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Chris S
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Zeno's paradox claims that you can never reach your destination or catch up to a moving object by moving faster than the object because you would have to travel half way to your destination an infinite number of times. The two conflicting elements in this paradox are: 1) We do reach destinations; and 2) we can't travel an infinite number of half way points to a destination. We can't prove that one element is incorrect by citing the other element (e.g., "I can travel infinite half way points because I actually do reach my destination."] This statement non-sensically states that one of two conflicting apparent facts is false because the other apparent facts is an apparent fact. We already know that both elements are apparent facts, and restating that one of them is an apparent fact doesn't demonstrate which element is false. To solve the apparent paradox, we must find the flaw in one of the elements. The flaw is in the notion that we can't travel an infinite number of half way points.

While it is true that there are an infinite number of "half way" points between a starting location and a destination at another location, these half way points are measurements of distance. Although there are an infinite number of measurements within a total distance, the total distance is not infinite. We can travel an infinite number of smaller measured distances while traveling a total distance that is not infinite. The task does not require us to travel an infinite distance. Thus, there is no actual paradox at all. The most important characters related to Zeno's apparent paradox are not the tortoise and its friends; it's the "elephants in the room": the orange and apples that shouldn't be compared.

Delta2

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In your formulation we can reject the paradox simply by rejecting premise 2:

2) we can't travel an infinite number of half way points to a destination.

There is no reason to accept that assertion. And if we did, why not just accept the simpler assertion that we can't traverse an infinite number of points in a finite amount of time? If we accept that assertion then any motion is impossible because any finite distance traversed corresponds to a real interval, which contains an infinite number of points.* There's no need to complicate things by talking about midpoints of hierarchically nested intervals, as Zeon does.

This all just stems from a dislike that many ancient greeks had for the concept of infinity. Mathematicians have long-since learned to accept the concept of infinity and develop ways to discuss it meaningfully, via people like Poincare, Weierstrass and Cantor.

Not all the ancients feared infinity. Lucretius (Stoic philosopher of ancient Rome) had an argument involving a javelin as to why the universe must be infinite (see here). The argument fails when you allow curvature of space (not a concept they had back then), but the point remains that he seemed unfazed by space being infinite.

* Indeed, the same argument can 'prove' that not only is it impossible to move, but it is also impossible for time to pass.

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Chris S
In your formulation we can reject the paradox simply by rejecting premise 2:

2) we can't travel an infinite number of half way points to a destination.

A paradox cannot be solved merely by choosing to reject one of the two elements. You must find a valid reason (logical flaw) in one of the two elements in the paradox.

". . . why not just accept the simpler assertion. . ."

Again, paradoxes can't be solved by deciding which element you like, and the simplicity of one element does not demonstrate the flaw in one element or prove the other element correct.

"This all just stems from a dislike that many ancient greeks had for the concept of infinity."

Zeno did not imply a dislike for the concept of infinity with the promulgation of his paradox. On the contrary, he gave infinity equal validity with the powerful logic that destinations can be reached in order to create the paradox. If Zeno didn't give infinity equal validity to the other element in the paradox, no paradox would exist.

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@Chris S Excluding a few special cases of interest only to metaphysicians, there are two sorts of paradoxes: veridical and falsidical. You can read about them here.
You appear to be presenting Zeno's argument as a falsidical paradox, ie as though it formally deduces a contradiction. So the onus is on you (or on Zeno) do present a formally valid logical argument that results in a contradiction.

The OP does not do that, but it appears that its argument takes statements (1) and (2) as premises (aka 'axioms' or 'postulates'), and then implies that one can deduce a contradiction from them - although it does not present the formal steps by which such a contradiction is reached.

Rejecting a premise when it leads to a contradiction is what mathematicians do. Most of mathematics would disappear if we did not do that. This practice only causes a problem when the premise we have to reject is one that seems intuitively true to us, so that we really, really don't want to have to reject it. An example of that is Russell's Paradox, a falsidical paradox that forced us to reject the axiom of unrestricted set comprehension, which kids are taught to use to define sets in school, so people really didn't want to have to reject it. But they had to, and did, so we got Zermelo-Frankel set theory, and the newer alternatives, instead.

So what I am doing is rejecting one of the OP's premises, saying I find it unintuitive and unnecessary. If you want to resurrect Zeno's paradox, you need to present a new logical version of it that uses premises that are more intuitively appealing - that people will want to accept. But there is no such version, because any rendition of Zeno's paradox will have in it somewhere - often hidden - a premise like the one I suggested above, or that every set of real numbers has a least element, and most mathematicians would regard such premises as unintuitive and unappealing.

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Zeno's paradox claims that you can never reach your destination or catch up to a moving object by moving faster than the object because you would have to travel half way to your destination an infinite number of times. The two conflicting elements in this paradox are: 1) We do reach destinations; and 2) we can't travel an infinite number of half way points to a destination. We can't prove that one element is incorrect by citing the other element (e.g., "I can travel infinite half way points because I actually do reach my destination."] This statement non-sensically states that one of two conflicting apparent facts is false because the other apparent facts is an apparent fact. We already know that both elements are apparent facts, and restating that one of them is an apparent fact doesn't demonstrate which element is false. To solve the apparent paradox, we must find the flaw in one of the elements. The flaw is in the notion that we can't travel an infinite number of half way points.

While it is true that there are an infinite number of "half way" points between a starting location and a destination at another location, these half way points are measurements of distance. Although there are an infinite number of measurements within a total distance, the total distance is not infinite. We can travel an infinite number of smaller measured distances while traveling a total distance that is not infinite. The task does not require us to travel an infinite distance. Thus, there is no actual paradox at all. The most important characters related to Zeno's apparent paradox are not the tortoise and its friends; it's the "elephants in the room": the orange and apples that shouldn't be compared.
Actually, a sharper version of Zeno's paradox is that ##1s## can never pass. You don't need to include the concept of motion, but may just focus on the passage of time:

##\frac 1 2 s, \frac 3 4 s, \frac 7 8 s \dots##.

There is an infinite sequence of times between ##0s## and ##1s##. This sequence never reaches ##1s##, so ##1s## never arrives.

Although, actually, I quite like an even sharper version:

##0s, 0s, 0s, \dots##

You can write down ##0s## an unlimited number of times, so there is nothing that obliges you to write down a time after ##0s##.

##0s, -1s, -2s \dots##

If you describe the events at those times, could you conclude that time moves backwards?

Looking at all these examples, one resolution to the paradox is that nature is not obliged to obey anything that you can write down. Instead, what you write down must obey what nature does.

WWGD and Lnewqban
Chris S
@Chris S Excluding a few special cases of interest only to metaphysicians, there are two sorts of paradoxes: veridical and falsidical. You can read about them here.
You appear to be presenting Zeno's argument as a falsidical paradox, ie as though it formally deduces a contradiction. So the onus is on you (or on Zeno) do present a formally valid logical argument that results in a contradiction.

The OP does not do that, but it appears that its argument takes statements (1) and (2) as premises (aka 'axioms' or 'postulates'), and then implies that one can deduce a contradiction from them - although it does not present the formal steps by which such a contradiction is reached.

Rejecting a premise when it leads to a contradiction is what mathematicians do. Most of mathematics would disappear if we did not do that. This practice only causes a problem when the premise we have to reject is one that seems intuitively true to us, so that we really, really don't want to have to reject it. An example of that is Russell's Paradox, a falsidical paradox that forced us to reject the axiom of unrestricted set comprehension, which kids are taught to use to define sets in school, so people really didn't want to have to reject it. But they had to, and did, so we got Zermelo-Frankel set theory, and the newer alternatives, instead.

So what I am doing is rejecting one of the OP's premises, saying I find it unintuitive and unnecessary. If you want to resurrect Zeno's paradox, you need to present a new logical version of it that uses premises that are more intuitively appealing - that people will want to accept. But there is no such version, because any rendition of Zeno's paradox will have in it somewhere - often hidden - a premise like the one I suggested above, or that every set of real numbers has a least element, and most mathematicians would regard such premises as unintuitive and unappealing.

I have provided my conclusion and rationale I believe resolves the apparent contradiction within the cited paradox, establishing this thread's topic: whether my conclusion and reasoning are correct or not. You are welcome to agree or present flaws in my logic and use mathematics if you wish; however, restating that you get to reject one of the contradictory elements merely based on the fact that one element appears "unintuitive and unappealing" to you does not demonstrate whether my conclusion or reasoning is flawed nor, more importantly, how it is flawed. Perhaps a mathematical analogy will be relatable to you: One side of an equation cannot be deemed unequal based on an observer believing that it is unintuitive and unappealing because mathematics does not use an observer's preference.

PeroK, Although I agree with your statement: (". . . nature is not obliged to obey anything that you can write down. Instead, what you write down must obey what nature does") it does not resolve the apparent paradox because it does not determine which of the contradictory elements is flawed. Nice quote though.

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PeroK, Although I agree with your statement: (". . . nature is not obliged to obey anything that you can write down. Instead, what you write down must obey what nature does") it does not resolve the apparent paradox because it does not determine which of the contradictory elements is flawed. Nice quote though.
Zeno's paradox is only kept alive by those who do not understand or choose not to learn modern mathematics. It has nothing to do with elephants, apples or oranges!

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jbriggs444 and phinds
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We can travel an infinite number of smaller measured distances while traveling a total distance that is not infinite.
This is simply your opinion which allows you to reject one of the elements. You have exhibited no particular logic.
A paradox cannot be solved merely by choosing to reject one of the two elements. You must find a valid reason (logical flaw) in one of the two elements in the paradox.

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This is simply your opinion which allows you to reject one of the elements. You have exhibited no particular logic.
If we accept that $$\sum_{n = 1}^{\infty} \frac {1}{2^n} = 1$$ then we can move on. There are still questions about why there is an arrow of time, but it's not Zeno's paradox that poses the real question for modern physics.

In a nutshell, Zeno assumed time to be continuous (infinitely divisible) but didn't have the knowledge of analysis or calculus to deal with that. Alternatively, if you assume time is discrete, then the solution is even simpler.

*now*, hutchphd and Lnewqban
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If we accept that $$\sum_{n = 1}^{\infty} \frac {1}{2^n} = 1$$ then we can move on.
Exactly.

Chris S
PeroK, Logic, rationalism, empiricism, and math have been successful methods for distinguishing between fact and fiction for millennia. Understanding only the later does not exclude the others from their ability and prevents discussion with those who are using them. Long live apples, oranges, and elephants!

Hutchphd, The logic is within my opinion. The logic that an infinite number of halfway measurements does not prevent traveling the distance to the destination is more logical than Zeno's logic that an infinite number of halfway measurements prevents traveling the distance to the destination. Zeno's notion that an infinite number of halfway measurements require traveling an infinite number of distances is more than "less logical"; it is false on a rational basis. Rationalism, logic, empiricism can be legitimate methods for determining fact from fiction and should not be dismissed as mere opinion just because there's no math involved. Although math is often a better method, it is not the only method and is also imperfect, subject to evolution. Birds fly; People die; and an infinite number of measurements between a given distance does not mean the distance is infinite. I can't claim otherwise by claiming those who assert these things are merely asserting opinions because there's no math. You don't require math for anything to be irrational even though math can also be a basis for describing or determining fact & fiction.

I hope a mathematical version of "infinite measurements between two points doesn't mean there is infinite distance between the two points" has arisen, or will arise, so everyone will be happy. In the meantime, I will espouse the logic on a rational basis until demonstrated otherwise. My hope was to have my logic tested to see if my solution to Zeno's paradox was correct, but we have been stuck on whether math is the only method of determining logical flaws. I thought the concept of infinite measurements within a given distance would be understood without having to provide and discuss math to discuss it.

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PeroK, Logic, rationalism, empiricism, and math have been successful methods for distinguishing between fact and fiction for millennia. Understanding only the later does not exclude the others from their ability and prevents discussion with those who are using them.
I can certainly pick you up there on a logical faux pas.

I said that Zeno's paradox is only kept alive by those we do not understand modern mathematics. That does not imply that you only need mathematics. In other words, you can only be fooled by Zeno if you know too little modern maths.

In general, if someone has no understanding of mathematics, they can look rather foolish when discussing things like the infinite division of a continuous variable.

Chris S
PeroK, Does the math that has been used against Zeno's paradox describe my perception that infinite measurements between two halfway points does not necessitate infinite distance between the two points?

Are you implying that the perception that infinite measurements between two halfway points not necessitating an infinite distance between the two points is foolish? [I am questioning the criticism of the perception, not any criticism of me personally.]

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PeroK, Does the math that has been used against Zeno's paradox describe my perception that infinite measurements between two halfway points does not necessitate infinite distance between the two points?
Yes, that's what those funny symbols mean:$$\sum_{n = 1}^{\infty} \frac {1}{2^n} = 1$$

The point is that in the 19th Century this sort of mathematics (called analysis) put calculus on a rigorous basis. This isn't a perception or some intuitive reasoning. Prior to that mathematicians had the notion that the above equation was true in some sense. The ideas of mathematical analysis put those notions on a firm foundation - in terms of what could be proved and what couldn't.

The other point worth making is that Zeno and the other ancient Greeks, however brilliant they were, did not have the understanding of the relationships between nature and mathematical models of nature. That's also part of the basis on which Zeno's paradox is no longer "a thing" in physics. The Greeks would never had analysed what they were doing in that way. These ideas date from Newton, really.

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PS in fact the simpest refutation of Zeno is that he simply postulated an invalid mathematical model of nature. One in which time (or motion) is impossible.

Although, it's more accurate to say that the model was okay, but Zeno didn't understand the mathematics that underpinned his model. I.e. he needed calculus (developed independently by Newton and Leibnitz in the 17th Century).

Chris S
PeroK, My goal of receiving feedback that confirms or denies my logic based perception has been accomplished by learning of the math based confirmation discovered centuries before I did. Next, I will invent a solution for turning oranges into juice. Thanks.

Delta2 and PeroK
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Dammit, while I was typing the thread ended

Actually, in physics it is not a paradox...

The logical refutation is that the second premise is false : there's no reason given for not being able to traverse an infinite number of points. T&F=F, so there's no "paradox"(which is defined as requiring more than one true statement (which is the bit you missed) ).

The mathematical refutation - if you choose to disambiguate math from logic - is (eventually) "Dividing by zero is undefined" which - thanks to memes and the propogation of basic arithmetic education - is much better known than what is effectively a 2,500 year old troll.

The philosophical refutation is that, while "infinity" is indeed a philosophical concept, ##1/2## is very definitely not : the "paradox" isn't philosophical in nature, see "mathematics".

Unless you mean "plurality", which is probably a concept in the philosophical realm ; I think it would be part of "set theory" in math.

But, in physics the speed of the measurer with his knotted string becomes so fast that Cherenkov radiation fries the traveller.

Who'da thunk.

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Delta2
The quickest way to go around Zeno's paradox: After 20 seconds Achilles is 200m from the starting line and the turtle is 120m from the starting line. So: After 20 seconds Achilles has passed the turtle with a good margin.

Oh - you wanted the exact time Achilles passed the turtle? That is not part of the paradox, but can be solved using only basic algebra,

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Zenon's paradox was in an era before infinite sums and integrals had been studied.

Now days anyone that has understand converging infinite sums (series) can resolve Zenon's paradox: We have infinite terms to sum (##\sigma=\sum_{n=0}^{\infty} a_n,##) but each term ##a_n## gets infinitely smaller (or in modern language ##\lim_{n\to \infty} a_n=0## so the total sum ##\sigma## can be finite).

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Zeno's paradox claims that you can never reach your destination or catch up to a moving object by moving faster than the object because you would have to travel half way to your destination an infinite number of times. The two conflicting elements in this paradox are: 1) We do reach destinations; and 2) we can't travel an infinite number of half way points to a destination. We can't prove that one element is incorrect by citing the other element (e.g., "I can travel infinite half way points because I actually do reach my destination."] This statement non-sensically states that one of two conflicting apparent facts is false because the other apparent facts is an apparent fact. We already know that both elements are apparent facts, and restating that one of them is an apparent fact doesn't demonstrate which element is false. To solve the apparent paradox, we must find the flaw in one of the elements. The flaw is in the notion that we can't travel an infinite number of half way points.
Part of the problem here is the second premise as you stated it is incomplete. You state one "can't travel an infinite number of half way points to a destination." "Why not?" is the obvious question.

If the implication is that an infinite number of half way points implies an infinite distance, which is the claim you considered, the premise is obviously false since everything isn't an infinite distance from where you started, so the premise can therefore be rejected out of hand.

If the implication is that traveling an infinite number of distances would take an infinite amount of time, you didn't really address this claim. This is related to the paradox that claims the hare can't catch up to the tortoise because by the time the hare gets to where the tortoise was, the tortoise has moved on.

Finally, the assertion could be that you simply can't complete an infinite number of tasks (as opposed to a finite number of tasks). I think someone who holds this position would reject your argument saying you begged the question.

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Finally, the assertion could be that you simply can't complete an infinite number of tasks (as opposed to a finite number of tasks). I think someone who holds this position would reject your argument saying you begged the question.
Then, you reject the continuum model, which is what Zeno should have done. You can't have it both ways: a continuum model for time with an infinite number of points and infinite indivisibility; and, an rejection of that hypothesis (by assuming that there cannot be an infinite number of instants). It's one or the other.

If Zeno had concluded that time must progress in finite discrete intervals (rather than be impossible), then that would at least be consistent.

russ_watters and sysprog
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Zeno's paradox is the perfect excuse to not have to do what you don't want to do because you simply can't.

Delta2 and russ_watters
Jarvis323
I think Zeno's paradox is more like a thought experiment, which may or may not lead to a paradox depending on the system that you apply it to. For example, if you had a system with continuous motion, but where some event or interaction is required to take place in order for an object at position x to advance to a position x+dx, and it is required that the events underlying the motion happen sequentially, so that the next can't start until the previous one finishes, and that each event has some minimum time it takes to occure, then you might have a true paradox.

weirdoguy, Motore, PeroK and 1 other person
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As Carl Sagan suggested in Cosmos, if the Greeks had been interested in doing experiments instead of philosophical arguments we would have interstellar spaceships by now. The paradox is they refused to believe what they saw in the world around them.

https://iep.utm.edu/zeno-par/

russ_watters, Delta2 and PeroK
BWV
This is sort of interesting in that there is a separate mathematical solution (limits) and a real world solution (you can only take discrete steps)

The St Petersburg paradox is a little different, as in reality no one has an infinite bankroll, and capping the payout size at, say, Jeff Bezos's wealth ($179 billion) reduces the formerly infinite expectation to$37, people care less about incremental levels of wealth above some level (utility) or they rightfully truncate and disregard extremely low probability events (say the 2^-100 probability of winning 2^100)

Mathematically the expectation is infinite, so there is not really a mathematical paradox

Delta2
Jarvis323
If Zeno had concluded that time must progress in finite discrete intervals (rather than be impossible), then that would at least be consistent.
It sounds like you and others believe that Zeno, and the Greeks overall, actually believed that motion is impossible.

I never really thought that to be the case, but maybe I'm wrong. What Zeno showed is that a set of currently (in his time) held beliefs led to a contradiction. This motivated people to question those beliefs in order to develop less problematic views. This motivated new research in mathematics, and deeper questioning about the nature of time and space. Coming up with obsurd yet logical sounding arguments based on sets of reasonable beliefs was highly popular at the time.

Jarvis323
I think Zeno's paradox is more like a thought experiment, which may or may not lead to a paradox depending on the system that you apply it to. For example, if you had a system with continuous motion, but where some event or interaction is required to take place in order for an object at position x to advance to a position x+dx, and it is required that the events underlying the motion happen sequentially, so that the next can't start until the previous one finishes, and that each event has some minimum time it takes to occure, then you might have a true paradox.
A modern application of the idea is seen in the quantum-zeno effect.

According to decoherence theory, the collapse of the wave function is not a discrete, instantaneous event. A "measurement" is equivalent to strongly coupling the quantum system to the noisy thermal environment for a brief period of time, and continuous strong coupling is equivalent to frequent "measurement". The time it takes for the wave function to "collapse" is related to the decoherence time of the system when coupled to the environment. The stronger the coupling is, and the shorter the decoherence time, the faster it will collapse. So in the decoherence picture, a perfect implementation of the quantum Zeno effect corresponds to the limit where a quantum system is continuously coupled to the environment, and where that coupling is infinitely strong, and where the "environment" is an infinitely large source of thermal randomness.

https://en.m.wikipedia.org/wiki/Quantum_Zeno_effect

https://iopscience.iop.org/article/10.1088/1367-2630/aaa38d

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Delta2
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To me, the crux of Zeno's arguments comes from the arrow paradox. It directly questions what is meant by motion and argues that if there is such a thing as a moment in time then a moving object must have some internal property that defines it as moving at each moment.

He argues that there is no way to distinguish the moving from the stationary because at an instant it is just where it is. So what about it is moving? He asserts that in fact there is no difference because motion is only detected over a finite time interval, not in an instant. But if there is no motion in the instant, what does it even mean to say that it is moving at all?

So the problem is with the idea of motion in an instant. What does this have to do with Achilles not being able to catch the tortoise? Achilles is clearly moving because he arrives at succeeding points in the sequence of positions over finite time intervals.

But what happens when he overtakes the tortoise? How much time does that take? At any previous time no matter how close to the moment that he has reached the tortoise, he has only traversed a finite number of the points in the sequence so no previous time interval is small enough. That means that convergence occurs instantaneously. It is motion in the instant. But motion in the instant has no meaning. This means that he can not cross the infinite number of positions in a finite amount of time.

One can take this as a proof that an infinite sequence can never be completed.

Ergo, motion must be an illusion since reason shows that it is not real.

-Zeno and other Greeks understood the idea of convergence in some non formal sense. Aristotle for instance spoke of the difference between potential and completed infinities. Zeno obviously saw that catching the tortoise was convergence. The argument that an infinite sequence can in fact be traversed because it can converge misses the point.

- One might think that if the motion is differentiable then one can assign an intrinsic property of motion in the instant, the instantaneous velocity vector. But this is an added structure and if one looks at it as a limit of average velocity over shrinking time intervals it is not something that is intrinsic to the moment but is rather a mathematical quantity derived from observable changes in position. Perhaps an intrinsic property might be a Leibnizian infinitesimal and I wonder if Leibniz considered this to be a solution to the paradox of the arrow.

But this said there is still non-differentiable motion for instance continuous Brownian motion. One still observes changes in position over finite intervals of time but there is no derivative. And there is no predictability. So in what sense is a Brownian particle moving?

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Delta2, vela and PeroK
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But this said there is still non-differentiable motion for instance continuous Brownian motion. One still observes changes in position over finite intervals of time but there is no derivative. And there is no predictability. So in what sense is a Brownian particle moving?
In a very real sense as you can see it under a microscope.

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In a very real sense as you can see it under a microscope.

Right. And how about a continuous space filling curve?

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Right. And how about a continuous space filling curve?
Not sure what you are getting at?

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Right. And how about a continuous space filling curve?
Just that one can also conceive of motions that can not be visualized.

graphking
Why no one think of the assumption that time is R (real numbers)? Is this assumption reasonable?

I saw some of you discussing 'motion', I think in Zeno's paradox he assumed the turtle and the man moving in a constant speed. Which is probably simple to define without the calculus.

And besides time, we are working in a space, which is R^1 (That is not very controversial, because the R is created based on modeling the space we live in, we called the model 'Euclid space'). And the moving objects are moving in only one direction, which makes it really simple if you want to know what the 'curve' that object moves over.

I remember I heard my teacher in the course 'real variable function' said in the start of the course: Zeno's paradox show that the Completeness Axoim for R is necessary. But I can't understand it. I think the crux is understanding the model that time is R, and if we want to use the completeness axoim, I guess we should understand time contains N, Z, Q step by step (but I cannot understand why time is pass by) That is what I am asking in this post, literally, what is time and why time is R?

P.S. I think most of you posters regard R obviously, regard 'time is R' is obviously...

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lavinia and Delta2
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I think the crux is understanding the model that time is R, and if we want to use the completeness axoim, I guess we should understand time contains N, Z, Q step by step (but I cannot understand why time is pass by) That is what I am asking in this post, literally, what is time and why time is R?
What do you mean by N,Z and Q?

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BWV
But this said there is still non-differentiable motion for instance continuous Brownian motion. One still observes changes in position over finite intervals of time but there is no derivative. And there is no predictability. So in what sense is a Brownian particle moving?
in this sense: