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# Student Challenges Basic Ideas of Time

Survey course, eh?

posted by ook at 8:17 AM on August 1, 2003

My impression was that it is substantially easier to get your work into some of those letters types of publications. It's not like he was accepted for publication in a peer reviewed journal.

My impression could be out of whack. They often are, but I still wouldn't give it any additional weight or abstract it too far from dorm room blathering just because somebody is publishing it.

posted by willnot at 9:03 AM on August 1, 2003

posted by raygirvan at 10:24 AM on August 1, 2003

Huh? I think you're confused about what constitutes infinity. And limits.

posted by bshort at 12:32 PM on August 1, 2003

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# Student Challenges Basic Ideas of Time

August 1, 2003 7:20 AM Subscribe

A bold paper published in the August issue of Foundations of Physics Letters seems set to change the way we think about the nature of time and its relationship to motion and classical and quantum mechanics. The work also appears to provide solutions to Zeno's paradoxes. (Via Kurzweilai.net. More inside...)

(Please note that the above was cribbed directly from Kurzweil's thoughts on the matter, as posted on his website.)

posted by Pinwheel at 7:24 AM on August 1, 2003

posted by Pinwheel at 7:24 AM on August 1, 2003

Geez, even the people on FARK.com were dissing this.

His paper (or at least this article) doesn't contain anything that's either novel or useful. I mean, damn, even my basic level physics class went over Hinesburg's uncertainty principle and all that. Showing how you couldn't measure position/velocity 'perfectly' or whatever.

and zeno's paradox has already been solved, afaik.

posted by delmoi at 7:43 AM on August 1, 2003

His paper (or at least this article) doesn't contain anything that's either novel or useful. I mean, damn, even my basic level physics class went over Hinesburg's uncertainty principle and all that. Showing how you couldn't measure position/velocity 'perfectly' or whatever.

and zeno's paradox has already been solved, afaik.

posted by delmoi at 7:43 AM on August 1, 2003

Zeno's paradox used to bother me terribly as a child. It's easy to understand the paradox, but I'm too damn stupid to understand the refutation.

Damn.

posted by Pericles at 7:46 AM on August 1, 2003

Damn.

posted by Pericles at 7:46 AM on August 1, 2003

Reminds me of this guy I used to know in college. "Ever think about ... time, man. No really, think about it." He too had come to the conclusion that time doesn't really exist.

posted by wobh at 8:02 AM on August 1, 2003

posted by wobh at 8:02 AM on August 1, 2003

This is a great article, both for the content of the theory proposed and for the "human interest" angle of this 27-year-old non-academic managing to overcome the cliqueness of physics academia and get his ideas reviewed, accepted by some authorities, and (soon) published.

If I understand his point conceptually, he's saying that Zeno's Paradox and some of the uncertainty of measuring objects' relative positions and speeds arises from the fact that there is literally no such thing as an "instant" in time. Any measurement one makes, no matter how precise, is a measurement of an object's position and speed over some

This does indeed sound like the sort of idea that one delves into in profound tones and fuzzy detail during dorm-room bull sessions. But since he's apparently been accepted for publication, I'm assuming that he's got some heavy-duty fact-checkable mathematical models to back this up. Which is ultimately what separates the physicists from the potheads when it comes to this stuff.

Anyway, as I understand it, he's not "writing off" the Heisenberg Uncertainty Principle, but instead asserting that

I think he goes further to assert that time is not a physical property of the universe, but rather a conceptual framework that we impose on our perceptions. That time is, in fact, an illusion -- but an illusion so necessary to our senses and thoughts that it's made it difficult for 2500 years to understand the world around us, resulting in Zeno's Paradox and other inconsistencies of physics.

It will be interesting to see the reaction as his ideas get more exposure and scrutiny. I'm certainly no expert: my understanding of this is pretty crude, and I wouldn't be surprised if it's eventually exposed as a flawed theory. But it's a bold and philosophically exciting set of new ideas, and bombshells like this are what make The March of Science such a messy but fascinating process.

posted by eyebeam at 8:13 AM on August 1, 2003

If I understand his point conceptually, he's saying that Zeno's Paradox and some of the uncertainty of measuring objects' relative positions and speeds arises from the fact that there is literally no such thing as an "instant" in time. Any measurement one makes, no matter how precise, is a measurement of an object's position and speed over some

*interval*of time. Even if this interval is exceedingly short, it can never be the "instant" that current physical models sort of take as an underlying given.This does indeed sound like the sort of idea that one delves into in profound tones and fuzzy detail during dorm-room bull sessions. But since he's apparently been accepted for publication, I'm assuming that he's got some heavy-duty fact-checkable mathematical models to back this up. Which is ultimately what separates the physicists from the potheads when it comes to this stuff.

Anyway, as I understand it, he's not "writing off" the Heisenberg Uncertainty Principle, but instead asserting that

*some*of the difficulty of accurate measurement is due to a misunderstanding of the nature of time and movement, rather than to quantum uncertainty. And he's asserting that this applies at the macro/classical level of observation as well, as Zeno illustrated in his Achilles vs. the turtle paradox.I think he goes further to assert that time is not a physical property of the universe, but rather a conceptual framework that we impose on our perceptions. That time is, in fact, an illusion -- but an illusion so necessary to our senses and thoughts that it's made it difficult for 2500 years to understand the world around us, resulting in Zeno's Paradox and other inconsistencies of physics.

It will be interesting to see the reaction as his ideas get more exposure and scrutiny. I'm certainly no expert: my understanding of this is pretty crude, and I wouldn't be surprised if it's eventually exposed as a flawed theory. But it's a bold and philosophically exciting set of new ideas, and bombshells like this are what make The March of Science such a messy but fascinating process.

posted by eyebeam at 8:13 AM on August 1, 2003

*even my basic level physics class went over Hinesburg's uncertainty principle and all that*

Survey course, eh?

posted by ook at 8:17 AM on August 1, 2003

A quick search o' the Net found this article (PDF) by Lynds, which claims to "build on the original article." No sign of the original, which in the world of theoretical physics is highly unusual (but not unheard of.) I'll hike over to the campus library later today and see if I can dig it up.

posted by Johnny Assay at 8:27 AM on August 1, 2003

posted by Johnny Assay at 8:27 AM on August 1, 2003

This is a link to a second paper of Lynds' on the Zeno paradox. What makes me kind of skeptical is that he doesn't even consider certain possible resolutions of the paradoxes i.e. through nonstandard analysis, which from the little I know this has been hailed as the final mathematical resolution of the Zeno paradoxes. Any mathematicians around?

wobh: "He too had come to the conclusion that time doesn't really exist."

Julian Barbour is a legitimate philosopher of science, accepted by the likes of Smolin and he shares the same opinion. This interview of his is fascinating.

posted by talos at 8:31 AM on August 1, 2003

wobh: "He too had come to the conclusion that time doesn't really exist."

Julian Barbour is a legitimate philosopher of science, accepted by the likes of Smolin and he shares the same opinion. This interview of his is fascinating.

posted by talos at 8:31 AM on August 1, 2003

ook: The Hinesburg uncertainty principle states that Felice, the Town's feline's quantum state is unknowable at any given time.

delmoi must have missed that basic level physics class when the teacher discussed the ideas of one Heisenberg.

posted by nkyad at 8:42 AM on August 1, 2003

delmoi must have missed that basic level physics class when the teacher discussed the ideas of one Heisenberg.

posted by nkyad at 8:42 AM on August 1, 2003

*But since he's apparently been accepted for publication...*

My impression was that it is substantially easier to get your work into some of those letters types of publications. It's not like he was accepted for publication in a peer reviewed journal.

My impression could be out of whack. They often are, but I still wouldn't give it any additional weight or abstract it too far from dorm room blathering just because somebody is publishing it.

posted by willnot at 9:03 AM on August 1, 2003

Hmmm, I assumed it was peer-reviewed.

I went looking for some information about 'Foundations of Physics Letters' but couldn't find anything about what their criteria is for publishing papers. However, I did learn that a subscription is US$730 for six bi-monthly issues. Good lord.

Willnot, can you enlighten me as to the difference between a 'letters' journal and a peer-reviewed journal? I'd like to be able to be a little more discriminating when I read these sorts of releases and articles.

posted by eyebeam at 9:15 AM on August 1, 2003

I went looking for some information about 'Foundations of Physics Letters' but couldn't find anything about what their criteria is for publishing papers. However, I did learn that a subscription is US$730 for six bi-monthly issues. Good lord.

Willnot, can you enlighten me as to the difference between a 'letters' journal and a peer-reviewed journal? I'd like to be able to be a little more discriminating when I read these sorts of releases and articles.

posted by eyebeam at 9:15 AM on August 1, 2003

willnot: I think that Foundations of Physics Letters *is* a peer-reviewed journal. It's just a really crappy peer-reviewed journal.

(Letters journals tend to want shorter articles than the other journals; they're not necessarily worse. e.g. PRL is no worse than the other Physical Review journals, and ApJL is no worse than the Astrophysical Journal.)

That being said, though, you're right. It's not too far from being dorm-room blather, even though it's "peer reviewed." Lots of bad stuff gets by the better journals, and lesser ones like this are catch-alls for all the crap that gets rejected several times.

posted by ptermit at 9:17 AM on August 1, 2003

(Letters journals tend to want shorter articles than the other journals; they're not necessarily worse. e.g. PRL is no worse than the other Physical Review journals, and ApJL is no worse than the Astrophysical Journal.)

That being said, though, you're right. It's not too far from being dorm-room blather, even though it's "peer reviewed." Lots of bad stuff gets by the better journals, and lesser ones like this are catch-alls for all the crap that gets rejected several times.

posted by ptermit at 9:17 AM on August 1, 2003

eyebeam: I think that you've got to be *very* discriminating when it comes to press releases. Anyone can write a press release and post it -- sometimes even journalists, especially those who don't know anything about science, get suckered.

A few warning signs with this one:

-- Incredibly hyperbolic language such as "ground breaking," "the science world's astonishment," "Lynds' work seems likely to establish him as a groundbreaking figure...."

-- Repeated appeals to authority (and not very good ones at that.) The only physicist of note is Wheeler, and look closely at what Wheeler says. He's being polite, admiring Lynd's boldness and saying that young people often cause revolutions. He says *nothing* about the theory itself.

-- Wearing criticism by the "establishment" and his lack of credentials as a badge of honor.

-- Lots of pseudoscientific gobbledygook. (Hard to spot unless you know it, but phrases like " the parameter and boundary of their respective position and magnitude are naturally determinable up to the limits of possible measurement..." don't make sense. "Parameter" and "boundary" mean things in physics; Lynds ignores their meaning and uses them as smokescreens.)

If this were really important, you'd see coverage in Science, Nature, Science News, and the New York Times, and eventually Scientific American, Discover, and some other monthlies. These publications have reporters who know science and write about it for a living... I suspect you'll hear deafening silence from all these places about this groundbreaking work.

posted by ptermit at 9:34 AM on August 1, 2003

A few warning signs with this one:

-- Incredibly hyperbolic language such as "ground breaking," "the science world's astonishment," "Lynds' work seems likely to establish him as a groundbreaking figure...."

-- Repeated appeals to authority (and not very good ones at that.) The only physicist of note is Wheeler, and look closely at what Wheeler says. He's being polite, admiring Lynd's boldness and saying that young people often cause revolutions. He says *nothing* about the theory itself.

-- Wearing criticism by the "establishment" and his lack of credentials as a badge of honor.

-- Lots of pseudoscientific gobbledygook. (Hard to spot unless you know it, but phrases like " the parameter and boundary of their respective position and magnitude are naturally determinable up to the limits of possible measurement..." don't make sense. "Parameter" and "boundary" mean things in physics; Lynds ignores their meaning and uses them as smokescreens.)

If this were really important, you'd see coverage in Science, Nature, Science News, and the New York Times, and eventually Scientific American, Discover, and some other monthlies. These publications have reporters who know science and write about it for a living... I suspect you'll hear deafening silence from all these places about this groundbreaking work.

posted by ptermit at 9:34 AM on August 1, 2003

Interesting. Too bad he actually uses 'would of' in the Zeno article Johnny Assay linked above.

posted by HTuttle at 9:37 AM on August 1, 2003

posted by HTuttle at 9:37 AM on August 1, 2003

Indeed, ptermit, often letters journals are also known for quick turnarounds (e.g. ecology letters with its month to weeks turnarounds) so they're great for rapid publication of interesting new results and often DO contain groundbreaking work, but the cost of such rapid turnaround is often some less quality articles will sneak by. More fuel for controversy in science, which, imho, is a good thing - its annoying to have your bold new argument, which may be crap, be held up in a one year review process - that's another year or two of research down the tubes!

posted by jearbear at 9:48 AM on August 1, 2003

posted by jearbear at 9:48 AM on August 1, 2003

Ptermit, thanks very much. One of the reasons I wouldn't make a very good scientist is that I get swept up in the "ooh, what a cool idea!" sense-of-wonder stuff, rather than applying the scientific approach of first being rigorously critical and then getting excited only if things pass the smell test. I appreciate your sharing some knowledge that will, I hope, help to keep my enthusiasm from shading into credulousness quite as often!

posted by eyebeam at 9:57 AM on August 1, 2003

posted by eyebeam at 9:57 AM on August 1, 2003

Back from the library, where I discovered that the August issue of Foundations of Physics Letters hasn't yet been published (or at least hasn't reached the University of Chicago yet.) In physics — hell, in any science — putting out a press release before the article appears is frowned upon.

Ptermit has the right idea, though — having a referee of a lower-tier journal say, "I'm afraid I am unwilling to waste any time reading further, and recommend terminal rejection," is not a badge of honour, and there's only one scientist whose statement is unambiguously positive. Note that even Profs. Grigson and Smart, who from his descriptions would be expected to be the most familiar with Lynds' work, aren't quoted as saying anything positive about his work.

Bottom line: a lot of sound and fury, certainly, but I'll be surprised if it signifies anything.

Also: here's a link to a page on the "nonstandard analysis" talos mentioned, and how they relate to Zeno's paradox.

posted by Johnny Assay at 10:00 AM on August 1, 2003

Ptermit has the right idea, though — having a referee of a lower-tier journal say, "I'm afraid I am unwilling to waste any time reading further, and recommend terminal rejection," is not a badge of honour, and there's only one scientist whose statement is unambiguously positive. Note that even Profs. Grigson and Smart, who from his descriptions would be expected to be the most familiar with Lynds' work, aren't quoted as saying anything positive about his work.

Bottom line: a lot of sound and fury, certainly, but I'll be surprised if it signifies anything.

Also: here's a link to a page on the "nonstandard analysis" talos mentioned, and how they relate to Zeno's paradox.

posted by Johnny Assay at 10:00 AM on August 1, 2003

Ford: Time is an illusion. Lunchtime doubly so.

Arthur: Very deep. You should send that in to the Reader's Digest. They've got a page for people like you.

posted by salmacis at 10:21 AM on August 1, 2003

Arthur: Very deep. You should send that in to the Reader's Digest. They've got a page for people like you.

posted by salmacis at 10:21 AM on August 1, 2003

**ptermit**: it sounds as if it'd score pretty well on the Crackpot Index.

posted by raygirvan at 10:24 AM on August 1, 2003

eyebeam: actually, a lot of scientists get swept up by the "ooh, cool" feeling, so don't think that it's a liability. It's an asset, but scientists can take that feeling about an idea, turn it into rigorous language, and (hopefully) test it. Obviously, laypeople don't often have the ability to do that themselves, so we have to rely on our sense of smell about scientific ideas. Often, the price of a sense of wonder is occasionally getting fooled. Better that way than lacking a sense of wonder, IMO.

raygirvan: Heh. I think you're right. Wonder whether Baez will mention this paper in his next mathematical physics roundup. (Incidentally, I think John Baez is related to Joan Baez -- they're cousins IIRC.)

posted by ptermit at 10:54 AM on August 1, 2003

raygirvan: Heh. I think you're right. Wonder whether Baez will mention this paper in his next mathematical physics roundup. (Incidentally, I think John Baez is related to Joan Baez -- they're cousins IIRC.)

posted by ptermit at 10:54 AM on August 1, 2003

I don't know anything about whether Lynds' paper is any good. (Although the press-release glorification seems suspicious.)

posted by tdismukes at 11:16 AM on August 1, 2003

**However**, Zeno's paradox is settled regardless. Take a calculus course, you'll learn that the infinite series 1/2 + 1/4 + 1/8 + 1/16 +... adds up to 1, not to infinity. It's not Zeno's fault calculus hadn't been invented yet, but Lynds doesn't have that excuse. (On the other hand, if he was to refute the existing proof of the limit of the infinite series, he would immediately become the most famous mathematician in the world. i.e. aint going to happen, folks)posted by tdismukes at 11:16 AM on August 1, 2003

Never having taken calculus, could somebody explain to me why the infinite series 1/2 + 1/4 + 1/8... adds up to 1. I can see it adding up to something so close to one that it may as well be 1, but it seems intuitively obvious to me that it can't possibly really be equal to 1.

posted by willnot at 11:40 AM on August 1, 2003

posted by willnot at 11:40 AM on August 1, 2003

the series, 1/2 + 1/4 + 1/8 can be represented as the function 1/2^x.

the limit as x approaches infinity is 1, meaning it will never actually get to 1, but it will get closer and closer as x increases. Essentially, we can be sure that the sum of the series is as close to 1 as we need it to be, by taking a large enough value for x.

posted by pemulis at 11:53 AM on August 1, 2003

the limit as x approaches infinity is 1, meaning it will never actually get to 1, but it will get closer and closer as x increases. Essentially, we can be sure that the sum of the series is as close to 1 as we need it to be, by taking a large enough value for x.

posted by pemulis at 11:53 AM on August 1, 2003

The key is an

posted by COBRA! at 12:11 PM on August 1, 2003

*infinite*series. A very, very large but finite series would add up to something so close to one that it may as well be 1, but a truly infinite series would hit 1.posted by COBRA! at 12:11 PM on August 1, 2003

*The key is an infinite series. A very, very large but finite series would add up to something so close to one that it may as well be 1, but a truly infinite series would hit 1.*

Huh? I think you're confused about what constitutes infinity. And limits.

posted by bshort at 12:32 PM on August 1, 2003

screw the infinite, I want to know what you can do with a large yet finite number of monkies

posted by jearbear at 12:33 PM on August 1, 2003

posted by jearbear at 12:33 PM on August 1, 2003

Possibly, but I'm pretty sure I've got it straight... I was trying to simultaneously super-simplify the explanation and piggyback on pemulis' explanation, and did a rotten job.

posted by COBRA! at 12:39 PM on August 1, 2003

posted by COBRA! at 12:39 PM on August 1, 2003

'The Straight Dope' gave a pretty decent layman's explanation of the infinite series solution to Zeno's Paradox in a recent column which can be found online here.

posted by eyebeam at 1:05 PM on August 1, 2003

posted by eyebeam at 1:05 PM on August 1, 2003

willnot: adding to the previous responses, the expression

"1/2 + 1/4 + 1/8 + ..." is shorthand for the more formal expression "the limit (as n-->infinity) of the sum of 1/2^n"

A limit is a point that a sequence approaches, even if the sequence never actually hits that point. For example, the limit of the sequence 1/2, 1/3, 1/4, 1/5 is zero -- each of the numbers gets closer and closer to zero -- even though none of the numbers themselves are *actually* zero.

Similarly, look at the sequence of the partial sums (i.e., 1/2, 1/2+1/4, 1/2+1/4+1/8, etc.) and you'll see that they get closer and closer to one. (To prove that one is the limit, calculus students use what's called an epsilon-delta proof, which I can go into if you're interested.) Therefore, the limit as (n-->infinity) of the sum of 1/2^n is 1, even though none of the partial sums themselves are *actually* one.

In other words, it's the destination, not the journey. And it may look fishy to you, but there's a bunch of formal theorems that give these ideas a solid grounding; for example, it's a theorem that in the reals, any Cauchy sequence (like the ones here) converges to a real number.

posted by ptermit at 1:09 PM on August 1, 2003

"1/2 + 1/4 + 1/8 + ..." is shorthand for the more formal expression "the limit (as n-->infinity) of the sum of 1/2^n"

A limit is a point that a sequence approaches, even if the sequence never actually hits that point. For example, the limit of the sequence 1/2, 1/3, 1/4, 1/5 is zero -- each of the numbers gets closer and closer to zero -- even though none of the numbers themselves are *actually* zero.

Similarly, look at the sequence of the partial sums (i.e., 1/2, 1/2+1/4, 1/2+1/4+1/8, etc.) and you'll see that they get closer and closer to one. (To prove that one is the limit, calculus students use what's called an epsilon-delta proof, which I can go into if you're interested.) Therefore, the limit as (n-->infinity) of the sum of 1/2^n is 1, even though none of the partial sums themselves are *actually* one.

In other words, it's the destination, not the journey. And it may look fishy to you, but there's a bunch of formal theorems that give these ideas a solid grounding; for example, it's a theorem that in the reals, any Cauchy sequence (like the ones here) converges to a real number.

posted by ptermit at 1:09 PM on August 1, 2003

Oh yeah. There's a good book that covers this in some detail, and it's aimed at people who don't have calculus. It's called

posted by ptermit at 1:15 PM on August 1, 2003

*Zero*; here's a review that mentions Zeno's paradoxes.posted by ptermit at 1:15 PM on August 1, 2003

"The key [to Zeno's Paradox] is an infinite series." (Cobra)

I have long assumed that the solution to Zeno's Paradox lay in Quantum Mechanics where, at the Quantum level, particles can travel distances in no time. So (I assumed) given that we are all (at one level) a bunch of ganged quanta, it would follow that - as the Paradox enters the realm of the extremely small and the extremely short - these sorts of paradoxical effects come into play.

So - if Achilles moves ten times as fast as the Tortoise (it's a very speedy tortoise) - then imagine: Achilles is only 1mm behind the tortoise, then 1/10mm, then 1/100mm, and so on.

But then again, I'm blissfully ignorant about that which I speak.

posted by troutfishing at 1:42 PM on August 1, 2003

I have long assumed that the solution to Zeno's Paradox lay in Quantum Mechanics where, at the Quantum level, particles can travel distances in no time. So (I assumed) given that we are all (at one level) a bunch of ganged quanta, it would follow that - as the Paradox enters the realm of the extremely small and the extremely short - these sorts of paradoxical effects come into play.

So - if Achilles moves ten times as fast as the Tortoise (it's a very speedy tortoise) - then imagine: Achilles is only 1mm behind the tortoise, then 1/10mm, then 1/100mm, and so on.

*Pretty soon, Achillles - or rather, the quanta which Achilles is composed of - can zap across Zeno's Paradox boundary in no time at all!*But then again, I'm blissfully ignorant about that which I speak.

posted by troutfishing at 1:42 PM on August 1, 2003

IIRC, the 1/2+1/4+1/8... series is actually Zeno's arrow paradox. He argues that before an arrow shot from a bow can hit a target it must first pass through a point halfway between the archer and the target. Before it can reach the halfway point, it must pass through the point 1/4 the way to the target, etc. The point of the paradox is not that the arrow will never reach the target (hit the limit of 1) but that it can never actually move in the first place because moving between two points implies that it crosses the infinite number of points that exist between the two points. The paradox leaves a person to choose from one of two 'unacceptable' positions:

YMMV, it's been a decade since I sat through that intro to philosophy course.

posted by Fezboy! at 2:35 PM on August 1, 2003

- There is no such thing as an infinite number of points
- Physical objects cannot move through space

YMMV, it's been a decade since I sat through that intro to philosophy course.

posted by Fezboy! at 2:35 PM on August 1, 2003

Continuing, the Achilles/tortise paradox hinges on the argument that, no matter how close Achilles gets to the tortise, he can never pass it because in the span of time that he would pass the point where the tortise is, the tortise moves ahead some further amount. Again, Zeno was arguing that time exists not as a continuous phenomena but comes in atomic units--we experience time sort of the same way we experience motion pictures. When we see the movie, it appears to flow as a continuous whole, but is actually made up of many discrete images that are shown in succession at a high rate of speed.

posted by Fezboy! at 2:43 PM on August 1, 2003

posted by Fezboy! at 2:43 PM on August 1, 2003

Fezboy - but there is another way to express the first objection (in your first comment): at the level of the very small, the idea of "points" has been shown to be a less than useful description. Xeno, of course, could not have known this. So, as you said, the paradox is rooted in the (disproved) notion that it is accurate, or useful, to describe time or space (at the level of the very short or very small) in terms of atomic (or irreducible - as would be more accurate in terms of how Xeno would have viewed it, I think) units.

posted by troutfishing at 5:32 PM on August 1, 2003

posted by troutfishing at 5:32 PM on August 1, 2003

Long time listener, first time caller.

As for the sum of 1/2 + 1/4 + 1/8 + ..., there's a pretty easy-to-understand trick to see why this comes to 1:

Suppose S = 1/2 + 1/4 + 1/8 + ...

Multiply each side by 2 to get

2*S = 2 * (1/2 + 1/4 + ...) = 2/2 + 2/4 + ... = 1 + 1/2 + 1/4 + ...

Then subtract S from each side. On the left, 2*S - S = S. On the right, the infinite terms of S cancel all of the terms except 1. So you're left with S = 1. Q.E.D.

Sure, it's easy to discount Zeno's paradoxes since they're so easily refuted with relatively simple math, but the essence of the paradoxes remain. I found the second link in the FPP especially interesting. Consider the series of mirrors in that link. What would happen to the photon in that situation? If you're tempted to discount the whole thing due to quantum mechanics, consider such a construction on a Euclidean plane. What happens to the curve? I've been thinking, and I can't figure it out. If I had my old differential equations textbook with me, I'd take a look at the inverse Lagrange transform of a curve that is zero everywhere except at -1/(n^2) where the curve is a unit spike alternating between up and down.

The second link also has a great examination of the Arrow Paradox. If you haven't read it, do. It's about 2/3s down the page. "It's worth noting that modern physics has concluded (along with Zeno) that the classical image of space and time was fundamentally wrong, and in fact motion would not be possible in a universe constructed according to the classical model." But, it goes on to explain, special relativity resolves this problem quite well.

(As an aside, that article makes the point that not only are Zeno's paradoxes paradoxes in and of themselves (excusing some of them or having been resolved), they create a larger paradox. The paradox of Achilles demonstrates that imagining space to be infinitely divisible creates a paradox. But then the paradox of the Arrow demonstrates a paradox from the assumption that time is discrete. This is Zeno's Paradox: time can apparently be neither discrete nor infinitely divisible. Fortunately for us (and the rest of the universe, I guess), there is an underlying assumption in the Paradoxes about how time relates to space. Removing this assumption (by applying special relativity) resolves the paradox to some extent.)

posted by samw at 12:47 AM on August 2, 2003

As for the sum of 1/2 + 1/4 + 1/8 + ..., there's a pretty easy-to-understand trick to see why this comes to 1:

Suppose S = 1/2 + 1/4 + 1/8 + ...

Multiply each side by 2 to get

2*S = 2 * (1/2 + 1/4 + ...) = 2/2 + 2/4 + ... = 1 + 1/2 + 1/4 + ...

Then subtract S from each side. On the left, 2*S - S = S. On the right, the infinite terms of S cancel all of the terms except 1. So you're left with S = 1. Q.E.D.

Sure, it's easy to discount Zeno's paradoxes since they're so easily refuted with relatively simple math, but the essence of the paradoxes remain. I found the second link in the FPP especially interesting. Consider the series of mirrors in that link. What would happen to the photon in that situation? If you're tempted to discount the whole thing due to quantum mechanics, consider such a construction on a Euclidean plane. What happens to the curve? I've been thinking, and I can't figure it out. If I had my old differential equations textbook with me, I'd take a look at the inverse Lagrange transform of a curve that is zero everywhere except at -1/(n^2) where the curve is a unit spike alternating between up and down.

The second link also has a great examination of the Arrow Paradox. If you haven't read it, do. It's about 2/3s down the page. "It's worth noting that modern physics has concluded (along with Zeno) that the classical image of space and time was fundamentally wrong, and in fact motion would not be possible in a universe constructed according to the classical model." But, it goes on to explain, special relativity resolves this problem quite well.

(As an aside, that article makes the point that not only are Zeno's paradoxes paradoxes in and of themselves (excusing some of them or having been resolved), they create a larger paradox. The paradox of Achilles demonstrates that imagining space to be infinitely divisible creates a paradox. But then the paradox of the Arrow demonstrates a paradox from the assumption that time is discrete. This is Zeno's Paradox: time can apparently be neither discrete nor infinitely divisible. Fortunately for us (and the rest of the universe, I guess), there is an underlying assumption in the Paradoxes about how time relates to space. Removing this assumption (by applying special relativity) resolves the paradox to some extent.)

posted by samw at 12:47 AM on August 2, 2003

This idea basically amounts to space/time being unexpressible as exact "points", doesn't it? You can't have an exact instant of time, only a period. Lines instead of points. Superstrings. I'm reading about the superstring theory in a book published in 1989 at the moment...

posted by Jimbob at 4:00 AM on August 2, 2003

posted by Jimbob at 4:00 AM on August 2, 2003

Wow, samw, when you wait over two years to make a comment, it's a doozy! Well done, sir, and thanks for the elucidation.

posted by languagehat at 7:55 AM on August 2, 2003

posted by languagehat at 7:55 AM on August 2, 2003

I can see from this thread why the world of physics is in such a mess. If this new view of time is sound, and I think it may be, it will help to answer a lot of difficult questions. But just for the moment consider two bits of history: Bohm's 1952 papers giving an alternative hidden variables or ontological interpretation of quantum mechanics, also published, I believe, in the Foundation of Physics letters, were totally ignored by the establishment for many years. In fact, his mentor J. Robert Oppenheimer, said of the papers "If we can't refute Bohm we must agree to ignore him." Many years later these papers were rediscovered and helped lead to an understanding of the EPR paradox and finally to a way of understanding matter in a way that no longer requires a classical limit or a collapse of the wave function. His point was that any theory is simply a way of seeing that may be able to aid our understanding in a way that another theory might not be able to. But any theory is just that, a way of seeing, it is not a statement of fact.

Second, it is worth considering that the past is gone, and the future hasn't happened yet. The present moment, therefore is a space between what doesn't exist and what doesn't exist. So what might it be?

posted by donfactor at 4:05 PM on August 2, 2003

Second, it is worth considering that the past is gone, and the future hasn't happened yet. The present moment, therefore is a space between what doesn't exist and what doesn't exist. So what might it be?

posted by donfactor at 4:05 PM on August 2, 2003

* slowing drum roll, lights dim *

posted by troutfishing at 9:04 PM on August 4, 2003

posted by troutfishing at 9:04 PM on August 4, 2003

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According to Lynds, the absence of an instant in time and determined relative position, and consequently also velocity, necessarily also means the absence of all other precisely determined physical magnitudes and values at a time, including space and time itself.

posted by Pinwheel at 7:22 AM on August 1, 2003