What are the mathematical definitions of

separation
dimensional separation
dimensional point
non-dimensional point

It is unusual to say that difference proves theory to be wrong. — Ludwig V

Why do you say this? Doesn't science proceed through the falsification of theories?

I would be happy to say, I think, that Zeno's application of the theoretical possibility of convergent series to time and space and the application in Thompson's lamp is a mistake. — Ludwig V

You don't go as far as me then. I say that infinite divisibility is a mistake.

But calculus does have uses in applied mathematics, doesn't it? — Ludwig V

Of course, we need to distinguish between truth as our goal, and pragmatics, which doesn't have any specific goal. Usefulness is relative to the goal, and the goal could be anything. I don't deny that calculus is extremely useful, but that usefulness may be misleading relative to the goal of truth.

Non-dimensional points which have a dimensional separation? H'm. — Ludwig V

Don't you agree, that this is the only way in which one point may be distinguished from another point, through a spatial, or dimensional, separation?

But then a boundary (between your property and your neighbour's) doesn't occupy any space, even though it has a location in the world and will consist of non-dimensional points. — Ludwig V

This is the way I understand boundaries between two pieces of private property. The boundary exists in theory as lines of two dimensions, or sometimes even three dimensions because elevations need to be accounted for. In theory, the line occupies no space. In practise though, the boundary becomes a fence, a disputed sliver, or something like that, and ends up actually occupying space.

So if you and I had a shared boundary, on paper the boundary would be described as occupying no space, you on one side, I on the other, and that would be the theoretical boundary. But in practise, there would be an area, known as the place of the boundary. Even if there is pins, and we stretched a string from pin to pin, the string occupies an area. And so does the pin occupy an area.

If there is a maximum number of divisions, then what is that maximum number?

That is, for what natural number n is it the case that 1/(2^n) is not a number?

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"non-dimensional points [...] dimensional separation"

Maybe start by defining 'non-dimensional point' and 'dimensional separation'.

/

If the discussion is about points in ordinary real 2-space or real 3-space then points are distinguished by being a different ordered tuple.

In 2-space, the point <x y> differs from the point <z v> iff (x not= z or y not= v).

In 3-space, the point <x y t> differs from the point <z v s) iff (x not= z or y not= v or t not= s).

If a particular line, say the ordinary horizontal axis, then <0 x> differs from <0 z> iff x not=z.

This is not the least bit baffling.

If there is a maximum number of divisions, then what is that maximum number? — TonesInDeepFreeze

That is an empirical matter:

Einstein’s General Theory of Relativity describes the properties of gravity and assumes that space is a smooth, continuous fabric. Yet quantum theory suggests that space should be grainy at the smallest scales, like sand on a beach.

One of the great concerns of modern physics is to marry these two concepts into a single theory of quantum gravity.

Now, Integral has placed stringent new limits on the size of these quantum ‘grains’ in space, showing them to be much smaller than some quantum gravity ideas would suggest.

...

Some theories suggest that the quantum nature of space should manifest itself at the ‘Planck scale’: the minuscule 10^-35 of a metre, where a millimetre is 10^-3 m.

However, Integral’s observations are about 10,000 times more accurate than any previous and show that any quantum graininess must be at a level of 10^-48 m or smaller.

My question was about mathematics not physics. Suppose there is a smallest number usable for a given application of mathematics. Then, must mathematics not allow smaller numbers? If mathematics must not allow smaller numbers, then how would that be rigorously enforced in a mathematical theory? Suppose p is the smallest number that is to be allowed. The ordinary operations are defined by:

For all x and y, x-y = the unique z such that y+z = x.

For all x and y, x/y = the unique z such that y*z = x.

What would be the definitions when there is a smallest number p?

Note that in informal contexts, we use a notion of 'undefined'. But in a fully formalized theory, we don't allow 'undefined' as it would violate the definitional criteria of eliminability, which is crucial for the requirement that the syntax be recursive.

My question was about mathematics not physics. — TonesInDeepFreeze

Then no, there is no smallest number.

But then I'm not sure what relevance this question has to the matter at hand?

An argument has been given that physics is impaired by an improper application of mathematics. So the question arises whether that argument extends to a claim that mathematics must be revised. The context is that paradoxes about motion involve application of certain mathematics to distance and time. So, if someone claims that the mathematics is to blame, then we would ask whether the mathematics itself (which holds that there is no smallest number) needs to be rejected, or whether the way in which the mathematics is applied needs to be rejected, or both.

It's not that complicated. Imaginary numbers have a use – even in electrical engineering – but I cannot have an imaginary number of apples in my fridge.

There's nothing wrong with maths, just sometimes an improper use of it. There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then it would suggest that there is a smallest unit of space and/or time – and that this isn't just a contingent fact about the physics of our world but something far more necessary.

Ordinarily, an argument is valid if and only if it is not possible for the premises to be true and the conclusion false. But a paradox has seemingly true premises, seemingly correct logic, but a false (indeed, often a contradictory) conclusion. So it is not clear what you mean by a "valid paradox", though perhaps you mean that the premises indeed entail a falsehood (or even a contradiction) or that the premises are true but entail a falsehood?

I understand the view that there is no smallest number but that there are smallest distances and durations. But I am asking whether some people here do believe there is a smallest number.

I'm not claiming that the implications are or are not complicated.

The analogy with imaginary numbers and apples is amiss in this regard: Yes, apples are counted by integers, not imaginary numbers, so indeed imaginary numbers are not the correct kind of number to count with. But distances and durations are measured by real numbers, so smaller and smaller real numbers are not a difference in the kind of number.

Why do you say this? Doesn't science proceed through the falsification of theories? — Metaphysician Undercover

Yes. The trouble is that the inapplicability of convergent series in certain situations does not, for my money invalid them in all situations.

I don't deny that calculus is extremely useful, but that usefulness may be misleading relative to the goal of truth. — Metaphysician Undercover

Well, it would be interesting to know what your criterion of truth is in mathematics, if a calculation procedure is effective and useful.

This is the way I understand boundaries between two pieces of private property. — Metaphysician Undercover

I agree with everything you say.

If the discussion is about points in ordinary real 2-space or real 3-space then points are distinguished by being a different ordered tuple.
In 2-space, the point <x y> differs from the point <z v> iff (x not= z or y not= v).
In 3-space, the point <x y t> differs from the point <z v s) iff (x not= z or y not= v or t not= s).
If a particular line, say the ordinary horizontal axis, then <0 x> differs from <0 z> iff x not=z. — TonesInDeepFreeze

Thank you. That's very clear.

There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then there is a smallest unit of space and/or time. — Michael

I'm puzzled. I thought you thought that Thompson's paradox was flawed and therefore invalid - as Thompson did, didn't he?

I understand the view that there is no smallest number but that there are smallest distances and durations. But I am asking about the views of others too. — TonesInDeepFreeze

That seems to be the result of some recent research. But I don't think it applies to mathematics as such, and perhaps one ought to wait and see whether anything else emerges from research.

I'm puzzled. I thought you thought that Thompson's paradox was flawed and therefore invalid - as Thompson did, didn't he? — Ludwig V

No, I think (as did he) that it successfully shows that supertasks are not possible.

What does "paradox is valid" mean? Does it mean that the premises indeed entail a contradiction. — TonesInDeepFreeze

Yes. If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible.

The analogy with imaginary numbers and apples is amiss in this regard: Yes, apples are counted by integers, not imaginary numbers, so indeed imaginary numbers are not the correct kind of number to count with. But distances and durations are measured by real numbers, so smaller and smaller real numbers are not a difference in the kind of number. — TonesInDeepFreeze

In this case the mistake is in the application of transfinite numbers.

Thomson says that there's a false premise, which is that infinitely many tasks cannot be completed in finite time. He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time.

[EDIT: I may have erred in the paragraph above. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.]

I think "valid paradox" is, at best, ambiguous and confusing.

But I am asking whether some people here do believe there is a smallest number. — TonesInDeepFreeze

That's a good question. As a relative simpleton who's been trying his best to follow along with the recent arguments, I would ask: why could you not take the smallest number (my rudimentary mind imagines something along the lines of 0.0[insert a bajillion zeroes here]1) and divide that by 2? And divide even that by 2? And so on? We can't "run out" of numbers, per se. For that is there design. Though I'm sure there reaches a "hard point", a threshold if you will where a certain degree of number fails to appear or exist anywhere in the known universe thus ceases to become of use or mention. I'm sure there's a term for that- somewhere. :chin:

Thompson says that there's a false premise, which is that infinitely many tasks cannot be completed in finite time. He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time. — TonesInDeepFreeze

So his paradox shows that the time between each task in a sequence cannot in principle be modelled by a geometric series, e.g. where the first task takes 1 minute, the second 30 seconds, the third 15 seconds, and so on, because if it were possible then it would be possible for an infinite number of tasks to be performed within 2 minutes.

I'd need to check Thomson's paper again to ascertain whether that properly describes the particulars of his view.

But, yes: The infinite sequence of durations converges to 0. And the marked time converges to 2 minutes. But it seems that Thomson is saying that it is in the nature of tasks that there is not an infinite sequence of them such that they are all completed in finite time.

If there is a greatest divisor, then there is a greatest natural number, call it 'g'. So then what is g+1? If one says addition is not allowed with g as a summand, then one needs to come up with a different definition of addition, which becomes very very complicated if we wish to still have addition in a formal theory.

Also quite inelegant. If, for arbitrary example, we say that g = 66589080980923842343287098023450390811321445645098760011287390453735490233999934393 is the largest natural number, then naturally a person would want to say, "My, that's awfully specific for mathematics that we would like to be most general."

Moreover, why should numbers be limited to only how many particles there are? Such a limitation would preclude the natural human inclination to ruminate on such things as, "Suppose we made a mistake and there are actually twice that number of particles" or even with g as all the possible finite combinatorial arrangements of the particles. Well, that entitles us to talk about numbers larger than g.

Ultrafinitists are welcome to try to convince me, but I am a tough customer when it comes to giving up my natural prerogative to add 1 to any number.

No, I think (as did he) that it successfully shows that supertasks are not possible. — Michael

Yes, that's what I thought. I think the concept of a valid paradox is a bit confusing.

Yes. If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible. — Michael

But space or time being infinitely divisible does not entail that supertasks are possible.

In this case the mistake is in the application of transfinite numbers. — Michael

That's the first I've heard of any use of transfinite numbers in this thread. I don't think they are relevant - more, I very much hope they are not relevant.

He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time. — TonesInDeepFreeze

How is that possible? Infinite means without limit.
Surely, the number of tasks you can complete in a given time depends on how long they take. If you want to perform an infinite number of tasks in a limited time, just define a task that takes the appropriate amount of time. in the puzzle, each task takes less time to perform, without limit. The trouble with Thompson's lamp is that no switch can function in an infinitely small time.

I am a tough customer when it comes to giving up my natural prerogative to add 1 to any number. — TonesInDeepFreeze

I'm sure it could count as a human right. But can we also stand up for the right to form the inverse of any natural number? (For clarity, forming 1/2 from 2, 1/3 from 3 and so on. (I'm not sure whether 0 or 1 need to be included here.)

That's the first I've heard of any use of transfinite numbers in this thread. I don't think they are relevant - more, I very much hope they are not relevant. — Ludwig V

If we're talking about an infinite number of tasks being performed then we are talking about a transfinite number of tasks being performed.

But space or time being infinitely divisible does not entail that supertasks are possible. — Ludwig V

I'm not talking about physical possibility. But even then, if space and time are infinitely divisible then motion is a physically possible supertask.

Yes, that's what I thought. I think the concept of a valid paradox is a bit confusing. — Ludwig V

Then forget the word "paradox". If Thomson's argument is valid then it proves that supertasks are impossible.

Then, must mathematics not allow smaller numbers? — TonesInDeepFreeze

Smaller numbers are not needed. This idea simply produces unnecessary complications. When you divide one thing into two, you get two things, not two halves. The idea that you get two halves when you divide one thing into two, rather than getting two new whole things, causes the problem being discussed in this thread

So, if someone claims that the mathematics is to blame, then we would ask whether the mathematics itself (which holds that there is no smallest number) needs to be rejected, or whether the way in which the mathematics is applied needs to be rejected, or both. — TonesInDeepFreeze

Yes, I do believe that the mathematics needs to be changed, for the reason given above. The issue, (as I stated earlier in the thread), is that division presupposes an entity or object to be divided. And, divisibility is dependent on the type of thing to be divided. Therefore, when it comes to division one standard does not fit all things, and the principles of division must be specifically designed for the different type of things to be divided.

Maybe some mathematicians like to think that "a number" is a type of thing, or object, and that there is no limit to the way that this type of object may be divided. But I think that's just a mistaken idea.

He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time — TonesInDeepFreeze

I've been thinking about this. My comment on this was wrong. Of course, one cannot complete infinitely many tasks in a finite time. "Complete" does not apply to infinite series, by definition.
On the other hand, what counts as one task. If one takes three steps, one completes three tasks. But that distance can be analysed in many different ways, so it could be represented as one task, or many.

I'm not talking about physical possibility. But even then, if space and time are infinitely divisible then motion is a physically possible supertask. — Michael

It depends on how you choose to analyse it.

Underphysician Metacover as the Baker:

Customer: I'd like a cherry pie, divided in two. I'm going to give one half to my niece and that other half to my yoga teacher.

Baker: That makes no sense. I would have to cut the pie, and then there would be two different things, not two halves of the same pie.

Gina: Excuse me?

Baker: You heard me.

Gina: I heard you. But I don't understand.

Baker: What don't you understand about the fact that when you slice something apart, there are never halves of anything, only two new things? If you still don't get it, then I suggest you read my posts at the 'Phil's Ossify For 'Em' website. It's a philosophy place where I write my posts showing that all of mathematics is wrong.

Gina: Well, I've studied philosophy and mathematics, and have not read anything like what you're saying.

Baker: Exactly. If you want to know what's really up, you have to come to me for it.

Gina: I just want you to cut the pie in half and put the separate halves in separate boxes. If necessary, I'll pay extra for you to do that.

Baker: There is no money in the world that would permit me to cut a pie in half. It's not a matter of money. It's that it is metaphysically, ontologically and mathematically impossible to do. Now I can slice a pie. But I can't call the pieces halves or sell them as halves. I can only sell you two new things that are not to be referred to as halves of anything. So would you like one new thing in its box and another new thing in its box?

Gina: Yes please.

Baker: Fine. But not if you call them halves!

Gina: Okay, I promise not to do that.

Later that day Gina arrives home and talks to her husband Ralph and son Timmy:

Gina: I had the most peculiar conversation with the new owner of the bakery downtown. He insists that he can't sell me two pie halves but only two different things that are made of pie.

Ralph: Yes, I know. He is a bit odd. Last week I asked for a baker's dozen of bagels and he gave me only twelve even though I reminded him that a baker's dozen is thirteen. He said that is a contradiction in terms and instead I need to ask for thirteen at the same price as for twelve and he'd do that. So I said, okay, just give me one more bagel to add to the twelve, since 12+1 is 13. Then he went into thing about how 12+1 is not 13, that numbers aren't even things of any kind, and that people are all wrong about the law of identity when they say things like "1+1 is 2". He even said that '1 is 1' is false because the first symbol '1' is not the second symbol '1'. Very strange fellow.

Timmy: I talked with him too. He has all kinds of very strong opinions about math, but he doesn't know anything at all about.

Meanwhile, still interested in knowing what the poster would claim to be the inherent ordering of the set whose members are the bandmates in the Beatles.

If the poster can't answer that question, then he lacks basis for his dogma that every set has an inherent ordering, which goes to the heart of his bizarre imaginings about mathematics.

I'm going to have a cup of tea. I shall divide it into parts 1/2, 1/4, 1/8 ... so it will last for as long as I want it to.

If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible. — Michael

Did Thomson make that argument? Was that part of his answer to the paradox?

Everyone knows that tea is taken at at the tea time hour and that one is not to dawdle still drinking it, not even hypothetically, not even gedankenishly, past the tea time hour.

Infinite means without limit. — Ludwig V

Not in mathematics.

The trouble with Thompson's lamp is that no switch can function in an infinitely small time. — Ludwig V

The lamp puzzle doesn't require anything to occur in an infinitely small amount of time.

And I don't think the discussions about a switch being moved or any aspect of the agency by which the light changes are relevant. It is missing the point to quibble about the mechanics of how the light changes. We need only take it for granted that it does change at the rate stated in the puzzle.

I'm sure it could count as a human right. — Ludwig V

A deep reading of American history reveals that the right to the arithmetic operations was to be enshrined in the Bill of Rights. But it failed to pass because the mid-Atlantic states feared that too much public exercise of arithmetic would allow citizens to become too number savvy and that would hamper the sports betting industry that was legal back then, especially in New Jersey.

the mistake is in the application of transfinite numbers — Michael

The only infinite number in the puzzle is the domain of the sequence.

It's not that complicated. Imaginary numbers have a use – even in electrical engineering – but I cannot have an imaginary number of apples in my fridge.

There's nothing wrong with maths, just sometimes an improper use of it. There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then it would suggest that there is a smallest unit of space and/or time – and that this isn't just a contingent fact about the physics of our world but something far more necessary. — Michael

So it seems your analogy is between misuse of imaginary numbers and misuse of infinite numbers.

I might not put it that way. But I do understand Thomson's point that there cannot be infinitely many task steps executed in a finite duration.* And I understand the different argument that the puzzle may dissolve if we allow that there is a shortest distance and shortest duration.

* EDIT: As I mentioned in an edit a few posts ago, I may have erred. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.

Mathematics doesn't say there is no limit to the ways objects may be divided.

Where does such a claim about mathematics even come from? What actual piece of written mathematics is claimed to say such an unfocused thing?

Rather, division of real numbers (which is the subject here) is merely by definition:

for y not= 0, x/y = the unique z such that z*y = x. And that is based on having previously proven that, for y not= 0, and for any x, there is a unique z such that z*y = x.

I know of no mathematician who wrote gibberish saying that "there is no limit to the ways objects may be divided".