1. ## Zeno's paradox and infinity

I was thinking of asking this in questions, but it's not really astronomy related (though definitely space related, depending on how you define the word!)

I've long been curious about some of these paradox. In particular, the one about the arrow never being able to go from point A to point B, if space is continuous. There are an infinite number of points in between, so it should never get there.

But actually, I was thinking about it, and came up with this. Well, if there is infinite space, there is also infinite time. So could you move an infinite distance in infinite time? I don't really know. What is perplexing, though, is that if the space is doubled in length, the time required also doubles. So it seems like you can multiply infinity by two, and it actually takes two infinities to travel it, or something like that.

Isn't it strange? If space is continuous, how does anything manage to happen?

2. Originally Posted by Jens
So could you move an infinite distance in infinite time?
What does it mean to move an infinite distance?

If you travel an infinite distance, then travel one additional day, then how far will you have gone in total?

3. I'm sorry, by "infinite" distance I meant "over an infinite number of points," since any distance can be sub-divided infinitely.

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5. Not necessarily fortunately.

The one I brought up above (I just looked it up) is apparently called the "dichotomy paradox", which is apparently a problem with a non-discrete universe.

If the units are discrete, then there is another paradox, which seems actually more intractable to me: the "arrow paradox." Suppose that time takes place in discrete increments. And then you have two arrows, one moving and the other at rest. If you look at any one of those discrete increments, what will look different about the two? If nothing looks different (since it is a moment of time), then how does one know that it should move and the other that it should stay still?

One solution that occurs to me is that there is a projector somewhere, and the projector is mapping the motion onto a screen. The problem is, how does the projector move?

6. Even if space and time are continuous, there's still no paradox, really.

Let's get our use of "infinity" straight. "Infinity," for our purposes, will be a very big number... something bigger in magnitude than all the real numbers. "Infinitesimal" will refer to something smaller in magnitude than any real number (but bigger than zero). We'll also say that you can multiply infinitesimal by infinity to get some real number.

Say you want to move one meter in one second. Now, Zeno might argue that you can never get there, because first you have to go half a meter, et cetera. The answer is that it only takes half a second to go that half meter. It only takes a quarter second to go a quarter meter. And so on, down to infinitesimal times and distances.

In fact, by the time you get down to infinitesimal distances (dividing by 2 infinity times), you're also dealing with infinitesimal lengths of time. In essence, it takes no real amount of time to move that tiny (unreal) distance. And if you move that unreal distance infinity times, then you get a real distance... and it takes infinity times infinitesimal = some real number amount of time.

Looking at it in slightly mathy terms, you move 1/infinity meters in 1/infinity seconds. Those are both infinitesimals. And when you solve for how long it takes to go 1 meter, you get (1/infinity seconds)/(1/infinity meters) x 1 meter = (1 * infinity seconds / infinity * 1 meters) x 1 meter = (1 second/meter) x 1 meter = 1 second, just as predicted. The infinities cancel nicely.

(The assumption is that those are the same infinity, of course. Properly, you'd do that with limits....)

So yes, you can certainly move infinite distance in infinite time, but even more importantly, you can move infinitesimal distance in infinitesimal time. If it took real time to move an infinitesimal distance, then there would be a problem. But as it stands, the infinities cancel out: the time gets shorter at some real multiple of the rate that the distance gets shorter, so the two reach infinitesimal at the same time.

7. Originally Posted by Jens
then how does one know that it should move and the other that it should stay still?
That's what physics is for. The glib answer is "conservation of momentum." But I really have no idea of what momentum "really" is or why it ought to be conserved.

8. The thing that's difficult about this for me is that, for example, when I look at the Wikipedia article, it seems to be saying something like: some people believe it is not a problem, but others think it is a problem. It doesn't make it seem like there is a real consensus on whether the paradoxes are problems at all.

9. Can you move? Yes? Then no problem.

The solution to the other paradoxes is almost always related to the solution I just gave. For instance, take Thomson's lamp (which I note is linked from the wiki entry). Is the lamp on or off after 2 minutes?

The question has no answer because it does not correspond to reality. Mathematically speaking, it has no answer because we are dealing with a divergent sequence. But there's no paradox with reality because you can't map the abstract mathematical problem to real life. You can't map the problem because it takes a finite amount of time to flip the switch.

See the analogy? In Zeno, the paradox arose because of the assumption that it would take a finite amount of time to move an infinitesimal distance. In Thompson, the paradox is because of the assumption that it would take an infinitesimal amount of time to flip the switch (akin to moving a finite distance).

Fortunately that doesn't happen in the real world, and the lamp ends up either on or off because at some point, you just can't flip the switch any faster.

I suppose that one could say that our universe enforces the law that finite change can occur only in finite (not infinitesimal) time. The somewhat complementary Zeno version is that motion occurs only when infinitesimal change is effected in infinitesimal (not finite) time.

10. Even though I agree with snarkophilus' explanation for how these paradoxes may be avoided with better mathematics, to me, what these so-called paradoxes really do is illustrate an extremely important principle of science, which has to do with the direction of flow of information. Does it make sense to come up with a concept, like space and time, and then try to reason from that concept about how the "real world" works? That's what is being done in these "paradoxes". But this is not what science is-- information in science flows in the opposite direction: it flows from the "real world" to the concepts we generate to try to understand it. So if you talk about motion, the question is not, "if space and time are this way or that way, how does motion happen", the question is, "given that motion happens the way we observe it, how can we conceptualize space and time to make sense of it". When framed this way, we see that the entire concept of "paradox" is falsely applied-- you either have an effective picture of how space and time are working, or you don't, in any given context. That is, the visualization you are using either works well to answer a particular question, or it doesn't, but it's never a source of bother beyond noting that a more sophisticated treatment might be needed to address more profound questions. There certainly is no way to generate anything paradoxical-- that would require confusing the reality for the concepts we generate to describe it.

11. This is the thing, though. Take a motion picture, for example. We see motion. But there really isn't motion (meaning motion in the movie itself). It's just a series of photographs, and we see it as motion due to our own perception. So I sort of wonder if Zeno's paradox could point to the possibility of a similar phenomenon taking place with our reality. I'm not sure how that impact on Ken's last statement. But we certainly should not start out with our perception of a moving picture, and assume from the reality that is presented, without looking at the process behind it.

12. Originally Posted by snarkophilus
In Zeno, the paradox arose because of the assumption that it would take a finite amount of time to move an infinitesimal distance.
More precisely, it arose from Greek mathematicians' mistaken belief that an infinite number of ever-smaller numbers could never add up to a finite number. Which is simply not true -- the series 1/2 + 1/3 + 1/4 + 1/5 + ... adds up to infinity, but the series 1/2 + 1/4 + 1/8 + 1/16 + ... adds up to a finite number, 1 to be exact. Once you accept that fact, all of Zeno's Paradoxes disappear.

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Jens,

To me, Zeno's paradox is fun because it shows how thinking logically
can give wrong answers!

It is a paradox only to someone who assumes that moving through
"an infinite number of points" (whatever that means) requires an
infinite amount of time. Or to put it another way, moving through
n points requires n seconds. There is actually no connection
between the number of points and the amount of time.

If it takes one second to move a distance of one meter, it takes
one second to move through all those points. It doesn't matter
if the number of points is 100 or 100 googolplex.

Some time ago I wrote the following for a web page that is still
unfinished. I want critical feedback.
Simple Answers to Eternal Questions

Zeno's Paradox: Can Achilles ever catch the tortoise?

Zeno asked: If Achilles and the tortoise run a race, and the
tortoise is given a head start, then Achilles must run for some
time before he reaches the point where the tortoise was when
they started. By then, the tortoise has moved ahead some
distance. Achilles must then run for an additional time to
reach that point. Again the tortoise will have moved further
ahead. And so on, without end. How can Achilles ever catch
up to the tortoise?

No problem.

Every increment is a smaller distance and shorter length of time
than the one before. The endless number of ever-smaller distances
add up to the distance from Achilles' starting point to the point
where he passes the tortoise. The endless number of ever-shorter
time periods add up to the time it takes Achilles to catch up to the
tortoise. Zeno's analysis simply looks at smaller and smaller pieces
of the interval remaining just before Achilles passes the tortoise,
and avoids ever looking at that point or beyond.

The process of dividing something into more and more pieces which
become ever smaller is the basis of calculus, invented by Isaac Newton
and Gottfried von Leibniz more than 2000 years after Zeno's time.
Calculus is needed to accurately describe changes in a thing when the
rate of change is itself changing. Although Zeno's paradox seems
rather silly, it introduced a very powerful mathematical idea
essential to the development of modern technology.
Is that all correct? Is it clear? Did I leave out something
important? I don't want to get into the question of discrete
versus continuous space and time, though.

-- Jeff, in Minneapolis

14. The apparent paradox is what happens when the way we look at things is too one-sided. We could say that it should be impossible to travel through an infinite number of points for any finite distance. But we could also say that since each point would then be infinitesimal, we should travel each point instantaneously, so that travelling through any number of such points should also be instantaneous, regardless of how far we've actually travelled. So if we are instantaneously travelling through an infinite number of points of zero dimension, how far can we possibly get? Well, zero distance divided by zero time gives us 0/0 (or (1/infinity)/(1/infinity), analogous to snarkophilus' post earlier). 0/0 gives us any real number, since any number can be multiplied by zero to give us zero (0/0=x, x*0=0), and we can therefore travel at any finite speed through any finite distance.

In fact, the idea of infinitesimal points is only a mathematical convenience anyway, as with expressing the gravitation of a spherical body as the representation of a single point. This is only possible because the actual radius of the sphere does not figure into the formula for gravity. Similarly, the idea of space as the compilation of infinitesimal points in a "connect the dots" sort of way is not valid. It is much more than that. Points really only signify a beginning and an end, as with boundaries and such. What lies between is something very different.

I'm not sure how it should really be thought about, though. I try mental experiments like this a lot. For instance, instead of just empty space, let's think about something substantial, like particles. We know that bodies are made up of atoms, which are made up of particles. Those particles can also be broken down into more particles, so what would the ultimate particle be? It would have to be something that cannot be broken down any further, so infinitely rigid. But if it is rigid, then it has an edge. If it has an edge, then it has structure. If it has structure, then something else must form this structure, so it is not the ultimate particle. If it has no edge, then it's concentration must taper off, and it is penetrable, both of which means it still has internal structure. If it is pliable, as a pure energy or a wave, then its structure changes, and so it must still be comprised of some more basic substance in order to do so, which would itself undergo the same dilemma.

It would appear, then, that our ideas about everything having a beginning and an end, whereby we define points in space and time, and that everything is built up something fundamental, may not be the case. It may be more in compliance to the way our brains operate, by starting with the most basic concepts we can comprehend and building up from this, with a beginning point and an end, while in reality no such boundaries might even exist.

15. Originally Posted by Jens
I'm not sure how that impact on Ken's last statement. But we certainly should not start out with our perception of a moving picture, and assume from the reality that is presented, without looking at the process behind it.
That's it exactly. Our goal is to make sense of what we see so that it works for us, and we can watch a movie. But we don't think it is 'paradoxical' if we later discover inaccuracies in the way we formed our picture-- we study the mechanism and build deeper pictures of what is going on that work on more levels. The approach Zeno is using is contrary to the way we do science-- it reverses the order of the logic. If Zeno came to see a modern movie, would he find it "paradoxical" that the objects "knew" where to go, even though each frame was just a static image? It's all a question of probing the mechanism that allows them to move-- in this case, the rules of how the movie gets made. Maybe he was just trying to say there is more going on than we understand-- that would indeed be an admirable and scientific recognition.

16. Originally Posted by Jeff Root
Did I leave out something
important? I don't want to get into the question of discrete
versus continuous space though.
But I think that's the whole issue-- I think that's what Zeno was concerned with. Even if one knows the mathematics of infinite sums, as Ilya correctly pointed out, you still need time to be perfectly subdividable to use those mathematics. We pretty much already know that time is not so subdividable (the uncertainty principle). My point is that the basic error is in trying to use a concept, like time, to learn about how reality works. That's backward-- we use reality to learn about how we need to make a functioning concept of time, and the function we achieve should never be expected to extend beyond where we have measured it. So Achilles does catch the tortoise, and the mathematics of infinite sums tells us just how long it will take, but it is not necessary for the details of what is happening be taken too literally-- mathematics applied to the real world is just a tool, it is not the same thing as the real world.

17. Originally Posted by grav
Points really only signify a beginning and an end, as with boundaries and such. What lies between is something very different.

I'm not sure how it should really be thought about, though.
Exactly-- no one does. There are a lot of scientific thinkers who have forgotten this basic truth, and have forgotten what science is as a result.

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Previous discussion.

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

19. Originally Posted by grav
It would have to be something that cannot be broken down any further, so infinitely rigid. But if it is rigid, then it has an edge. If it has an edge, then it has structure.
I don't know... does a singularity have an edge? It's just a point.

I am under the impression that all fundamental particles are just points, really. We just measure sizes for them because of the strength of the forces we necessarily use for those measurements.

For that reason, I don't think it's fair to say that what's between two points is somehow different from the two points themselves. Also, the mathematics would break down if that were the case.

20. Originally Posted by snarkophilus
I am under the impression that all fundamental particles are just points, really. We just measure sizes for them because of the strength of the forces we necessarily use for those measurements.
This comment exposes just the kind of commonly altered view of what science is that I was talking about above. It's no fault of yours, this is very often the way science is expressed and talked about, but it is not what modern science is, and has been since Galileo. If you analyze your statement, you'll see that what you are in effect doing is taking a concept ("point") and saying that its "reality" trumps the measurements-- the measurements are somehow at fault. But that reverses the logic of modern science. We use concepts to understand measurements-- not the other way around!

21. Originally Posted by Disinfo Agent
Great Russell quote, I think he might have been referring to the way Zeno is exposing the limitations of mathematical thought about reality. If so, the point Zeno is making is not that we can't use mathematical thinking to gain understanding of how reality functions, but simply that we should never take that understanding too seriously, because if we do, we stumble into the limitations of our own words and concepts. I just wouldn't call that "paradoxical"-- to me we should always have expected that!

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Originally Posted by Ken G
We use concepts to understand measurements-- not the other way around!
This I think is a very concise statement that has so much relevance to what science is about. It seems a far more satisfactory approach to define concepts in this way, rather than giving a "reality" to them as if they somehow exist as entities in themselves that are independent from the act of observation/measurement.

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Zeno might also be illustrating that you can only get so far in math before you have to come to terms with the ugly fact that logarithms are different from polynomials.

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Originally Posted by Ken G
Great Russell quote, I think he might have been referring to the way Zeno is exposing the limitations of mathematical thought about reality. If so, the point Zeno is making is not that we can't use mathematical thinking to gain understanding of how reality functions, but simply that we should never take that understanding too seriously, because if we do, we stumble into the limitations of our own words and concepts. I just wouldn't call that "paradoxical"-- to me we should always have expected that!
Not that I'm an expert on ancient Greek philosophy or anything, but what's often argued is a little more subtle than that. There was an ongoing debate among Greek philosphers about the nature of reality. One thing which concerned them from early on was the dichotomy between "appearance" and "truth", or if you prefer between empiricism (what we feel and touch is what there is) and idealism (genuine knowledge can only be attained through reason; our physical sensations are misleading). Sound familiar?

As you have probably guessed, Zeno was part of an idealist school, the Eleatics. They are said to have believed that the real world behind the curtain of our sensations was quite different from what we experience, and in particular that all change was an illusion. Well, movement is a kind of change. Zeno probably came up with these paradoxes to confound his empiricist opponents, suggesting that the very concept of motion led to logical (and/or physical) contradictions.

I am not persuaded that his logic is airtight, but I have to respect an argument which has made so many generations stop and scratch their heads for a while.

25. Originally Posted by Len Moran
It seems a far more satisfactory approach to define concepts in this way, rather than giving a "reality" to them as if they somehow exist as entities in themselves that are independent from the act of observation/measurement.
Yes, quite so-- and the very definitions of modern science, which is founded on empiricism, supports your view.

26. Originally Posted by Disinfo Agent
As you have probably guessed, Zeno was part of an idealist school, the Eleatics. They are said to have believed that the real world behind the curtain of our sensations was quite different from what we experience, and in particular that all change was an illusion. Well, movement is a kind of change. Zeno probably came up with these paradoxes to confound his empiricist opponents, suggesting that the very concept of motion led to logical (and/or physical) contradictions.
That's quite interesting, thank you. It sounds now like he was saying that if you try to break motion up into measureables, you can't make sense of it, so it has to arise from something more ideal. I would note that the entire debate between empiricism and idealism is missing the possibility (or if you ask me, likelihood) that neither might be correct-- that scientific understanding of natural phenomena might require a kind of synthesis of both approaches (as used by modern science), while the "truth" might involve neither (as both are constructs of limited human intelligence).

And on the matter of trying to decide which philosopher was "right", I think it is far more productive to think of philosophy like the way we think of zoology-- the highest goal of philosophy is to label and explore the characteristics of all possible ways of thinking about a problem. That process has value, independently from the identification of which one is "correct"-- I doubt the latter is possible at all.

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Originally Posted by Ken G
-- the highest goal of philosophy is to label and explore the characteristics of all possible ways of thinking about a problem.
There is an old joke more or less to this effect.
The head of the Physics department approaches the university principal for more funding; various expensive pieces of equipment are required if the Physics department is to keep up with other centres.
"But why can't you be more like the Mathematics department?" asks the principal. "All they ever ask for is paper, pencils and erasers. Or Philosophy: they just ask for the paper and pencils."

Grant Hutchison

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Originally Posted by Ken G
I would note that the entire debate between empiricism and idealism is missing the possibility (or if you ask me, likelihood) that neither might be correct-- that scientific understanding of natural phenomena might require a kind of synthesis of both approaches (as used by modern science), while the "truth" might involve neither (as both are constructs of limited human intelligence).
Yes. I think that was the great conceptual breakthrough which eventually led to modern science: stop trying to find absolute answers about methodology, but see how far you can tentatively go, using all methods at your disposal. This implied a loss of certainty (which nowadays is sometimes forgotten), but turned out to be so much more fruitful in the end!

Originally Posted by Ken G
And on the matter of trying to decide which philosopher was "right", I think it is far more productive to think of philosophy like the way we think of zoology-- the highest goal of philosophy is to label and explore the characteristics of all possible ways of thinking about a problem. That process has value, independently from the identification of which one is "correct"-- I doubt the latter is possible at all.
I very much agree. There is a tendency among certain modern authors, especially those who come from a science background, to read the history of philosophy as though it were the history of a scientific discipline; where you get one thinker who says something, and then another one comes along and says "Wait, that's not how it is..." and proves him wrong, and then a third thinker proves the second wrong... and that's amazingly naive. It's just not how philosophy works!

You can take words that Plato or Aristotle wrote millennia ago, and they're still relevant. Of course, these modern "scientological" retellers rarely dare to claim that Plato has been "shown wrong". Sometimes they say it about Aristotle, with some reason because he did dabble in the natural sciences, and there he made many mistakes. But it's mostly more modern philosophers which they claim to be able to refute. For me, refuting a point of view seems like a rather peculiar notion.

29. Originally Posted by Ken G
If you analyze your statement, you'll see that what you are in effect doing is taking a concept ("point") and saying that its "reality" trumps the measurements-- the measurements are somehow at fault.
I do not think that it is. What I was trying to express is that experiments have shown that the most fundamental particles don't have a size or internal structure. How long is a top quark? Well, it doesn't have a measured length. What's the measured radius of an electron? It's very close to zero, at most.

I understand the distinction you are making, and it is a very important one. I've made it already in this thread (albeit in a slightly different form). But I think it's also important to note that mathematics is devised to correspond to reality, and it happens quite often that the generalizations we create using that math can predict new measurements.

For instance, while you can't say for certain that an electron really is smeared out across space the way its wavefunction suggests, that's a really useful way of thinking about it. It's how the electron behaves, so in that sense, that smearing is how the electron really exists. I usually don't see the need to make that distinction because it's implicit that all of the world is that way. And very often it's useful to think of the electron as a gas of varying density (or whatever else your mathematical model might suggest), because you can gain new insights into its behaviour from that -- and then do more experiments.

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Zeno's paradox, as classically described, implies completely continuous space and time. Its purpose is to prove not that space or time is discrete, but that motion is an illusion to begin with. Some people still hold to that idea, but for this explanation, I'll say that motion is not an illusion.

Zeno formulated his paradoxes without having any knowledge of calculus, which shows us that the sum of an infinite series can converge to a finite value. Example: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ..... = 2. The use of this information resolves the paradox, as we are not compelled to count by powers of 1/2 to reach 2, even though 2 is the sums of the powers of 1/2. We can just count to 2 in the normal manner, and be confident that we really did get there.

Now, this is really only a geometric problem; that is, it is pure math, not necessarily involved in physical reality. As such, it does not prove that our space is continuous. However, it does show that our space could be continuous without encountering the paradox in question.

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