Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this. — Ryan O'Connor

Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.

Would you agree a wise intellectual warrior should first know his enemy before striking?

The Numberphile video on Zeno's paradox expresses concern about what is at the end of an infinite series with no final term. The mathematician said he wanted a *physicists* to explain it to him

The Numberphile video on Zeno's paradox expresses concern about what is at the end of an infinite series with no final term. The mathematician said he wanted a *physicists* to explain it to him — Gregory

He asks whether actual time and actual space can be divided according to the math of ZP.

Oh you watched it, good. He said it "melts the brain" to think of an infinite series going to a destination without a final term. Math, I think you're saying, explains it in its own ways, but there are aspects which still trouble philosophers

Oh you watched it, good. He said it "melts the brain" to think of an infinite series going to a destination without a final term. — Gregory

I found the vid at https://www.youtube.com/watch?v=u7Z9UnWOJNY . I didn't want to watch the whole thing so I skipped to 12:00 and he said that "in the physical world" it's not the same, but I couldn't find the exact quote about brain melting. As I mentioned to @Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series, as the Numberphile guy says; and it's unsolved physically, as he also seems to agree.

In most discussions online people seem to believe the problem is solved by calculus, but that's not true. Only the mathematical problem is solved. The physical problem remains, and even moreso today, because we know about the Planck scale. Below a certain point, our physical theories are not applicable to space or time. The tortoise or the arrow or whatever only have to iterate through 35 or so steps before reaching the point where nothing sensible can be said.

At some point, you can't sensibly divide a distance in half, nor an interval of time. Physics doesn't allow us to get a sensible answer. We don't know whether space and time are arbitrarily divisible; but we DO know that our current understanding of physics doesn't let us sensibly speculate.

I left it out because it is a nonrestrictive clause. Further, any necessary termination is for a reason external to the process itself, usually to make an approximation. — tim wood

I don't know what you're talking about tim. To say "X will necessarily be terminated" seems very restrictive to me. Obviously, the cause of termination of the process is external, that's Newton's first law. But how's that relevant?

If A to B is seen as infinite fractions (half, quarter, eighth, ect) and as one travels it the mover changes color every fraction, he is not a definite color by the time he reaches B. He is kind a blur. But he can reach B because, on the flip side, A to B is a finite segment and must be finite in actuality. So there is the infinite approach and the finite way to see it. For some of us the tension between the infinite and finite here is strained and we feel there is and can only be something off about this. But as you say, it's telling us that there are aspects of reality that transcend us, and oddball paradoxes like this are just like jokes put into reality by God

its demonstratable that space is infinitely packed. — Gregory

It is? Can you point me to the demonstration? This may be going off on a tangent, but the Bekenstein bound poses an upper limit on the information that can be contained within a given volume. Or are you talking about mathematical space being infinitely packed?

Quantum physicists descriptions of reality are not necessarily accurate. They have to fit true philosophy. — Gregory

I believe that physics, math, and philosophy all inform each other. Physicists don't have it all figured out, but philosophers need to update their views based on the latest advancements in physics (and math).

Modern physics only says how stuff work. Do this, that happens. That sort of thing. The electrons might be doing what the protons are said to do or the other way around. For every wave there could be a hidden particle. We don't need pilot theory to know this, but if that's works for ye that' fine. My point is much of what QM says is false because they are clothing their findings in the language of an unprovable philosophy. They can predict things, but that is all QM can do.

As for matter, it's spatial? Yes. Is all space infinitely divisible? We can't conceive it as discrete, so our natural lights say yes. So your computer is infinitely packed. This is intuitive for me at least. That objects are in a sense very finite is also true, but General Relativity goes well with infinitely parted objects in the sense I mean that. GR is a great theory, and not just because it has my initials :) Lastly, there are different kinds of density and ZP is about the relationship between volume and mass. Until you grasp this you'll be in the dark of QM topology

Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.

Would you agree a wise intellectual warrior should first know his enemy before striking? — jgill

Since both you and @fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question. I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5? If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well.

In the Numberphile on Zeno's Paradox, James Grime says the following:

"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."

Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process?

I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants.

This is Ryan O'Connor under a new account. Hopefully, you'll be able to @mention me now.

Why not? Explain to me exactly why someone can't put one foot in front of the other and take a step. How does the mathematical theory of the real numbers preclude anyone from doing that? — fishfry

Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask?

Why do you think the modern theory of the real numbers prevents anyone from walking? — fishfry

My problem is that the theory of real numbers implies that continua are made up solely of points. To me, this means that to move from x=0 to x=1, Atalanta must sweep over infinite points (she must complete a supertask). Instead of saying that there are infinite points, I'm saying that she only needs to sweep over a finite number of things. For example, in the image below, Atalanta needs to sweep over only 4 line segments. It’s a task that does not require any supernatural powers.

And -- a question that I keep asking you and that you never answer -- why does any mathematical theory have anything to do with physics? — fishfry

Math and physics are not necessarily related, but as you point out, we move all the time. I take that to mean that nature has found a way to avoid the traps of Zeno's Paradoxes in the physical universe. As such, nature may provide some clues on how to solve Zeno's Paradoxes in the mathematical universe. The cheap solution is to say that continua are discrete. I think nature has found a far more elegant solution.

I genuinely can not understand your point. How does the mathematical theory of the real numbers prevent anyone from taking a step? — fishfry

Let me try from a different angle. Let us say Atalanta lives in a perfect resolution simulation. As such, as she traverses from x=0 to x=1, she must at some instant in time be at each of the intermediate points. If we take out all the fancy graphics of the simulation, the computer is essentially just outputting a list of coordinates from 0 to 1. But we know that the real numbers are not listable (countable). There must be something wrong with the story. Do you agree?

I would say that when you draw the real number line, that's a visual depiction of the mathematical real line, which itself is an abstract object that cannot be depicted. — fishfry

I'm not trying to trick you by using pixels on your screen or atoms on my paper. Take my drawing and 'idealize it'. I know that a truly 1D object cannot be depicted (it takes up no area after all). We should move to 2D to make it depictable but you didn't like my earlier graphs so to keep things simple, let's just assume that we're looking at it with our mind's eye. This is a representation of how I see the idealized image:

I see 4 line segments with the numbers corresponding to the gaps in between the line segments. Points vanish upon idealization.

Fine. What of it. How would anyone's interpretation of that picture prevent them from walking or allow them to walk?......This is your thesis. Don't you see how silly it is?.......Before Dedekind had his clever idea, were people able to walk? And then the day he published, they couldn't? Isn't what you are saying patent nonsense? — fishfry

One can hold incorrect views on the laws of nature, but they will nevertheless follow the true laws. Similarly, one can hold incorrect views on the nature of continua which imply that motion is impossible, but they will still be able to walk. What I'm saying is that your views imply that motion is impossible. I'm not saying anything about your ability to walk.

According to the Copenhagen interpretation, we have no idea what happens when we're not looking. According to Many Worlds, everything happens. How is this relevant to the conversation? — fishfry

Many worlds provides a different interpretation on wavefunction collapse, but it still holds that in between quantum measurements what exists is a superposition, not a definite state. QM is only relevant if it informs our solution to Zeno's Paradox. I'm bringing it up because without QM my view sounds absurd. But you're right, I don't know the correct interpretation of QM. I also don't know what exactly entails a quantum measurement.

Has it occurred to you that perhaps you are not personally possessed of the ultimate truth about how the universe works? — fishfry

Of course. Truth comes out with formalized theories. I'm merely sharing an atypical intuition, which has neither been proven right or wrong. Ideas are cheap, including mine.

Now you're just being silly, since if you claim 1/3 is the smallest positive real number I'll just divide it by 2 (using the field axioms) and note that 0 < 1/6 < 1/3. Your claim stands refuted. — fishfry

No, I'm not being silly. 1/3 is the smallest non-zero number in this system.



In this system 1/6 is the smallest non-zero number.



Once we place another point on that line, we have a distinctly different system. This system is composed of 6 points and 5 line segments. In this system 1/6 is the smallest non-zero number. You have refuted nothing.

As I mentioned to Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series — fishfry

As I mentioned to jgill, James Grime in the Numberphile video suggests that the solution kind of involves the completion of infinite tasks. Is this the view you hold?

My point is much of what QM says is false because they are clothing their findings in the language of an unprovable philosophy. They can predict things, but that is all QM can do. — Gregory

No doubt, QM is in dire need of philosophical progress. But we can't downplay the value of physics. Experiments and corresponding models ground our reasoning.

Is all space infinitely divisible? We can't conceive it as discrete, so our natural lights say yes. So your computer is infinitely packed. — Gregory

Infinitely divisible does not imply infinitely packed. We can cut a 'mathematical' string indefinitely but never will the string be in infinite pieces.

I've been arguing that to be infinitely divisible means that it has infinite parts. These seem identical to me

Sorry, I forgot to mention that this is Ryan O'Connor with a new account.

I've been arguing that to be infinitely divisible means that it has infinite parts. These seem identical to me — Gregory

These are not identical. A 'mathematical' string is infinitely divisible because I can cut it to no end. I end up with infinite parts only when done cutting it, but am I ever done cutting it?

The parts are always there!

Since both you and fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question. — Ryan O'Connor

@keystone, I'll reply to this one before the other one. I had no trouble @-ing you so at least that worked. I'll continue to quote you as Ryan here but I assume that if you quote me back you'll be on your @keystone account and everything will be clear.

I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5? — Ryan O'Connor

As I have asked you several times, is this a mathematical or a physical scenario? If mathematical, the answer is yes. And @jgillI made the same point. By the intermediate value theorem, a continuous function passes through all intervening points between one value and another. And I assume continuity is one of your assumptions here. Is it? Essentially you're asking about the nature of continuity.

If this is a physical scenario, the answer is that one, nobody knows for sure; and two, the question is meaningless with respect to contemporary physics because we cannot reason sensibly below the Planck length. Does she hop from quark to quark? No physicist would regard that as a meaningful question.

By the way I looked up [url=https://www.thoughtco.com/greek-mythology-alanta-1525976[/url], and having been a student of Greek mythology way back in the day I was pleasantly informed. She's the goddess of running. The link I gave, which is better than her Wiki link, is full of good stories.

So: I say again, as I have said several times: Mathematically there is no question that she passes through every point indexed by a real number. Physically, the question is open in general, and meaningless in current theory. Even a goddess can't run between quarks.

If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well. — Ryan O'Connor

You're perfectly well conveying your refusal to read what I'm writing. If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1]. If this is a physical thought experiment, the question is metaphysically open and physically meaningless. So which question are you asking?

In the Numberphile on Zeno's Paradox, James Grime says the following:

"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly." — Ryan O'Connor

That's right. Mathematically the sum of an infinite series is defined as the limit of the sequence of partial sums; and the limit of a sequence is defined as the number the sequence gets (and stays) arbitrarily close to. And the Numberphile guy says exactly that.

Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process? — Ryan O'Connor

Depends on whether this is a mathematical or physical thought experiment. The physical question is certainly not resolved by the math, but I've stated that many times already.

I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants. — Ryan O'Connor

If you would stop conflating math and physics, and avoiding answering whether you are asking a mathematical or physical question, all would be clear.

On the other hand, it's perfectly obvious that motion does occur in the real world (unless we live in a block universe or we're all programs running in the great computer in the sky or brains in vats or I'm just dreaming all this.

philosophers need to update their views — Ryan O'Connor

Aren't you the one saying you agree with Aristotle? He believed that the reason bowling balls fall down is that they're made of "stuff" and so is the earth, and like attracts like. Aristotle needs an update too.

Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask? — keystone

No, I'm suggesting that the mathematical concept of the real line doesn't apply to the true nature of physical space; or that if it does, this fact lies far beyond contemporary physics. Even physics doesn't claim it does. Can't we stop here and nail down this point?Why are you deliberately ignoring this point, which I have made to you over and over? Nobody knows if space is infinitely divisible; and everyone knows that we don't know this.

Human beings and even goddesses can not traverse the mathematical real line because the latter is a mathematical abstraction. They can only traverse physical space, and nobody knows if physical space is like the mathematical real line or not. And we DO know that contemporary physics can not address the question.

There is a dynamism in the resting of matter, something that Heraclitus glimpsed and could only call "fire". Perhaps he knew of Zeno type paradoxes and referred to it specifically in his union of opposites (SEP article is great on him). One thing we know for sure is that infinity and " the finite " are opposites

I've been arguing that to be infinitely divisible means that it has infinite parts. — Gregory

The parts are always there! — Gregory

No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude.

Mathematically there is no question that she passes through every point indexed by a real number. — fishfry

This is one particular mathematical model of the interval--the dominant modern one, to be sure, but not the only one. Again, it is not mathematically necessary to treat a spatial interval as somehow consisting of unextended points. We can understand them instead as denoting locations in space, not constituents of space. As such, she does not really "pass through" them, we just just use them to track her progress.

Although the parts exist together, they exist even though they are not separated. A continuum is an infinite substance. Nothing potentially has parts. It HAS the parts whether they are separated or not

One foot does not potentially have two united six inches. The 12 inches are there

Nothing potentially has parts. It HAS the parts whether they are separated or not — Gregory

I obviously disagree. Again, a true continuum has no definite parts except those that we deliberately mark off within it for a particular purpose. It is infinitely divisible, but not actually divided.

One foot does not potentially have two united six inches. The 12 inches are there — Gregory

Neither the foot nor the inches are "there" unless and until we mark them. They are arbitrary units of length for measuring things, not intrinsic to space itself.

The continuum could easily be a dog instead of a segment. He has legs regardless of whether you point them out or cut them off

The continuum could easily be a dog instead of a segment. — Gregory

Responses like this are why I rarely bother jumping into these discussions.

There seems to be no reason the segment has to be parted into infinity in order to contain infinity. That seems irrelevant to me

A one-dimensional line does not "contain" anything. We can mark it with as many zero-dimensional points as we can imagine, but those points are not parts of the line itself.

No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude. — aletheist

If this is true, what you describe here, then it is impossible that "a true continuum is infinitely divisible. If marking points on a continuous line does not constitute dividing it, then there is nothing to indicate that the continuous line is divisible at all. And if dividing it once would break it's continuity, then a continuum cannot be infinitely divisible because dividing it once would prove it to be discontinuous.

Therefore it is a contradiction to say "a true continuum is infinitely divisible". We ought to say instead, "if it were divisible it would not be a true continuum". A true continuum is indivisible.

It is infinitely divisible, but not actually divided. — aletheist

Again, this is contradiction. If it cannot actually be divided, then it is false to claim that it is divisible, in any sense.

A true continuum is indivisible. — Metaphysician Undercover

In itself, yes; but we can still "divide" it at will to suit our purposes. That is what I mean when I say that the whole is real and the parts are entia rationis, creations of thought. For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space.

This is one particular mathematical model of the interval--the dominant modern one, to be sure, but not the only one. Again, it is not mathematically necessary to treat a spatial interval as somehow consisting of unextended points. We can understand them instead as denoting locations in space, not constituents of space. . — aletheist

If you view real numbers as locations where points might live, then she passes through each location. I don't see how this changes anything. Also I've previously referenced the constructive, hyperreal, and Peircian concepts of the real line, so I don't think I can be accused of narrow mindedness on the topic. Unless you reject the intermediate value theorem, my point stands. And even the constructivists have their own version of IVT, which they make work by restricting functions to being computable ones. Some version of the IVT is always valid regardless of one's model of the real line.

As such, she does not really "pass through" them, we just just use them to track her progress — aletheist

Distinction without a difference. IMO. She passes through each point or she passes through each location that is the address of a point. Not sure what you are getting at.

Points are the limit that infinite divisions approach. Space does lead to a pure contradiction. Our minds are not precise enough to understand it. Every part can be divided by the mind endlessly.