No Present: Debate on Time's Definition

[Evil]

ok...i am quite tired now so pardon me for any errors...let goes
my agruement is that there is no present becos of two reasons.
firstly there is still no clear cut definition for time.hence we cannot define present.it is not an qulaitive value.
second, when one say present it have alredi become a past n so there is no present.
so to sum it all up there is no present no now, but onli past n future becos i tink tis 2 area overlap..
anyone with any other agruements?

 

[hypnagogue]

There is quite obviously a subjective experience of time, at least, that involves perception of the present. It is simply what you are perceiving right now, at this instant.

The subjective experience of the present has also been called the specious present, since it actually encompasses a short duration of time rather than an instananeous point in time. A proof of this is simple. TV presents dynamic images to us by consecutively refreshing its image in a series of horizontal lines from the top of the screen to the bottom, but when we watch TV we do not notice this line-by-line refresh; rather we just see a single, coherent image. So photons that impress the new image on us first from the top of the screen and later from the bottom of the screen are actually perceived as striking our eyes at the same point in time. This brief duration of time where two non-simultaneous events are perceived as simultaneous is the specious present.

 

[Evil]

so am i right to sae that the present is nothing more than an experience and therefore it does not exist?

 

[hypnagogue]

Originally posted by Evil
so am i right to sae that the present is nothing more than an experience and therefore it does not exist?

Well first of all, it depends on what you mean by "present." Are you talking about the subjective, specious present which actually encompasses a brief duration of time or are you talking about an instaneous moment in time?

Second, who says that just because something is an experience that it doesn't exist? Perhaps you mean it doesn't exist in an objective sense, but that is something else entirely.

In any case, when people talk about the present they are almost always talking about the specious present. Assuming you mean the present in your argument to be an instantaneous moment in time, I still don't see a problem with the notion. I think your difficulty arises by combining the two: you cannot reconcile how what is really a brief duration of time can be durationless, for good reason. They can't be reconciled. But this is just because we are really working with two definitions: the specious present, and the instantaneous present. Once we recognize these differences our problem vanishes.

For instance, you say:

second, when one say present it have alredi become a past n so there is no present.

In this instance, the recitation of the word "present" occurs over a brief duration of time that we perceive (roughly) as the specious present. Since this perceived 'present' is actually a duration of time, many moments (instananeous presents) will pass by in the duration of one specious present. But these are two distinct things, so there really is no paradox.

 

[Evil]

wow
man that took a while to digest haha
ok when i thought of the present i think it as an instaneous moment.so perhaps defining the present as an experience does not work here since it does not exisit
second u mentioned the present as the instaneous moment which mean it cannot be measured by time since it is well, instantaneoous. since it cannot be measured by time, wouldn't it be apporiate to classify the present as non exisitent ?
this argument is kinda fun so keeping on sending in yr agruements...=>

 

[Canute]

Originally posted by hypnagogue
In this instance, the recitation of the word "present" occurs over a brief duration of time that we perceive (roughly) as the specious present. Since this perceived 'present' is actually a duration of time, many moments (instananeous presents) will pass by in the duration of one specious present. But these are two distinct things, so there really is no paradox. [/B]

Do 'instantaneous presents' exist? I thought Zeno (and more recently Peter Lynds) showed us that they didn't?

 

[Evil]

arh... didnt see that...can u tell mi about it?

 

[Canute]

Briefly the argument is that if spacetime is a continuum them 'instants' are infinitessimals and don't exist. There is just Hypnagogue's 'specious present' (which is not made out of instants).

 

[selfAdjoint]

Where do you get that points don't exist in a continuum ("infinitesimals" is a red herring)? Th3e curve in the plane intersects points and it's perfectly reasonable to ask _which_ points. That's the basis of coordinates and equations. So the world line of your life passes theough the points (here-now) of spacetime, in the spacetime model.

 

[Canute]

Wasn't the calculus invented specifically to get around the problem of the infinities and infintessimals that crop up in continuous curves and continuous space?

 

[Mentat]

Originally posted by selfAdjoint
Where do you get that points don't exist in a continuum ("infinitesimals" is a red herring)? Th3e curve in the plane intersects points and it's perfectly reasonable to ask _which_ points. That's the basis of coordinates and equations. So the world line of your life passes theough the points (here-now) of spacetime, in the spacetime model.

Aside from this, most of the candidates for the QG theory require that a Plank's length be the smallest possible unit of spacetime, and thus spacetime itself is quantized.

 

[Canute]

Originally posted by Mentat
Aside from this, most of the candidates for the QG theory require that a Plank's length be the smallest possible unit of spacetime, and thus spacetime itself is quantized.

So spacetime is quantised because we our theory depends on it? That's not a strong argument. How do you get around Zeno's objections?

 

[Mentat]

Originally posted by Canute
So spacetime is quantised because we our theory depends on it? That's not a strong argument.

General Relativity and Quantum Mechanics are some of the strongest theories ever to exist. They've got experimental, mathematical, and observational proofs behind them, and so are respected as being "correct" (for the most part). So, if the unification of those theories (which have proven themselves correct at all times in their respective arenas) requires quantized spacetime, then that is what must be postulated.

How do you get around Zeno's objections?

There was a thread about Zeno's paradoxes, and they all seemed to be completely non-sensical by the end of the thread...what exactly is Zeno's objection against instantaneous presents? (btw, the quantized spacetime would allow for finite amounts of spacetime as the most discreet units thereof, and thus instantaneous presents aren't required at all for these theories.)

 

[Canute]

Originally posted by Mentat
There was a thread about Zeno's paradoxes, and they all seemed to be completely non-sensical by the end of the thread...what exactly is Zeno's objection against instantaneous presents? (btw, the quantized spacetime would allow for finite amounts of spacetime as the most discreet units thereof, and thus instantaneous presents aren't required at all for these theories.) [/B]

I don't think that's quite right. My guess is that if space is quantised then it entails that time is also.

Zeno's paradox of the race between Achilles and the tortoise is a 'reduction ad absurdam' argument against the idea that motion is quantised. To him this was illogical because it gave rise to paradoxes.

It is not hard to see what he was getting at. If you make the tortoise go as slow as possible, (one quanta of distance (P-length if you like)in one instant of time), and Achilles go faster, then either the race becomes non-computable or you have to accept that at any instant Achilles is not at any particular position.

This is not quite his argument, but it's equivalent. It suggests that the fabric of reality, whatever it is, is a continuum, and that paradoxes arise from trying to quantise it.

So far, whenever I work it out, he turns out to be right. But I may be missing something.

 

[Iacchus32]

Why is it that we can only remain conscious within the present? Seems to me that's all we really have ... Consciousness and the Present Moment.

Could it be that consciousness is the cause of which everything else is the effect? Wow! How much easier could it be to say that everything emanates from the Mind of God?

 

[hypnagogue]

Originally posted by Canute
I don't think that's quite right. My guess is that if space is quantised then it entails that time is also.

Zeno's paradox of the race between Achilles and the tortoise is a 'reduction ad absurdam' argument against the idea that motion is quantised. To him this was illogical because it gave rise to paradoxes.

It is not hard to see what he was getting at. If you make the tortoise go as slow as possible, (one quanta of distance (P-length if you like)in one instant of time), and Achilles go faster, then either the race becomes non-computable or you have to accept that at any instant Achilles is not at any particular position.

This is not quite his argument, but it's equivalent. It suggests that the fabric of reality, whatever it is, is a continuum, and that paradoxes arise from trying to quantise it.

You have it kind of backwards. Zeno devised his paradox to show that motion is impossible. That's what makes it a paradox: the logical conclusion of the argument contradicts our everyday perception of the world.

And if anything, Zeno's paradox goes against the idea that space is a continuum, not the other way around. His paradox relies on the notion that space is infinitely divisible. If you discount this notion by assuming that space is quantized, then the paradox breaks down.

 

[THANOS]

Instead of just there is no present. For we have not the technology to figure that out yet. Why not just state that the present that we precept is false.

 

[hypnagogue]

In what way is it 'false'? I suppose it gives us a false impression of simultaneity, since non-simultaneous events can be perceived as simultaneous in the specious present. But otherwise our perception of the present cannot be said to be 'true' or 'false' any more than the color red could be called true or false.

 

[Canute]

Originally posted by hypnagogue
You have it kind of backwards. Zeno devised his paradox to show that motion is impossible.

Illogical, not impossible. Zeno was perfectly aware that motion was possible.

That's what makes it a paradox: the logical conclusion of the argument contradicts our everyday perception of the world.

Exactly. Therefore the fault lies somewhere with his scenario, not with the world itself.

And if anything, Zeno's paradox goes against the idea that space is a continuum, not the other way around. His paradox relies on the notion that space is infinitely divisible. If you discount this notion by assuming that space is quantized, then the paradox breaks down.

Now I'd say that you have it kind of backwards. His paradox remains a paradox as long as you assume that space is divisible, whether infinitely, or only as far as some fundamental quanta. The paradox only goes away once you say that space is a continuum.

The confusion arises here because if space is a continuum then it really is infinitely divisible, and therefore any finite division is arbitrary and leads to paradox.

 

[hypnagogue]

Originally posted by Canute
Illogical, not impossible. Zeno was perfectly aware that motion was possible.

No, impossible.

Zeno argues that it is impossible for a runner to traverse a race course. His reason is that “motion is impossible, because an object in motion must reach the half-way point before it gets to the end” (Aristotle, Physics 239b11-13).

from http://faculty.washington.edu/smcohen/320/zeno1.htm

Zeno himself didn't have a proper solution to the paradox, nor did he seek one. The paradox suited his philosophy perfectly. He was a member of the Eleatic school of thought, whose founder, Parmenides, held that the underlying nature of the universe was changeless and immobile. Zeno's puzzles appear to have been in support of Parmenides' argument; in showing that change and motion were paradoxical, he hoped to convince people that everything is one-- and changeless. Zeno really did believe that motion was impossible, and his paradox was this theory's chief support.

from Zero, by Charles Seife

Now I'd say that you have it kind of backwards. His paradox remains a paradox as long as you assume that space is divisible, whether infinitely, or only as far as some fundamental quanta. The paradox only goes away once you say that space is a continuum.

The confusion arises here because if space is a continuum then it really is infinitely divisible, and therefore any finite division is arbitrary and leads to paradox.

But Zeno's paradox does not deal with finite divisions. It deals with infinite divisions. Infinite divisions are only possible on a continuum; if space is quantized, then there comes a point where it cannot be divided any further. This point of indivisibility is where Zeno's paradox would break down, since it requires that you be able to make further divisions of space indefinitely.

There were other schools of thought. The atomists, for example, believed that the universe is made up of little particles called atoms, which are indivisable and eternal. ...the indivisible kernels of matter in atomic theory got around the problem of Zeno's paradoxes. Because atoms are indivisible, there is a point beyond which things could not be divided. Zeno's hair-splitting doesn't go on ad infinitum. After a number of strides, Achilles would be taking tiny steps that can't get any smaller; eventually he would have to hurdle an atom that the tortoise doesn't. Achilles would finally catch up to the elusive turtle.

from Zero, by Charles Seife

 

[Evil]

but ppl wat of the present?
is it exisitent??

 

[Canute]

Originally posted by hypnagogue
No, impossible.

Impossible if space is quantised, perfectly possible if it isn't.

From your link -

"Zeno argues that it is impossible for a runner to traverse a race course. His reason is that “motion is impossible, because an object in motion must reach the half-way point before it gets to the end” (Aristotle, Physics 239b11-13).

Why is this a problem? Because the same argument can be made about half of the race course: it can be divided in half in the same way that the entire race course can be divided in half. And so can the half of the half of the half, and so on, ad infinitum.

So a crucial assumption that Zeno makes is that of infinite divisibility: the distance from the starting point (S) to the goal (G) can be divided into an infinite number of parts.

Zeno assumes divisibility and the paradox follows. If you don't assume this then there is no paradox.

Zeno argues that it is impossible for a runner to traverse a race course. His reason is that “motion is impossible, because an object in motion must reach the half-way point before it gets to the end” (Aristotle, Physics 239b11-13)

Exactly. An object in motion cannot ever be at a point, which was Zeno's point.

Zeno himself didn't have a proper solution to the paradox, nor did he seek one. The paradox suited his philosophy perfectly. He was a member of the Eleatic school of thought, whose founder, Parmenides, held that the underlying nature of the universe was changeless and immobile. Zeno's puzzles appear to have been in support of Parmenides' argument; in showing that change and motion were paradoxical, he hoped to convince people that everything is one-- and changeless. Zeno really did believe that motion was impossible, and his paradox was this theory's chief support.

Zeno knew that he could move, he wasn't insane. His argument was that this relative movement was founded on something which was changeless and unquantised.

But Zeno's paradox does not deal with finite divisions. It deals with infinite divisions.

Nooo. It deals with infinitely small finite divisions.

Infinite divisions are only possible on a continuum; if space is quantized, then there comes a point where it cannot be divided any further. This point of indivisibility is where Zeno's paradox would break down, since it requires that you be able to make further divisions of space indefinitely.

The point of indivisibility is where the paradox becomes a paradox. I agree that infinite divisions are only possible in a continuum, but this is because in a continuum there is an endless regress of divisions. The truth is that in a continuum all division are arbitrary, there aren't any real divisions. With no divisions the race can go ahead.

There were other schools of thought. The atomists, for example, believed that the universe is made up of little particles called atoms, which are indivisable and eternal. ...the indivisible kernels of matter in atomic theory got around the problem of Zeno's paradoxes. Because atoms are indivisible, there is a point beyond which things could not be divided. Zeno's hair-splitting doesn't go on ad infinitum. After a number of strides, Achilles would be taking tiny steps that can't get any smaller; eventually he would have to hurdle an atom that the tortoise doesn't. Achilles would finally catch up to the elusive turtle

I don't know the writer but this is the opposite of the truth. If space is 'atomised' then Zeno is right, motion is impossible. Yet we know that motion is not impossible. It follows that space is not quantised.

Try it this way:

Assume that Achilles and the tortoise are particles one indivisible quanta of space in diameter. Let particle T go at one quanta of space per indivisible instant of time. Let particle A go at ten quanta per instant. You can see immediately that particle T has an indeterminate location, having become in effect become ten quanta in length. The consequences of this are paradoxical.

As Zeno says, the idea that sapce or time is quantised in inconsistent with our obeservations of motion. Peter Lynds argues similary here:

http://www.peterlynds.net.nz/

Regards
Canute

 

[Iacchus32]

Yes, time and space comes together in the here and now, and that is The Present.

Indeed, how much easier could it be to say that everything emanates from the Mind of God?

 

[hypnagogue]

Originally posted by Canute
Zeno assumes divisibility and the paradox follows. If you don't assume this then there is no paradox.

Zeno assumes infinite divisibility-- that one can divide space into smaller portions indefinitely. Infinite divisibility only is possible on a continuum.

Zeno knew that he could move, he wasn't insane. His argument was that this relative movement was founded on something which was changeless and unquantised.

Zeno explicitly states that motion is impossible. He regarded motion as an illusion, similar to how apparent motion of objects on a movie screen is also an illusion created by a succession of static picture frames-- see Zeno's arrow paradox.

Nooo. It deals with infinitely small finite divisions.

The divisions are finite in length, but there are infinitely many of them.

The point of indivisibility is where the paradox becomes a paradox. I agree that infinite divisions are only possible in a continuum, but this is because in a continuum there is an endless regress of divisions. The truth is that in a continuum all division are arbitrary, there aren't any real divisions. With no divisions the race can go ahead.

I don't know the writer but this is the opposite of the truth. If space is 'atomised' then Zeno is right, motion is impossible. Yet we know that motion is not impossible. It follows that space is not quantised.

No, you still have it backwards. Zeno argues that Achilles can never reach the finish line because before he reaches it, he must reach the halfway point-- and before he reaches the halfway point, he must go halfway to the halfway point-- and so on, ad infinitum. So there is an infinite series of divisions: for every segment of length L, Zeno argues that first Achilles must cross L / 2. In fact, not only can Achilles never catch the Tortoise-- he can never get started in the first place!

In the atomist paradigm, there is a fundamental unit of space that cannot be subdivided into smaller parts, call it L₀. Zeno's paradox says that Achilles cannot cross L₀, because before he can do that he must cross L₀ / 2. But in the atomist conception, L₀ / 2 does not exist. At this point Zeno's inductive argument breaks down, since his inductive step is blocked. Achilles does not have to cross L₀ / 2, since by definition he cannot-- he simply crosses L₀ all at once. So the infinite regress of Zeno's paradox is circumvented.

Try it this way:

Assume that Achilles and the tortoise are particles one indivisible quanta of space in diameter. Let particle T go at one quanta of space per indivisible instant of time. Let particle A go at ten quanta per instant. You can see immediately that particle T has an indeterminate location, having become in effect become ten quanta in length. The consequences of this are paradoxical.

I assume you meant A has an indeterminate length, having become in effect ten quanta in length. But this is not really the case. In the atomist conception, continuous motion does not exist. T "jumps" from one quantum of space to the next in one indivisible unit of time. A does the same, except instead of jumping 1 unit of space per 1 unit of time, he jumps 10 units of space per 1 unit of time. The fundamental "jumpiness" is the same in both cases-- at one instant, A and T are located at a certain quantum of space, and in the next instant, they are discontinuously located in the next. This is perhaps counter-intuitive, but not inherently paradoxical or logically inconsistent.

 

[Canute]

Originally posted by hypnagogue
Zeno assumes infinite divisibility-- that one can divide space into smaller portions indefinitely.

Exactly. That's what leads to the paradox.

Infinite divisibility only is possible on a continuum.

Quite agree. Infinite divisibility means that at the limit you get a mathematical point, which doesn't exist. Therefore a continuum is not made of quanta, and that's why it's a called a continuum. (Of course you can arbitralily divide it up any old how).

Zeno explicitly states that motion is impossible. He regarded motion as an illusion, similar to how apparent motion of objects on a movie screen is also an illusion created by a succession of static picture frames-- see Zeno's arrow paradox.

Agree. Zeno's example shows that if space is quantised then motion is impossible or illusory.

The divisions are finite in length, but there are infinitely many of them.

if they are infintitely small then they don't exist, hence 'infinitessimals'.

No, you still have it backwards. Zeno argues that Achilles can never reach the finish line because before he reaches it, he must reach the halfway point-- and before he reaches the halfway point, he must go halfway to the halfway point-- and so on, ad infinitum. So there is an infinite series of divisions: for every segment of length L, Zeno argues that first Achilles must cross L / 2. In fact, not only can Achilles never catch the Tortoise-- he can never get started in the first place!

Exactly. So the idea that Achilles is ever at precisely the half way point in some particular instant is incoherent.

In the atomist paradigm, there is a fundamental unit of space that cannot be subdivided into smaller parts, call it L₀. Zeno's paradox says that Achilles cannot cross L₀, because before he can do that he must cross L₀ / 2. But in the atomist conception, L₀ / 2 does not exist. At this point Zeno's inductive argument breaks down, since his inductive step is blocked. Achilles does not have to cross L₀ / 2, since by definition he cannot-- he simply crosses L₀ all at once. So the infinite regress of Zeno's paradox is circumvented.

Well yes, if you want to assume something even more paradoxical.

I assume you meant A has an indeterminate length, having become in effect ten quanta in length. But this is not really the case. In the atomist conception, continuous motion does not exist. T "jumps" from one quantum of space to the next in one indivisible unit of time. A does the same, except instead of jumping 1 unit of space per 1 unit of time, he jumps 10 units of space per 1 unit of time. The fundamental "jumpiness" is the same in both cases-- at one instant, A and T are located at a certain quantum of space, and in the next instant, they are discontinuously located in the next. This is perhaps counter-intuitive, but not inherently paradoxical or logically inconsistent. [/B]

Fair enough. If you want to believe that motion is discontinuous and that an object in motion spends most of its time in some never never land between locations then the paradox is solved. I agree that this is the only alternative to believing Zeno, but personally I find it paradoxical and logically inconsistent as well as counterintuitive.

In your explanation how do you explain where the object is between 'instants', and how does it cross the gaps in no time at all?

 

[hypnagogue]

Originally posted by Canute
Exactly. So the idea that Achilles is ever at precisely the half way point in some particular instant is incoherent.

I think Zeno's paradox is more involved with motion than location. It's not so much the idea that Zeno is located at a particular point that is singled out in Zeno's paradox-- it's that he must perform an infinity of tasks in order to get anywhere.

Fair enough. If you want to believe that motion is discontinuous and that an object in motion spends most of its time in some never never land between locations then the paradox is solved. I agree that this is the only alternative to believing Zeno, but personally I find it paradoxical and logically inconsistent as well as counterintuitive.

The object does not spend time in never-never land between locations-- it is always located somewhere. There is just a discontinuous transition between locations. And for what it's worth, this is our best working theory for how subatomic particles behave.

But believing in quantized space is not the only way to alleviate Zeno's paradox. The problem in the paradox centers on the infinite number of tasks Achilles must perform in order to travel from one point to another. There are two direct objections here: either Zeno has to travel an infinite distance, or he has to take an infinite amount of time to perform his infinite amount of tasks.

We can rule out the former objection, since the limit of the summation of all his infinite steps approaches a whole number, not infinity. For instance, if Achilles has to travel 1 unit of space to reach the goal, then he needs to first cross 1/2 unit of space in the first step, then 1/4 in second step, then 1/8 in the third, and 1/2ⁱ units in general in the ith step. But the sum [tex]\sum_{i=1} 1/2^i[/tex] approaches 1 as i approaches infinity, so Zeno has to cross a finite distance in spite of his infinite amount of tasks.

Similarly, Zeno only needs a finite amount of time to do his infinite number of tasks. Assume as before that Zeno needs to travel over 1 unit of space to reach the goal, and furthermore assume that Zeno travels at a constant velocity of 1 unit of space per 1 unit of time. Then he needs 1/2 unit of time to reach the halfway point, 1/4 unit of time to reach the 1/4 point, and so on. As before, in the limit as the number of tasks approaches infinity, Zeno only needs 1 unit of time to do his infinite number of tasks.

There is also the approach of trying to derive a logical contradiction from Zeno's paradox. Rather than go into this myself, I refer again to what is an excellent resource on the paradox:
http://faculty.washington.edu/smcohen/320/zeno3.htm

So we can assume space is an infinitely divisble continuum and still survive Zeno's paradox.

In your explanation how do you explain where the object is between 'instants', and how does it cross the gaps in no time at all?

It isn't somewhere between instants. It is always located at one location or another. When it moves, there is a discontinuous jump (quantum leap) from one location to another. To assume it is located somewhere inbetween in the meantime is to basically continue assuming continuous motion in continuous space.

As for how this all refers back to the initial question about instants, we can safely say that Zeno's paradox does not rule out the notion of instantaneous points in time. Further investigations in physics may rule out the notion of truly durationless points in time, but I don't think these can be ruled out purely on a logical basis.

 

[Mentat]

Originally posted by Canute
I don't think that's quite right. My guess is that if space is quantised then it entails that time is also.

Of course it would entail time also...but these would not be instaneous or infinitesimal units, but finite, measurable, quanta.

Zeno's paradox of the race between Achilles and the tortoise is a 'reduction ad absurdam' argument against the idea that motion is quantised. To him this was illogical because it gave rise to paradoxes.

It is not hard to see what he was getting at. If you make the tortoise go as slow as possible, (one quanta of distance (P-length if you like)in one instant of time), and Achilles go faster, then either the race becomes non-computable or you have to accept that at any instant Achilles is not at any particular position.

This is not quite his argument, but it's equivalent. It suggests that the fabric of reality, whatever it is, is a continuum, and that paradoxes arise from trying to quantise it.

So far, whenever I work it out, he turns out to be right. But I may be missing something.

Perhaps you should state the actual paradox, in Zeno's words, so that I can work on it's actual form.

 

[Canute]

Originally posted by hypnagogue
I think Zeno's paradox is more involved with motion than location. It's not so much the idea that Zeno is located at a particular point that is singled out in Zeno's paradox-- it's that he must perform an infinity of tasks in order to get anywhere.

What Zeno specifically argued is not the point. There are different ways of analysing the race.

The paradox of motion is equally a paradox of location. It just depends which way you look at it. It is also a pardox of time.

The object does not spend time in never-never land between locations-- it is always located somewhere. There is just a discontinuous transition between locations. And for what it's worth, this is our best working theory for how subatomic particles behave.

I didn't know that, I've never looked into the orthodox quantum mechanical view on this. This is our very best scientific theory of the motion of subatomic waves? Wow.

I can't agree with any of your analysis of the divisibility of time or space. You've 'renormalised' the infinities by sleight of hand.

So we can assume space is an infinitely divisble continuum and still survive Zeno's paradox.

You haven't quite convinced me yet.

As for how this all refers back to the initial question about instants, we can safely say that Zeno's paradox does not rule out the notion of instantaneous points in time. Further investigations in physics may rule out the notion of truly durationless points in time, but I don't think these can be ruled out purely on a logical basis. [/B]

Well, I can only say I don't agree. I think Zeno's paradoxes have lasted so long because he had a point.

 

[Canute]

Originally posted by Mentat
Perhaps you should state the actual paradox, in Zeno's words, so that I can work on it's actual form. [/B]

Call it my paradox then. I don't want to take responsibility for Zeno's argument, which I don't know well. It's the same problem underneath.

 

[Jeebus]

Originally posted by Canute
Call it my paradox then. I don't want to take responsibility for Zeno's argument, which I don't know well. It's the same problem underneath.

Zeno notes that if physical objects exist discretely at a sequence of discrete instants of time, and if no motion occurs in an instant, then we must conclude that there is no motion in any given instant. (As Bertrand Russell commented, this is simply "a plain statement of an elementary fact".) But if there is literally no physical difference between a moving and a non-moving arrow in any given discrete instant, then how does the arrow know from one instant to the next if it is moving? In other words, how is causality transmitted forward in time through a sequence of instants, in each of which motion does not exist?

 

[hypnagogue]

Originally posted by Canute
I didn't know that, I've never looked into the orthodox quantum mechanical view on this. This is our very best scientific theory of the motion of subatomic waves? Wow.

Yes. Assuming energy is transmitted continuously actually leads to some much more grave paradoxes in physics than Zeno had in mind-- for instance, the paradox of why we aren't burned to a crisp by looking at a fireplace.

I can't agree with any of your analysis of the divisibility of time or space. You've 'renormalised' the infinities by sleight of hand.

What have I renormalized, and what is my sleight of hand? It's math, pure and simple.

Well, I can only say I don't agree. I think Zeno's paradoxes have lasted so long because he had a point.

Zeno's paradoxes lasted so long because it took a while for calculus to be invented. Using calculus, we see that Zeno can cross a finite distance in a finite time, even if it can be theoretically broken down into infinitely many subdivisions. Specifically, we can show that in the limit as the number of steps approaches infinity, the size of the steps trails off quickly enough that their sum approaches a finite number rather than infinity (likewise for the time needed to complete the tasks).

Zeno clearly did not understand that an infinite sum can have a finite value, as evidenced by his argument against plurality (see http://faculty.washington.edu/smcohen/320/zeno2.htm). His argument here basically was that "All objects are composed of infinitely many parts; all parts must have some finite size; therefore, all objects are infinitely large." But calculus shows us concretely that infinite sums can in fact have finite values. This simple fact alleviates both Zeno's paradox and his argument against plurality.

 

[hypnagogue]

Originally posted by Jeebus
Zeno notes that if physical objects exist discretely at a sequence of discrete instants of time, and if no motion occurs in an instant, then we must conclude that there is no motion in any given instant. (As Bertrand Russell commented, this is simply "a plain statement of an elementary fact".) But if there is literally no physical difference between a moving and a non-moving arrow in any given discrete instant, then how does the arrow know from one instant to the next if it is moving? In other words, how is causality transmitted forward in time through a sequence of instants, in each of which motion does not exist?

This is quickly becoming a thread about Zeno. I propose we re-name this thread, "there is no paradox."

In any case, you are referring to Zeno's arrow paradox. Zeno's arrow paradox is solved at http://faculty.washington.edu/smcohen/320/ZenoArrow.html .

 

[Canute]

Originally posted by hypnagogue
Yes. Assuming energy is transmitted continuously actually leads to some much more grave paradoxes in physics than Zeno had in mind-- for instance, the paradox of why we aren't burned to a crisp by looking at a fireplace.

Well, you've got a choice. You can have the paradox in physics or you can have it in real life. Take your pick. I tend to think motion is possible and that it is mathematical physics that can't handle the infinities involved in dealing with a continuum.

What have I renormalized, and what is my sleight of hand? It's math, pure and simple.

To eternally approach a finite number is not the same as being a finite number.

Zeno's paradoxes lasted so long because it took a while for calculus to be invented. Using calculus, we see that Zeno can cross a finite distance in a finite time, even if it can be theoretically broken down into infinitely many subdivisions.

Calculus just hides the infinties. That's what it's for.

Specifically, we can show that in the limit as the number of steps approaches infinity, the size of the steps trails off quickly enough that their sum approaches a finite number rather than infinity (likewise for the time needed to complete the tasks).

It approaches a finite number for all eternity. At what particular arbitrary point do you assume that it arrives?

Zeno clearly did not understand that an infinite sum can have a finite value, as evidenced by his argument against plurality (see http://faculty.washington.edu/smcohen/320/zeno2.htm). His argument here basically was that "All objects are composed of infinitely many parts; all parts must have some finite size; therefore, all objects are infinitely large." But calculus shows us concretely that infinite sums can in fact have finite values. This simple fact alleviates both Zeno's paradox and his argument against plurality. [/B]

No offense, but I feel that Zeno (and Russell) gave this more thought than you think they did.

Thanks for the link, but it does not appear to be a solution, more a missing of the point. Jeebus summed up the problem very well. How would you answer his post?

 

[hypnagogue]

Originally posted by Canute
Well, you've got a choice. You can have the paradox in physics or you can have it in real life. Take your pick. I tend to think motion is possible and that it is mathematical physics that can't handle the infinities involved in dealing with a continuum.

Physics is real life.

Again, Zeno's paradox relies on space being a continuum to draw its paradoxicalness. Discretely quantized space is not the central figure to Zeno's paradox, and further there is nothing inherently paradoxical about motion through quantized space. Counter-intuitive != paradoxical.

To eternally approach a finite number is not the same as being a finite number.

Calculus just hides the infinties. That's what it's for.

The point is that an infinite numbers of steps in the race does not imply infinite distance or an infinite amount of time to complete the steps. This is not a matter of hiding infinity, but solidly reasoning about it.

It approaches a finite number for all eternity. At what particular arbitrary point do you assume that it arrives?

So you are arguing that [tex]\sum^{\infty}_{i=1} 1/2^i[/tex] does not equal 1? If you are not, then my argument stands. If you are, I would like to see your proof to the contrary.

No offense, but I feel that Zeno (and Russell) gave this more thought than you think they did.

As I have explained, Zeno clearly did not understand that an infinite sum could be finite. This is no fault against him, since he couldn't have known without inventing calculus himself. But his paradox depends on this incorrect notion.

Thanks for the link, but it does not appear to be a solution, more a missing of the point. Jeebus summed up the problem very well. How would you answer his post?

Jeebus brought up Zeno's arrow paradox, which is un-paradoxed in the link I replied with. The central idea is that motion is only defined on an interval of time. If an object changes its physical location during a given interval of time, it has been in motion; if it does not, it is at rest. For an arrow at a given instant of time, we cannot infer whether it is in motion or at rest, since by necessity we need to define these terms with respect to an interval of time.

 

[Canute]

Oh well. No point in going through it again. We'll have to disagree.