Zeno's Paradox - real or just a play on words?

[Can'tThinkOfOne]

I've heard the following argument from one professional philosopher about the impossibility of motion (it's a version of Zeno's paradox):

Assume that motion is possible. If an object is to move from point A to point B, then it has to move through an infinite number of points to get to B. This means that in order for the object to complete its journey in a FINITE amount of time, at each point the object can only spend ZERO time, since there's an infinite number of points, and anything other than zero multiplied by infinity will give an infinite result. But being at some place for zero time means you haven't been at that place at all (if you were on Mars for zero time yesterday, then that means that you weren't there at all). I.e. the object has not been at any of the points on its trajectory. So, we have a logical contradiction, and therefore motion is impossible.

This argument seems like a play on words more than anything else to me, but I'd like to hear your thoughts on this.

 

[Hurkyl]

But being at some place for zero time means you haven't been at that place at all

Why would you think that?

Incidentally, this very example is a heuristic proof that the quoted statement is false. (assuming you accept the other hypotheses)

 

[Gelsamel Epsilon]

Zeno's Paradox is more a play on our concepts imho.

 

[Can'tThinkOfOne]

Why would you think that?

Incidentally, this very example is a heuristic proof that the quoted statement is false. (assuming you accept the other hypotheses)

I'm not the one who's thinking that. This argument is from a philosophy prof. He seems to argue that the universe is like a movie, i.e. a sequence of stationary states, and motion is illusory. I was looking for a solid refutation if there is one. It seems to me that even if motion IS illusory, and the universe IS a sequence of stationary states (doesn't quantum mechanics suggest that space and time might be quantized?), the described paradox doesn't have much to do with it and doesn't prove or disprove anything.

 

[HallsofIvy]

It's hard to "refute" something that has one error after another! All you can really do is assert that what he is saying isn't true and he may simply deny that you are correct.

As Hurkyl pointed out, the statement "But being at some place for zero time means you haven't been at that place at all" is just not true. If it was your philosophy teacher who said that, ask him what justification he can produce for it. The same confusion occurs in students with the distinction between discrete and continuous probability. As long as there are only a finite number of events, a probability of 0 means it can't happen. With a continuous probability distribution, that isn't true. If a number is chosen, from 0 to 1, at random (uniform distribution) then each individual number has probaility 0- but obviously some number must be chosen.

 

[RVBuckeye]

The first known Greek philosophers were constantly trying to explain reality as being composed of one "thing". Some thought the ultimate "stuff" was water, some air, some numbers, others fire. This type of thinking, that reality is composed of one ultimate thing, is precisely why Zeno baffled so many people with this paradox.
Later philosophers reached the conclusion that reality was not composed of one thing but many. These were known as Pluralists, and most notable of them was Democritus. (who first suggested atoms) Not only that, motion and empty space were real. To them, it was inertia that needed explaining.
The first philosophers didn'tmake a distinction between sense data and reason (later known as empircism and rationalism)

 

[Can'tThinkOfOne]

As long as there are only a finite number of events, a probability of 0 means it can't happen. With a continuous probability distribution, that isn't true. If a number is chosen, from 0 to 1, at random (uniform distribution) then each individual number has probaility 0- but obviously some number must be chosen.

I haven't thought of that argument. This is perhaps somewhat irrelevant, but since algebraic numbers are countable, does this mean that the probability of getting any non-transcendental number is zero? Anyway, what does it even mean to pick a random number on a line? You could use a physical process, such as throwing a dart, but you won't be able to measure the dart's position exactly, so you won't even be able to tell if the number the dart hit is rational or irrational.

As far as the impossibility of motion is concerned, would I be correct in assuming that, while Zeno's arguments are fallacious, space and time MIGHT very well be quantized and all motion might be "jerky" (like a movie, a sequence of frames)? But it's just that Zeno's paradox proves nothing and is irrelevant to the question of the possibility of motion?

 

[Hurkyl]

I'm not the one who's thinking that.

I wondered about that -- take my response as what I would say to the person who is.

Pretty much every Zeno-like (pseudo)paradox I've seen boils down to someone making the mistake of treating the infinite as if it were finite.

But it's just that Zeno's paradox proves nothing and is irrelevant to the question of the possibility of motion?

Zeno's pseudoparadox certainly proves something: it demonstrates a contradiction inherent in a collection of assumptions... so one of those assumptions must be false. (But it doesn't say which assumption is false -- as I stated above, it's going to be the assumption about the behavior of the infinite)

 

[RVBuckeye]

Pretty much every Zeno-like (pseudo)paradox I've seen boils down to someone making the mistake of treating the infinite as if it were finite.

That's certainly one way of looking at it, and I'm not trying to disagree. But I think it's a little deeper than infinities. As I stated above (#6), those pre-socratic philosophers, being monists as they were, had the most trouble trying to imagine how one essential thing could move from a place where this substance is, to where it isn't. (Since there could be no such place as "where it isn't")

Hence, Pluralism

 

[Can'tThinkOfOne]

Anyhow, does quantum mechanics support the view that the universe is "jerky", and space and time are quantized (so motion would be discontinuous)?

 

[selfAdjoint]

Anyhow, does quantum mechanics support the view that the universe is "jerky", and space and time are quantized (so motion would be discontinuous)?

No; what QM says is that states evolve smoothly ("Unitary") and our observations of them are discontinuous. This is the minimal interpretation of the formalism. Interpreting the nature of the discontinuity ("Collapse of the wave function", splitting of the universe in MWI, etc.) is more or less a Roschach blot. You pays your money and takes your choice.

Quantum gravity says that space (area and volume) is quantized, which means they have states that behave according to the QM formalism above, not that they are discontinuous. QG says nothing about time, indeed that is a recognized problem with QG, they have a great deal of trouble saying anything cogent about time in their formalism.

 

[Mycroft7]

There are at least two ways of dealing with Zeno's Paradoxes, which really, any professor of philosophy (or mathematics, for that matter) certainly ought to be aware of.

First, says Aristotle, there is the distinction between potential infinite and actual infinite. Potential in this case meaning that space is infinitely divisible, and actual meaning it is made of an infinite number of parts. It is the former which is more accurate. For one thing, a line is not "made up" of points. A point has zero magnitude, or, by Euclid's definition, "that which has no part." So there's no such thing as a part with zero magnitude, because that's not a part. If a line is made up of anything, it is smaller segments of line, each of which is indefinitely divisible. Yet infinite divisibility does not mean you have to approach it as an infinity. It is, after all, a finite distance, and can be seen as one entity. You can, in the imagination, divide it into more than one, but you can also NOT divide it. Further, no matter how many times you divide it, it will still be a FINITE number of times.

Newton deals with the problem even more simply. Just as space is infinitely divisible, so is time. So even if you were to posit an actual infinity of infinitely many infinitely small distances to traverse, you have a complementary infinite time in which to do it. :-)

 

[Hurkyl]

I have lots of minor issues, but the biggest problems I have with your rebuttal are:

Yet infinite divisibility does not mean you have to approach it as an infinity

Zeno doesn't suppose that you "have to" -- only that you "can".

Further, no matter how many times you divide it, it will still be a FINITE number of times.

Only if we limit ourselves to iterating the process "take one thing and divide it into two parts". That is not the only way we can "divide" something in our analysis.

potential infinite and actual infinite

I really frown on this terminology, for two reasons:

(1) I've never seen anyone attempt to rigorously define them
(2) The way they are used appears to be hiding a fundamental misunderstanding of the infinite.

In particular, I think by "potential infinite" that you mean something like "unbounded, but finite".

A class of things is said to be unbounded, but finite if every individual thing in that class has finite "size", but there is no finite upper bound on how "large" they can be.

 

[selfAdjoint]

In particular, I think by "potential infinite" that you mean something like "unbounded, but finite"

The way I've seen it used, "potentially infinite'" is an imprecise, culture-loaded approximation to "unbounded" simpliciter. It does not at all imply finite. For example the integers are described as potentially infinite, meaning you can always find a bigger one, but every one you find will be finite. Actually infinite would be if you could actually exhibit an infinite integer.

 

[-Job-]

If there's an infinity of points between A and B then they must be infinitely small. The time necessary to travel through something infinitely small, at some constant speed, must be infinitely small.

The time necessary to go from A to B is the infinite sum of infinitely small values, which is not infinite at all.

One infinity pulling one way to make the value infinitely large and another pulling another way to make the value infinitely small cancel out.

 

[DaveC426913]

If there's an infinity of points between A and B then they must be infinitely small. The time necessary to travel through something infinitely small, at some constant speed, must be infinitely small.

The time necessary to go from A to B is the infinite sum of infinitely small values, which is not infinite at all.

One infinity pulling one way to make the value infinitely large and another pulling another way to make the value infinitely small cancel out.

As Hurkyl points out here: "Pretty much every Zeno-like (pseudo)paradox I've seen boils down to someone making the mistake of treating the infinite as if it were finite."

You cannot use finite mathematics on infinities and get sensical answers.

 

[Mycroft7]

Zeno doesn't suppose that you "have to" -- only that you "can".

Ah, but he does. If you didn't have to treat it as infinite, then there would be no paradox. But the paradox lies in phrasing it such that you do. In order to go the whole distance, first you must go half, etc.

I really frown on this terminology, for two reasons:

(1) I've never seen anyone attempt to rigorously define them
(2) The way they are used appears to be hiding a fundamental misunderstanding of the infinite.

In particular, I think by "potential infinite" that you mean something like "unbounded, but finite".

A class of things is said to be unbounded, but finite if every individual thing in that class has finite "size", but there is no finite upper bound on how "large" they can be.

I used those terms because they are the closest translations to what Aristotle used. "Potentially infinite" does basically mean unbounded, yes. And yes, "actual infinite" here is a misunderstanding, as if one could have "infinity parts" or something, where each part is equal to zero. But any number of "zeros" added together, even "infinity zeros," will yield nothing but zero. Really, I've always felt that Zeno, rather than trying to say that there is actually no such thing as continuous motion, was merely pointing out fundamental flaws in the way "infinity" was conceived at the time, since obviously there is motion. It can also be viewed as an intuitive demonstration of the impossibility of performing arithmetic functions on "infinites," i.e., that there is really no such thing as an "infinite number."

 

[Hurkyl]

Ah, but he does. If you didn't have to treat it as infinite, then there would be no paradox.

If the theory can derive a contradiction, then there is a serious problem with the theory -- ignoring a paradox doesn't make it go away.

Also, it looks like you keep changing your mind about whether you're applying "can" and "must" to the analysis, or to what Achilles is actually doing.

Assuming the theory is correct, then if I can analyze the problem to show that Achilles must have passed by the half-way point, then Achilles must have done so. It doesn't matter whether or not I actually do analyze the problem in such a way. To put it simply:

{The theory can be used to predict P} ==> {P must actually be true}

is a valid implication, and not merely

{I used the theory to predict P} ==> {P must actually be true}

I used those terms because they are the closest translations to what Aristotle used.

Just because the ancient Greeks didn't have the benefit of the modern treatment of mathematics doesn't mean we should similarly handicap ourselves.

I've always felt that Zeno, rather than trying to say that there is actually no such thing as continuous motion, was merely pointing out fundamental flaws in the way "infinity" was conceived at the time, since obviously there is motion.

Okay -- but unless we're doing a historical survey, shouldn't we be more interested in how the infinite is treated today? When someone is confused about Zeno's paradoxes, wouldn't telling them about the ancient Greeks' problems just confuse them more?

It can also be viewed as an intuitive demonstration of the impossibility of performing arithmetic functions on "infinites," i.e., that there is really no such thing as an "infinite number."

and wrong. Today, we know of at least two different ways of doing so:

(1) We have algebraic structures that contain infinite quantities. (e.g. extended reals, hyperreals, ordinals)

(2) We have infinitary algebraic operations. (e.g. infinite sums from calculus, or the union of an infinite collection of sets, or the sum of an infinite collection of ordinals)

 

[Mycroft7]

If the theory can derive a contradiction, then there is a serious problem with the theory -- ignoring a paradox doesn't make it go away.

Also, it looks like you keep changing your mind about whether you're applying "can" and "must" to the analysis, or to what Achilles is actually doing.

Assuming the theory is correct, then if I can analyze the problem to show that Achilles must have passed by the half-way point, then Achilles must have done so. It doesn't matter whether or not I actually do analyze the problem in such a way. To put it simply:

{The theory can be used to predict P} ==> {P must actually be true}

is a valid implication, and not merely

{I used the theory to predict P} ==> {P must actually be true}

Obviously I haven't been clear. I pretty much agree with all of the above. Zeno is saying that one CAN approach the problem in this certain way, and therefore one MUST approach it in this way, because, as you say, ignoring a paradox doesn't make it go away. Aristotle and Newton, each in their own way, are specifically NOT ignoring the paradox, but showing its analysis to be flawed.

Just because the ancient Greeks didn't have the benefit of the modern treatment of mathematics doesn't mean we should similarly handicap ourselves.

Okay -- but unless we're doing a historical survey, shouldn't we be more interested in how the infinite is treated today? When someone is confused about Zeno's paradoxes, wouldn't telling them about the ancient Greeks' problems just confuse them more?

Maybe, but that would hardly be as much fun. Anyway, I was trying to show that this professor fellow is either grieveously mistaken in his conclusions are just messing with his students, since the conclusion in question, that there cannot be continuous motion, was dealt with 25 centuries ago. (Not that that necessarily settles the question definitively, but in order to support the paradox as valid one would also have to give an adequate account of why these generally accepted solutions are flawed.) The emphasis lies in the fact that modern mathematics isn't even required.

and wrong. Today, we know of at least two different ways of doing so:

(1) We have algebraic structures that contain infinite quantities. (e.g. extended reals, hyperreals, ordinals)

(2) We have infinitary algebraic operations. (e.g. infinite sums from calculus, or the union of an infinite collection of sets, or the sum of an infinite collection of ordinals)

All examples of thinking around the problem, yes. An infinite sum, for example, IS treating the "infinite" in the "all at once" way which Aristotle suggests. By demonstrating what a sum must ultimately approach if one were to continue adding successive terms, one does not actually perform the operation of addition an infinite number of times, which, just like Achilles, you could never actually do.

 

[CPL.Luke]

this s really just a simple mechanics problem.

so there is an object trveling from point A to point B, there is some distance (x) from point A to point B.

as was stated in the original post there is an infinite number of points between A and B, let's start by analyzing one of those infinitely small points dx.

it takes some infinitessimal amount of time dt to cross this point, let's add in a constant v to get the units working (note this also represents the rate at which the object crosses the distance dx.

algebraicly we can now right

dx=v dt

we can now some up the amount of time it takes to cross each of those points across all of the points on the interval dx (otherwise known as integrating)

so we get that the total amount of time to cross all of those points is

integral of (dt) = t

or integrating the original equation

x=vt

so it takes some finite amount of time to cross the distance x, even though there is an infinite number of points between points A and B.

the real problem in the op is the assumption that it takes 0 seconds to cross an infinitesimal point, when in fact it takes an infinitessimal amount of time to cross an infinitessimal distance.

 

[selfAdjoint]

CPL, that calcuus excercise depends on the branch of mathematics called measure theory, whic underpins why those calculus operators obey the laws they do.

 

[CPL.Luke]

yes but how does that effect the problem at hand? this is just an integral over an interval in the real number system.