What is nothing?

[Chip]

Los Angeles Times science writer K. C. Cole has written that "nothing" is the most elusive entity in the universe. Whenever science comes closer to defining an absolute void "something new is discovered in its place." How something (perhaps everything) can come from "nothing" seems to be one of the central motivating questions behind a host of concepts; including string theory, quintessence, dark matter, vacuum energy, multiple dimensions, accelerated expansion, and surely more. Any ideas and thoughts?

[Donnie B.]

Why is there air? [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

Historically, the question "Why is there something rather than nothing" has been more in the realm of Philosophy or Theology than science. However, a number of other issues have migrated from the first two to the last; so maybe this conundrum will eventually yield to empirical investigation.

I'm not too optimistic, though. Science isn't all that good at those "why" questions. Not as good as with "How", "What", "Where", "When", or "How much" questions, anyway.

[Ben Benoy]

Why is there air? To fill balls with. Duh.

Anyway, I don't know that I would say that string theory, dark matter, et al. are really there to try to explain "why" there is something instead of not. I thought that they were models of "how" the things that do exist work the way they do. Unless you have something pretty convincing, I'm going to keep on believing that. Especially about dark matter, which I thought was just matter we hadn't detected, but that ought to be there.

Er.. I think that's all. Far be it from me to just ramble on and on about nothing in particular after I finish my point. Er... doh!

Ben Benoy

[Mr. X]

What is nothing?

Nothing is the absence of anything. [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

Then there's also this. Enjoy! [img]/phpBB/images/smiles/icon_lol.gif[/img]

Main Entry: noth·ing
Pronunciation: 'n&-thi[ng]
Function: pronoun
Etymology: Middle English, from Old English nAn thing, nAthing, from nAn no + thing thing -- more at NONE
Date: before 12th century
1 : not any thing : no thing <leaves nothing to the imagination>
2 : no part
3 : one of no interest, value, or consequence <they mean nothing to me>

[Chip]

[quote]
On 2001-11-04 22:08, Mr. X wrote:

What is nothing?
Nothing is the absence of anything.

Ah, but that's the problem. When science tries to define, detect, measure, or observe the space where there is total void, such as in the huge empty "bubbles" between galaxy clusters they always detect something. (Though very tenuous.) The fabric of space itself has a kind of texture on the large scale, and when they look on the tiniest subatomic scale, there seems to be something akin to a vibration behind the cold, empty vacuum. (I am not being scientific with these descriptions.) There is a lot of emptiness out there, away from the stars and gas, between the galaxies. But, there is always some kind of background radiation, curvature, or occasional cosmic ray. There seems to be no absolute nothingness. I find that interesting. [img]/phpBB/images/smiles/icon_eek.gif[/img]

[Kaptain K]

I think that *Nothing*, like many other concepts, is a mathematical construct that can only be approximated in the "real world". [img]/phpBB/images/smiles/icon_wink.gif[/img]

[Mr. X]

On 2001-11-05 03:49, Kaptain K wrote:
I think that *Nothing*, like many other concepts, is a mathematical construct that can only be approximated in the "real world". [img]/phpBB/images/smiles/icon_wink.gif[/img]

Joking or not, this seems to me to be the most appropriate answer!
Same with always walking half the previous walked distance towards a wall and eventually hitting the wall, when mathematically you shouldn't.

I guess nothing of our universe is as perfect as the mathematics that define them. That is why it sucks (gravitationally, of course). [img]/phpBB/images/smiles/icon_wink.gif[/img]

[GrapesOfWrath]

On 2001-11-05 07:31, Mr. X wrote:
Same with always walking half the previous walked distance towards a wall and eventually hitting the wall, when mathematically you shouldn't.

Hey, no one jokes about mathematics*. It is not true that "mathematically you shouldn't." The mathematics works out the smae as in the real world, for the most part-- mathematically, you also hit the wall. As I say in another thread about the twin paradox (or is it the rockets and rope paradox?), a paradox usually only arises because we analyze something from a different point of view, and forget something in our analysis, making the results incompatible and seemingly paradoxical.

In this particular paradox, each time you walk halfway to the wall does not take the same amount of time, each time gets shorter in proportion to the distance. The paradox arises only when we imagine, or assume, that each of the infinite number of halfway steps take a fixed amount of time--which would then have it require an infinite amount of time to hit the wall. Clearly, that assumption is our problem, not the mathematics.

*OK, OK, everybody does. Sheesh. (Why didn't the guy mathematician get along with his girlfriend? He could've handled base two, but couldn't get past base one.)

[James]

*OK, OK, everybody does. Sheesh. (Why didn't the guy mathematician get along with his girlfriend? He could've handled base two, but couldn't get past base one.)

*Groan* That was bad.

[Donnie B.]

(Why didn't the guy mathematician get along with his girlfriend? He could've handled base two, but couldn't get past base one.)

At which point, the girlfriend gave up on him. Hell, even she knew that base one was useless! [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

[Donnie B.]

On 2001-11-05 07:31, Mr. X wrote:

On 2001-11-05 03:49, Kaptain K wrote:
I think that *Nothing*, like many other concepts, is a mathematical construct that can only be approximated in the "real world". [img]/phpBB/images/smiles/icon_wink.gif[/img]

Joking or not, this seems to me to be the most appropriate answer!

I'd agree with that assessment. There's a whole lot of nothing out there (and in there, too, at the subatomic level) -- but it's mighty dirty nothing.

Now, there's a concept: "a large amount of nothing."

Oh, by the way, wouldn't it be fair to say that there's really no place in the universe that truly contains nothing? Aren't there electromagnetic fields everywhere? Isn't gravity acting on everything all the time? Just because gravity is intangible doesn't make it 'nothing'. Just a thought...

[GrapesOfWrath]

On 2001-11-05 08:43, Donnie B. wrote:
Oh, by the way, wouldn't it be fair to say that there's really no place in the universe that truly contains nothing? Aren't there electromagnetic fields everywhere? Isn't gravity acting on everything all the time? Just because gravity is intangible doesn't make it 'nothing'. Just a thought...

With general relativity, gravity is the shape of spacetime. In other words, it's another fictitious force, like coriolis or centrifugal.

(Why was the mathematician suspicious of his therapy group? Because one of them wasn't a commuter, so things just didn't add up.)

[Kaptain K]

Oh, by the way, wouldn't it be fair to say that there's really no place in the universe that truly contains nothing? Aren't there electromagnetic fields everywhere? Isn't gravity acting on everything all the time? Just because gravity is intangible doesn't make it 'nothing'. Just a thought...

And don't forget virtual particles that appear from and disappear into the "quantum foam" in Planck time and "zero point energy".

[Donnie B.]

On 2001-11-05 09:08, GrapesOfWrath wrote:

On 2001-11-05 08:43, Donnie B. wrote:
Oh, by the way, wouldn't it be fair to say that there's really no place in the universe that truly contains nothing? Aren't there electromagnetic fields everywhere? Isn't gravity acting on everything all the time? Just because gravity is intangible doesn't make it 'nothing'. Just a thought...

With general relativity, gravity is the shape of spacetime. In other words, it's another fictitious force, like coriolis or centrifugal.

All right, if you choose to use that formulation (but isn't there such a thing as a graviton? Well, never mind...)

Even so, every point in the universe is being affected by gravity; that is, even a place where there's "nothing" has a gravitational "shape".

[Donnie B.]

On 2001-11-05 09:10, Kaptain K wrote:

Oh, by the way, wouldn't it be fair to say that there's really no place in the universe that truly contains nothing? Aren't there electromagnetic fields everywhere? Isn't gravity acting on everything all the time? Just because gravity is intangible doesn't make it 'nothing'. Just a thought...

And don't forget virtual particles that appear from and disappear into the "quantum foam" in Planck time and "zero point energy".

I was assuming that these were what K.C.Cole was referring to in the OP. Guess I should have remembered the old adage about assuming... [img]/phpBB/images/smiles/icon_smile.gif[/img]

[Mnemonia]

On 2001-11-04 22:08, Mr. X wrote:

What is nothing?

Nothing is the absence of anything. [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

And that would make it something ("the absence of anything".)

Nothing can not "be" anything, nor can it "not be" anything. You can not compare nothings, nor can you assign it properties. Nothing simply does not exist in those terms.

What you should say is that space is empty (for the most part.) Space has shape and can be filled with something. Nothing would have no shape and no volume if it were real.

So yes it is more of a philosophical view than scientific, becuase science can not describe the unreal.

(I hope that made sense to somebody.)

[Wiley]

On 2001-11-05 11:21, Mnemonia wrote:

On 2001-11-04 22:08, Mr. X wrote:

What is nothing?

Nothing is the absence of anything. [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

And that would make it something ("the absence of anything".)

Nothing can not "be" anything, nor can it "not be" anything. You can not compare nothings, nor can you assign it properties. Nothing simply does not exist in those terms.

What you should say is that space is empty (for the most part.) Space has shape and can be filled with something. Nothing would have no shape and no volume if it were real.

So yes it is more of a philosophical view than scientific, becuase science can not describe the unreal.

(I hope that made sense to somebody.)

Couldn't we do a variation of this theme?

Define "anything" as any set that contains at least one element. Then let's define "everything" as the set comprising all "anything"s. And "nothing" would the complement of this set "everything".

Of course, the empty set would not be in "everything" as its not "anything" - the empty set is defined as the set with no elements. However the empty set is a subset of "everything" by the theorem: the empty set is a subset of every set. This implies that the empty set is not "everything" but is a subset of "everything". Uh-oh, contradiction.

Wiley - whapped in the head by Bert Russell once again

[Mr. X]

On 2001-11-05 08:09, GrapesOfWrath wrote:
Hey, no one jokes about mathematics*. It is not true that "mathematically you shouldn't." The mathematics works out the same as in the real world, for the most part-- mathematically, you also hit the wall. As I say in another thread about the twin paradox (or is it the rockets and rope paradox?), a paradox usually only arises because we analyze something from a different point of view, and forget something in our analysis, making the results incompatible and seemingly paradoxical.

In this particular paradox, each time you walk halfway to the wall does not take the same amount of time, each time gets shorter in proportion to the distance. The paradox arises only when we imagine, or assume, that each of the infinite number of halfway steps take a fixed amount of time--which would then have it require an infinite amount of time to hit the wall. Clearly, that assumption is our problem, not the mathematics.

I think you misunderstand the nature of the problem. Time was an afterthought in that. The main part is the distance. Going half the previous distance to a mathematical point many times consecutively never gets you to that point. Which point is the first point after a point? None. There is never a first point. There is always the possibility of "inserting", if you wish, another point in between. As for the "real" world I don't know, since I don't live there. [img]/phpBB/images/smiles/icon_wink.gif[/img]

And what is teratology? That's written in your profile, but I don't understand what it is. Is it real?

[Wiley]

On 2001-11-05 15:51, Mr. X wrote:

On 2001-11-05 08:09, GrapesOfWrath wrote:
Hey, no one jokes about mathematics*. It is not true that "mathematically you shouldn't." The mathematics works out the same as in the real world, for the most part-- mathematically, you also hit the wall. As I say in another thread about the twin paradox (or is it the rockets and rope paradox?), a paradox usually only arises because we analyze something from a different point of view, and forget something in our analysis, making the results incompatible and seemingly paradoxical.

In this particular paradox, each time you walk halfway to the wall does not take the same amount of time, each time gets shorter in proportion to the distance. The paradox arises only when we imagine, or assume, that each of the infinite number of halfway steps take a fixed amount of time--which would then have it require an infinite amount of time to hit the wall. Clearly, that assumption is our problem, not the mathematics.

I think you misunderstand the nature of the problem. Time was an afterthought in that. The main part is the distance. Going half the previous distance to a mathematical point many times consecutively never gets you to that point. Which point is the first point after a point? None. There is never a first point. There is always the possibility of "inserting", if you wish, another point in between. As for the "real" world I don't know, since I don't live there. [img]/phpBB/images/smiles/icon_wink.gif[/img]

And what is teratology? That's written in your profile, but I don't understand what it is. Is it real?

The mathematician in me wants to answer Zeno with position is function of time, x(t). And we can create an inverse function t(x) which is a function of position. So if we look at the limit of x at it approaches a point x0, we can also look at the limit of t(x) as x approaches x0. Unless you have a golden watch, Zeno's paradox is not really a paradox.

Of course, the pragmatist in me wants to answer I'm gonna take this Louisville Slugger and whomp the side of your head. If you are correct, the bat will never reach your head. This is known as proof by contradiction, or put your money where your mouth was. [img]/phpBB/images/smiles/icon_smile.gif[/img]

[Mr. X]

I'm gonna take this Louisville Slugger and whomp the side of your head. If you are correct, the bat will never reach your head.

Um, don't do that, please. [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

You're right at some point, since you inadvertantly brought something to my attention. That would be a geometric series , i.e. 1/2+1/4+1/8+1/16+... with (I call it "r") abs(r)<1 and hence converges (to 2). [Edit] (The series would start at n=1, not 0, hence it would converge to 1, not 2. Or if you prefer to start it at 0 feel free to do so) [Edit]

However, it still leaves the fact that if you geometrically always take the point halfway between one and another you can never reach that point. Just like you can't ever get to zero by dividing.

But I'm still not conviced. There's something missing. If you use the limit as you said, the exact value you want is excluded, you're only working very close to the point. Agreed you get increasingly close but never reach it. The point isn't there but studying the limit we never care whether it's there or not.

I think I could do the same with my argument on the series. Take any number of terms you want and add them up, you could never reach 2, because that would mean that there is a last term to the series (there isn't) and that you could eventually go over the 2. <u>Then there is the matter of an infinity of terms which adds up to a round sum.</u> ***Note at the bottom***

So to reach the point wouldn't you need to add up an inifnity of halves of the preceding distance?

[Edit] Just discovered who that guy Zeno was, sorry for looking like an idiot, and...

Okay, okay, found something:

Similarly, it takes an infinite number of time intervals for Achilles to catch the tortoise, but the sum of these time intervals is a finite amount of time.

Hence my assumption is right that it takes an infinite number of halves to add up.

However, I see now why some were talking about time. But how come can it take a finite amount of time to go an infinite number of intervals? Would that be that because the infinite number of intervals would take an infinitely small time to go over, hence it would become the limit of something that would be "undetermined" (excuse my english), a 0*infinity. And since it isn't a real zero but a tendency zero we can't assume it would equal 0, but from what I can see you have been trying to tell me, it would go towards 0 after further investigation, or towards a finite amount of time, C.Q.F.D.! (Well, not really because I didn't actually do it, and I guessed, but what the hey.).
[Edit]

Keep in mind I'm talking about "imaginary realm" only, so the the baseball bat will be quite unecessary (Phew!). [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

[Edit] Also, I have brought to my own attention (yay!) that what "I" just "solved" isn't really the problem I was talking about in the beginning! It got twisted around with that Zeno guy and small nuances, but it's as good as the other one! See you around! [Edit]

Help me Ben Benoy! Save us please!
[No need anymore!]

[Edit] I also managed (with the help of others, including Wiley, no less [img]/phpBB/images/smiles/icon_smile.gif[/img] ) to steer this whole astronomy topic in an entire other mathematics related talk, which is no small feat [img]/phpBB/images/smiles/icon_wink.gif[/img], yet was probably more interesting. You may now go back to the other, more boring, talk [img]/phpBB/images/smiles/icon_smile.gif[/img] [Edit]

**NOTE**

Then there is the matter of an infinity of terms which adds up to a round sum.

But of course! The infinite number of halves of the previous to be added are infinite and are therefore equal to sum towards which the series converges! Therefore the infinite amount of halves add up to a finite distance that can therefore be crossed by a moving body with a finite speed in a finite amount of time! (Unless someone wants to add some relativity trick I am unaware of!) Requiring of course that the speed be in the appropriate direction, yada yada yada...

Well I guess this makes C.Q.F.D. x2. I'd hate to be Zeno right now! [img]/phpBB/images/smiles/icon_smile.gif[/img]
**End of note**

[Edit]Shove it, Zeno![Edit]

Edited many a times for specifications, new finds and of course typos and the whole text in bold because of a missing tag!

<font size=-1>[ This Message was edited by: Mr. X on 2001-11-05 19:20 ]</font>

[BADad]

To Mr. X's question about Grapes of Wrath occupation: I haven't heard the term "teratology" in years. There was this really scary book called "The Cadaver of Gideon Wick", Alexander Laing, 1934, that discussed the study of monstrosities - unusual forms of embryos or babies. Does Grapes of Wrath really do this?

[JF]

On 2001-11-04 22:08, Mr. X wrote:

What is nothing?

Nothing is the absence of anything. [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

Then there's also this. Enjoy! [img]/phpBB/images/smiles/icon_lol.gif[/img]
Incorrect. Nothing is the absence of everything. -JF

Main Entry: noth·ing
Pronunciation: 'n&-thi[ng]
Function: pronoun
Etymology: Middle English, from Old English nAn thing, nAthing, from nAn no + thing thing -- more at NONE
Date: before 12th century
1 : not any thing : no thing <leaves nothing to the imagination>
2 : no part
3 : one of no interest, value, or consequence <they mean nothing to me>

[Chip]

Just out of curiosity, does anyone know of research that has attempted to determine theoretically how thin the vacuum of space can get? There seems to be a considerable amount of matter spread out ever more thinly within and far beyond a given galaxy. However, I read once years ago a proposed figure for the very tenuousness of space far between galaxies. It was something like "one atom per cubic centimeter." (I'm giving this as an example of how the measurement was presented. It wasn't actually that figure, and I don't know how the actual figure was arrived at, even if theoretical.)

Chip

[David Hall]

I'm about as far from being a mathematician as can be <font size=-1>(this is a warning that I really don't know what I'm talking about here)</font>, but I've been thinking about it and I have an observation to make about the paradox of the infinite halves. Please tell me if I get anywhere close to having a legitimate point here. [img]/phpBB/images/smiles/icon_smile.gif[/img]

Doesn't the paradox require a continuous rate of movement? I.E, the action has to be something that can be infinitely divided up to begin with. In such a case, I can see a mathematical paradox. But in most real-life situations, movement is not continuous, but is broken up into discrete units. Take, for example, footsteps.

As long as you are approaching the wall, you would have a constant velocity of x centimeters per unit of time, the distance and time it takes to make one stride. There is no way to break that down into halves. One step per second is one step per second. You can't say that the man suddenly is taking only half a step. In the end, there would always be a final point in which the distance to the wall is less than the length of one pace, and therefore the wall would be reached in that one final step, no matter what kind of halving goes on mathematically.

I think it's likely that all physical movement has to follow this discrete-interval type of movement, if only down to the Planck length. [img]/phpBB/images/smiles/icon_smile.gif[/img] In which case, the paradox becomes impossible in physical terms.

Ok, That's my thought. Now everybody is welcome to come piling onto me and show me what a real moron I am. What am I doing wrong here? Don't hold back. I'm ready for a whuppin. [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

[Chip]

On 2001-11-05 23:40, I wrote: [img]/phpBB/images/smiles/icon_wink.gif[/img]
Just out of curiosity, does anyone know of research that has attempted to determine theoretically how thin the vacuum of space can get? There seems to be a considerable amount of matter spread out ever more thinly within and far beyond a given galaxy. However, I read once years ago a proposed figure for the very tenuousness of space far between galaxies. It was something like "one atom per cubic centimeter." (I'm giving this as an example of how the measurement was presented. It wasn't actually that figure, and I don't know how the actual figure was arrived at, even if theoretical.)

Chip

Well, I found this info on "ScienceNet:"

"In the air we breathe, there are about 25 billion billion molecules of gas per cubic centimetre. Scientist can create partial vacuums that have ten thousand molecules per cubic centimetre.
It is not an absolute vacuum in outer space. The density of interstellar matter is extremely low. It averages roughly just 1 atom or molecule per cubic centimetre. Matter of this low density is far more tenuous than the best vacuum that we can make in laboratories here on Earth. While much of the mass of the galaxy is concentrated in stars, a significant fraction of this mass resides in the vast regions of interstellar space. Observations show that most of this material is hydrogen either in atomic form or combined as H2 molecules. It has been found that the average density of H atoms in all forms in the interstellar medium near the Sun is 2 x 10-24g cm-3 and the average density of material in stars is 1.8 x 10-23g cm-3.

Giant molecular clouds of hydrogen are associated with star formation. Although they are dense at about 106 particles per cubic centimetre, they are about 10 trillion times less dense than air. Nevertheless, the extinction in these clouds is so great that we cannot see through them at visual wavelengths."

Quoted from ScienceNet:
http://www.sci-ctr.edu.sg/ScienceNet..._sol00706.html

Singularities, black holes, nothing, two slits - one photon, thinking too much about this stuff, I'm going to end up talking to myself! [img]/phpBB/images/smiles/icon_smile.gif[/img]

[Mr. X]

By the way, when I say "But of course" this isn't a usual expression for me, so whatever it got you thinking about me is most likely false! [img]/phpBB/images/smiles/icon_smile.gif[/img] Most likely. Bear in mind that I am somewhat strange and that weird stuff might turn out to be accurate. [img]/phpBB/images/smiles/icon_wink.gif[/img]

I haven't heard the term "teratology" in years. There was this really scary book called "The Cadaver of Gideon Wick", Alexander Laing, 1934, that discussed the study of monstrosities - unusual forms of embryos or babies. Does Grapes of Wrath really do this?

Cool! [img]/phpBB/images/smiles/icon_cool.gif[/img] Or maybe it should be this: [img]/phpBB/images/smiles/icon_razz.gif[/img] minus the smile!

Is it real? Does he study abominations?

BADad, could you tell me more about this book? I saw cadaver, then I saw medical student, teratology, how could it NOT be good?

BADad, I also found this on the 'net:

somewhat of a mystery, with maybe a touch of horror, and the final plot twist was an unusual take on the evil twin (the evil twin was growing out of Gideon's chest but no one knew until Gideon was dead).

Is that the one?

[Edit]Apparently it still exists (Teratology).www.teratology.org. "Interesting" to say the least.[Edit]

<font size=-1>[ This Message was edited by: Mr. X on 2001-11-06 17:15 ]</font>

[Mr. X]

And, sorry Dave, I can't whup since my english skills aren't quite equivalent to yours, and you may go around me 5 times in your fancy language while I'm still trying to understand the first thing you said.

If you could dumb it down with the terms and the syntax it would really help me!

[Wiley]

On 2001-11-06 06:42, David Hall wrote:
I'm about as far from being a mathematician as can be <font size=-1>(this is a warning that I really don't know what I'm talking about here)</font>, but I've been thinking about it and I have an observation to make about the paradox of the infinite halves. Please tell me if I get anywhere close to having a legitimate point here. [img]/phpBB/images/smiles/icon_smile.gif[/img]

Doesn't the paradox require a continuous rate of movement? I.E, the action has to be something that can be infinitely divided up to begin with. In such a case, I can see a mathematical paradox. But in most real-life situations, movement is not continuous, but is broken up into discrete units. Take, for example, footsteps.

As long as you are approaching the wall, you would have a constant velocity of x centimeters per unit of time, the distance and time it takes to make one stride. There is no way to break that down into halves. One step per second is one step per second. You can't say that the man suddenly is taking only half a step. In the end, there would always be a final point in which the distance to the wall is less than the length of one pace, and therefore the wall would be reached in that one final step, no matter what kind of halving goes on mathematically.

I think it's likely that all physical movement has to follow this discrete-interval type of movement, if only down to the Planck length. [img]/phpBB/images/smiles/icon_smile.gif[/img] In which case, the paradox becomes impossible in physical terms.

Ok, That's my thought. Now everybody is welcome to come piling onto me and show me what a real moron I am. What am I doing wrong here? Don't hold back. I'm ready for a whuppin. [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

(Wiley puts pine tar on his Louisville Slugger [img]/phpBB/images/smiles/icon_smile.gif[/img])

There is no mathematical paradox for either the continuous or the discrete case. There appears to be a paradox because Zeno forgets that we are dealing with a two dimensional problem: space and time. Zeno takes the limit with respect to position but neglects the corresponding limit with respect time; hence, he becomes confused.

Graph the function x = v t. Take the limit as x approaches 1. Let x = 1/2,3/4, and 7/8, the corresponding values of time are t = 1/(2v), 3/(4v), and 7/(8v). As x approaches 1, t approaches 1/v. Coming back to the baseball bat example, as the bat approaches a point in space, it also approaches a point in time. As the bat approaches the position of impact with Mr. X's head [img]/phpBB/images/smiles/icon_smile.gif[/img], it also approaches the time of impact. We never go beyond this time.

As for the discrete case, if there is a minimum time step or spatial step, we can not take a limit. An infinite sequence will only approach a finite value if the the sequence tends towards zero. Even if the smallest spatial step is the Plank length, we can not take a limit and Zeno's paradox is meaningless. Your footsteps analogy is sound.

[Mr. X]

Argh, my way is better! Admit it Wiley! [img]/phpBB/images/smiles/icon_smile.gif[/img]
I won't rest until you admit it! [img]/phpBB/images/smiles/icon_biggrin.gif[/img] Even if I didn't do anything! [img]/phpBB/images/smiles/icon_smile.gif[/img]

[Mr. X]

Since Wiley isn't being very supportive of my wild claims, I'll just shut up! [img]/phpBB/images/smiles/icon_smile.gif[/img]