Maybe. I'm still mulling it over.

If, as I've suggested earlier, you think of mathematics as the long working out of how to join two sorts of intuitions into a single enterprise (number or count, on the one hand, and something like shape or space, on the other), then Zeno's paradox is a kind of speed bump there, and indicates that this will not be so simple as we might have hoped.

I think that's one reason so many of us grew up thinking calculus somehow "solves" the paradox, or overcomes it, because calculus does represent some kind of completion of a very long process of drawing together various fragments of mathematics.

But something else we might say comes oddly out of the discussion above, about how the reals cannot be counted, and the standard alternative, if we're casting about for a different metaphor, in English anyway, would be that they must be measured. (If you're not a count noun, you're a mass noun.) Somewhere I suggested that "measure" is the first step in joining the two strands of intuition, number and shape, and that's obviously true. But it's also true that a distinction persists. So — to get to it — we don't count distances; we measure them. But the whole structure of Zeno's analysis, despite relying on dividing by 2 and all that, tends toward counting, as if it's an attempt to force distance into the procrustean bed of counting. The whole procedure seems intended to undermine the idea of measuring at all through a perverse insistence on the model of counting. (If that's not clear, I can take another swing at it.)

As for the supertask business, it's the framework that interests me. Zeno insists, apparently reasonably, that to carry out the task of traveling from A to B, you must perform a series of actions — indeed you could say this about anything. It's hard to imagine an alternative, but it's quite an odd thing really. Everything is done by carrying out certain steps one at a time, in order? That's demonstrably false for a great number of things we do. The universe is a concurrent place, and we are concurrent beings. In order to walk, you don't move a leg, then an arm, then the other leg, then the other arm. If you tried to walk by performing a number of actions sequentially, you'd fail.

The most interesting case is probably thinking itself, because we know for a fact that the brain is a massively parallel system, and yet we have put enormous effort down through the generations into shaping that mass of simultaneous activity into something linear and sequential. We get logic that way, and human speech, which is one damned word after another, linearly. We are very proud of our linear triumphs, but it is, so far as I can tell, impossible to say whether that linearity is an illusion.

In short, what interests me about the paradox has less to do with "infinity" and more to do with "sequentiality".

I read you as basically saying there's a higher truth missed by Zeno. Per tradition, his point was exactly that: that the way we picture the world, the way we commonly think, is missing something.

If you go back and look at one of the paradoxes, they're pretty simple. Looking at in terms of truth it starts here:

Isn't it true that in order to get from point A to point B, you have to travel half the distance between them?

Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? If you say no to that, you've already ejected yourself from common sense. If you say yes to it, you're on your way to being ejected from common sense because there's a convergent series of points between. Either way: common sense has a problem.

There are two kinds of people: ones who can tolerate a threat to common sense, and those who can't. I think the first category is usually non-linear thinkers.

Maybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke. — Srap Tasmaner

When the measuring stick needs to be measured, it's time to throw away the measuring system completely, and devise anew. Otherwise paradoxes are produced, like Russell's.

Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. — Srap Tasmaner

Of course, an infinite collection by any standard definition of "collection" is nonsense. A collection consists of things which have been collected, not things designated as collectible. And that's problems arise in set theory, "collection" becomes a designated collectible type, rather than the collection itself.

This is how the concept of "the empty set" creates paradoxes like Russel's. A "collection" with no items is not a collection at all. It is only a criterion for collection, therefore an abstract 'type" distinct from an item. Allowing for an empty set means that "the set" itself is not the collection of things (or else an empty set would not be a set), but "the set" is the abstract type, which describes the things to be collected. The things themselves, therefore, the elements of the set, must be categorically distinct from the sets, or else the empty set is the contradictory notion of a collection of nothing. Failing to follow this categorical distinction, which is necessitated by "the empty set", and allowing that a set might itself be an element of a set, produces problems.

But if the collection consist of things designated as collectible, and there is none of them, then it makes sense to talk about an empty set. However, this leaves cardinality as completely unjustified because the elements are just possibly collected, and therefore not counted.

Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? — frank

I think that what Srap is saying is that we cannot reduce motion to a succession of truths. That's what Aristotle demonstrated as the incompatibility between being and becoming. If change is represented as a succession of different states of being, one after the other, then there will always be the need to posit a further distinct state, in between any two. Then we have an infinite regress, without ever accounting for what happens between two states, as the change, or "becoming" which occurs as the transition from one to the next.

So if motion is described as getting from A to B, A and B are the two points of being, you are at A, then you are at B. Since they are not the same, there is distance between, and we can posit a middle point. You are at C. Then we posit a point of being in between A and C. You are at D. Notice, we've reduced motion to "being at a point which is different from the previous point". But this produces an infinite regress without ever addressing the real issue of how you get from one point to the other, what happens in between. This is the real nature of motion, what happens in between, and it cannot be represented as being at a designated point.

This is the real natu — Metaphysician Undercover

Ok. All I know is that it's common sense that if you're driving from Washington DC to Alaska, you will, at some point, be in British Columbia. Those who claim this view is wrong should at least acknowledge that what they're saying sounds bizarre.

Ok. All I know is that it's common sense that if you're driving from Washington DC to Alaska, you will, at some point, be in British Columbia. Those who claim this view is wrong should at least acknowledge that what they're saying sounds bizarre. — frank

I would say the opposite is the case, what you say sounds bizarre. You are representing driving through British Columbia, as being in British Columbia at some point. What does "at some point" even mean in this context? You use it because it's an acceptable figure of speech, but taken literally, it doesn't fit. So what does it really mean?

would say the opposite is the case, what you say sounds bizarre. You are representing driving through British Columbia, as being in British Columbia at some point. What does "at some point" even mean in this context? You use it because it's an acceptable figure of speech, but taken literally, it doesn't fit. So what does it really mean? — Metaphysician Undercover

Being in British Columbia usually entails waking up in your car with a Canadian citizen tapping on your window to see if you're ok. You roll your window down and try to do a Canadian accent so they don't know you're American, at which point they just stare at you. Does that explain it?

Does that explain it? — frank

No, you described a long process, and the problem is with the use of "at some point". How does a process occur at a point?

No, you described a long process, and the problem is with the use of "at some point". How does a process occur at a point? — Metaphysician Undercover

I think you just need some more of whatever mind altering substance you have available. Then you'll get it.

So that's deflationary nominalism. It's a minority view. — frank

@frank A Great quip for a reader lost in the woods. I think I was losing the grasp of the argument presented, due to the multiple citations of the author in the text, some of which I have none or only cursory experience.

fucksake.

What, if anything, in the supposed paradoxes of motions from Zeno, is not answered by limits, infinitesimals and calculus?

I suggest that what does remain is not a problem about motion, space, or time, but about conceptual confusion over infinity, divisibility, and description.

The key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1

Velocity is not defined at an instant by a finite displacement, but as: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

I think you just need some more of whatever mind altering substance you have available. Then you'll get it.

I think it’s time to play Tom Waites; The Piano has been Drinking.

Joking aside, Zeno’s paradox is an anomaly, just like infinity, they’re both anomalies thrown up by thinking. They don’t apply to the real world.

Maybe there are two kinds of worlds, one where nothing happens (I can’t describe that world), or a world where everything is in motion, everything is happening (moving), such that there isn’t anything that isn’t happening (moving). Now we’ve solved the problem of how to get from A to B, but we can’t stop now, we’re just going to have to keep moving (happening) now, add infinitum.

For God’s sake, who the hell decided to set everything in motion.

The key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1 — Banno

You mean the key is to put an end to the infinite sequence by rounding off. That's what we've done with pi for thousands of years. But if you think that this puts an end to the infinite sequence, and solves Zeno's paradoxes, you misunderstand.

This obviously works in practise. But the paradoxes are theoretical, they always have been, and they've always been irrelevant to practise. "Limits, infinitesimals and calculus" change the practise, but have no affect toward answering the paradoxes, which remain unchanged despite the changes in practise.

You mean the key is to put an end to the infinite sequence by rounding off. — Metaphysician Undercover

No. Nothing to do with rounding.

Your failure to understand mathematics is not our problem.

I'll repeat, since you did not address the issue.

It is a difference between theory and practise. In theory, the sum approaches the limit. In practice the sum is the limit. The latter can be understood as "rounding off". Failure to recognize this is to misunderstand.

It is a difference between theory and practise. In theory, the sum approaches the limit. In practice the sum is the limit. The latter can be understood as "rounding off". Failure to recognize this is to misunderstand. — Metaphysician Undercover

It's not an ordinary sort of rounding off, though. The difference between the limit and the sum is an infinitely small number. We could say that this solves Zeno's paradox as along as space and time actually conform to the calculus framework. I think the average scientist would agree that they do conform, but there is still room to reject the calculus angle.

A series limit isn't a literal sum of an infinite series, unless the number of summed terms that are non-zero is finite. E.g.

The infinite sum of (1,1,1,1,0,0,0,0....) = 4, as is its limit.

The infinite sum of the geometric series (1,0.5,0.25,...) is technically undefined, for in this case, every partial sum S(n) is non zero, since S(n) = 2 - 0.5^(n-1).

Sure, we can call 2 the "infinite sum", but it isn't an infinite sum in any literal sense of the word, rather 2 is the limit of the geometric series.

A limit isn't defined as a position on a sequence, but is defined as a finite winning strategy in a finite game, that involves cutting off the tail of an infinite sequence at a position δ, such that the height of the tail is within a prespecified distance ε to a prespecified value called "the limit", as per Cauchy's (ε, δ) definition of a limit.

E.g we have a sequence game S. Eloise first chooses a value v, then Abelard chooses a positive rational ε, then Eloise chooses a natural number δ in response. Abelard is now tasked with choosing an n greater than δ such that |S(n) - v| >= ε, otherwise Abelard loses. If Eloise's choice of v can guarantee her victory over Abelard, then we say that the limit of S is v.

There is no approaching the limit when proving a limit, for a proof of a winning strategy for Eloise doesn't involve multiple rounds of Abelard choosing ε1 then Eloise choosing δ1, then Abelard backtracking to choose ε2 < ε1 then Alice choosing δ2 > δ1 etc. Rather, a proof of a limit is just an inductively defined function ε -> δ established in two steps, for mapping Abelard's possible choice of ε to Eloise's choice of δ.

This is really cool, but I'm not convinced.

The gist of it is that there is a dominant strategy iff the sequence has a limit. If you countenance classical mathematics, you do an existence proof; if you don't, you do a constructive one. And then you have an answer about the game-theoretic application.

I guess what I don't get is that if you want to go the other way—to actually define the limit as a strategy—then you still have to start with an account of how to construct a δ given an ε. You seem to be doing the same thing but saying it's for a different reason.

The limit as strategy is interesting, and it's cool that it can be presented that way, but it looks like you still end up doing exactly the same math (of your preference) you would if you just presented it as a bare question, does this sequence converge? What am I missing?

The difference between the limit and the sum is an infinitely small number. — frank

"Infinitely small number" really has no meaning in this context. If the formula is applied to spatial distance, as in the Zeno paradox, it means infinitely short distance, not infinitely small number.

We could say that this solves Zeno's paradox as along as space and time actually conform to the calculus framework. I think the average scientist would agree that they do conform, but there is still room to reject the calculus angle. — frank

I don't agree. I think the average scientist would say that it doesn't make sense to talk about infinitely short distances. So if they round something off to zero it wouldn't be an infinitely short distance which is being rounded off, because the limitation of practise would require rounding off before infinitely short distance (whatever that actually means) is reached.

For example, when I use pi I round off to 3.14. Some scientific applications might request something more precise, but really the precision of the outcome is relative to the precision of the actual measurement. But, it's never an infinitely short amount which is being rounded of. So in the other example, are the measurements such that you are rounding 1/2+1/4+1/8 +1/16 to 1, or are you rounding 1/2+1/4+1/8+1/16+1/32+1/64 to 1? In the first case, 1/16 would be lost, as rounded to zero. In the second case 1/32 would be lost. The smaller the size becomes, the more difficult it becomes to measure it, and the required precision is application dependent.

The infinite sum of the geometric series (1,0.5,0.25,...) is technically undefined, for in this case, every partial sum S(n) is non zero, since S(n) = 2 - 0.5^(n-1). — sime

The fact that no partial sum equals 0 does not imply anything about whether the limit exists, or what it is. Limits routinely exist even when no term (or partial sum) ever equals the limiting value.


  The infinite sum of a series is defined as the limit of its partial sums (when that limit exists):

$\sum_{k=1}^{\infty} a_k := \lim_{n \to \infty} S_n$

For the geometric series

$1 + \tfrac12 + \tfrac14 + \cdots$

the partial sums are

$S_n = 2 - \left(\tfrac12\right)^{n-1}.$

Since

$\left(\tfrac12\right)^{n-1} \to 0 \quad \text{as } n \to \infty,$

it follows that

$\lim_{n \to \infty} S_n = 2.$

The fact that every partial sum is non-zero is irrelevant; convergence depends on the existence of the limit, not on whether any partial sum equals the limiting value.

Even on this “game” interpretation, the geometric series trivially has a winning strategy for every
ε. So by your own account, the limit exists.

The latter can be understood as "rounding off". — Metaphysician Undercover

No.

There is no principled theory/practice gap here. “Approaches the limit” and “equals the limit” are not in tension. Introducing “rounding off” does not correct or deepen the mathematics—it changes the subject.

Maybe this is won't help, but "rounding" is something you do when all you need is an approximation.

It's not that the adjacent members of a sequence become "infinitely close": they become "arbitrarily close", and so the series (in this case, the sum of the members of the sequence) becomes arbitrarily close to — well, that's the thing, to what? And that's your limit.

We are indeed talking about approximation, and therefore approximation of some value. It turns out we can imaginatively construct a "perfect approximation" which just is the value we are approximating. If you can get arbitrarily close to it, you can figure out what you're getting close to.

The difference between the limit and the sum is an infinitely small number. — frank

For a convergent series the sum is defined as the limit. There is no residual “infinitely small difference” between the sum and the limit. The sum is the limit. Partial sums are less than the limit, but their difference goes to zero in the standard real number system.

For a convergent series the sum is defined as the limit. There is no residual “infinitely small difference” between the sum and the limit. The sum is the limit. Partial sums are less than the limit, but their difference goes to zero in the standard real number system — Banno

There is a branch of mathematics called Numerical Analysis which, among other tasks, attempts to predict how far out one has to go in an infinite expansion to achieve an approximation of the limit to a specified degree of accuracy. I wrote some papers about this topic concerning continued fraction expansions. For example:

https://www.researchgate.net/publication/303490331_An_error_estimate_for_continued_fractions

There are various infinite expansions beyond sums: Infinite series, infinite products, infinite compositions, infinite continued fractions, etc. as well as infinite sequences arising from other algorithms.

It's not that the adjacent members of a sequence become "infinitely close": they become "arbitrarily close", and so the series (in this case, the sum of the members of the sequence) becomes arbitrarily close to — well, that's the thing, to what? And that's your limit. — Srap Tasmaner

That's exactly when rounding off is employed, when things are designated as "arbitrarily close". How have you done anything other than described a case of rounding off?

Interesting. A worthy topic - a more intricate form of "rounding off"? :wink:

I'll defer to your experience. My understanding is that what I said holds for classical convergence in Real Analysis.

I'll defer to your experience. My understanding is that what I said holds for classical convergence in Real Analysis — Banno

Keep up the good work!

How have you done anything other than described a case of rounding off? — Metaphysician Undercover

It's the difference between saying (1) here is an approximation of the value that is within some tolerance you have specified (or precision, or significant digits, whatever), and (2) here is a value that is within any tolerance you might specify, however small. For (2) to be possible, I must be offering you the actual value.

I know you're not a foundations guy, but I for one would appreciate a rap on the knuckles if I get the basics wrong.

(Decades, it's been decades since I did this in school. I could look everything up on wiki, but it's more fun to see if I can piece back together stuff I used to actually know.)

:smile:

Back a few pages I began a bit on the definition of a limit. I got as far as completeness and the least upper bound. Every nonempty set of real numbers that is bounded above has a least upper bound in ℝ, the smallest real number that is greater than or equal to every element of the set. It's the existence of this number that guarantees the existence of a limit when one uses the sequence in a calculation... if it's a monotone increasing sequence that is bounded above...

But as you found, the interesting stuff is the variations on these themes. The thread is focused on a small, very specific region of maths, and mostly failing to get a good handle on even that.

I don't agree. I think the average scientist would say that it doesn't make sense to talk about infinitely short distances — Metaphysician Undercover

But an electron is conceived as a point. It doesn't have any length or height. Isn't that the same as the idea of an infinitesimal in math?