Zeno revisited

Larry,

You may be right that many things can be sorted out with enough effort. But sometimes one must digest and assimliate accumulated information before drawing conclusions or moving on. Sometimes alas, people have to agree to disagree.

As to those who think that there can be no disagreements in philosophy (which subsumes the field of logic), and that all who disagree with them are "illogical" I can only suggest that they read in the history of philosophy. Check out Copleston for example. Polemic has a long and venerable tradition, both sides often exceedingly logical and learned, yet in disagreement nonetheless. The same can be said for science, though in that case many reasonable theories and the associated factual evedence is often drowned out or silenced outright by those who control the "mainstream," or "establishment" paradigm.

Thanks, Tom and the others who replied to the original post.

Neil

<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 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|-
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">






On a number line from 1 to 5, since infinite divisibility is allowed, there are an infinite number of points between 1 and 2, an infinite number of points between 2 and 1, 3 and 1 , 4 and 1, and 5 and 1. So which distance is larger? If you say 5 and 1 then there are more points between 5 and 1 than the other pairs. But if that is the case then the distance between the other pairs must not be infinitely divisible, no?

<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>
On a number line from 1 to 5, since infinite divisibility is allowed, there are an infinite number of points between 1 and 2, an infinite number of points between 2 and 1, 3 and 1 , 4 and 1, and 5 and 1. So which distance is larger? If you say 5 and 1 then there are more points between 5 and 1 than the other pairs. But if that is the case then the distance between the other pairs must not be infinitely divisible, no? <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
Contratulations! You have just undone centuries of mathematics. Hello, Nobel?...


JR

<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>
Sarcasm is not a valid rebuttal. Show me my ignorance instead.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
Infinity is not a number or quantity, so it is nonsensical to ask which infinity is bigger. There are actually hierarchies of infinities in mathematics, but that is not relevant to your question.


JR

[123...] "Sarcasm is not a valid rebuttal. Show me my ignorance instead."

You really are going to have to do some reading to see what is going on with the concept of infinity. This forum is good for a lot of things, but teaching someone the basics is not one of them. That takes books (plural). And time.

The books mentioned above by Dr. Van Flandern are a good way to get up to speed in a hurry.

Once you have been through a good book or two, this forum is then a great place for testing your new knowledge and clearing up any points of confusion. Or even challenging some part of it.

===

And you are right, sarcasm is not the most helpful response. But I 've got to tell you, if it weren't for the DELETE key I'd have done it myself a few times.

One of them was funny as hell, but not really very nice. Nor was it a valid rebuttal. I appologize for the intent even though I never published it.

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 />On a number line from 1 to 5, since infinite divisibility is allowed, there are an infinite number of points between 1 and 2, an infinite number of points between 2 and 1, 3 and 1 , 4 and 1, and 5 and 1. So which distance is larger? If you say 5 and 1 then there are more points between 5 and 1 than the other pairs. But if that is the case then the distance between the other pairs must not be infinitely divisible, no?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The mathematics of infinities is a very interesting subject. It has specific rules, it has good reasons for those rules, and the set of rules is consistent; i.e., no contradictions (such as your example above) can arise. You will really enjoy learning it.

Sticking to the lowest order of infinity (there are higher orders, as jrich mentioned), here are a few basics, where "inf" = infinity, "con" = any positive finite constant, and "indet" = indeterminate (can be anything):

inf+con = inf
inf-con = inf
inf*con = inf
inf/con = inf
inf^con = inf
inf+inf = inf
inf-inf = indet
inf*inf = inf
inf/inf = indet
inf^inf = a higher order of infinity

When you asked above "which distance is larger?", distance is not based on counting points or else all line segments would have infinite length. The number of points in all line segments, however large or small, is infinite. And by the above rules, the difference between the number of points in any two line segments is indeterminate.

So when you say "the distance between the other pairs must not be infinitely divisible, no?", the answer is indeed "no". They are all infinitely divisible, and that is consistent with the rules for the mathematics of infinities.

If your next question is "why", that is where Larry's advice about reading a book or two, and my earlier recommendation about a visit to your local library or ordering a book from the internet, come into play.

The math of infinities is a <font color="orange"><i>big</i></font id="orange"> subject. [8D] -|Tom|-