact that motion is possible it is possible to prove that there can be no smallest interval of time and resolve the paradox in the process.

But a logical proof will not satisfy some here, who will insist regardless that reality may differ from what is intellectually demanded. They are correct and I can offer no argument to disuade them. However, if logic fails in this instance, it may fail in others, and how are we to know what is true and what is fantasy. The implicit assumption in scientific endeavours is that reality follows the rules of logic, or stated differently, that the rules of logic are sufficient to understanding reality. Nevertheless, almost all knowledge may be impossible to discover due to limits imposed by nature, NOT logic. An illogical reality is undiscoverable, but those who wish to try are welcome to it. Unfortunately for humanity, you will have plenty of company.


JR

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by tvanflandern</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</i>
<br />If a series never ends, there can be no sum either.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I gave an example of an infinite series with a finite sum: [1/2 + 1/4 + 1/8 + 1/16 + ...] = 1. This is the infinite series: Summation (1/2^n) from n = 1 to infinity.

I also explained how we know the sum is correct. There are mathematical ways to prove this, but the short story can be revealed by trial and error: The sum of these terms will always eventually exceed any real number less than 1 as n grows larger. But the sum can never exceed 1 no matter how large n gets.

Another way to see this is a reverse one-to-one correspondence. Consider Zeno's paradox that someone attempting to cross a street must first reach the half-way point (the first term in our series), then the point half-the-remaining-distance to the other side (the second term in our series), then ... Each series term corresponds to another half-the-remaining distance point along the way when crossing the street. There are an unlimited (i.e., infinite) number of such points and of corresponding terms in our series. Yet the distance across the street is finite, represented by the number 1 for the series sum.

OK, I've offered some further explanations for the position I stated. But you haven't yet provided an explanation why you declared "there can be no sum". The summation of infinite series is a well-established branch of mathematics. I have on my shelves a whole book devoted to nothing else: "Summation of series" by L.B.W. Jolley, Dover, 1961. I recommend a library visit, which will have many less-technical books on this than the one I mentioned. I highly recommend another Dover book, Gamov's "One, Two, Three...Infinity", which I read and appreciated in high school. (The internet cannot yet compete with books and classrooms for serious learning in most areas, although we can see the day coming when it probably will.)

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">That would be like trying to calculate the life expectancy of our universe.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I don't see the connection, or why you mentioned this at all. If it is still relevant, please explain. -|Tom|-
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

If a series has an infinite number of elements, how can you add all of them together at once?

With this in mind, what I meant in reference to the life expectancy of the universe is that if you started out assuming the universe cannot cease to exist, and say we add time to it in the manner of the converging geometric series, it does not change the fact that the universe can't cease to exist. It makes as much sense to find the sum of an infinite series as it does to determine the life expectancy of a presupposed eternal universe.

You can say the universe is getting older and older but never how old it can be. Likewise, you can say the geometric series used as a resolution to Zeno's paradox is getting closer and closer to 1 but the limit (or sum) cannot exist by the very supposition of an infinite process.

[123 ...] "If a series has an infinite number of elements, how can you add all of them together at once?"

You can't.

Literally, you can not do this. Nor can you do it one element at a time.

Physically, all of the elements can not be summed.

===

Why would you think you need to?


???,
LB

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</i>
<br />If a series has an infinite number of elements, how can you add all of them together at once?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">We do not need to add them all. Take the example I gave you: [1/2 + 1/4 + 1/8 + 1/16 + ...] = 1. A general formula for the sum of any finite number of terms (n) in this series is (1 - 1/2^n). For example:
n = 1 sum = 1/2
n = 2 sum = 3/4
n = 3 sum = 7/8
n = 4 sum = 15/16
...

Because that formula is valid for <i>any</i> n, we can find the limiting sum as n --&gt; infinity. And that limiting sum is 1. This reflects the fact that we cannot make the sum exceed one by trial and error, no matter how many terms we add. It also reflects the fact that the series represents the pieces of a line segment with a length of (say) one foot, and the sum of of all the pieces cannot exceed the length of the whole segment, one foot.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">With this in mind, what I meant in reference to the life expectancy of the universe is that if you started out assuming the universe cannot cease to exist, and say we add time to it in the manner of the converging geometric series, it does not change the fact that the universe can't cease to exist. It makes as much sense to find the sum of an infinite series as it does to determine the life expectancy of a presupposed eternal universe.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I still do not see the analogy despite your explanation. To make the analogy work, we would have to take a finite time interval; say, one second. Then if we divide the second into an infinite number of sub-intervals (1/2 second plus 1/4 second plus 1/8 second plus ...) and sum them together, we get back to one second only after we have added all infinity of the sub-intervals. The whole cannot exceed the sum of its own parts. You can't do that with the age of an eternal universe because you can't take fractions of that age such as one-half.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">you can say the geometric series used as a resolution to Zeno's paradox is getting closer and closer to 1 but the limit (or sum) cannot exist by the very supposition of an infinite process.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Okay, that is a clear statement of where the problem lies. You have not yet studied calculus, which depends for its existence on our being able to take limits. The idea in differential calculus is that a quantity that changes with some independent variable (such as space or time) nonetheless has a specific and definite value at an infinitesimal point. And we can evaluate that value exactly by taking a limit as the change in the independent variable near that point approaches zero. The idea in integral calculus is that we can take an infinitesimal element of some function (such as an area or a volume) and calculate exactly the limiting sum of all such elements as the number of elements approaches infinity.

You will surely recognize that calculus is the tool behind most of the engineering marvels of the modern world. So no one educated in calculus questions that it works as advertized. But rather than asking you to take a crash course in calculus so you can better understand infinities and limits, I will again recommend Gamow's book, "One, two, three ... infinity". It was reprinted in 1988, so copies can still be found on the internet. I would expect it to be cheap, probably in paperback for under $10. (I'm guessing.) And it is an interesting read for anyone who wants to understand these subjects.

It's an easy way to get up on the learning curve fast, and for just $10 or so and the time it takes to read a few chapters of a book. You will then find yourself more or less on a level playing field with many others in this forum, some of whom presume you already have this background and are mystified by your comments. -|Tom|-

The problem as I see it is in viewing time as an iteration of discrete events over a certain span. The iteration is analogous to counting, but it is not the same as the summing of members of the infinite series. Summing is not counting. Some in this MB hold the iteration view and Tom counters with the infinite series. But the infinite series does not demonstrate countability and thus those who hold the iteration view are unmoved. But the point of the infinite series is to show that the iteration view is not the only valid one, not to reconcile iteration with infinite divisibility. One cannot use the non-iterability of infinite time as a disproof, because infinite time is excluded in the premise of iterability. It is exactly the same as "proving" that the number of fractions between 0 and 1 is finite because the infinite series cannot be counted.

Now if one defines "time" as interactions of forms of substance, then it is clear that infinite divisibility of time demands infinite interactions. To have infinite interactions there must be infinite forms which requires infinite substance. In this way the MM is logically consistent. If one dimension is infinite, logically all must be. This is why finite universe models are logically inconsistent, since it is trivial to prove that at least one dimension is not finite.


JR