Another math question.

[BRPORSCHE]

zeno's paradox : suppose that a man wants to cross to the far wall of a room that is 20ft across. First, he crosses half the distance to reach the 10ft mark. next, he crosses halfway across the remaining 10ft to arrive at the 5ft mark. dividing the distance in half again, he crosses to the 2.5ft mark, and continues to cross the room in this way, dividing each distance in half and crossing to that point. because each of the increasingly smaller distances can be divided in half, he must reach an infinite number of "midpoints" in a finite amount of time, and will never reach the wall. explain the error in zeno's paradox.

Guys I haven't a clue to how to even start it. I do understand that you can dividing a number and it gradually gets smaller. Very close to 0, but does it ever reach zero? Please braintrust need some help here on some late night hwk.

[Aerkuld]

This is going to sound odd, but the man can never get to the far wall so the journey does not have a finite time.

[trekkor]

The person walking will just step to zero.

The math divisions will never get you there.


Why do you ask?


KT

[Danny_Ocean]

A suggested problem with using calculus and mathematical series to try to solve Zeno's paradoxes is that these solutions miss the point. To be precise, while these kinds of solutions specify the limit point of infinite series, they do not explain how such a series can actually ever be completed and the limit point be reached. Thus, calculus and mathematical series can be used to predict where and when Achilles will overtake the tortoise, assuming that the infinite sequence of events as laid out in the argument ever comes to an end. However, the problem lies exactly with that assumption, as Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, an infinite number of physical events need to take place, which seems to be impossible in and of itself, independent of how much time such an act would require if it could actually be done.

Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned. To be precise, Zeno *started* with the assumption that a finite interval can be split into infinitely many parts, and then argued that it is impossible to move through such a landscape. For calculus and other series-based solutions to make the point that the sum of infinitely many terms can add up to a finite amount therefore merely confirms Zeno's assumption about the landscape (geometry) of space, but does nothing to answer Zeno's question of how we can actually (dynamically) move through such a space.

Put a different way, when these kinds of solutions tell us that Achilles passes the tortoise 10/9 meter after the tortoise's starting point, they assume that Achilles can actually reach that point, but Zeno questioned that Achilles can actually ever get to that point. Similarly, when we are told that Achilles passes the tortoise 10/9 seconds into the race, it is assumed that time can actually flow to that point, but once again we get the same problem: If there are an infinite number of time points between t = 0 and t = 10/9, how can t = 10/9 ever be reached? How, indeed, can time flow at all if it is assumed that between any two time points there are infinitely many other time points that, at least under our naive conception of time, have to occur one after the other? Thus, all these kinds of solutions presuppose that Zeno's difficulties have already been solved when trying to resolve the paradox. Which is to say, they beg the question and therefore don't resolve anything at all.

An unfortunate complication among these kinds of discussions is that many treatments of Zeno's paradox present Zeno's reasoning in such a way that calculus and series-based solutions really do work as objections to that reasoning. To be precise, Zeno's reasoning is often presented as arguing that because there are an infinite number of tasks to be done, it will take an infinite amount of time to complete all these tasks, and the calculus and mathematical series based solutions are now perfectly correct in objecting to that argument by pointing out that the sum of an infinite number of time intervals can add up to a finite amount of time. However, such a presentation of Zeno's argument makes the argument into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler argument that does not at all consider any quantifications of time. This much simpler argument simply states that for Achilles to capture the tortoise an infinite series of physical events need to be completed, which is logically impossible. The calculus and mathematical series based solutions offer no insight into this much simpler, much more stinging, paradox.[2]

The following thought experiment can be used to illustrate the fact that time is irrelevant to the paradox. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle at all, then apparently Achilles must have stopped counting, and we could ask Achilles what the greatest number he counted to was. But of course this is nonsense: there is no greatest number, and Achilles can never stop counting. So, Achilles can't catch up with the Tortoise, whether he has finite time or infinite time to do so.


continued...

.

[Danny_Ocean]

The situation, then, is this. Any proper variant of Zeno's paradox, such as the above thought experiment, provides a mathematical and logical account of the physical process of movement through space (or time), and argues that it is impossible for Achilles to win (or move at all). So, in order for this paradox to be resolved, one needs to either show something wrong with the math or logic (which calculus and series-based solutions do for the improper variant, but not for the proper variant), or show why this mathematical analysis cannot be used in our physical world. As suggested below, maybe space and time are not so that between any two points one can always find another point, which would indeed prevent this analysis to go through, and possibly our naive conceptions of space and time are mistaken in other ways as well. But calculus and series-based proposals do not challenge any of our conceptions of space and time in any way, as they are purely mathematical analyses that say nothing about the nature of space and time at all. Hence, these kinds of solutions do not resolve the paradox in this second way either. In short, there is nothing in calculus or series-based solutions that prevent the infinite sequences to crop up that lead to the whole paradox. So, as such, they do not resolve the paradoxes.


Issues with the issues with the proposed calculus-based solution:

If we more closely examine the thought experiment, it is clear that Achilles naming the positions "first", "second", and so forth, is a nonphysical/mathematical act rather than a physical act; as an illustration, try getting your friend to say the word "Bob" on the 1/2 second mark, then the 1/4 second mark, and so on... you just can't do it. Consequently, the "counting process" is a mathematical process, while the "catching up with the turtle" is a physical process. As with most attempts to peddle Zeno's paradoxes, the central element is the conflation of these two processes. But they are simply not to be identified. The mathematical "counting process" goes on to infinity, and this is never something one could complete. However the physical "catching up with the turtle" process is something that can be completed. This is shown by an elementary application of limiting process theory, with time as a parameter.

These considerations (one must divorce the mathematical and physical processes at hand) also apply to the paradox as given in the "much more stinging" form: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible". Here the word finish has been confusingly used for both the physical process and the mathematical process in an effort to conflate the two.

The issue with the statement "Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned." is similar. They (the vast majority) do not assume that one can finish the limiting (mathematical) process, and they do not need to. To finish the physical process it is not required to finish the associated mathematical (limiting) process. The two processes are completely different in nature, and divorcing the two is essential if one is to resolve the paradox.

The mention of time does not make the paradox into a strawman, and telling someone they can't mention time in their solution is extremely unfair, because the problem is posed in the form of the physical, and consideration of time is implicit in any consideration of the physical. Just because someone worded the problem without using the phrase "time" does not make it illegal to use the word "time" in the solution.


Are space and time infinitely divisible?
Another proposed solution to some of the paradoxes is to consider that space and time are not infinitely divisible. Just because our number system enables us to give a number between any two numbers, it does not necessarily follow that there is a point in space between any two different points in space, and the same goes for time.

If space-time is not infinitely divisible (and thus not perfectly continuous), it is "discrete" (composed of “lumps” and “jumps”, as is experimentally observed in the field of quantum physics e.g. electron orbitals jumping from one level to another). This means that motion is, at the smallest physical level, a series of jumps from one quantum space-time coordinate to the next, each occurring over distance and time intervals that are not divisible into smaller measures.

Thus the total number of quantum jumps made while traversing from point A to point B is finite, and therefore there is no paradox.


Does motion involve a sequence of points?
Augustine of Hippo was the first to posit that time has no precise "moments," in his 4th century C.E. text, Confessions. In Book XI, section XI, paragraph 13, Augustine says, "truly, no time is completely present," and in Book XI, section XV, paragraph 20, Augustine says "the present, however, takes up no space."

Some people, including Peter Lynds, have proposed a solution based on this ancient premise. Lynds posits that the paradoxes arise because people have wrongly assumed that an object in motion has a determined relative position at any instant in time, thus rendering the body's motion static at that instant and enabling the impossible situation of the paradoxes to be derived. Lynds asserts that the correct resolution of the paradox lies in the realisation of the absence of an instant in time underlying a body's motion, and that regardless of however small the time interval, it is still always moving and its position constantly changing, so can never be determined at a time. Consequently, a body cannot be thought of as having a determined position at a particular instant in time while in motion, nor be fractionally dissected as such, as is assumed in the paradoxes (and their historically accepted solutions).


Conceptual and semantical approaches
Another approach is to deny that our conceptual account of motion as point-by-point movement through continuous space-time needs to match exactly with anything in the real world altogether. Thus, one could deny that time and space are ontological entities. That is, maybe we should acknowledge our Platonic view of reality, and say that time and space are simply conceptual constructs humans use to measure change, that the terms (space and time), though nouns, do not refer to any entities nor containers for entities, and that no thing is being divided up when one talks about "segments" of space or "points" in time.

Similarly, one can say that the number of "acts" involved in anything is merely a matter of human convention and labeling. In the constant-pace scenario, one could consider the whole sequence to be one "act," ten "acts," or an infinite number of "acts." No matter how the events are labeled, the tortoise will follow the same trajectory over time, and all of the acts will be "finished" by the time the tortoise reaches the finish line. Thus, the labeling of acts is arbitrary and has nothing to do with the underlying physical process being described and that it is possible to "finish" an infinite sequence of acts.

From the philosophical standpoint of Bergsonian space-time, the paradox is resolved as follows. The steps of the paradox as presented above can be summarised as:

There are an infinite number of positions defined by any finite movement.
Let movement from one position to the next be called an 'act'.
An infinite number of acts cannot be completed in a finite amount of time.
An infinite number of acts cannot even be started.
Thus movement cannot be started or completed.
Movement is an illusion.
Moving backwards, any claims about the nature of illusions or acts are intrinsically claims about the nature of experience. According to Bergson's conception of time, all moments of time are comprised of a mixture of both a 'snap-shot' extrinsic property and a durational intensive property, which are irreducible to one another.[2]

The arrow paradox makes an argument that considers only time as a measurable, extensive, homogeneous construct that can be modeled spatially (the above diagram of it with lines being a good example). Thus a conclusion concerning the nature of experience is not warranted by an incomplete proof of only partial properties of time. The point is that 'acts' are experiential in nature.


The notion of different orders of infinity:
Some people state that the dichotomy paradox merely makes the point that the points on a continuum cannot be counted — that from any point, there is no next point to proceed to. However, it is not clear how this comment resolves the paradox. Indeed, as one variant of Zeno's paradox would state: if there is no next point, how can one even move at all? Also, it is not clear what this comment has to do with different orders of infinity: the rational numbers are countable, i.e. of the same order of infinity as the natural numbers, but on the rational number line, there is for any rational number still no next rational number either.


Status of the paradoxes today:
Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Most philosophers, and certainly scientists, generally agree with the mathematical results.


continued...

[Danny_Ocean]

Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this, he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's

[aigel]

Where are you in your math? What level?

The only way you arrive is if you are allowed to take an infinite amount of steps. Otherwise you will not reach the wall one by adding 1/2, 1/4, 1/8 etc. If you are able to write sums you basically should be able to show that for n from 1 to infinity the sum over 1 devided by n equals one. You can even write a little computer program to see how quickly it converges.

I looked for some info online and this one is excellent:

http://members.aol.com/kiekeben/zeno.html

George

[Danny_Ocean]

Feel free to use this for your paper...I won't be responsible, however, for any plagarism charges leveled by your professor.

[BRPORSCHE]

Danny awesome stuff!!

[Danny_Ocean]

Originally Posted by BRPORSCHE

Danny awesome stuff!!

Thanks. That took me a while to copy/paste.

Wikipedia is your friend...

[Aerkuld]

I suppose the improbability of the whole situation comes down to a matter of scale. The half-way points would end up getting so ridiculously small that they would become meaningless to the man moving them so that the distance between him and the far wall would become negligable.

Look at the problem a different way to eliminate the scale problem.
Imagine the distance the man is attempting to travel is 8ft and the man's stride is 4ft. It should take him two paces to reach the far wall. But if he shrinks to half his size with each step forward and his step is reduced accordingly then he can never reach the wall, thus the journey is infinite (not allowing for the fact that he would eventually get to the point where he would need to reach a sub-molecular size). For the time in this instance you could say that he takes 1 second per step regardless of length. Now it's more obvious that the time is infinite.

[svandamme]

me thinks Zeno is an idiot for coming up with the idea of walking only half the distance, instead of just walking the goddamn' 20 feet in one go like normal people do.

mathematical problem my ass, it's just an mismatched analogy of a math problem with a real world example that just doesn't work

[john70t]

Eventually the person and the wall would be so close that dynamic molecular forces would occur.

[svandamme]

fancy word for touching? his nose to the wall?

less then 25 steps, if you can call those last ones "steps", not even a shuffle, more like , leaning over another 1/100th of a mm, and even that requires a big stretch of imagination, because i'de like to see anybody having so much control over his body movement that he can choose to move any bodypart for just 1/100th, let alone the whole body

so mathematical infinity, becomes < 25

as such the analogy of this theoretical math problem and this practical real life example doesn't work one bit

[island911]

Originally Posted by BRPORSCHE

...
Guys I haven't a clue to how to even start it. I do understand that you can dividing a number and it gradually gets smaller. Very close to 0, but does it ever reach zero? Please braintrust need some help here on some late night hwk.

Yeah, but if you plot the change you get a nice asymptotic (sp?) function. --those are very useful for understanding calculus limits. If you don't understand that whole "walking to the wall' thing, you will have a tough time understanding how you can mathimatically find the slope of curved line at any given point. And, if you don't understand that, you will have a tough time having faith/confidence in solving first derivitives based on rules . . . which is the basis for understanding/doing second, third derivitives.... integration... then on to all the other fun nonlinear fun.

btw, it all has good use. FOr example a torque/HP graph will have more meaning if you understand calculus. Same for even your typical graph of 0-140mph. Lots of extra info is there for those who understand.

Anyway, I think what you are looking for is that predictable asymptotic (sp?) function (should have been learned in pre-calc)

[Hetmann]

The man has a finite amount of time because he will eventually die. Given that he can only travel half the distance with each "trip", he will never reach the wall.

[Aerkuld]

What's on the far wall that the man wants anyway?

[1fastredsc]

Island911 got it on the head. This sounds like the primer in calculus before they introduce limits and the idea of "approaching" a value and evaluating what it should do at that value without every getting there. This idea helps develop how it's possible to evaluate something being divided by zero even though normally with math that is not possible. For instance the limit as x approaches 0 of sin(x)/x is a good example. With normal math you cannot divide by zero and therefore cannot evaluate this expression. However when you use zeno's idea, then you say well, i won't actually get to zero instead i'll see what happens as i get ever closer to zero. sin(x)=0 when x is zero, therefore as you approach zero it "looks" like that expression is equal to 1 since they are both converging to the same number as x goes to 0. This idea pays off huge when you are taught what the definition of a derivative is which is everything in calculus.

[1fastredsc]

Oh, and the error would be based on how many times you add by half. If you cross half way an infinite amount of times in theory your error is zero. However anything less than infinity will bring an error. As wayne said that's the difference between engineers and mathematicians, mathematicians live in theory, engineers set a tolerance, run a program, and say that 19.9999 feet is good enough .

[GO DAWG GO]

The explanation and parallels are right out of Wikipedia...


This is theoretical malarkey and good for very little without some sort of agreement to finite acception and tolerance. Wayne is reasonable...

Kind of Silly... Like divide 100 by 3 and trying to derive a concise conclusion