The sequences may approach 200m and 12:00:20, but because there is no sensor on the 200m finish line neither 200m nor 12:00:20 will display on the screen. — Michael

Assuming you maintain a constant speed, you will pass 200m at 12:00:20, as you point out. That is also the point I was after.
One might think that the screen will display the penultimate distance and time. But that is not defined. Nor is the distance and time before that defined. Nor is the distance and time at any other stage defined. You had this discussion a while ago, as I remember it. You can't count backwards because the argument doesn't give you the information you need to do so.
Ordinary arithmetic will not give you an answer at the limit, or at any other specified stage counting forwards, but not at any stage defined by reference to the last stage of the series. I didn't read the case you proposed carefully enough. I had in mind the Achilles case. My bad.

Yes, ultra-finitists reject mathematical induction as a proof method, but that is a rather extreme position. — SophistiCat

Thanks for the confirmation. I don't think that position is at all plausible. But are there any non-extreme positions around this topic?

I was referring to how folk who are unfamiliar with specialist terms that are based on words in the ordinary language try to make sense of those terms: they interpret them in light of the more familiar senses of the words. Naturally, this doesn't always work. Misinterpretations happen even in familiar contexts, and they are all the more likely in an unfamiliar domain. And as with neologisms, some just aren't going to like the coining for one reason or another, even when they understand the context. But that alone shouldn't be a barrier to understanding and accepting specialist terms. — SophistiCat

I think most people in this day and age can cope with specialist jargon. Many of them speak one of the many jargons available. There's an additional problem here, that the context is so startlingly different from common sense.

For the sake of the argument the sensors just exist at their locations. — Michael

To avoid the problem , you just assume the impossible. There is a limit to the number of sensors which can exist in that space, depending on the size of the sensors, Because a sensor takes up space. Or, are you assuming that an infinite number of sensors can fit in a finite space?

To avoid the problem , you just assume the impossible. There is a limit to the number of sensors which can exist in that space, depending on the size of the sensors, Because a sensor takes up space. Or, are you assuming that an infinite number of sensors can fit in a finite space? — Metaphysician Undercover

They can if they are infinitely small. Is it possible that you can imagine that? Is there any argument that will settle the issue either way?

To avoid the problem , you just assume the impossible. There is a limit to the number of sensors which can exist in that space, depending on the size of the sensors, Because a sensor takes up space. Or, are you assuming that an infinite number of sensors can fit in a finite space? — Metaphysician Undercover

The sensors are two dimensional with a width and height but no length. If spacetime is continuous and infinitely divisible, as is assumed, then an infinite number of two dimensional sensors can fit within finite space.

But if you prefer then we can stipulate that only one sensor exists at a time, the next placed only when the previous has been passed.

A thought experiment like this is perfectly appropriate in philosophy. See for example Bernadete's Paradox of the Gods which is similar in kind to mine.

They can if they are infinitely small. Is it possible that you can imagine that? Is there any argument that will settle the issue either way? — Ludwig V

Obviously, a sensor cannot be infinitely small. It consists of components.

If spacetime is continuous and infinitely divisible, as is assumed, then an infinite number of two dimensional sensors can fit within finite space. — Michael

That is not necessarily the case. A sensor is a material object, space and time are not material objects. There is no necessity that the limitations of a material object are the same as the limitations of space and time. In the end, it's all conceptual, and the problem is in making the conception of an object consistent with the conceptions of space and time.

What these "supertasks" show us is that there is a disconnect between the conceptual structures of mathematics and the concepts of the empirical, natural philosophy, (science). The problem is compounded when mathematicians assume that their conceptions are objects, and these supposed objects get integrated into the work of scientists so that the boundary between the two incompatible conceptual structures is lost. This is the case in quantum physics, where the influence of mathematics allows for a non-dimensional object in the physical world, virtual particles. The purely imaginary concepts of mathematical objects is allowed to penetrate the theories of physics to the point where physicists themselves cannot distinguish between the real and the imaginary.

But if you prefer then we can stipulate that only one sensor exists at a time, the next placed only when the previous has been passed. — Michael

Come on Michael. Fire Ologist explained the problem with "placing", and you said, we could assume that they are already placed. Now I show you the problem with "already placed", and you say we can assume placing. What's the point in switching back and forth, when both are shown to be problematic? Move along now.

A thought experiment like this is perfectly appropriate in philosophy. — Michael

Sure, and the purpose of such thought experiments is to determine the underlying conceptual problems. If someone denies that the exposed conceptual problems are problems, then the purpose of the thought experiment is defeated.

Come on Michael. Fire Ologist explained the problem with "placing", and you said, we could assume that they are already placed. Now I show you the problem with "already placed", and you say we can assume placing. What's the point in switching back and forth, when both are shown to be problematic? — Metaphysician Undercover

We can assume that they simply exist in their places or we can assume that they are placed just before the runner reaches the next designated distance.

What these "supertasks" show us is that there is a disconnect between the conceptual structures of mathematics and the concepts of the empirical, natural philosophy, (science). — Metaphysician Undercover

I agree. I am trying to prove this by accepting the assumptions of those who believe in supertasks and then showing that their assumptions entail a contradiction. This is how refutation by contradiction works, and is going to be more convincing than an argument that denies their assumptions outright.

The purely imaginary concepts of mathematical objects is allowed to penetrate the theories of physics to the point where physicists themselves cannot distinguish between the real and the imaginary. — Metaphysician Undercover

Yes. I am neither physicist nor mathematician, and I'm not sure that a bystander like me has a proper basis for an opinion. But after the discussions on this thread, I understand the point.
I think there's another bugbear at issue here - the idea that whatever can be imagined is at least logically possible.

We can assume that they simply exist in their places and are two dimensional or we can assume that they are placed just before the runner reaches the next designated distance. — Michael

Why don't you just stick to the mathematics? If we ask about any specific stage of the series, we can calculate exactly what time, as you show:-

Say we run at a constant speed. We pass the 100m sensor at 12:00:10, the 150m sensor at 12:00:15, the 175m sensor at 12:00:17.5, and so on. — Michael

That corresponds to what your screens will show. That's all perfectly clear and correct.
It is also perfectly clear that we cannot place screens at each stage, even with your modifications, because we cannot complete the series. That is reflected in the fact that we cannot calculate the distance and time of the last stage, or the penultimate stage, or the one before that.
These so-called thought experiments are just distracting fairy tales.

We can assume that they simply exist in their places or we can assume that they are placed just before the runner reaches the next designated distance. — Michael

As Fire Olo pointed out, if they are placed, you never get finished placing them, if it were the case that you could carry out what is prescribed. So the runner can never get past them all. And if they already exist in their places, there is the problem I pointed to, the sensors, being material objects cannot physically fit in the space as prescribed.

I agree. I am trying to prove this by accepting the assumptions of those who believe in supertasks and then showing that their assumptions entail a contradiction. This is how refutation by contradiction works, and is going to be more convincing than an argument that denies their assumptions outright. — Michael

I know that's what you're trying to do, but you haven't succeeded in that way. And I think you misunderstand where the true contradiction lies, and that's what misleads you into thinking that you ought to be able to prove some other contradiction.

The contradiction is actually within the assumptions which you accept. As I've said since the beginning, the contradiction is between the premises of the prescribed supertask, and your assumption, that the amount of time which serves as the limit which the supertask approaches, will actually pass. In other words, if you accept that the prescribed supertask can carried out, than you must deny the possibility that the limiting amount of time will ever pass. The supertask makes it impossible for that amount of time to pass. And, vise versa, if you accept that the limiting amount of time will pass, then you deny the possibility of carrying out the supertask. It's actually quite simple, and Fire Ologist demonstrates a very clear understanding of this situation, where the two conceptual frameworks ( the conditions of the supertask, and the condition of the limiting amount of time passing) are simply incompatible.

I think there's another bugbear at issue here - the idea that whatever can be imagined is at least logically possible. — Ludwig V

It's better stated that distinct things which are logically possible, may be mutually exclusive. So we might allow that whatever is not self-contradicting is logically possible, but one logical possibility might be incompatible with another. When logical possibilities are incompatible, there is not necessarily one specific method which we would use to choose one over the other. For example, we might choose the most useful one, or we might choose the one which is most consistent with empirical observation. The two are not always the same, and that appears to be the issue here. Infinite divisibility is probably the most useful, but it is incompatible with empirical observation, as these paradoxes show.

So we might allow that whatever is not self-contradicting is logically possible, but one logical possibility might be incompatible with another. — Metaphysician Undercover

That seems reasonable. But the question arises whether we can imagine something that is logically impossible. Philosophical practice says no, we can't (thought experiments) and yes, we can (reductio arguments). I suppose if two contradictory statements follow from a single premiss, we can conclude that the premiss is self-contradictory. But then, that's not always obvious, as in this case.

Infinite divisibility is probably the most useful, but it is incompatible with empirical observation, as these paradoxes show. — Metaphysician Undercover

I'm not convinced of that. I think that the confusion develops from not distinguishing between "+1" as a criterion for membership of the set of natural numbers and as a technique that enables to generate them in the empirical world.
When we consider the first use, we think of the entire set as "always already" in existence; when we consider the second, we get trapped by the constrictions of time and space in the world we live it. The difficulties arise because it seems on the one hand that we can never specify the entire set by means of applying the algorithm and yet we can prove statements that are true of the entire set. This oscillation between the abstract and timeless and the concrete and time/space bound is very confusing, and, what's worse, it (the oscillation) encourages us to think that an infinite series can be applied to the physical world in just the same way as an ordinary measurement.
I'm channelling Wittgenstein here. I don't think finitism can make sense of this, but I'm deeply sympathetic to his approach to philosophy.
That's all wrong, of course. It's only an attempt to point towards an approach.

That seems reasonable. But the question arises whether we can imagine something that is logically impossible. Philosophical practice says no, we can't (thought experiments) and yes, we can (reductio arguments). I suppose if two contradictory statements follow from a single premiss, we can conclude that the premiss is self-contradictory. But then, that's not always obvious, as in this case. — Ludwig V

I believe this involves the distinction between imagining and saying. We can say contradictory things like "square circle", but can we imagine such things? Imagining involves a sense image, and this is where the difficulty arises because imagination defers to empirical data. So mathematics uses a technique where terms are defined, and the sense image is not necessary. For instance, a nondimensional point, infinite divisibility, etc.. These things cannot be imagined.

So the issue is not whether things can be imagined, but whether they can be defined so as to coherently fit into a conceptual structure without contradiction. In this way mathematics removes itself from imagination, and the empirical world associated with it.

I'm not convinced of that. I think that the confusion develops from not distinguishing between "+1" as a criterion for membership of the set of natural numbers and as a technique that enables to generate them in the empirical world.
When we consider the first use, we think of the entire set as "always already" in existence; when we consider the second, we get trapped by the constrictions of time and space in the world we live it. The difficulties arise because it seems on the one hand that we can never specify the entire set by means of applying the algorithm and yet we can prove statements that are true of the entire set. This oscillation between the abstract and timeless and the concrete and time/space bound is very confusing, and, what's worse, it (the oscillation) encourages us to think that an infinite series can be applied to the physical world in just the same way as an ordinary measurement.
I'm channelling Wittgenstein here. I don't think finitism can make sense of this, but I'm deeply sympathetic to his approach to philosophy.
That's all wrong, of course. It's only an attempt to point towards an approach. — Ludwig V

I don't see the relevance of "+1". The supertasks described here involve an endless division, not adding one in an endless process. These two are completely different. The formula for "+1" involves no limitations of space or time, so there are no restrictions and it can simply continue forever, without any inconsistency with empirical observation. The supertasks however, start with a defined space and time, and start dividing that specified section.

It is this, the idea of dividing a definite section of space and time, indefinitely, which creates the problem. What i think, is that the assumptions which provide for a definite section, also deny the possibility of indefiniteness. So for example, assume "one hour". To validate this measurement a beginning and end point is required. The assumed beginning and end point allow for the specified "one hour" and these points cannot be arbitrary because "one hour" is an empirically defined period of time. If the points are not arbitrary, they must inhere within time itself, therefore possible division would be dependent on those points, and could not be indefinite. In other words, a "definite section" relies on nonarbitrary points, but this is incompatible with infinite divisibility.

Imagining involves a sense image, and this is where the difficulty arises because imagination defers to empirical data. — Metaphysician Undercover

At first sight, that seems to be true. But is the impossibility of imagining a round square based on trying to imagine such a thing and failing? We frequently (in the context of sf fiction, for example, imagine faster-than-light travel between the stars. What picture could possibly constitute imagining it? Or consider @Michael's two-dimensional sensors?
(I won't bother with the psychologists' empirical claim that people differ in the extent to which they actually make a picture when they imagine something.)

So the issue is not whether things can be imagined, but whether they can be defined so as to coherently fit into a conceptual structure without contradiction. ..... In this way mathematics removes itself from imagination, and the empirical world associated with it. — Metaphysician Undercover

But people frequently disagree about whether a specific proposition is self-contradictory and/or incoherent or not - as in this thread.

I don't see the relevance of "+1". The supertasks described here involve an endless division, not adding one in an endless process. — Metaphysician Undercover

The problem for me, then, is that I do not see a relevant difference between "+1" and "<divide by>2" or "divide by>10". (The latter is embedded in our number system, just as "+1" is embedded in our number system).

So mathematics uses a technique where terms are defined, and the sense image is not necessary. For instance, a non-dimensional point, infinite divisibility, etc.....In this way mathematics removes itself from imagination, and the empirical world associated with it. — Metaphysician Undercover

... so you don't see a relevant difference, either. I agree with you that the problem arises in applying mathematics to the physical world, specifically to space and time.

It is this, the idea of dividing a definite section of space and time, indefinitely, which creates the problem. — Metaphysician Undercover

But if that's your problem, you ought to have a difficulty with "+1", because there are an infinite number of non-dimensional points between my left foot and my right foot whenever I take a step. Or are you thinking that "+1" involves adding a physical object to a set of physical objects?
If you don't have a problem with that, I can't see why you have a problem with a infinite convergent series.
There are real practical difficulties with the idea that a cheese can be cut up into an infinite number of pieces (which could then be distributed to an infinitely large crowd of people). I don't deny that. But dividing the space that the cheese occupies into an infinite number of pieces is a completely different kettle of fish. It doesn't involve cutting anything up and, hopefully, not imagining cutting anything up either.

We frequently (in the context of sf fiction, for example, imagine faster-than-light travel between the stars. — Ludwig V

I would not call that "imagining". Like the "round square" it's simply a case of saying without imagining. An author can say that the space ship moves from here to there in a time which implies faster than the speed of light, but to imagine faster than the speed of light motion requires imagining a material body moving that fast. That body moving that fast, could not be seen, and therefore cannot be imagined.

Or consider Michael's two-dimensional sensors? — Ludwig V

Much of what is said in this thread, and supertasks in general, involve this problem, saying things which cannot be imagined. It's easy to say things which cannot be imagined, and we justify these things through logical possibility, but when logical possibility conflicts with empirical principles, then we have a problem.

The problem for me, then, is that I do not see a relevant difference between "+1" and "<divide by>2" or "divide by>10". (The latter is embedded in our number system, just as "+1" is embedded in our number system). — Ludwig V

They are completely different principles. You're comparing apples and oranges, and saying 'my comparison is relevant because they are both fruit'. We are not talking about "our number system" in general, because that is not the problem. We are talking about a very specific problem which is infinite divisibility, not the general "fruit" (number system) but the specific apple (infinite divisibility).

I agree with you that the problem arises in applying mathematics to the physical world, specifically to space and time. — Ludwig V

The problem is exactly what @Michael has been insisting on, the assumption that space and time are continuous. This supports the principle of infinite divisibility. The problem though is that space and time are conceptions abstracted from empirical observation, how material things exist and move, and the (unimaginable) mathematical conception of an infinitely divisible continuum is not consistent with the empirical data. Hence the Zeno type paradoxes.

But if that's your problem, you ought to have a difficulty with "+1", because there are an infinite number of non-dimensional points between my left foot and my right foot whenever I take a step. Or are you thinking that "+1" involves adding a physical object to a set of physical objects? — Ludwig V

Why does "+1" need to imply anything other than counting? There is nothing between one and two in the act of counting, yet they are distinct. We need to account for that distinction. What separates one from two? When we describe this principle of separation we also provide ourselves with the basis for division.

If you don't have a problem with that, I can't see why you have a problem with a infinite convergent series. — Ludwig V

The problem is not "infinite convergent series". That is a misrepresentation which has occurred over and over again on this thread. The "infinite convergent series" is a particular mathematical idea which has emerged from a proposed solution to the problem of infinite divisibility. The problem arises when people believe that the infinite convergent series is the necessary outcome of the problem of infinite divisibility instead of seeing it as one possible representation.

If you don't have a problem with that, I can't see why you have a problem with a infinite convergent series.
There are real practical difficulties with the idea that a cheese can be cut up into an infinite number of pieces (which could then be distributed to an infinitely large crowd of people). I don't deny that. But dividing the space that the cheese occupies into an infinite number of pieces is a completely different kettle of fish. — Ludwig V

Why do you say this? The cheese is an imaginable, sensible object. The conception of "the space that the cheese occupies" is completely dependent on, and therefore abstracted from that empirically observed cheese. Why therefore, do you conclude that we can do something more with the space than we can do with the cheese?

I would not call that "imagining". Like the "round square" it's simply a case of saying without imagining. An author can say that the space ship moves from here to there in a time which implies faster than the speed of light, but to imagine faster than the speed of light motion requires imagining a material body moving that fast. That body moving that fast, could not be seen, and therefore cannot be imagined. — Metaphysician Undercover

OK. That seems clear enough for now. I won't argue about words.

The problem though is that .... the (unimaginable) mathematical conception of an infinitely divisible continuum is not consistent with the empirical data. — Metaphysician Undercover

What empirical data do you have in mind?

The problem is exactly what Michael has been insisting on, the assumption that space and time are continuous. This supports the principle of infinite divisibility. — Metaphysician Undercover

The problem arises when people believe that the infinite convergent series is the necessary outcome of the problem of infinite divisibility instead of seeing it as one possible representation. — Metaphysician Undercover

You seem to be saying in the first quotation that the assumption that space and time are continuous gives rise to the problem of infinite divisibility and in the second that the problem of infinite divisibility gives rise to the problem of infinite convergent series. I must be misunderstanding you. Can you clarify?
But I agree with you that the convergent infinite series is a possible representation of certain situations. (I would call it an analysis, but I don't think the difference matters much for our purposes.) All I'm saying is that it doesn't give rise to any real problems unless you confuse that representation with the cutting up of a physical object.

Why therefore, do you conclude that we can do something more with the space than we can do with the cheese? — Metaphysician Undercover

Because the cheese is a physical object and the space is not an object and not physical. You seem to be saying the same thing here:-

The problem though is that space and time are conceptions abstracted from empirical observation, how material things exist and move, and the (unimaginable) mathematical conception of an infinitely divisible continuum — Metaphysician Undercover

By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum.

When we describe this principle of separation we also provide ourselves with the basis for division. — Metaphysician Undercover

And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting is endless. The surprise is entirely due to mistaking non-dimensional points for a physical object - thinking that we can separate them, rather than distinguish them.

The problem arises when people believe that the infinite convergent series is the necessary outcome of the problem of infinite divisibility instead of seeing it as one possible representation. — Metaphysician Undercover

Although I don't agree there is a problem with "infinite divisibility", another procedure you might mention is described by Tannery's theorem, which concerns series in which each term changes as the series progresses. In the extreme case, a series in which each term converges to zero as described will itself converge to zero. I.e., infinite summable to zero.

(I extended this idea to composition theory some time back Generalizations . . .)

Although you and I don't agree on the soundness of established mathematics, I do enjoy reading what you have to say.

And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting is endless. — Ludwig V

There can be no counting to begin with.

Mathematicians don't claim that the mathematical sense of 'countable' corresponds to the everyday sense of counting a finite number of objects. The use in mathematics is a certain technical sense:

S is countable if and only if either S is finite or there is 1-1 correspondence between S and the set of natural numbers.

And in that mathematical context it is not the case that 'infinite' and 'uncountable' mean the same.

Technical fields of study often have special definitions. Quibbling about that is pointless.

It was claimed that certain ideas in physics are mixed up because of importation of certain mathematics. What are some specific examples of published work in that regard?

What empirical data do you have in mind? — Ludwig V

The physical evidence, is that we cannot just keep dividing something forever. There is nothing which provides us with the capacity to keep dividing it. The continuum of mathematics is not consistent with any sense evidence.

You seem to be saying in the first quotation that the assumption that space and time are continuous gives rise to the problem of infinite divisibility and in the second that the problem of infinite divisibility gives rise to the problem of infinite convergent series. — Ludwig V

I would say more, that the assumption of infinite divisibility gives rise to the idea of continuity, and the idea of continuity supports the idea of an infinite convergent series. So, to state it simply, the infinite convergent series is the result of, or produced by, the way that calculus deals with continuity. And we need to deal with continuity because we assume that some things (space and time) are infinitely divisible.

The root problem, I claim, is infinite divisibility. From this is derived the concept of "continuity", "continuum", and calculus deals with the continuum by applying the infinite convergent series. Since infinite divisibility is a bogus concept, the whole thing is a problem.

But I agree with you that the convergent infinite series is a possible representation of certain situations. (I would call it an analysis, but I don't think the difference matters much for our purposes.) All I'm saying is that it doesn't give rise to any real problems unless you confuse that representation with the cutting up of a physical object. — Ludwig V

It's not an analysis, but a hypothesis. Infinite divisibility is a theory. And, it does give rise to real problems, as is the case when the representation is not true,

Because the cheese is a physical object and the space is not an object and not physical. You seem to be saying the same thing here:- — Ludwig V

The point is that "space" as a concept, and "time" as a concept, are both derived from our experiences of sensing the world. Kant was wrong to say that these concept are somehow prior to, as necessary for sense experience. Since these concepts are derived from our experiences, then whenever they differ, or are incompatible with our experiences, they are faulty.

I recognize and uphold the difference between the physical things, and the concept, but I also affirm that when the concepts do not conform, there is a problem. So, cheese is like any other physical object, it is not infinitely divisible. The concept of "space" allows for infinite division, but that's inconsistent with the physical world, which "space" is supposed to provide a representation of, so there is a problem.

By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum. — Ludwig V

Speak for yourself.

The surprise is entirely due to mistaking non-dimensional points for a physical object - thinking that we can separate them, rather than distinguish them. — Ludwig V

What do you mean? What is this difference between distinguishing and separating?

Although I don't agree there is a problem with "infinite divisibility"... — jgill

Well.do you know of anything that's actually infinitely divisible?

It was claimed that certain ideas in physics are mixed up because of importation of certain mathematics. What are some specific examples of published work in that regard? — TonesInDeepFreeze

Th Fourier transform and the resultant uncertainty principle.

Specific article or text? What specific theorem of mathematics is used to derive physics that's mixed up?

By the way, related to the claim that mathematics is fundamentally errant by its notion of sets without inherent order, I'm still interested in what is supposed to be the inherent order of the set whose members are all and only the bandmates in the Beatles. Without an answer to the question, the claim that every set has exactly one inherent ordering is unsustained. I've asked the question many times, but it has not once been addressed.

There can be no counting to begin with. — jgill

I'm surprised. Could you explain why?

The continuum of mathematics is not consistent with any sense evidence. — Metaphysician Undercover

That's odd. The surfaces of the objects around me look as if they are continuous.

By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum.
— Ludwig V
Speak for yourself. — Metaphysician Undercover

You said:-

So mathematics uses a technique where terms are defined, and the sense image is not necessary. For instance, a nondimensional point, infinite divisibility, etc.. — Metaphysician Undercover

If spacetime is continuous and infinitely divisible, as is assumed, then an infinite number of two dimensional sensors can fit within finite space. — Michael

Only if space is infinitely divisible and they are not physical sensors. And you say in the quote below that a sensor is a material object.

That is not necessarily the case. A sensor is a material object, space and time are not material objects. There is no necessity that the limitations of a material object are the same as the limitations of space and time. In the end, it's all conceptual, and the problem is in making the conception of an object consistent with the conceptions of space and time. — Metaphysician Undercover

Well, do you know of anything that's actually infinitely divisible? — Metaphysician Undercover

What do you mean by "actually"? Take any natural number. It can be divided by any smaller natural number. The result can be divided by that same number again. Without limit.

What do you mean? What is this difference between distinguishing and separating? — Metaphysician Undercover

Whenever concepts are defined in relation to each other, they can be distinguished but not separated. Distinguishing is in the head, separation is in the world. Examples of inseparable distinctions are "up" and "down", "north" and "south" (etc.), "convex" and "concave", "clockwise" and "anti-clockwise", "surface" and "object" (in cases such as tables and chairs).

The surfaces of the objects around me look as if they are continuous. — Ludwig V

Isn't that surface itself an edge, a discontinuity? And isn't it true, that what you see (sense) is actually a discontinuity, and you think it to be a continuous surface? I suppose, that you might think that within the confines of the edge, there is continuity, but look closer, and you'll see colour changes, texture changes, and other deformities which indicate discontinuity within the surface.

Only if space is infinitely divisible and they are not physical sensors. And you say in the quote below that a sensor is a material object. — Ludwig V

Is this directed at me, or Michael? I maintain that a sensor is a material object consisting of components. The proposition of a non-physical sensor is incoherent.

What do you mean by "actually"? Take any natural number. It can be divided by any smaller natural number. The result can be divided by that same number again. Without limit. — Ludwig V

What I meant by "actually", is what can be carried out in practise. Your example is theory. Anything is infinitely divisible in theory. You see an object and theorize that it can be endlessly divided. But practise proves the theory to be wrong. That is the point in me asking the question of whether you know of anything which is actually infinitely divisible. This would provide evidence that the theory is not false. However, in reality, all attempts have led to a falsification of that theory of infinite divisibility. So what you have stated as your example is just a false theory.

Whenever concepts are defined in relation to each other, they can be distinguished but not separated. Distinguishing is in the head, separation is in the world. Examples of inseparable distinctions are "up" and "down", "north" and "south" (etc.), "convex" and "concave", "clockwise" and "anti-clockwise", "surface" and "object" (in cases such as tables and chairs). — Ludwig V

Aren't you making a category mistake here? If separation is in the world, and distinguishing is in the head, then your examples up/down etc., are examples of distinctions, not separations. It is a category mistake to talk about these as "inseparable" by the terms of your definitions, separable and inseparable would apply to the category of things in the world, while distinguishable and indistinguishable apply to what's in the head.

Here is your statement again. I'll ask the question in a different way, using the definitions you've provided.

And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting is endless. The surprise is entirely due to mistaking non-dimensional points for a physical object - thinking that we can separate them, rather than distinguish them. — Ludwig V

What is "the principle of distinction between non-dimensional points on a line"? How would you distinguish one point from another, except by location? But location is dimensional, and actually a separation. There actually is no distinction between one point and another, they are all exactly one and the same, by definition, each point is the very same point as another point, if they are supposed to be things. The only thing which makes them not the same is a dimensional separation, the idea that they are supposed to be at different locations in the world.

Isn't that surface itself an edge, a discontinuity? And isn't it true, that what you see (sense) is actually a discontinuity, and you think it to be a continuous surface? I suppose, that you might think that within the confines of the edge, there is continuity, but look closer, and you'll see colour changes, texture changes, and other deformities which indicate discontinuity within the surface. — Metaphysician Undercover

H'm. I thought you would throw the results of sub-atomic physics at me - that apparently solid object is mostly empty space. But you are right. You are also right that the surface of an object is a discontinuity - a border - between the object and the rest of the world. But my point is that you cannot peel the surface of an object off, in the way that you can peel a skin off it. We can distinguish between a surface, with all its irregularities, and the object, but we cannot separate them.

Is this directed at me, or Michael? I maintain that a sensor is a material object consisting of components. The proposition of a non-physical sensor is incoherent. — Metaphysician Undercover

I'm sorry. I get confused sometimes about who said what. I'm glad we agree on that.

Aren't you making a category mistake here? If separation is in the world, and distinguishing is in the head, then your examples up/down etc., are examples of distinctions, not separations. It is a category mistake to talk about these as "inseparable" by the terms of your definitions, separable and inseparable would apply to the category of things in the world, while distinguishable and indistinguishable apply to what's in the head. — Metaphysician Undercover

You are right. I should have put the point differently - something along the lines you used.

What I meant by "actually", is what can be carried out in practise. Your example is theory. Anything is infinitely divisible in theory. You see an object and theorize that it can be endlessly divided. But practise proves the theory to be wrong. — Metaphysician Undercover

So we are closer than we seem to be. The difference between theory and practice is well enough known. It is unusual to say that difference proves theory to be wrong. 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. But calculus does have uses in applied mathematics, doesn't it? I imagine that physics will come up with some interesting ideas about time and space; at the moment it all seems to be speculation, so I'm suspending judgement about that.

The only thing which makes them not the same is a dimensional separation, the idea that they are supposed to be at different locations in the world. — Metaphysician Undercover

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

There can be no counting to begin with. — jgill
I'm surprised. Could you explain why? — Ludwig V

Real numbers are uncountable.

Real numbers are uncountable. — jgill

I see. Why can't I count with natural numbers?

Real numbers are uncountable. — jgill

I see. Why can't I count with natural numbers? — Ludwig V

"principle of distinction between non-dimensional points on a line" does not specifically speak of natural numbers. Language play.

Language play. — jgill

You had me going there. :smile:
So if I had said "And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting with natural numbers is endless", you would have agreed?

So if I had said "And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting with natural numbers is endless", you would have agreed? — Ludwig V

No. If "the points on a line" correspond to integers or rational numbers, yes. Way too vague.

No. If "the points on a line" correspond to integers or rational numbers, yes. Way too vague. — jgill

Fair enough. Should I be talking about a bijection between the non-dimensional points on a line and the set of integers?

Should I be talking about a bijection between the non-dimensional points on a line and the set of integers? — Ludwig V

You can use that term, but only if you are more specific about "points on a line" and specify natural numbers or rational numbers corresponding to these points. That's it.

You can use that term, — jgill

Do you mean "bijection"?

only if you are more specific about "points on a line" — jgill

Do you mean how they are to be identified?