CoffmanPhDIs space-time discrete or continuous?

[onemind]

But it never actually gets to the limit.

 

[arunbg]

From the wikipedia article,

Issues with the proposed calculus-based solution

A suggested problem with using calculus to try to solve Zeno's paradoxes is that this only addresses the geometry of the situation, and not its dynamics. It has been argued that the core of Zeno's paradoxes is the idea that one cannot finish the act of sequentially going through an infinite sequence, and while calculus shows that the sum of an infinite number of terms can be finite, calculus does not explain how one is able to finish going through an infinite number of points, if one has to go through these points one by one. Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself, independent of how much time such an act would require.

Another way of putting this is as follows: If Zeno's paradox would say that "adding an infinite number of time intervals together would amount to an infinite amount of time", then the calculus-solution is perfectly correct in pointing out that adding an infinite number of intervals can add up to a finite amount of time. However, any descriptions of Zeno's paradox that talk about time make the paradox into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler inherent paradox that does not at all consider any quantifications of time. Rather, this much simpler paradox simply states that: "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". The calculus-based solution offers no insight into this much simpler, much more stinging, paradox.

A thought experiment used against the calculus-based solution is as follows. 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 in finite time, the counting process will be complete, and we could ask Achilles what the greatest number he counted to was. Here we encounter another paradox: while there is no "largest" number in the sequence, as for every finite number the turtle is still ahead of Achilles, there must be such a number because Achilles did stop counting.

 

[neurocomp2003]

To cleanup that quote of arunbg:
3 concepts - time, space,action (inregards to the paradox).
the time between [0,1] is 1 time-metric (doesn't matter which metric scale)
the space between [0,1] is 1 space-metric (doesn't matter which metric scale)
the action(dynamics) is the event of going from point Ai to point Ai+1

So to my understanding of onemind' grasp of the paradox.

the traveller going from 0-1 space in 0-1 time should never reach either because they must perform inf# of actions. So according to onemind(IMO), he's asking how do we justify using math in physics. Which i would answer they're ain't no other TOOL. As for the paradox itself...it would be resolved with a stepsize or fundamental speed. But if space was truly continuous, you would still have to account for velocity.

 

[Hurkyl]

But it never actually gets to the limit.

What does it matter if any of its partial sums are unequal to the limit?

 

[Hurkyl]

The calculus-based solution offers no insight into this much simpler, much more stinging, paradox.
...
there must be such a number because Achilles did stop counting.

It's not very stinging at all; this statement is flawed. If Achilles counted as stated in the article, then he finished counting; he went through every natural number.

Of course, if he was counting in ordinal numbers instead of natural numbers, then he passes the turtle as he says \omega.

 

[arunbg]

Hurkyl, I don't quite understand what you are trying to say, can you please clarify a bit more.
If Achilles finished counting, he must have stopped at some number, but there seems to be no end to the counting process, what is wrong with the paradox?

 

[Hurkyl]

Hurkyl, I don't quite understand what you are trying to say, can you please clarify a bit more.
If Achilles finished counting, he must have stopped at some number, but there seems to be no end to the counting process, what is wrong with the paradox?

If Achilles counts as in the Wikipedia article, then Achilles said every natural number.

As you're aware, there is no largest natural number. Correspondingly, there is no instant in time where Achilles says the last number.


Assuming it takes Achilles one second to finally pass the tortise, then the time in which Achilles is counting spans the interval

[0, 1)​

The final instant at time 1, when he reaches the tortise, occurs after Achilles has said every natural number. The period of time over which Achilles catches the tortise is the interval

[0, 1]​

which is larger.

 

[onemind]

What does it matter if any of its partial sums are unequal to the limit?

Because that is the whole point of the paradox. It never gets there because it goes for eternity and there will never be any discrete set of partial sums that equal the limit. Thats a fine concept for mathematicians but it makes no sense in the real world because in the real world things do get to the limit and do not go for eternity like infinite sequences.

I think Hurkyl has been blinded by science and can't come to terms that this really is a paradox outside his ability to explain hence the never ending tedious explanations.

 

[neurocomp2003]

but somethings cannot be broken down into lamens terms,
and these are the subtle differences between pure math and applied math.

Also this type of paradox is one of the pains of implementing virtual simulations.

 

[Saladsamurai]

I think Hurkyl has been blinded by science and can't come to terms that this really is a paradox outside his ability to explain hence the never ending tedious explanations.

So what is the point of this thread? You have asked a question in a mathematics forum, but your disposition is that of one whom refuses mathematical proof. Where is this going? I am not trying heat things up; however, I do not see anyone converting to the other side after this one...

 

[onemind]

So what is the point of this thread?

Well, i was new to zenos paradox a couple of days ago which was exposed to me in a beginner calculus book and not being a mathematician i was just curious to how mathematicians explain this paradox.

You have asked a question in a mathematics forum, but your disposition is that of one whom refuses mathematical proof.

This is what i meant by:

I mean, is math in this case just a simplification in order to deal with this problem but doesn't represent the true physical reality of movement?

In my original post.

Now i know.

Thanks to all that took their time to share their views.

 

[Hurkyl]

(For concreteness, I'll suppose that the tortoise runs 1 meter per second, Achilles runs 10 meters per second, and the tortoise initially had a 9 meter head start)

Because that is the whole point of the paradox. It never gets there because it goes for eternity

No it doesn't: it only goes for 1 second. (Given the numbers I stated above)

You are correct in that Achilles does not pass the tortoise during the sequence of events:
(1) Achilles covers the initial 9 meters in 0.9 seconds, and the tortoise advances 0.9 meters.
(2) Achilles covers the next 0.9 meters in 0.09 seconds, and the tortoise advances 0.09 meters.
(3) Achilles covers the next 0.09 meters in 0.009 seconds, and the tortoise advances 0.009 meters.
...

but this entire sequence of events only covers the time span that begins at the zero second mark, and extends up to (but not including) the one second mark.

That's hardly an eternity.


Of course, during this sequence of events, Achilles does not catch the tortoise. And given just this sequence of events, we cannot prove that Achilles ever catches the tortoise.

That's one of the reasons why we would postulate that time is a continuum, and that motion is continuous.

Postulating that time is a continuum proves that there is a one second mark. (assuming that the universe doesn't cease to exist)

Then, postulating that motion is continuous proves that Achilles reaches the turtle exactly at the one second mark.

I think Hurkyl has been blinded by science and can't come to terms that this really is a paradox outside his ability to explain hence the never ending tedious explanations.

Did you consider the possibility that I just might know what I'm talking about?

 

[onemind]

Did you consider the possibility that I just might know what I'm talking about?

Of course, but after listening you give the same explanation over and over without considering where i am coming from i came to the conclusion that you are blinded by science.

I think i will go with Einstein on this one rather than Hurkyl.

 

[Cane_Toad]

Paradoxes like Zeno's are fun, but slight of hand. My only use for Zeno's paradox is when there's one slice of pizza left, and a bunch of hungry but polite friends who always cut the last slice in half. "Anybody want a slice of Zeno's pizza?", you call out from the fridge, and then everybody knows there was a pizza slice, but you've already eaten it.

It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther;

Like most of the variations, the devil is in the [omitted]details. It says nothing about "how far", or what "some period of time" is.

If it becomes a strictly mathematical bifurcation process, then of course it will never stop. This is only the implied idea behind the paradox, which is then deceptively intermixed with the real world.

These are silly, ancient word-play. There are lots of better paradoxes out there "i.e. special relativity" if you want to ask a serious question about how paradoxes, math, and reality relate. The main usefulness of Zeno's paradox is the introduction to the concepts of infinity and limits.

As for "continuous", there is continuity in the objective world, or else you're going to assert that because we can't be omniscient, that language is useless. Concepts are approximations with respect to the real world, and everybody agrees that a cat is a c-a-t, and the breed can be safely ignored. Either you agree it's a cat, or choose to invent your own private language, or take a vow of silence. Saying, "we can't know whether that's really a cat because we can never know all the depths of catness", is just more silly word play, in this case, indicating an incomplete exposure to epistemology and abstraction.

 

[Hurkyl]

Of course, but after listening you give the same explanation over and over without considering where i am coming from i came to the conclusion that you are blinded by science.

I think i will go with Einstein on this one rather than Hurkyl.

I can't fix your problems by myself: you have to cooperate.

For example, the distinction between the duration of an interval of time and the number of points in an interval of time has come up several times in this thread, brought up by several people. And yet you have not indicated you recognize they are different, nor have you indicated that you think they are the same.

 

[onemind]

the distinction between the duration of an interval of time and the number of points in an interval of time has come up several times in this thread, brought up by several people. And yet you have not indicated you recognize they are different, nor have you indicated that you think they are the same.

I said finite infinty.

I disagree with cane toad that this concept is merely semantic but of course i agree that it is not useful.

The whole concept of unit size makes no sense when dealing with infinity, only in a relative real world sense.

 

[Cane_Toad]

..
I disagree with cane toad that this concept is merely semantic but of course i agree that it is not useful.

Ok, but which? I said that the Zeno related paradoxes where mostly semantic, but your point regarding infinity, continuum, and humans seems to go deeper.

 

[onemind]

Ok, but which?

Zenos paradox. I don't see what is semantic about contemplating finite infinity which is basically what zenos paradox is minus the greek analogy of achilles and the turtle.

 

[neurocomp2003]

if only there was an online interactive animation sequence of fractals(particular mandelbrot and the koch snowflakes).

onemind: u don't believe a fundamental stepsize would play in reality
yet you used the terminology "continuous/continuity"...would you care to elaborate on your understanding of this terminology, and how you would describe "physical/reality" concept "motion"?

 

[Cane_Toad]

Zenos paradox. I don't see what is semantic about contemplating finite infinity which is basically what zenos paradox is minus the greek analogy of achilles and the turtle.

Wikipedia:
Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction.

I said the paradox is semantically based. The greater concept is not, but the way he went about trying to make his proof, and the resulting paradox, is via the turtle, arrow, whatever, and was more rhetorical than earnest. To win a debate via reductio ad absurdum, one seeks the most outlandish example, not the closest fit, as Zeno did. Thus it wasn't a contemplation, at least overtly, and wasn't really about infinity per se, but about whether things were divisible in nature/reality. Indirectly, and perhaps accidentally, and centuries later, we come finally to infinities and calculus.

The paradox is childish, but the reflections it sparked are not.

 

[onemind]

yet you used the terminology "continuous/continuity"...would you care to elaborate on your understanding of this terminology, and how you would describe "physical/reality" concept "motion"?

I can't explain it.

All i know is that in reality, i can move an object from A to B but i don't know how. I understand how to get around this problem with the use of calculus to create useful models of reality and realize that the models are not reality.

Hence my statement a few pages back that i will die ignorant like everyone else :)

And for the record, i am not trying to be a smart ass and accept the possibility that you guys know something i don't and am unable to understand because of my own lack of mathematics education but i doubt it.

 

[onemind]

The paradox is childish, but the reflections it sparked are not

Well i guess i am a child then because zenos paradox still baffles me.

 

[Cane_Toad]

It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead.

Only the parts of the story that support the paradox are adequately stated. It doesn't say that Achilles has to stop at each point, which is left as an implication(?). If he doesn't than there is no paradox.

The reaction time of Achilles each time he stops isn't specified, so:

Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.

If Achilles does stop at each point, he is in essence forever chosing to allow the turtle to stay ahead, which is no paradox either.

The other implication is that Achilles *must* go only 1/2 the distance, and no further, but we'll let that slide, since that's the good part.

Zeno could simply have said, if you halve something and halve the result, and repeat, would you ever have nothing left? This is the clear idea behind it all, but instead Zeno chooses a clever wording that creates an apparent paradox out of a straightforward concept, the paradox being "Achilles can never reach the turtle". There is no paradox in the previous statement of infinite reduction, only the question of whether you think there is an indivisible limit or not.

So, Zeno's "proof" only served to cloud and confuse, which was what he was attempting, since he was in a charged debate, not actually trying to find the answer.

On the other hand,

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...
— Bertrand Russell, The Principles of Mathematics (1903)1

So, if you're still in awe of the paradoxes, you can count him on your side.

Personally, I think he's just being contrarian.

Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite.

So much for being the "foundation of a mathematical renaissance" after two thousand years. Not only that, but it was the "failure" of his proofs which stood the test of time, so giving him the credit for the success in the area of infinities is accidental at best. Remember he was trying to show the falacy of infinitely divisible space in reality.

The paradoxes lasted this long because they were cute, and had real world objects in them doing odd things. Zeno was just an early spin doctor, a master of sound bites. He didn't invent the concept, just stuck words around it. People thought they were fun, so they were retold, and of course they mystified the layman. Aesop's fables lasted for the same reason, but they were a lot better.

Blah, the more I think about it, the more I dislike him. The world is full of disingenuous people doing things like that, using their brilliance to warp ideas to suit their purposes, and is very much the worse for it.

 

[onemind]

Only the parts of the story that support the paradox are adequately stated. It doesn't say that Achilles has to stop at each point, which is left as an implication(?). If he doesn't than there is no paradox.

The reaction time of Achilles each time he stops isn't specified, so:

What in the world are you talking about mate? Reaction time of achilles? I think your reading too much into the story.

Forget the story and look at it the way in terms of point A to B as i outlined in the first post.

Talk about disingenuous people using ideas to suit them.

 

[arunbg]

Onemind, if you look at Hurkyl's posts, you would see that nowhere is he stating that there is no paradox at all. He is simply trying to clarify how mathematics has resolved the paradox to some extent.
For eg, it is believed that Zeno was ignorant that the sum of the infinite geometrically decreasing time intervals taken by Achilles to reach the previous positions of the tortoise, was in fact finite.

But although the total time taken to catch up is finite(which is real worldish), it takes him an infinite no. of steps(which is non-real worldish), this is the heart of the paradox today. I believe this still remains a paradox.

 

[Cane_Toad]

What in the world are you talking about mate? Reaction time of achilles? I think your reading too much into the story.

Forget the story and look at it the way in terms of point A to B as i outlined in the first post.

You clearly ignored my point: either you pay attention to the words, as if they were intended as a serious question, or you ignore them, and treat it as a mathematical problem.

As I understand it, Zeno's intent wasn't a pure mathematical problem. Taken as a real-world problem it has many flaws. Taken as a mathematical problem, it isn't a paradox, it just introduces a few concepts about infinity and limits.

You're making it a paradox by introducing poorly defined time elements in your A to B post. There were many replies about this.

Talk about disingenuous people using ideas to suit them.

Honestly, what did I write all that for? Show that you were at least paying attention by posting something displaying effort on your part. Wise-ass remarks are unwelcome if you haven't earned the right.

 

[onemind]

Thanks Arnbg.

If that is the case then this wouldn't have went on for 4 pages of the same explanations.

 

[Hurkyl]

But although the total time taken to catch up is finite(which is real worldish), it takes him an infinite no. of steps(which is non-real worldish), this is the heart of the paradox today. I believe this still remains a paradox.

Well, (actual) paradoxes aren't matters of opinion: either you have, or you have not exhibited an argument that derives a contradiction from a specified set of hypotheses. What hypotheses do you think lead to a paradox? What contradiction is derived? What is the proof?

 

[neurocomp2003]

the paradox: there are infinite amount of steps(action) to get to the bounding conditions(time/space bounds)?

 

[Hurkyl]

the paradox: there are infinite amount of steps(action) to get to the bounding conditions(time/space bounds)?

What is the contradiction? What statement is both proven and disproven?

And (IMHO less importantly), what are the hypotheses, proof, and disproof[/size]?