I remember a friend telling me once that mathmatically, an arrow shot through the air should never reach its target. The reason, she said, is this: if the arrow was shot from 10 feet away, it must pass the halfway point: 5 feet. Then, it must pass the halfway point again: 2.5 feet. Then it must pass the halfway point again: 1.25 feet. Then, it must pass the halfway point again, and again and again - eternally. As anyone with a calculator can see, you can divide by two forever (or until your calculator gives up and shows an E).

Obviously, reality shows us differently. The arrow DOES eventually hit its target. If what my friend said is correct, then math is not always truth.

I suspect I must be missing something. What is it?

(I was going to post this question in this thread, but then I decided it might be off topic. If it turns out to be relevant, I'll link to it in the other thread.)

This message has been edited. Last edited by: DorianGreyed,

I believe this one is known as Xeno's (sp)paradox.
The time to travel each small distance is also decreasing as the distance goes to zero so does the time to travel that ever smaller decreasing distance.. The arrow does get to the target.
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04-26-05, 09:52 PM
gerry
Yes, this is called in the mathematics a convergent infinite series, whereas in this case
5 + 2.5 + 1.25 + 5/8 + 5/16 + 5/32 + 5/64 etc = exactly 10.
If the arrow is travelling at a modest 10 ft per second, it reaches its target in exactly 1 second. It takes 1/2 second to travel the first 5 feet, 1/4 second to travel the next 2.5 feet, 1/8 second to travel the next 1.25 feet, etc., and again using the convergent series ,
1/2 + 1/4 + 1/8 ..... = exactly 1 second. The mystery to me still lingers, however, in that it travels the last infinitesimal distance in 0 seconds, and that still boggles my mind, I think this is where string theory, Planck's lenght, and quantum mechanics enters the picture?
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04-26-05, 10:34 PM
methos
Yes, this is called Zeno's paradox and no, it is not really an example of math failing to describe reality. Zeno liked coming up with "logical" arguments that conflicted with reality. The key, as Gerry suggests, is that Zeno does not include time in the description.
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04-27-05, 02:29 PM
Sarai
I have total faith that you three are correct, so please don't view this as an argument, because there's no way I'm arguing about math with anyone, let alone three experts! Smile However, I admit that I (a mathmatical idiot) still don't understand exactly why this isn't evidence of math describing reality incorrectly.

I did a search on the internet and have found several sites that basically agree with what you're saying, but it still isn't clear to me. I see what Gerry is saying about how the distances add up to 10 and the time adds up to one second, but I'm still stuck on the fact that the number of half points is, mathmatically, infinite. Perhaps I'm uneducable... but if anyone wants to try to help me out, I'm all ears! Smile
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04-27-05, 06:38 PM
gerry
You make a very good point, Sarai. While it takes 1 second exactly for the arrow to travel 10 feet (at a rate of 10 feet/sec), the infinite series of 'half step' periods ...
1/2 + 1/4 + 1/8 +1/16 + .....etc. adds up to
0.9999999999999999999999999999999999999+ seconds with an infinite number of 9's after the decimal point.....so many 9's that that after an infinite number of 'half steps', the total time converges (in the limit) to 1.0 second exactly. But this seems to indicate that infinity, and its inverse, the infinetisimal 1/infinity, must exist! I leave this to others to explain. In the real world, however, the arrow can only get a finite distance from the wall before it actually 'touches' it....this tiny tiny distance is the Planck Length of about 10 to the minus 33 centimeters...after that, it makes a quantum leap to the wall, and the classic math says nothing about that. But now we're into the Physics, and I'm probably over my head as well as yours! Maybe Peteeo or Methos can explain.
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04-27-05, 07:04 PM
frankvan
See if this makes any sense. Here's how I think of it.The only way we can measure distance is along a straight line, the distance between two points. However, when we want to measure distance along a curve we need to devise some way to break that curve up into a series of smaller and smaller straight lines, measure each increment, and then total them. How accurate that distance measure will be depends on how closely that series of straight lines approximates the actual curve. So the series that gerry illustrates doesn't actually ever reach an end, even though he calls it a 10, it is actually 9.99999999999999ad infinitum. It is called 10 because it can approach 10 more and more closely without ever reaching it. 10 is its limit. And hence the time it takes to reach the end distance at 10 feet per minute is 59.999999+ seconds, never actually using up a complete minute or whatever speed is assumed. Without this business of infinite division of distances the mathematics of calculus could not exist. To use a familiar example: Think of the quantity Pi in the Pi r squared, etc formulae. We only approach as closely as possible to the actual relationship between the straight line diameter and the curved line circumference. How closely depends on how many decimal places you choose to go with 3.14159265+++.
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04-27-05, 09:08 PM
Professor
Zeno's paradox is one example of many confusing issues that were not really resolved by mathematicians until the concepts of limits and convergence of series was formalized and made logically rigorous in the modern era (the past 3 centuries or so), paving the way for calculus and other forms of analysis, as already pointed out by frankvan.

A similar classic conundrum concerned the fleet-footed Achilles, running a race against a tortoise who is given a head start. The ancient Greeks reasoned that Achilles should never catch up to the tortoise because he would have to cover half the distance between them, then half again, etc. Basically the same dilemma in a different form.
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04-27-05, 10:14 PM
methos
Don't worry, Sarai, I'm not convinced I understand it either, but don't tell anyone Wink.
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04-28-05, 11:26 AM
Sarai
Oh no! Are you all telling me that in order to understand this I have to study calculus (which, unfortunately, I opted not to take when I was in school), and even then, I might not understand it? NOOOOOOOOOOOOOOO!

Seriously, is there any way an almost 30-something lady can hope to learn enough calculus to understand this problem?
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04-28-05, 12:58 PM
frankvan

quote:
Seriously, is there any way an almost 30-something lady can hope to learn enough calculus to understand this problem?



Seriously, I had to learn a few things beyond advanced calculus and differential equations, and I'm not sure I even know what the problem is, let alone understand it. You're still a child and obviously a rather bright one. I'm sure you can learn anything that interests you. Wink
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04-28-05, 02:30 PM
Sarai
You know, it has just occured to me that this problem can be applied to all movement and all time.

For example, to type these words, my fingers have to get halfway from where they are before I strike the key and the key itself, then halfway between the halfpoint and the key, and then halfway between that point and the key, etc into eternity.

Likewise, if it is currently 12:00, and I want to live to 12:01, I have to live half of that time first (30 seconds), then half of that time (15 seconds), then half of that time (7.5 seconds), then half of that time (3.75 seconds), then half of that time (1.875 seconds) etc into eternity, thus making the one minute an eternal, impossible minute to complete because divided by an eternity of half points.

Based on this logic, nothing should ever be able to move and time should not actually advance. And yet we do move and time does advance.

So, now that I've completely confused myself, what about calculus should I try to learn? Although Frank has inspired me to think I might have a chance (thanks, FrankVan! Smile), it would be nice to have some direction to know what about calculus I need to understand to resolve this apparent logical problem. So I'm going to try to teach myself (gulp) some basic calculus. No promises that I'll succeed! Anyway, is there a website or book that anyone might recommend? Any suggestions about what, exactly, I should be trying to learn about calculus?

This message has been edited. Last edited by: Sarai, 04-28-05 04:04 PM
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04-28-05, 04:24 PM
frankvan

quote:
into eternity, thus making the one minute an eternal, impossible minute to complete because divided by an eternity of half points.



Not an impossible minute, not into eternity, just a period of time, however small, that can be divided into an infinite number of pieces. How many numbers can you write from 1 to infinity? You can divide the space between the number 1 and the number 2 into how many pieces? Is one infinite number bigger than the other. 1/4 is smaller than 1/2 and 1/100 smaller yet, but how large a number can you put in the denominator? And when you reach the largest conceivable number how infinitesimal is the value of the fraction? Now when you add all of those bits together do they add up to anything more than the numerator? I don't think you need to learn calculus, just get your head around this idea, Zeno was not up to date. To study calculus you need to be pretty good at the pre-requisite algebra, trigonometry, analytic geometry, and satisfied with a less than perfect precision.IMHO
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04-29-05, 11:08 AM
Kendor
So how does the arrow even get to the first halfway point? How does it begin moving at all? Doesn’t it first have to travel one quarter the distance, after having traveled one eighth the distance, after traveling one sixteenth…

This is not how reality works.

Here’s the simple answer. The arrow has to travel the entire first half, and then the entire last half. As soon as you start calling out secondary and tertiary, (and so on), halves, you enter into the realm of quarters, eighths, sixteenths, etc. So then we say, “The arrow has traveled the first three quarters and then will complete the last quarter.” Or “It has traveled the first 7 eighths and must travel the last eighth.” (Then to sixteenths, thirty seconds, etc.). You must reference only one of these units to explain the completion of the arrow’s traverse.

That’s my story and I’m sticking to it.
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04-29-05, 02:53 PM
frankvan
One more attempt to refute Zeno and simplify the paradox's solution. He is trying to relate time and distance which relates to speed or velocity. If a projectile, an arrow, or a runner, Achilles travels at say, 30 mph and over a distance of 60 miles, in order to determine the time it takes to complete the journey we divide the distance by the speed: 60mi/30mph= 2 hrs. 60/30 can only keep the same value as a fraction if we perform the same operation on both numerator and denominator. You can NOT divide the numerator by 60 unless you divide the bottom by 60 at the same time. If you do that, you get 1mile/1/2mph=2 hours - same as before. Zeno is performing an illegal operation by performing an operation on the top half of a fraction and leaving the bottom alone and pretending that the fraction is still the same. Wrong for the same reason that you can't divide by zero, or add a series of asymptotes and say that the sum equals some value - it only approaches that value as a limit. WinkBTW,jusork, How do you get a tertiary half??
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04-29-05, 03:16 PM
Kendor

quote:
Originally posted by frankvan:
BTW,jusork, How do you get a tertiary half??



That's part of my point, frank. (Kendor, not jusork)

In the original post the first half is half way twixt two points. The second half is actually 3/4 the way, and the third, (tertiary), is 7/8ths. Not halves at all of the original distance, but smaller and smaller distances that will never complete the span when perpetuated this way mathematically. To me there's no paradox at all.
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04-29-05, 06:24 PM
Sarai
You know, as we discuss this, I'm becoming more and more skeptical. I think this IS an example of math describing reality incorrectly.

Frank, your explanation explains how we go from one point to another, but I'm not sure that it actually shows a mathmatical error on Zeno's part. In his case, both parts of the fraction are referring to space (or in my other example, time). We do this often in math and it is not wrong. For example, I can measure a 1/2 cup of flour - I don't have to take time into consideration. I can measure anything without taking time into consideration, and by Zeno's logic, all things that can be measured are made up of an infinite number of points. Time or no time, infinity cannot (logically) be traversed.

Your point, Frank, shows that math can be correct, but I think Zeno's point shows that math can also be incorrect, even if there is nothing wrong with the math itself.

The point is that math CAN describe reality incorrectly.

This message has been edited. Last edited by: Sarai, 04-29-05 07:54 PM
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04-29-05, 06:48 PM
Sarai

quote:
Originally posted by Kendor:

This is not how reality works.



I agree. That is why I am beginning to think math can describe reality incorrectly.

quote:
Here’s the simple answer. The arrow has to travel the entire first half, and then the entire last half. As soon as you start calling out secondary and tertiary, (and so on), halves, you enter into the realm of quarters, eighths, sixteenths, etc. So then we say, “The arrow has traveled the first three quarters and then will complete the last quarter.” Or “It has traveled the first 7 eighths and must travel the last eighth.” (Then to sixteenths, thirty seconds, etc.). You must reference only one of these units to explain the completion of the arrow’s traverse.



Whether you call them eights, sixteenths, millionths, or googlionths, there still end up being an infinite number of points between any two places.

Finally, Kendor, you said that this implies that arrows can't move at all. Yes, it I agree es - I also think it implies that time can't progress (see above post).
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04-29-05, 06:54 PM
Sarai
1) It is impossible to travel an infinite number of points and arrive at the end, because there is no end to infinity.

2) There are an infinite number of points in the distance between any two places (or in the time between any two moments). The fact that you can divide by two infinitely demonstrates that there are, in fact, an infinite number of half-points from one place to another.

3) Therefore, nothing can travel from one place to another (nor can time progress from one moment to the next).

Can anyone disprove the above points without referring to empirical evidence? If the only thing that proves the math is wrong is our experience, then math in itself is, in fact, not truth. It is simply a tool that can give truths about reality, but that can also give us very false results about reality. Thus, this is an example of perfectly good math describing reality incorrectly. Math is fallible.
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04-29-05, 09:10 PM
jusork

quote:
Originally posted by Sarai:
Math is fallible.



Or maybe math is fallible in a finite world, but not fallible in itself, which doesn't exist physically in this world, but exists in a non-finite, eternal, timeless logic. And logic can also be applied to the nature of the universe. The math that shows itself in nature is only a reflection of the truth. 1 + 1 exists in the math place of truth and, hey!, that piece of logic in math is also logical in the finite world. Math is a truth, but it's not the same thing as reality. Math goes by math logic which comes in both infinite and finite. The universe goes by finite logic, which sometimes uses the finite math logic. It just describes reality. Math isn't an element in reality, but reality is an element in math. Is there anything in reality that math can't describe?

So yeah, I guess it would be math describing reality incorrectly...because math doesn't exist hand in hand with reality. Math is still a truth. Remember, everything doesn't revolve around the universe. Big Grin

I'm not a mathmatician so I could be missing something, but it seems to make sense.

Hey, Frank!
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04-30-05, 07:11 AM
gerry
Zeno's Paradoxes - Is this an example of math describing reality incorrectly?
Math is just a tool for the science of the real world. Supposing a frog jumps in half leaps to a wall 10 feet away, that is, its first jump is 5 feet, its next, 2.5 feet, then 1.25 feet, etc. Let's also say that it takes him one minute to make each jump, no matter what the distance of the jump (this allows it to rest in between jumps). How long does it take him to reach the wall? The math says it will take an infinite amount of time for the frog to reach the wall....that is, the frog will never reach it. In reality, the frog, once it gets to say 1/4 inch from the wall, will be unable to jump the next 1/8 inch, because its feet are too big, its motor control too slow, etc., and instead, on its last 1/4 inch jump, will end up smashing into the wall. The science will describe it exactly if you really get into it deep enough...the math will just help you along.
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04-30-05, 08:47 AM
frankvan
If one accepts Zeno's logic, in which we can never go from point A to point B because we divide the intervening space into an infinite number of halfway points between A and B forever without end, what happens when we divide the distance between point A and point B+, (B+ being a point just beyond point B)??

Whether or not it is a case of math describing a reality unrealistically or Zeno talking absolute rubbish, I have never heard of any math that uses the concept of infinity as though it was something tangible and attainable or measureable, only an indication that we are talking about a convenient but undefined value. I have never seen it used in any context other than an arrow and infinity symbol to indicate "as a variable grows toward an arbitrary limit (the symbol which I can't make on this keyboard' or as tha variable trends toward 0.As Gerry says, math is a tool. But it is a very disciplined and truthful tool which, to my knowledge, never indulges in describing reality inaccurately. I have, however seen many instances in which people have mis-applied math in order to arrive at patently ridiculous conclusions. Achilles can beat the tortoise, the frog can jump as far as the wall and even beyond it, depending on the height, And I will defend, with my life if need be, etc. etc. Wink
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04-30-05, 12:11 PM
methos
My problem with this whole line of argument is that we are really discussion a philosphical or logical argument, rather than a mathematical one. However, if we take logic as a part of mathematics, which it is sometimes considered, I'll have to agree with Sarai that this is math failing to describe reality.

As far as I can think of, there are two types of failures of math to describe reality.



One is where the mathematical description is incomplete.

In this case, Zeno considers the movement in space without the movement in time. Sarai's argument similarly takes the movement of time without considering the movement through space. It is only when you take them simultaneously that it works (it doesn't take infinite time to move a finite distance, even by the arbitrary half-step method, if you consider that the change in time is gettign ever smaller with each step as well).

Another example of this sort is Newton's laws of motion. In certain cases (high speeds), they fail to describe reality. Einstein and Lorentz's equations for relativity fill in the missing pieces for a more complete description of reality.



The second case is a bit trickier. It is a functional mathematical system that may or may not have anything to do with our reality.

Relativity doesn't describe reality properly under all conditions, either. Theories such as string theory make use of other math to try to complete the description. For now, we don't know whether this math is describing our reality or a fictional one.
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04-30-05, 12:15 PM
Sarai
I agree that it is a ridiculous conclusion, Frank,which is why I think we should be careful - math CAN describe reality incorrectly.

I now feel quite confident that mathmatically, it is true that there are an infinite number of points between any two points (recognizing that infinite cannot be fully defined). Thus, I think we must be careful not to put too much faith in math alone.

So now I'm going to post in the original thread that brought me to this question. I think I've changed my mind about the nature of math. Thanks, everyone! Smile
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04-30-05, 12:22 PM
Sarai
Methos - Sorry, I posted at the same time as you did, and didn't see your post. I think what you're saying makes sense, and fits with the idea that math is a tool, not "truth."

Thanks again. Smile
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04-30-05, 12:30 PM
Sarai

quote:
Originally posted by jusork:

quote:
Originally posted by Sarai:
Math is fallible.



Or maybe math is fallible in a finite world, but not fallible in itself, which doesn't exist physically in this world, but exists in a non-finite, eternal, timeless logic.



Illusory- I'm responding to you in the other thread. I think we should move this discussion over there.
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05-02-05, 01:07 PM
FlyingHellfish
Maybe I'm missing something, but I'm failing to see where the mathematics is not accurately reflecting what is actually happening in this "real world" experiment. What I see is a paradox posed in a certain way such that the casual reader will not think to implement the appropriate mathematical model used to describe it properly, which is hardly the same thing as no such model existing at all. It's what I call a logical illusion-kind of like an optical illusion for the mind Smile

In my mind, mathematics describes it perfectly well; the problem arises when we try to visualize what is happening at the extremes, or in this case, as the arrow approaches an infinitesimally small distance from its target. This coincides with the elapsed time getting infinitesimally closer to the calculated time to impact. The mathematical reason (not based on simple observation) why the arrow does in fact reach its target is that the distance and time until it makes contact approach 0 (or, put differently, 1/infinity) at the same relative rate.

EDIT: Oops...I missed the link to where the discussion was continued Smile
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05-03-05, 04:21 PM
Professor
FH: Ditto to what you said. Smile

I made a related posting about Zeno's paradox here -- there are multiple math threads getting entangled!
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05-03-05, 07:08 PM
Sarai
Sorry about that - I have made 3 questions related to a similar subject. I have even more, but plan to wait a while so that I can start a new thread without causing massive chaos! Smile

By the way, for any other unlucky souls who come across this thread without seeing the others... Sarai finally saw the light and came to understand why it is possible to travel an infinite distance in a finite amount of time. It is because time is being divided infinitely along with the distance.

I'm still not 100% clear on how it is possible to get from one point in time to the next considering the infinite number of moments between two points in time, but in addition to creating a chaotic number of threads, I think people are getting ready to throw rotten tomatoes at me, so I'll give it a rest for a while. Eek

Thanks again, everyone!

This message has been edited. Last edited by: DorianGreyed,