That is true only of analysis, not of actuality. This seems so obvious, that it simply astounds me that you cannot see that.

That is true only of analysis, not of actuality. This seems so obvious, that it simply astounds me that you cannot see that. — John

No, it actually does pass through the half way point, and so on. If it didn't then it wouldn't be continuous motion, it would be discrete.

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous.

So to assume that it is infinitely divisible is to assume that something is capable of dividing it infinitely. — Metaphysician Undercover

Here's the mistake you have been making for years, Meta.

If motion is discrete, it's not motion as we understand it to be. As object A "moves" from discrete point 1 to point 10, what is the time lapse between 1 and 2? Does A go out of existence during the lapse, and how do we claim A maintains identity during teleport and reappearance?

You can't just offer discrete movement as a solution to the paradoxes associated with analog movement without also explaining how discrete movement really works. It might be there's no coherent explanation to something as basic as movement, just like there's not with causation.

Anyway, discrete movement is an obvious adoption of the computer graphics model imposed on reality. Identity of a computer graphic over time is preserved by the underlying programming, which is a quite literal deus ex machina. If we're going to insert Deus, I suppose anything is possible, including analog movement.

I was simply responding to John's claim that motion doesn't involve an object actually going through the half-way point. My point is that if this was true then it wouldn't be continuous motion.

But as for an actual account of discrete motion, I believe there's Atomic electron transition or "quantum jump".

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous. — aletheist

MU is right that it has to be more complex than that. Talk of actually counting smuggles in the necessity of the maker of the infinesimal divisions or Dedekind cuts.

For there to be observables, there has to be an observer. Or for the semiotician, for there to be the signs (the numeric ritual of giving name to the cuts), there has to be a habit of interpretance in place that allows that to be the ritualistic case. Which is why the number line itself is just a firstness or vagueness. In the ultimate analysis it is the raw possibility of continua ... or their "other", the matchingly definite thing of a discontinuity.

So infinity and infinitesimal describe complementary limits - one is the continuum limit, the other the limit on bounded discreetness, the limit of an isolate point.

Thus counting presumes an observer then able to stand inbetween. The counter can count forever because the counter also determines the cuts that pragmatically "do no violence" to the metaphysics, at least as far as the counter is concerned.

My point is thus that an observerless metaphysics is as obtuse as an observerless physics, or theory of truth, or observerless anything when it comes to fundamental thought.

I know. Your position was simply atracking his, but I'm accepting your criticism and now asking for a proof of your alternative. If your alternative fails, and we know there is movement of some sort, then it might be analog movement is just as reasonable as an any.

Are you conceding that discrete movement is nonsensical? If so, why'd you offer it as an possible solution to Zeno?

The proof, if there is any, would be the fact that continuous motion is illogical (and so impossible). If this were so then it must be either that discrete motion is possible or that motion is impossible. Unless there's a type of motion that's neither continuous nor discrete?

But I'm not really interested in defending the notion of discrete motion. What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point.

You mentioned the relevance of transversing the Planck scale. And while I applaud taking the physical facts seriously, in fact any exactness of location results in a complementary uncertainty about momentum (or equivalently, duration).

So if you talking about a physical continuity on the Planck scale, your attempt to mark the first location would already then have your fixed point transversing the whole distance to its resulting destination.

It is like the way a photon is said to experience no time to get where it is going. Travelling at c means the journey itself is already described by a vector - a ray rather than a succession of points.

So in the real world, locating your starting point is subject to the uncertainty relation. The Planck scale is the pivot which prevents you reaching your goal of exactitude by diverting all your measurement effort suddenly in the opposite direction. In effect you so energise the point you want to measure that it has already crossed all the space you just imagined as the context that could have confined it.

Zeno definitely does not apply in quantum physical reality.

What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point. — Michael

I'm not sure there's a way to show that. A person either understands the paradox or not. It's not complicated.

What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical — Michael

All that would imply is that you chose the wrong logic.

That is true only in a formal sense; there is no actual halfway point it goes through.

This is the assumption that I'm showing to be false. Each movement from one point to the next is a tick. — Michael

I am afraid that you just can't get past the concept of counting, or rather to see it in its context. There's no point in me trying to explain it to you now, because I would just be repeating myself. But later, when you are no longer engaged in defending your position, I suggest that you acquaint yourself with the basics of set theory and calculous.

You might think that mathematics is this very specialized discipline that is only relevant to solving certain technical problems, but it's not. Mathematics is relevant to any abstract thought, metaphysics included. It expands your conceptual apparatus and gives you the tools for dealing with complex concepts in a systematic, disciplined way.

When you become familiar with the foundations of mathematics and see how concepts such as sets and numbers are built upon each other, perhaps then you will see what we have been trying to tell you. You might still resist the concept of a continuum on physical or metaphysical grounds, but at least you will be doing it with the clear understanding of its logical structure.

If motion is discrete, it's not motion as we understand it to be. As object A "moves" from discrete point 1 to point 10, what is the time lapse between 1 and 2? Does A go out of existence during the lapse, and how do we claim A maintains identity during teleport and reappearance?

You can't just offer discrete movement as a solution to the paradoxes associated with analog movement without also explaining how discrete movement really works. It might be there's no coherent explanation to something as basic as movement, just like there's not with causation.

Anyway, discrete movement is an obvious adoption of the computer graphics model imposed on reality. Identity of a computer graphic over time is preserved by the underlying programming, which is a quite literal deus ex machina. If we're going to insert Deus, I suppose anything is possible, including analog movement. — Hanover

I think that you are making some unconscious metaphysical assumptions here. Why does continuous motion preserve identity and discrete motion does not? You can construe your idea of identity this way, but this construal doesn't have the force of logical necessity - it is just one possibility among many.

You ask how discrete motion "really works." What do you mean by this question? Do you understand how continuous motion "really works?"

But I'm not really interested in defending the notion of discrete motion. What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point. — Michael

An alternative view is to accept Zeno's paradoxes as lessons of the impossibility to deduce how reality behaves a priori, from mathematics or any other way.

What can happen is determined by the laws of physics alone, and if they say that an uncountable infinity of points in space will be traversed, then that is what will happen. Something mathematically infinite in the process may have occurred, but that involves nothing physically infinite.

It seems that Zeno's mistake is to assume that a particular mathematical notion of infinity somehow determines what can and cannot happen in reality.

Plank's Length — Banno

Indeed. I do not see the relevance of being 5'11", unless as part of an argument to say that Max Planck was infinitely tall or something, or couldn't grow, and was born fully formed.

Indeed. I do not see the relevance of being 5'11", unless as part of an argument to say that Max Planck was infinitely tall or something, or couldn't grow, and was born fully formed. — bert1

Appealing to a degree of granularity to space probably makes matters worse. How do you get from one step to the other if there is nothing in between?

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. — aletheist

That's right. It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. Why would you accept the contradictory premise, that something which is impossible to count is countable? That makes no sense to me. This is the basic nature of infinity, that it is not countable. To believe otherwise is very clearly to believe a contradiction. The notion of infinity may be useful, but it's a fiction, a useful fiction.

That's right. It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. Why would you accept the contradictory premise, that something which is impossible to count is countable? That makes no sense to me. This is the basic nature of infinity, that it is not countable. To believe otherwise is very clearly to believe a contradiction. The notion of infinity may be useful, but it's a fiction, a useful fiction. — Metaphysician Undercover

Countable infinities are precisely those which can be put into one-to-one correspondence with the integers. This is a definition, and no, no one expects you to count them all.

The real numbers, or any finite interval on the real line, cannot be put into one-to-one correspondence with the integers. This type of infinity is much larger than the countable infinity, and is the first in a sequence of uncountable infinities.

While there are many cardinalities (could be infinite number of them as far as I know) the distinction the countable and uncountable infinities is the most important.

Countable infinities are precisely those which can be put into one-to-one correspondence with the integers. This is a definition, and no, no one expects you to count them all. — tom

That's the point, they are not countable, so to call them "countable" is just a name, a label, it doesn't mean that they are actually countable. You might differentiate natural numbers from real numbers by saying that one is countable and the other not, but that's just a name, in actuality neither are countable.

That's the point, they are not countable, so to call them "countable" is just a name, a label, it doesn't mean that they are actually countable. You might differentiate natural numbers from real numbers by saying that one is countable and the other not, but that's just a name, in actuality neither are countable. — Metaphysician Undercover

Try counting the real numbers between 0 and 1.

Try counting the real numbers between 0 and 1. — tom

It can't be done, but that doesn't mean that the natural numbers are countable. Neither real nor natural numbers are actually countable, because of the nature of infinity. One has no beginning point, the other has no ending point, but neither, as an infinity, is actually countable.

It can't be done, but that doesn't mean that the natural numbers are countable. Neither real nor natural numbers are actually countable, because of the nature of infinity. One has no beginning point, the other has no ending point, but neither, as an infinity, is actually countable. — Metaphysician Undercover

Countable and uncountable infinities are different.

You said you couldn't count a subset of the reals.

Do you think you might be able to count a subset of the integers?

Do you think you might be able to count a subset of the integers? — tom

Yes of course, but a subset of integers is not infinite. The difference here is with respect to the thing being counted, what is within the set, real numbers versus integers, one is assumed to be divisible, the other is not. It is not a difference in the infinity itself. With respect to the infinity itself, one is no different from the other.

Yes of course, but a subset of integers is not infinite. The difference here is with respect to the thing being counted, what is within the set, real numbers versus integers, one is assumed to be divisible, the other is not. It is not a difference in the infinity itself. With respect to the infinity itself, one is no different from the other. — Metaphysician Undercover

OK, so you can count integers, but you cannot count the real numbers, even in a tiny subset. There is an uncountable infinity of reals within any subset - hence it is a continuum. The countable infinities do not have this property. They are different and one is at least infinitely bigger than the other.

OK, so you can count integers, but you cannot count the real numbers, even in a tiny subset. There is an uncountable infinity of reals within any subset - hence it is a continuum. The countable infinities do not have this property. They are different and one is at least infinitely bigger than the other. — tom

No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity?

What is the case is that as you say, the real numbers represent a continuum, while the integers represent discrete, indivisible units. So there is a fundamental difference between what each represents. The continuum is assumed to be infinitely divisible, and also there is assumed to be an infinite number of discrete units. The meaning of "infinite" remains the same, so there is no difference between these two infinities. There is a difference between what "infinite" is being assigned to, division or addition.

No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity? — Metaphysician Undercover

Why don't you just look it up, or Google it? Plenty of stuff on cardinalities, countable and uncountable infinities, the diagonalization argument, Cantor ...

If you're basing this on set theory, there may be an argument that one infinity can be bigger than another. The argument may have been explored more thoroughly than I'm aware within the field of meta/mathematics, but I'm not personally 100% convinced; it takes some liberties with the characteristics of infinity.

Set theory isn't the only way to think of numbers, though. Another approach is to say that, numbers don't exist per se, but there are an agreed set of rules and methods for developing relationships (what you might call "arithmetic") between a finite set of symbols (what you might call "digits") - and these symbols can potentially be arranged in an infinite number of combinations (what you might call "numbers").

So with a base 10 numeral system and the rules of arithmetic with which we're already familiar, it's not granted that the numbers 0, 0.5 and 1 'exist'. Rather, the process of dividing 1 by 2 produces the arrangement of symbols, '0.5'.

My point being, it's naive to build a whole world view around one arbitrary concept - and therefore, greater questions about infinity aren't so easily answered by set theory alone.

(I rushed this post because my food was delivered half way through; I hope it makes any sense at all)

(I rushed this post because my food was delivered half way through; I hope it makes any sense at all) — Efram

I love the Khan Academy with pizza

https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/infinity/v/proof-infinities