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.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
Thanks for the confirmation. I don't think that position is at all plausible. But are there any non-extreme positions around this topic?Yes, ultra-finitists reject mathematical induction as a proof method, but that is a rather extreme position. — 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.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
For the sake of the argument the sensors just exist at their locations. — Michael
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
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? — Ludwig V
If spacetime is continuous and infinitely divisible, as is assumed, then an infinite number of two dimensional sensors can fit within finite space. — Michael
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
A thought experiment like this is perfectly appropriate in philosophy. — 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? — Metaphysician Undercover
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
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.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
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:-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
That corresponds to what your screens will show. That's all perfectly clear and correct.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
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
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 think there's another bugbear at issue here - the idea that whatever can be imagined is at least logically possible. — Ludwig V
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.So we might allow that whatever is not self-contradicting is logically possible, but one logical possibility might be incompatible with another. — 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.Infinite divisibility is probably the most useful, but it is incompatible with empirical observation, as these paradoxes show. — 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. — Ludwig V
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
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?Imagining involves a sense image, and this is where the difficulty arises because imagination defers to empirical data. — Metaphysician Undercover
But people frequently disagree about whether a specific proposition is self-contradictory and/or incoherent or not - as in this thread.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
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).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
... 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.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
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?It is this, the idea of dividing a definite section of space and time, indefinitely, which creates the problem. — Metaphysician Undercover
We frequently (in the context of sf fiction, for example, imagine faster-than-light travel between the stars. — Ludwig V
Or consider Michael's two-dimensional sensors? — Ludwig V
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
I agree with you that the problem arises in applying mathematics to the physical world, specifically to space and time. — Ludwig V
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
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
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
OK. That seems clear enough for now. I won't argue about words.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
What empirical data do you have in mind?The problem though is that .... the (unimaginable) mathematical conception of an infinitely divisible continuum is not consistent with the empirical data. — Metaphysician Undercover
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
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?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
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:-Why therefore, do you conclude that we can do something more with the space than we can do with the cheese? — Metaphysician Undercover
By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum.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
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.When we describe this principle of separation we also provide ourselves with the basis for division. — 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
And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting is endless. — Ludwig V
What empirical data do you have in mind? — Ludwig V
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
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
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
By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum. — Ludwig V
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
Well.do you know of anything that's actually infinitely divisible?Although I don't agree there is a problem with "infinite divisibility"... — jgill
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
I'm surprised. Could you explain why?There can be no counting to begin with. — jgill
That's odd. The surfaces of the objects around me look as if they are continuous.The continuum of mathematics is not consistent with any sense evidence. — Metaphysician Undercover
You said:-By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum.
— Ludwig V
Speak for yourself. — Metaphysician Undercover
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
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.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. — 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.Well, do you know of anything that's actually infinitely divisible? — 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).What do you mean? What is this difference between distinguishing and separating? — Metaphysician Undercover
The surfaces of the objects around me look as if they are continuous. — Ludwig V
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
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
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
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
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.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
I'm sorry. I get confused sometimes about who said what. I'm glad we agree on that.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
You are right. I should have put the point differently - something along the lines you used.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
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.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
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.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
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
Should I be talking about a bijection between the non-dimensional points on a line and the set of integers? — Ludwig V
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