Zeno's Paradox: MORE

Cecil's Columns/Staff Reports
1

Re: Cecil’s column http://www.straightdope.com/columns/030711.html

While playing with the Zeno Paradox mentioned in Cecil’s column some years ago, I came to a strange conclusion. If we solve for the distance the tortoise is ahead of Achilles at any point in the race, and we express this as D=ID-nt(V1-V2) where D equals the distance the tortoise leads Achilles, ID equals the initial separation distance, n equals the number of equal but infinitesimal time periods of the race at some given time, t is the infinitesimal time period considered as a unit of time, V1 equals Achilles velocity and V2 equals the tortoise’s velocity, it is easy to see that as n (time) increases, D (the tortoise’s lead) decreases. This is exactly as you would observe.

Zeno’s example requires us to consider smaller and smaller time periods, which is why in this formula, n is a number of infinitesimal time periods and it will require an infinite number of them in order for Achilles to catch that creature (D=0 when n=infinity).

However, as Cecil notes, empirical evidence shows that Achilles will not only overtake the tortoise, but will actually pass it. This requires the value of D to decrease to 0 and then turn NEGATIVE as the tortoise’s lead becomes a lag., and continue with an increasing absolute value as a negative number. Since the only other variable in our formula is n, the only way this can happen is if n also turns to a negative number AFTER REACHING INFINITY. At this point, n is negative with a high absolute value, and continues negative with a decreasing absolute value.

So as time progresses, D decreased to and through zero to become a negative number with an initially low and increasing absolute value and n increases to and through infinity to become a negative number with an initially high and decreasing absolute value.

I’ve done a lot more thinking on this and there are some very interesting questions and scenarios that develop.

Can we conclude that number systems are circular in nature and that like time, infinity is meaningless.?

There really is no substitute for the pleasures of an infinite series (assuming of course that infinity has any meaning).

2

This could be way off topic, but I have always been amazed/amused by the concept that ALL the positive numbers lie between 0 and 1, if you put them in the denominator of a fraction with 1 as the numerator. Same could be said for negative numbers of 0 to -1.

Which further suggests that the “change of state” nature of binary 1 and 0 (as in “on” vs. “off”) can be extrapolated to cover the Big Bang, among other less universe shaking concepts.

And if your premise is valid then there may be another “change of state” between 1 and -1.

Wouldn’t it be neat if we could replace infinities with 1, 0, and -1?

3

“Interesting questions and scenarios that develop” between a man and a turtle in a foot race?

4

You don’t have to be amazed anymore, since it’s simply not true. Take 1/2 as a simple counterexample. It’s positive, yet 1/(1/2) is not between 0 and 1.

**

…and taking -1/2 as a simple counterexample, shows this isn’t true either.

**

In a sense, we already can. Not only that, but also (1)^2, (1)^3, etc., more specifically aleph null, aleph 1, aleph 2, etc. Look into cardinal and ordinal numbers sometime.

5

Darrell, you’re right: I should have used “integer” instead of “number” in my comment. And the point that amazed/amused me had less to do with the lack of awareness of the various infinities mentioned or alluded to, but the peculiar natures of zero and one.

From nothingness to somethingness is a vastly more significant concept than oneness to twoness. Coming into existence in cases like the Big Bang out of supposed nothingness is very hard to relate to. Not impossible; very hard.

6

geezergeek, I believe that your problem is that you are trying to use infinitesimals as if you would ordinary numbers.

Infinitesimals are not part of the real numbers, and are usually excluded from standard math because of the problems they cause.

There are specialized branches of math in which the peculiar properties of infinitesimals are made to work.

The infinitesimal problem is why calculus needed to be developed and why the Greeks could not solve these paradoxes.

7

Returning to geezergeek’s question about the possible circularity of number systems:

I’ve noticed something of the sort with the old Cartesian Graph (the x-axis / y-axis thing). A vertical line on the graph is said to have “no slope.” Reflection reveals that such a line has, in fact, infinite slope [start at a horizontal “zero slope” line and work up from there until you achieve enlightenment]. Now here is the interesting thing: a vertical line has both infinitely positive slope and infinitely negative slope. Hmmmm.

8

Nope, this is as wrong as most of your pronouncements on math have been.

Slope is delta y divided by delta x. But in this case delta x = 0.

When you divide by 0 you get an undefined amount, definitely not infinity. That’s why vertical lines are said to have no slope and not infinite slope.

9

Most of my pronouncements?

Gee, ya mean I actually got something right once?

10

OK, please bear in mind that I have no ral understanding of either mathematics or physics, but can we not say that because distance and time are not infinitely divisible (Planck lentght and time) this is moot anyway?

Somebody please correct me if I am wrong

11

The Planck length and time are physically defined as the measures beyond which our current classical physics does not give good answers. That doesn’t necessarily mean that time and space can’t actually be sliced smaller than that. (It might under some theories, but not under others.)

But in mathematical terms time and space can be sliced as fine as anyone can conceive. The rules of nature do not apply.

12

geezergeek: number systems ARE circular.

Examples: modular and field mathematics.

13

After getting the above cheap shot out of my system, I thought over the matter a bit and realized that Exapno Mapcase’s rejoiner “Slope is delta y divided by delta x. But in this case delta x = 0.” actually supports my position since, arguably, the quotient of division by zero is “infinity”.

Since my insight here isn’t readily apparent, I’ll try to explain it as well as I can on an interface not made for math. (There is nothing special about the specific numbers used here. The thing to look for is a trend.)

12/6=2

12/2=6

12/1=12

12/0.3= 40

12/0.006=2,000

12/0.00003=400,000

et cetera

Now, this moronic moose sees a pattern here. As the divisor gets smaller, the quotient gets larger. Seems to me that if you carried this progression on “forever”, you would “reach”:

12/0=infinity

And so I assert, with some confidence, that a line on a Cartesian graph with a delta x of zero has infinite slope.

(No way, you say? Then let me humbly suggest you carry out the above progression to the point that you can demonstrate a different conclusion. Forgive me if I don’t hold my breath.)

OK, I realize that by the rules of this particular game one will eternally approach infinity but never actually get there. What I am doing is permitting myself the luxury of a little common sense and making the small “leap of logic” required to reach the endpoint to which the number line points but is constitutionally incapable of arriving at. The problem is in the rules. If, instead of starting from zero, we started from “the infinite number” and worked out from there, then zero would be the poor orphan condemned by the gods to eternally journey home but never get there.

14

What you have just proven (or rather, demonstrated: It’s not rigorous enough to be a full proof) is that the limit of 1/x as x approaches zero from the positive direction is positive infinity. This is true, and is accepted by mathematicians. It is likewise true and accepted that the limiof 1/x as x approaches zero from the negative direction is negative infinity. But there is nothing in mathematics which says that the limit from the left must equal the limit from the right, nor that the limit of a function must equal the value of the function. In fact, neither of those is true in this case: The limit from the right is different from the limit from the left, and the function has no value at all at that point, much less being equal to infinity.

For another example, suppose I have a very simple function. I define it so that f(x) = 1 iff x > 0, f(x) = -1 iff x < 0, and f(x) = 0 iff x = 0. Now, if we take some sample values of x: f(1) = 1, f(0.1) = 1, f(0.000000000001) = 1, and so on. We can also go from the other side: f(-1) = -1, f(-0.1) = -1, and so on. Does this mean that -1 = 1, and that this is the value of the function at x=0? Of course not! We know that -1 does not equal 1, and we said right from the outset that f(0) = 0.

15

What about the projective plane?

16

In terms of REAL NUMBERS, there is indeed “no” slope for a vertical line, since there is no real number that equals this slope.

This is what is meant when we say a vertical line has “no” slope. If for any two points on a vertical line we perform the calculation according to the definition of slope of a line, we will get a denomninator of 0, hence the slope is undefined. Note this is quite different from saying a line has 0 slope, as does a horizontal line.

There is nothing wrong, however, with referring to the slope of a vertical line as being (unsigned) infinity for the reasons alluded to, so long as we understand, as most do I suppose, that infinity is not a real number.

Also, to quickly comment on anoterh poster’s remarks concerning limits: if the limit from the left is not the same as the limit from the right, then INDEED, the limit does not exist. Given the function that we were given:

f(x) = 1 x>0
= -1 x<0
= 0 x=0

Then the limit of f(x) as x approaches 0, does not exist. We can talk about the limit from the right, or the limit from the left, but that’s not the same thing. Nor does this have ANYTHING, necessarily, to do with the value of f(0). unless of course if we desire f to be continuous at f(0). Even then, there is no value you could assign to f(0) that can make that happen. f is dsaid toi have an essential (nonremoval) discontinuity at f(0). this is just a fancy way of saying the discontinuity can’t be patched with any finite number of points.

My point, as applied to this discussion, is simply to demosontrate that limiting values have nothing to do, really, with how we define the actual functional values. For instance, some heavyweight mathematicians define 0^0 to be 1 although the function z=x^y has an essential discontinuity at x=y=0. (no, I do NOT want to delve into this topic, it has been beat to death already). My point is simply that limiting values have absolutely nothing to do with the definition of f(0) in the aforementioned function. f(0) was clearly defined to be “0” although lim x–>0 f(x) does not exist.

17

Ok then I would have to ask, why would you consider only “integers” when any number, between 0 and 1 noninclusive, is NOT an integer?

18

if 12/0 = infinity
then 12 = infinity x zero
so 0 + 0 + 0 +… is equal to 12?
Obviously the sum of infinite zeros is zero not 12. For a vertical line wouldn’t it be incorrect to call the slope infinite when the slope is undefined(or defined as no slope) because it involves dividing by zero?

19

Well, what about it?

20

Darrell, read Zeldar’s posts again. You’re leaving out the step where he puts the integer in the denominator.

1/1 1/2 1/3 etc.