It doesn't seem intuitive to me at all that space divides to infinity and yet has a finite limit. To my mind that is a direct contradiction, like a round triangle — Gregory

If I remember correctly, Pythagoras or one of those other Greek math guys calculated pi by dividing up a circle into uniform triangular pie slices. It seems intuitive to me, and apparently did to him, that as the size of the slices increase in number and decrease in size, the sum of the area of the triangles approaches the area of the circle with as much precision as we want. Carrying that one step further into calculus using the limit at infinity seems - intuitively - natural and logical.

The two are admittedly modeled as points, which works if you consider say their centers of gravity or their most-forward point. But by your assertion, do you mean that the tortoise is never at these intermediate points, only, the regions between? — noAxioms

I mean that if the tortoise is moving it is never at a point. This is because time is continuously passing, therefore motion is continuous too. So the closest thing we could truthfully say is that it is passing a point. To be at a point would require a stoppage in time. There is no time when a moving thing is at a point because that would a stoppage of time, which is a matter of removing the thing from time. That's the point of Zeno's arrow paradox.

Sorry to find a nit in everything, even stuff irrelevant to the OP, but relativity theory doesn't say this. In the frame of Earth, Earth is stationary. There's noting invalid about this frame. — noAxioms

I didn't say it's not valid I said that it's not true. Obviously the earth is not stationary. So that frame in which the earth is stationary is not true, it's an arbitrary (untrue) assumption, made for some purpose.

Carrying that one step further into calculus using the limit at infinity seems - intuitively - natural and logical — T Clark

When you "imagine" infinite points on a segment you are not really imagining an infinity. I realize that the infinity gets smaller and smaller, but it still never ends and hence should have no finite boundary. Each digit of pi corresponds to a slice of space, so infinite space makes finite object, a contradiction, so says the Eleatics. What is intuitive for me is to say there are discrete steps, but it's impossible to explain that geometrically. Infinity seems necessary as a tool, not as a truth

"Zeno’s paradoxes have received some explicit attention from scholars throughout later centuries. Pierre Gassendi in the early 17th century mentioned Zeno’s paradoxes as the reason to claim that the world’s atoms must not be infinitely divisible. Pierre Bayle’s 1696 article on Zeno drew the skeptical conclusion that, for the reasons given by Zeno, the concept of space is contradictory. In the early 19th century, Hegel suggested that Zeno’s paradoxes supported his view that reality is inherently contradictory"
Internet Encyclopedia of Philosophy

Addition:

"Loop Quantum Gravity (LQG) is a theory that attempts to reconcile general relativity and quantum mechanics by proposing that spacetime is not continuous but rather composed of discrete, fundamental units at the Planck scale, forming a 'fabric' of spacetime loops". Google AI

This is a reinterpretation of General Relativity, which traditionally had continuums in its maths. It's just hard to image space that can't be divided. There is something missing it seems when we try to reason about it. God knows. Food for thought. Weren't the Greek Atomists a reaction to Zeno?

So there might be a point that the paradox breaks-down as you move from physics to maths. An infinite geometric series in maths is inapplicable to a physically real distance. — Nemo2124

Isn't this an example where the false premises lead to wrong conclusions? Even if the argument appears valid in the form, it cannot reflect the true reality of the world.

When you "imagine" infinite points on a segment you are not really imagining an infinity. — Gregory

Who are you to tell me what I am or am not imagining. Just because you can't imagine something infinitely large or infinitesimally small doesn't mean the rest of us can't. We can imagine things that don't or even can't actually exist.

I realize that the infinity gets smaller and smaller, but it still never ends and hence should have no finite boundary. Each digit of pi corresponds to a slice of space, so infinite space makes finite object, a contradiction, so says the Eleatics. What is intuitive for me is to say there are discrete steps, but it's impossible to explain that geometrically. Infinity seems necessary as a tool, not as a truth — Gregory

Points are infinitesimally small in three dimensions, lines in two dimensions, and planes in one dimension, yet we use them all the time in mathematics and physics. They can be used as effective, accurate models of the behavior of actual observable phenomena. Why are the infinitesimals we are discussing any different?

Why are the infinitesimals we are discussing any different — T Clark

Because as Berkeley said they are ghosts of departed quantities. You are adding up nothing and arriving at something. Why do you post on this forum at all if we could all have magical powers in the mind you don't have?

The mathematical interpretation of Zeno's paradox seems straightforward to me. Evaluating limits makes the so-called paradox disappear. What is illogical about that? And what does this have to do with calculus. Representing a continuum as an infinite series of infinitesimals seems like a good model of how the universe works, simple and intuitive. — T Clark

Zeno's dichotomy paradox corresponds to the mathematical fact that every pair of rational numbers is separated by a countably infinite number of other rational numbers. Because of this, a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p. Hence calculus does not say that f(x) moves towards L as x moves towards p.

Zeno's dichotomy paradox corresponds to the mathematical fact that every pair of rational numbers is separated by a countably infinite number of other rational numbers. Because of this, a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p. Hence calculus does not say that f(x) moves towards L as x moves towards p. — sime

I don't understand how "...every pair of rational numbers is separated by a countably infinite number of other rational numbers." implies "a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p." It's a model for goodness sake.

I'm going to leave it at that. I'm not a mathematician and I've carried this as far as I can. You can have the last word.

Despite successive attempts to resolve this paradox, it seems as if the tortoise still edges-out Achilles. — Nemo2124

Achilles takes one step. That’s a physical event.

You need to add concepts to this picture to say whether he has moved a fraction of some other step, approached some other limit infinitely, or already won the race. You can not add other physical steps (like infinitely smaller fractional actual steps) to the step that was already the subject of inquiry without denying the existence of the step in the first place. So is there a step, a motion to discuss, or not?

If Achilles can’t catch the tortoise, the tortoise can’t move either, and there is no paradox, because there is no race.

There’s something like a category issue going on here to fabricate the paradox, and mess with the betting odds at the racetrack.

Measured distances, fractions thereof, and infinity, are concepts. Mental things. We grasp physical things with our hands. We don’t grasp infinity like that, ever. Achilles’ stride and the tortoises’ pace need have nothing to do with any of those concepts. Strides and pacing are physical things.

If I move ten centimeters, I can be said to have moved one-tenth of a meter. Or I can be said to have moved one whole decimeter. So was this a fraction or whole motion? Does that motion have infinite parts or no parts?

These are concepts, mental constructs, we can only assert apply to physical things. Only by first positing a conceptual scheme in which one meter is equal to ten decimeters can I then name something “one tenth of a meter.” And only by positing a whole meter (or going the whole distance conceptually) do I fix the denominator that names the decimeter 1/10th meter (“10” here meaning “whole one”). You don’t get fractions before wholes; you take wholes and divide them, to conceptualize fractions.

So the race had to be over before anyone could tell you at what point Achilles moved one tenth of a distance, or any fractional distance.

There is no fraction of a physical thing - it is only made a fraction conceptually by relating that whole thing to some other whole thing and seeing the relation is fractional according to your conceptual relational scheme.

You cut an apple in half. You can say you only have a fraction of a whole apple. But let’s say you never saw fruit before, and someone hands you a single “half-apple” - you would have one whole thing in your hand and no means to determine it relates to some other “half”. You would have a whole thing. And physically, that’s all there ever is. The determination of whole versus half of that whole requires concepts, not physical steps or physical processes.

So to reasses who conceptually wins and loses a non-conceptual physical race, one would have to wait until it is physically over before one could properly conceptualize the fractions and partial movements that can be said to make up that whole race.

All of these measurements are post-hoc measurements asserted of some external thing. And for this paradox, they are post-hoc concepts turning a physical thing into a conundrum for those concepts, not for the spectators of the race.

No one ever actually moves one-tenth of any distance. They move an actual, finite, whole distances. In every move.

So, unless Achilles brakes his heel and drops out, the tortoise always loses the physical race. That has nothing to do with any math nor provides any more information about the paradox.

The paradox is really just the irony that it is impossible for the smartest people in the universe to explain a simple motion )which it truly is). Or, it takes sheer genius to prove through concepts, that motion right before your eyes can’t happen.

Why do you post on this forum at all if we could all have magical powers in the mind you don't have? — Gregory

Here, wait a second, I'm going to imagine infinity... There, satisfied? Want me to do it again? It's not a magic power, it's just imagination.

Nuff said.

Is your point that Zeno treats motion as a series of steps, while both physics and maths treat it as continuous?

I'll go along with that.

Hence calculus does not say that f(x) moves towards L as x moves towards p. — sime

Rubbish. :roll:

Is your point that Zeno treats motion as a series of steps, while both physics and maths treat it as continuous?

I'll go along with that. — Banno

More or less in the case of Zeno. Mathematics is often said to resolve the paradox in terms of the topological continuity of the continuum, by treating the open sets of the real line as solid lines and by forgetting the fact that continuum has points, meaning that the paradox resurfaces when the continuum is deconstructed in terms of points.

In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.



Mathematical limits are proved in two steps using mathematical induction - which obviously does not involve a literal traversal of each and every rational number in order, which leads nowhere. (The proof of a limit is intensional, whereas the empirical concept of motion is extensional).

Here, wait a second, I'm going to imagine infinity... There, satisfied? Want me to do it again? It's not a magic power, it's just imagination.

Nuff said — T Clark

You must think finitism is repressive or something. Anyway, explain how a two meter segment placed parallel to a one meter segment is longer when you have a one-to-one correspondance between points, but when you angle the two meter segment to form a triangle they line up the same. Thank you

your point that Zeno treats motion as a series of steps, while both physics and maths treat it as continuous? — Banno

Continuous means infinitely dense, which in turn would be either then discrete or an infinite series of steps

Continuous means infinitely dense — Gregory

No, it doesn't.

...by treating the open sets of the real line as solid lines and by forgetting the fact that continuum has points, — sime

A continuous path is not reducible to a mere sequence of points; rather, it is a unified whole in which limits make sense without requiring traversal of individual points.

Treat it as points, or as a continuum, but not both.

Why can't you just divide the "unified whole" if it's not discrete?

While I'm not at all sure of the wisdom of answering here, you can divide it - into bits of the continuum, which are themselves continuous.

A general rule: if the description you give of something that happens says that it can't happen, you are using the wrong description.

So does this subdividing result in a series of discrete steps, or goes on forever? Since it's continuous, it must sink down in there forever, which doesn't make any sense since it is a finite segment. See?

See — Gregory

I see you haven't understood. I doubt I can help. If it is continuous, the by that very fact it is not discreet.

I didn't say the continuous was discrete. I said the continuous doesn't make sense because spatial infinity squished into a finite size makes no sense.

Like when "experts" say the universe is infinite and expanding. That's called mental masturbation. A bad habit

(The proof of a limit is intensional, whereas the empirical concept of motion is extensional). — sime

Heuristics may not be precise, but its value can be substantial in an introductory course. I've never come across this sort of philosophical detail in elementary calculus, which includes lots of motion. "f(x) approaches . . . as x approaches . . ." is common language in math. But, whatever you say.

Like when "experts" say the universe is infinite and expanding. That's called mental masturbation. A bad habit — Gregory

While the "experts" might say something like that, the experts don't. Space is expanding, but saying the universe is expanding implies that it has a size, which it doesn't if it isn't bounded.

I said the continuous doesn't make sense because spatial infinity squished into a finite size makes no sense. — Gregory

Zeno did not describe infinite space squished into finite something. It was never spatial infinity.

These comments will also not help you Infinity isn't a hard concept to grasp, but giving it a bound when by definition there isn't one is always going to run into trouble.

While the "experts" might say something like that, the experts don't. Space is expanding, but saying the universe is expanding implies that it has a size, which it doesn't if it isn't bounded — noAxioms

There are many physicists who say the infinite-in all-directions-universe is expanding into hyper-dimensional multiverses. I would say Aristotle was right in writing that the spiritual (Heaven) surrounds the finite universe and that's that. He was wrong though about Zeno's arguments and the Greek atomists were on to something

Zeno did not describe infinite space squished into finite something. It was never spatial infinity.

These comments will also not help you Infinity isn't a hard concept to grasp, but giving it a bound when by definition there isn't one is always going to run into trouble. — noAxioms

Zeno said the apparent finite distance was really a series of infinite steps; hence infinite inside finite. It's not that hard to grasp. Infinity as an idea is sound only when it is used to refute itself

A koch snowflake has a finite area but an infinite boundary. Odd, that. Very nice.

The koch snowflake is just sending a finite boundary into infinity. It can't exist. How do you prove that all the even numbers are equal to the whole numbers? Well they line up a few numbers and send it to infinity, not realizing that you can do the same trick with an uncountable infinity. Take all the points on a segment and line each one up one at a time to the whole numbers. Walla they both get sent off into hhe infinite universe. Equal! (The problem i put to T Clark illustrates how geometrically this all doesn't make sense.)Take an orange and cut it in half. Then cut one of the halves in half and do this forever, lining them up largest to smallest. What is the smallest? Obviously something discrete otherwise the orange's partsv would go out infinitely into an infinite universe. Mathematical nfinity swallows itself and there's nothing that save it