I ask again; if I'm to count every rational number between 1 and 2, which number do I start with?

Or more relevantly, if I'm to pass through every rational-numbered coordinate between the start point and the end point, which coordinate do I pass through first?

There is if motion is continuous, which is a premise of the argument that gives rise to the paradox. If motion is discrete then the paradox wouldn't arise. — Michael

You have it exactly backwards - the paradox only arises by insisting that space is made up of infinitely many points, and time is made up of infinitely many instants. When we recognize that both space and time are continuous, the paradox dissolves - there are no intermediate points that I have to "touch" while moving from defined point A to defined point B in a finite interval of time, just like I do not need to count any intermediate numbers in order to get from 1 to 2.

The problem here is equating continuity with infinite divisibility, as if space and time consisted of infinitely many points and instants, respectively. The reality is that there are no actual points, just continuous space; and there are no actual instants, just continuous time. An infinitesimal distance can be traveled in an infinitesimal interval of time. A finite distance can be traveled in a finite interval of time. — aletheist

I think you are onto a good idea and your argument makes sense; the paradox arises because Zeno slips in between reality and the abstract language we need to use to discuss certain aspects of reality, mathematics.

You have it exactly backwards - the paradox only arises by insisting that space is made up of infinitely many points, and time is made up of infinitely many instants. When we recognize that both space and time are continuous, the paradox dissolves - there are no intermediate points that I have to "touch" while moving from defined point A to defined point B in a finite interval of time, just like I do not need to count any intermediate numbers in order to get from 1 to 2. — aletheist

If space is continuous then we can plot infinitely many points in it, so I don't understand your objection. And I don't understand your distintion between infinite divisibility and continuity. As explained here, "While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit. This means that the process of dividing it into ever smaller parts will never terminate in an indivisible or an atom—that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are divisible without limit or infinitely divisible."

If space is continuous then we can plot infinitely many points in it, so I don't understand your objection. — Michael

We can plot infinitely many points, but we do not have to plot any points between the two of interest. In other words, there are infinitely many potential points between any two actual points, but the only other actual points are the ones that we arbitrarily define. We can count from 1 to 2 in one step, in two steps, or in any other discrete number of steps; it is entirely up to us, and there is certainly no requirement to count all of the rational numbers in between.

And I don't understand your distintion between infinite divisibility and continuity. — Michael

A true continuum is infinitely divisible into smaller continua; it is not infinitely divisible into discrete individuals. For example, a line is infinitely divisible into smaller lines; it is not infinitely divisible into points. There is also a distinction between being infinitely divisible (potentially) and infinitely divided (actually). We are talking about the former, not the latter.

I answered you. There are an infinite number - choose any you want.

I answered you. There are an infinite number - choose any you want. — Banno

I can't choose any I want. I have to count every rational number (in sequential order) between 1 and 2. I'm not allowed to just skip ahead to some arbitrary point.

So you are claiming that there is no way to systematically list the rational numbers between 1 and 2?

But http://www.math-only-math.com/to-find-rational-numbers.html

We can plot infinitely many points, but we do not have to plot any points between the two of interest. In other words, there are infinitely many potential points between any two actual points, but the only other actual points are the ones that we arbitrarily define. We can count from 1 to 2 in one step, in two steps, or in any other discrete number of steps; it is entirely up to us, and there is certainly no requirement to count all of the rational numbers in between. — aletheist

The task is to count every rational number between 1 and 2.

And so by the same token, assume that I overlay a region of space with a coordinate system that contains every rational number between 0 and 1. Given an object that starts at 0 and is supposed to move towards 1, what rational-numbered coordinate does it pass through first?

I have to count every rational number (in sequential order) between 1 and 2. I'm not allowed to just skip ahead to some arbitrary point. — Michael

You are already being arbitrary by only counting all of the rational numbers between 1 and 2. What is your excuse for not counting all of the real numbers between 1 and 2 - i.e., also including irrational numbers?

So you are claiming that there is no way to systematically list the rational numbers between 1 and 2? — Banno

I'm not claiming anything. I'm asking you for the first rational number I would count if my task was to sequentially count every rational number between 1 and 2.

You are already being arbitrary by only counting all of the rational numbers between 1 and 2. What is your excuse for not counting all of the real numbers between 1 and 2 - i.e., also including irrational numbers? — aletheist

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first?

The task is to count every rational number between 1 and 2. — Michael

No, the task is to move from point A to point B. You are mathematically modeling it as counting every rational number between 1 and 2. I am challenging the fundamental assumption of your model. I can count from 1 to 2 without counting any other numbers in between. Likewise, I can move from point A to point B without touching any points in between, because there are no (actual) points in between.

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

My coordinate system only uses the integers. After all, units of physical measurement are completely arbitrary.

No, the task is to move from point A to point B. You are mathematically modeling it as counting every rational number between 1 and 2. I am challenging the fundamental assumption of your model. I can count from 1 to 2 without counting any other numbers in between. Likewise, I can move from point A to point B without touching any points in between, because there are no (actual) points in between. — aletheist

No, when I say that the task is to count the rational numbers between 1 and 2 I'm not talking about the task to move from point A to point B; I'm talking about the task to count the rational numbers between 1 and 2.

My coordinate system only uses the integers. After all, units of physical measurement are completely arbitrary. — aletheist

OK, but I'm asking about my coordinate system. What rationally-numbered coordinate does the object pass through first?

Again your picture is muddled. The task you set was to count the rationals; now you have slid from that to finding the first rational.

There is no first rational between 1 and 2. But there are exactly a denumerably infinite number of rationals between one and two.

Again your picture is muddled. The task you set was to count the rationals; now you have slid from that to finding the first rational. — Banno

To count the rationals in sequential order I first have to count the first rational.

There is no first rational between 1 and 2. But there are exactly a denumerably infinite number of rationals between one and two.

If there's no first rational between 1 and 2 then how do I sequentially count the rationals between 1 and 2? Are you admitting then that such a task is impossible, and so that your claim that "if the length of time it took to count a number reduced as the size of the number, we would be able to count the rational numbers in a finite time" is premised on a flawed understanding of what it means to count the rational numbers?

No, when I say that the task is to count the rational numbers between 1 and 2 I'm not talking about the task to move from point A to point B; I'm talking about the task to count the rational numbers between 1 and 2. — Michael

The only reason you brought this into the conversation was as a (mistaken) model of moving from point A to point B, which is the subject of the thread. I frankly have no interest in counting the rational numbers between 1 and 2. It reflects the misconception that a true continuum is made up of infinitely many individuals, which is not the case; even the real numbers do not exhaust it.

To count the rationals in sequential order I first have to count the first rational. — Michael

You dropped the italicised bit in.

One can count the rational numbers without putting them in sequence.

Just list the fractions between one and two; 3/2, 4/3, 5/3, 5/4, 7/4...

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

The whole purpose of any discrete coordinate system is to facilitate measurement. The smallest rational number that is greater than 1 cannot be identified unless you specify a finite tolerance, so a viable coordinate system using (all of) the rational numbers is impossible. Yet we can and do routinely create coordinate systems using integers, fractions, and decimals down to whatever small (but still finite) increment suits the data. Again, the only actual points on a line are the ones that we define.

You're overthinking it.

½ + ¼ + ⅛...=1 — Banno

How do you get =1 here Banno? It appears to me, like no matter how far you go you'll always be a fraction short of 1. Have you got a cheat?

A true continuum is infinitely divisible into smaller continua; it is not infinitely divisible into discrete individuals. — aletheist

The story in a nutshell. Points are a fiction here. The reality being modelled is the usual irreducibly complex thing of a vector - a composite of the ideas of a location and a motion...

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

....and the corollary is that what is being counted is not points but (Dedekind) cuts. The numbers count the infinite possibility for creating localised and non-moving discontinua.

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

The cut bounds the continua in question. So the continua has already been "traversed" in the fact there is this first cut. You are then asking how near the other end of the cut continua can be brought in the direction of the first cut in question. The answer is that it can be brought arbitrarily close. Infinitesimally near.

So you are creating difficulties by demanding that continua be constructed by sticking together a sequence of points. However there is no reason the whole story can't be flipped so that we are talking about relative states of constraint on a continuity - or indeed, an uncertainty - when it comes to the possibility of some motion, action, or degree of freedom.

It appears to me, like no matter how far you go you'll always be a fraction short of 1. — Metaphysician Undercover

Convert 1/3 to a decimal, then multiply it by 3. Is the result 1, or an infinitesimal fraction short of 1?

It just doesn't work that way. You can never count to the end of an infinite series of numbers. — Michael

The only reason we cannot do it is because we could never count infinitely fast which is what we would need to be able to do to complete the series. When it comes to traversing though, luckily we don't have to take account of each segment as we traverse it. You are confusing analysis with actuality; there is no actual infinite number of segments that we need to traverse when we move from one place to another.

Convert 1/3 to a decimal, then multiply it by 3. Is the result 1, or an infinitesimal fraction short of 1? — aletheist

Can't be done. What does that have to do with my question?

Can't be done. — Metaphysician Undercover

Of course it can, students have to do it in math class all the time. You can also do it on a calculator.

What does that have to do with my question? — Metaphysician Undercover

@Banno's example was an infinite series, so you have to keep adding smaller and smaller fractions. When you have done so infinitely many times, you get the result of 1. Likewise, if you carry the outcome of my example out to infinitely many decimal places, you get the result of 1. If his sum is always a fraction short of 1, then my product is always a fraction short of 1. Yet everyone agrees that 1/3 x 3 = 1 (exactly).

Not again. Not interested.

So you are assuming that you can be done doing something an infinite amount of times? Sounds like a falsity to me.

Because you know you're wrong.

No; because history shows you cannot understand mathematics.

https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_⋯

So you are assuming that you can be done doing something an infinite amount of times? — Metaphysician Undercover

Not at all. We can reason about infinity without actually doing anything an infinite number of times. If someone (God, perhaps) were to add up @Banno's infinitely many fractions or carry out my multiplication to infinitely many decimal places, then the result would be 1 in either case.