"One half" in practise does not have the same meaning as "1/2" in theory. — Metaphysician Undercover

I hate to be difficult, and I'm not really disagreeing, just amplifying. But I would like to add that if the pipe is cut in half lengthways, neither half is a (newly individuated) pipe. You have two gutters (or that is what I call them). And that if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours.

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. — TonesInDeepFreeze

Yes. I discovered that after the tea-time hour, it turns into a grumpy tortoise.

too much public exercise of arithmetic would allow citizens to become too number savvy — TonesInDeepFreeze

Since then, however, it has been discovered that citizens will still get themselves into a hopeless muddle even if they practice all day. So the betting industry is safe.

We need only take it for granted that it does change at the rate stated in the puzzle. — TonesInDeepFreeze

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

Yes. I was careless.

But I do understand Thomson's point that there cannot be infinitely many task steps executed in a finite duration. — TonesInDeepFreeze

But here's my problem. If I take one step, do I execute one task, or many? The argument of the paradox is that in order to take my step, I either must execute infinitely many tasks in a finite duration or fail to complete (or even begin) my step. I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations.

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

I think everyone agrees that there's misuse of something going on here. There's disagreement about what is being misused and how.

However, whatever you mean by 'complete', there are infinite series that have a sum. — TonesInDeepFreeze

I never meant to deny that.

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.

Only came to mind as I composed my reply to your earlier post. — Wayfarer

OK. Thinking on one's feet is allowed.

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.

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.)

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.

And, well, I think you can guess the problem lies (as we have been talking about a limitation on mathematic modelling). — ssu

And secondly, the result here isn't generally accepted or public knowledge. Just look at the references, videos or writings about LD. The usual idea is that since we have quantum physics, LD isn't happening because the physics isn't all Newtonian. But that's it. — ssu

Well, perhaps I'm ahead of the curve, for a change.

let's just call it "Philosophy". — Gnomon

I'll buy that. I'm sure we can get along and maybe occasionally agree to disagree. Most topics in philosophy seem to have only contested definitions, so there's nothing new here.

[/quote]

On the contrary, I think classical philosophy has always demanded something of that approach. I’m thinking for example of Pierre Hadot’s ‘Philosophy as a Way of Life’. — Wayfarer

It would have been helpful if you had mention Hadot in the first place. Philosophy as a way of life is a recognizable topic within philosophy. I've never been convinced by any proposals I've seen. So I fall back on Socrates. As you point out, for him the search was the philosophical way. I think that many of us do that. Some people give up, but it is hard to know whether that's because they have found their answers or because they have despaired of finding any. Some people don't seem to be bothered by the question at all.

According to Hadot, one became an ancient Platonist, Aristotelian, or Stoic in a manner more comparable to the twenty-first century understanding of religious conversion, — IEP

Well, light-bulb moments do occur in secular contexts. The term metanoiais quite rare, but seems to be used in quite ordinary contexts, and ancient Greece didn't discuss religious conversion in this sense, so far as I'm aware. However, metanoia isn't mentioned in any Ancient Greek philosophical work, or so Liddell & Scott tell me. I can't help feeling that both Plato and Aristotle would have insisted on rational persuasion as the only sound basis for philosophy. It is mentioned in Acts and Hebrews, but I assume that's the religious meaning.

The difficulty is, that to even attempt to name or indicate something beyond the contingent or constructed, brings it within the scope of a ‘community of discourse’ which is once again one of social construction and language. — Wayfarer

Well, you can't expect to name or indicate something without a social context and a language. I think language does quite well in dealing with the world. I doubt it would survive if it did not.

But I recall an instruction I read once, that the student (‘prokopta’, or ‘preceptor’) can become aware of certain kinds of evidential experience in their quality of life as a consequence of right realisation, although for obvious reasons that is not necessarily something ascertainable in the third person. — Wayfarer

H'm. I doubt that would stand up to even the mildest philosophical scrutiny and suspect that it would carry with it great moral dangers. But if it makes them happy and they do no harm, who's to complain?

It follows from wanting to adopt degrees of belief (which Manuel did), except for hinge propositions such as logical laws and such (the existence of God is no such proposition non-presups would agree). — Lionino

Well, I'm not fond of degrees of belief. But there are certainly ways we can qualify our commitment to what we believe. I don't think it is impossible to accept a logical law hesitantly or doubtfully. I read somewhere that Van Til's presupposition is not that God exists, but that the Bible is true.

ThEn you can't discard the pOsSiBiLity of a green donkey behind Jupiter! — Lionino

Yes, I can. There is no evidence that it is possible that there's a green donkey behind Jupiter.

The debate happens when people concede to theists the definition of 'atheist' "explicitly stating the non-existence of God", instead of the normal "not believing because there is no reason to believe": — Lionino

I think that some religious people will be quite happy to engage in debate with you on the basis that you need reasons to believe. But I suppose that does mean accepting the burden of proof. I would be absurd for an atheist to accept the burden of proof, because proving that something doesn't exist is much, much harder than proving that it does.

It isn’t robust relativism that leads to skepticism, but Idealism and empiricism, by not realizing that the practices of meaning we find ourselves enmeshed within are already real and true, already of the world, absent of any need to valid them on the basis of conformity to anything outside of these already world-enmeshed practices , ‘beyond the contingent’. — Joshs

I'm more or less with you on this, though I'm doubtful about what "beyond the contingent" means. But why do you classify that as relativism?

Our understanding doesn’t evolve by more and more closely approximating some foundational content but by using our past world-engaged practices to construct more intricately relational forms of understanding. — Joshs

Yes, I think that's about right. Foundationalism seem to provide endless questions, rather than a secure foundation.

No matter what the device says, we are free to choose the other option. — Ruckavicka above

Yes. That's why we could only ever conclude from the LD that prediction is not control and though the D may be said to determine, at least sometimes, in the sense of "discover", it cannot be said to determine in the sense of "control".

The effects of feedback of predictions on future action is very well known in economics, isn't it? Though perhaps it is the self-fulfilling prophecy that is more to be feared. That said, I do think that feedback loops in general are very important in understand how the body works, so they will contribute substantially to our understanding how we are able to do what we want - without being incompatible with determinism. That's the only way to go, in my opinion.

The limitation is essential part of logic, yet it's not understood as to be so. — ssu

I'm not quite sure what you mean. But I think that the important logical part of this is that the future is unlike the past in the sense that a prediction is not really true or false, but fulfilled or not. Compare G. Ryle "Dilemmas" Lecture II 'It Was To Be'. But I'm not aware that it has been discussed in the context of the symmetry of past and future in science.

Your pitch in is welcome, indeed. But you must have expected to encounter disagreement - you've been involved here for quite a while.

The difficult point about religious doctrines, in particular, is that they generally demand certainly qualities of character. — Wayfarer

I'm happy to agree that religious beliefs, on the whole, are not empirical - although Christ's Resurrection is often claimed (isn't it?) to be a historical (empirical) fact. But the idea that believing them requires certain qualities of character looks like an empirical claim to me.
I don't have any proper empirical evidence, but anecdotally, I've found all sorts of people hold religious beliefs and not all of them are particularly virtuous in any conventional sense. Scandals occur in amongst religious believers as well, you know.

Your quotation doesn't seem to deal with religious doctrines specifically, but with truth in general. There is a good deal of philosophical discussion of epistemological virtues and there is much to be said for that idea. However, I'm not aware of any specific arguments that everybody needs a radical transformation of character in order to know anything.

I'm not aware of purity as a moral virtue. Could you define it?

I do hope that isn't a version of the old argument that atheists are wicked. I thought we had got way beyond that.

As best i can bring myself to adopt a label, its emotivism. — AmadeusD

I always resist labels. They are supposed to be shorthand for complex views, but in practice they enable people to pigeon-hole where they have arguments prepared. It saves thought, which is almost always a bad thing. The objective/subjective distinction is another example of the same kind.

There is nothing coherent about claiming a belief and not knowledge unless you also claim the thing cannot be known — AmadeusD

That seems paradoxical. But if one believes on faith, especially in the case of religious belief, one may well believe that what one believes cannot be known, on the assumption that knowledge requires evidence and proof.

I think that "believe" has special connotations that get neglected in philosophical discussion, and perhaps in ordinary discourse as well. It can be used to declare trust or confidence in something. This is particularly prominent in religious discourse. If one believes on faith, it is not really appropriate to claim knowledge (because, perhaps, one also acknowledges that rational proof is not available).

But this is incompatible with the widespread idea that belief implies one is not certain (and knowledge implies one is certain). I'm happy to assert that that is not the case, but I doubt if anyone will pay any attention.

But I would say that a belief must be capable of being true and most people think that religious doctrines are true or false.

And when you just talk about limitations to modelling and forecasting, the debate can avoid drifting to metaphysical questions. — ssu

Well, I prefer it because it is so much easier to understand what is being said. But people seem to believe in it, and I can't work out why. The encyclopedias are not much help.

This is a good point. Free will is quite a loaded term, especially when you juxtapose free will with determinism. I think that's one of the problems here. — ssu

Quite so. But nobody seems to be interested in teasing out the complexities. It's all Freedom (capital F) and never free (attention to context and cases.) What are the differences between addiction and preference? Can people who do something in a temper plead provocation? Can a sincerely held, but completely unjustified, belief excuse a crime? (I thought the person I killed was an alien invader). And so on. Endless real questions.

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?

Do you consider philosophy to be an ideal "language game" of no importance in the "real" world? — Gnomon

That's a difficult question to answer. Language-games are not well-defined entities. They are mostly useful as heuristics the "battle against the bewitchment of our intelligence by means of our language." Some of those bewitchments are very important. How effective philosophy is in neutralizing them is hard to discern. I don't justify philosophy any more than I justify science or art. All of them are worthwhile for their own sake, though one always hopes to be fighting on the side of the angels.

I was using physical indeterminacy as a parallel analogy to the philosophical question of Freedom vs Determinism. — Gnomon

Do you see any relationship between physical freedom (mathematical value) and mental freedom*3 (metaphysical value)? :smile: — Gnomon

Not really. I think that freedom is contextually defined, except where it is inapplicable. In each context, one needs to understand what counts as a constraint or compulsion, and that can be different.
If determinism is true, freedom and constraint or compulsion are inapplicable, at least in physics and similar sciences. On the other hand, there are ordinary language uses of "free" that do give a sense to saying that insensate objects are free or constrained, but philosophy seems unwilling to recognize them.
I try not to mention metaphysics, since I don't know what it means. :smile:

Some scientists inferred that the mind of the scientist could play the role of a Cause in the experiment. — Gnomon

Well, if you are really desperate, it's worth considering. I'm surprised the parapsychologists haven't got in there years ago. It's really a wild west out there.

Indeterminacy is a mathematical concept ; whereas Freedom is a human feeling, derived from lack of obstacles to Willpower*2. — Gnomon

If indeterminacy is a mathematical concept, then so is determinism. At last, we'll get an answer. Oh, wait, mathematicians don't agree about anything, either.
Being able to do what you want to do is not a bad definition of freedom. But then, are those choices necessarily free? It seems that sometimes they are not, so it's not enough. There's something about needing to be in good physical and mental health, living in a healthy society if one is to be free.
Willpower is very problematic concept for me; it is metaphysically loaded and poorly defined, even though there are ordinary language uses that are unobjectionable. It is awkward that if someone is trying very hard to achieve something, we say that they are determined to do it.

Perhaps the brain does not operate in a "classical" way. — Gnomon

Now there's something to agree with, so long as it isn't taken to have metaphysical implications.

I think there is a real problem about understanding how physics relates to human action, and the blanket free or determined is very unhelpful. At this stage and for our purposes, it is the detailed analysis of cases that will help us most.

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?

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?

Real numbers are uncountable. — jgill

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

So one could argue that free will (or interaction) is a limit to making models, extrapolation or forecasting, but it doesn't refute determinism. — ssu

No, I think that our limits to modelling, extrapolation and forecasting do not show anything about free or constrained choices, because actions are a different category or language-game from events. For a start, they are explained by references to purposes and values, which have no place in theories of physics, etc. BTW, I think that the concept of free will is hopelessly loaded with metaphysical assumptions, and it would be much better to talk about freedom, free choices or free actions.

Determinism is not absolute. So, why assume human choices are forbidden by the gapless Chain of Cause & Effect? — Gnomon

Any events that are not determined by cause and effect are indeterminate. Freedom (or at least the philosophical version of it) is a language-game distinct from physics, etc.

Personally, I don't think human Life, or Culture, is incompatible with scientific explanation. — Gnomon

Nor do I. On the contrary, I think that scientific explanation is a part of human life of culture.

Does ordinary life require a whole different way of thinking in the same sense that we need to think of large numbers of air molecules as thermodynamics, because we simply can't perceive such a gargantuan number, much less calculate all the interactions that will take place between all of them within the space they occupy? — Patterner

I don't know about "in the same sense", because the cases are very different. But along the same lines, yes.

Is everything in this reality deterministic, — Patterner

I don't think that the idea that everything in this reality is deterministic is an empirical hypothesis. It is a completely different kind of proposition. Think of it as a research programme that defines what questions can be asked about phenomena and when they have been answered. Does that help?

But the necessity for Observer choices --- in experimental set-up, and interpretation of evidence --- resulted in "a whole different way of thinking". — Gnomon

H'm. I probably don't know enough to evaluate that. But I would have thought that observer choices in setting up experiments and interpreting evidence have always played an essential role in science. Though it is true that scientists have mostly assumed that it is possible to observe phenomena without affecting them, and that only becomes inescapably false at the sub-atomic level.

We cannot logical deduce or find out answers to metaphysical questions. If we could, they wouldn't be metaphysical. — ssu

I like that. Can we stop talking about it now?

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?

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).

Our senses and our abilities are of course limits to us, but that actually is quite a different thing. — ssu

Yes, they limit us, but the also, at the same time, they give us opportunities.

"The "strongest" system where everything is provable is with sytem where 0=1". — ssu

"From a contradiction, anything you like follows." Calling that strength is a bit counter-intuitive. But I'm not going to argue.

And before he or she thinks that you are attacking the whole idea of determinism, it should be told that the issue in the limitations of modelling that determinism, not the determinism itself! — ssu

So you are saying that the world is deterministic, even though our models will never demonstrate that?

That you did make choices isn't relevant for the determinist model: your choosing to throw the pillow is just given. — ssu

Yes. Physics doesn't have the conceptual apparatus to describe or even acknowledge choices. Ordinary life requires a whole different way of thinking.

But you hopefully understand that it's different to model this when it hasn't happened, especially you know about the model before you have thrown it. — ssu

Yes. Past and future are different, even if physics can't acknowledge the fact.

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.

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.

Can you define "freedom"? Freedom from what? — Patterner

Like Augustine and most of Socrates' interlocutors, I can participate in normal life, but that doesn't mean I can give a definition. The language-games around actions are unbelievably complicated and very difficult to summarize. The same is not true of events.

Can you give me an example of a free action? — Patterner

Tempting. But it wouldn't be a clear case. Almost everything we do can be described as free from some perspectives and not free from others. I scratch my nose because it is itching. Free or not free? What about going go to work in the morning? or signing a mortgage contract to buy a house/car? or asking a question?

The important point is that we can meaningfully ask the question about actions, but not about gas leaks or car breakdowns or pandemics or the weather. That's because actions can be free or constrained. The weather is neither free nor constrained - at least in philosophical discussion, which tries to impose a radical distinction.

Also, by "reasonably reliable", do you mean the casual network is not always reliable? If that is what you mean, can you give an example of it not being reliable? — Patterner

I had an attack of realism, remembering that my car is pretty reliable, but does break down sometimes. Causal determinism is at work either way. The distinction depends on my perspective as a human being. Do we have the same perspective on eclipses? Perhaps, perhaps not.

The question is this: Did I let go of the pillow in exactly the way I did because all the constituents of my brain - whether we examine them as particles and physics, or molecules and chemistry, or structures and biology, or whatever - acted in the only ways each of them could, all purely physical interactions driven by the physical laws?
Did I throw the pillow because all the constituents of my brain acted in the only ways each of them could, all purely physical interactions driven by the physical laws? — Patterner

The answer to those questions is yes. But the questions are asked in the context of the glass breaking and so lead us to neglect the conceptual difference between the glass breaking (an event) and my throwing the pillow (an action, normally).

If the answer is Yes, then we are not choosing things any more than the glass is choosing to break exactly as it does, or the debris is choosing to come to rest exactly as it does after an avalanche. We merely have awareness of things that the glass and mountain lack. — Patterner

So you are an epiphenomenalist?
I don't agree. If the answer is yes, that doesn't justify your conclusion. It defines our problem. If the answer is no, that also doesn't justify any conclusion. It also defines our problem.

Most traditional arguments against Fatalistic Determinism are based on Morality. — Gnomon

Yes, and they are less than persuasive for that reason. However, I think that while fatalistic determinism is easy to confuse with causal determinism, it does not pose the same problems. (I'm assuming you mean by "fatalistic determinism" what I think you mean - the ancient form that did not appeal to science and causality, but to logic and metaphysics) Roughly, Laplace's demon is a version of fatalistic determinism and easier to refute on logical grounds than causal determinism.

It's a feature of Nature that the human mind may be able to exploit in order to impose its will on Nature. — Gnomon

If we think of it like that, we are making a mistake. The human mind is a product of Nature and part of it. Or, to put it another way, to think of Nature as something to exploit perpetuates the practices that have landed us with climate change. Worse than that, although we can and do exploit Nature in some ways, Nature also imposes itself on us - witness climate change and antibiotic resistance. It has to be a balance.

Lorenz's equations have already been used to explain why the weather is unpredictable. Maybe, in time, they will also reveal why the human mind is unpredictable. — Gnomon

Yes, I'm aware that there are many examples of systems and situations that reveal that the systems at work in the world are much more complex and much less predictable than our classical models have recognized. They do give us a basis for thinking that human life may be, in the end, not incompatible with scientific explanation. But they do not get us there, any more than simple randomness gets us there. I think that the research into self-constituting autonomous systems, feedback loops and ideas like Conway's Game of Life are much more to the point.

I don't think that unpredictability is a significant phenomenon here. Volcanoes and football matches, not to mention the weather, are all unpredictable. But no-one thinks that free will is involved.

The determinism holds. But it shows that this determinism isn't at all a limit here. — ssu

Well put. Though perhaps we might say that the causal network is sometimes a limit on what we can do, and sometimes an opportunity to achieve what we want to achieve. Which it is, depends on the context of what we value, what we want, what we need on different occasions. So our attitude to the fact (insofar as it is a fact, as opposed to an aspiration) of causal determinism depends on us, not on what the facts are.

But that's the incredible thing: there isn't the influence or a controlling force with determinism! — ssu

Yes. I'm impressed by your articulation of this argument. It is very tempting to think that the causal network in our world imposes things on us; we forget that it also enables us to do the things that we want to do, or at least some of them. With respect to our values and desires, it is neutral.

Now, does this deterministic view of there being your answer 1038, 1039 and 1050 limit what you can write? No. Could they be forecasted? Again no, this isn't simple extrapolation from what has become for. — ssu

Yes. This is essentially the argument against fatalistic or logical determinism, but chimes with the neutrality of the causal network.

"Chance" and "fate" that has an effect on our lives while others don't, — ssu

I don't think that "Chance" or "fate" have an effect on our lives. "Chance" is just a basket into which we put events that we don't have an explanation for. "Fate" is another basket into which we put the things that actually happen, whatever the explanation may be.
It's important that "determine" or "determined" or "determinism" has more than one meaning. It can mean "fixed" or "exact"; it can mean "discover" or "reveal"; it can mean "control" or "influence".

they are trying to convince someone who believes in free will of the strength of their arguments – that a free willist will consider all the evidence and, in the end, choose to believe that determinism makes the most sense. — Thales

Yes, that is self-contradictory. But you don't seem to recognize that the importance of this. Insofar as we are rational, calculating (in the widest sense) animals, with goals and preferences, what we do needs to be explained in particular ways, which are not the same as the ways that we explain the way the world works. There are different, but related, language games here; our problem is to understand how they are related.
I think we can begin to get a handle on this by thinking about why we consider that computers can do calculations. A physical process can output the result of a calculation. It is clearly possible, but how?

Which leaves gaps (junctions?) in the chain of causation for the exercise of personal willpower to choose (decide) the next step. — Gnomon

The orthodox articulation of the debate requires either positing free will as a magical kind of cause that is causally determined and/or a gap in causality that allows this unique kind of event to occur. Neither is at all plausible.

And that just shows how meaningless the idea is. Because you have to make decisions. That determinism says that with probability 1 you make or abstain from making a decision has no value, because it doesn't give you any more information. — ssu

This is a promising approach, but nonetheless seems to leave our supposedly freely made decisions vulnerable to the apparently controlling force of determinism. It may not give me any information, but it will certainly influence the attitude of others to my decision, and may even influence my own attitude to my own decision.

The determinist model of you making a choice doesn't help you. — ssu

I think it's more like to hinder me. (There's a classic argument against fatalism, that it tends to make us lazy, since the causally determined outcome will happen "whatever I do" or at least whatever I do will make no difference. I realize it's a muddle, but still...)

For the hard-core determinist, there's no difference between causes and "actions" performed by "agents". But of course this making the division between causes and actions does understand they have to thought of differently. — ssu

That's certainly true. The practical syllogism, which models rational decision-making about action, is quite different from the paradigm syllogism. Practice syllogism require values, desires &c and lead to action. Neither is true of the paradigm syllogism.
But that's the point. Thinking about actions (people) is a different language game from thinking about events. But it's not a matter of two different kinds of event, but a different way of thinking about some events. Most philosophers leave the argument there, but that won't do. We have to understand how actions can be (need to be) explained in two apparently incompatible ways - as actions, and as events with causes.

If that's the key problem - and I'm sure it is - then here are a couple of out-of-the-box thoughts.
(This will only be a starting-point.)

1 Freedom is not opposed to determinism; it requires it. We could not act freely if the causal network was not (reasonably) reliable.

2 The idea that freedom does not apply to causally-determined events is a prejudice of philosophers, in pursuit of their idea of the special cause for actions. Ordinary language is perfectly happy to call such events free, and paying attention to that illuminates what free should mean. Why do we speak of bodies being in free fall? Why do require our wheels and cranks to spin freely? In both cases, we are thinking of what happens if nothing intervenes to prevent it in the context of our lives in the world. Free fall is usually a disaster, so we approach the phenomenon from that perspective - not from the dispassionate perspective of the scientist. Freely spinning wheels and cranks are important for the proper functioning of various machines - again, the human, practical perspective.

3 The (correct) idea that our brains and bodies are subject to causality is only half the story. The preparation and weighing up of our decisions may have a causal network behind it. If it works properly, we can act freely. When it goes wrong, we don't. In just the same way, when a computer carries out a calculation, success follows when the system works properly and failure follows when it doesn't. It's not about being caused or not. Admittedly, this requires interpreting or explaining the action in the light of the human beings involved in it, and this is not the same view of the world as the dispassionate examination of the workings of the machine (that metaphor is, of course, seriously misleading) that is the universe.

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.

Thanks for the thought. As it happens, I do understand. If one tries to respond to everything, it quickly becomes too much (and I, at least, get muddled). Even if one limits oneself to actual mentions, it still gets too much - especially if some of the replies overlap. But it is nice to get an acknowledgement.

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.

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?

I presume that 'the sims' are the humans in the simulation.
The hypothesis is that the sims are us, so tautologically they're as self-aware as you are.
If 'the sims' is a reference to the simulation software, program, or process, well that's a different answer since people are not hypothesized to be any of those things. — noAxioms

So the humans are entities created by the software? Then how are they not real people and not simulations of anything?

Particles interact and do their thing. Your experience is a function of matter interactions — noAxioms

Quite so. But my experience is real experience, not a simulation of experience. So the people "inside" your software are real people.

how is such a "choice" is of no any greater value or interest than is the final arrangement of the rocks and dirt when an avalanche settles? — Patterner

You are missing a trick here. Sure, the final arrangement rocks and dirt is not of any interest. But the outcome of the causal sequence of events in a calculating machine is of interest, because it instantiates a calculation, because we arranged it that way. Again, there is a causal sequence from the keys you press to my reading what you write, and that is extremely interesting. In their various ways all causal sequences are of some interest, but some are more interesting than others. The causal sequences in my brain are much more like those in a computer than they are like the final outcomes of an avalanche.

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.

That doesn't help now an outdoors event planner that is looking arranging something for summer 2026. — ssu

The problem is that people don't distinguish between different ideas about determinism. Saying "there are certain days next year when it will rain" and saying "It isn't possible to identify which days will have rain for next year" are very different claims. Both are true. Both can be described as determined or not determined. Though actually, in a case like that, we would retreat to probabilities.

Determinism doesn't say much. — ssu

That's true. But I think that's because everyone is treating it as an empirical hypothesis, forgetting that not all propositions are empirical hypotheses. Effectively, determinism defines what a complete and final explanation of an event (past, present or future) would be. It's a "regulative ideal", to steal a phrase.

Anyway, this is just a specialist term. It doesn't have to "make sense" to be cogent and useful. — SophistiCat

There isn't a problem with specialist terms. But "cogent and useful" is both cogent and useful as a definition of "make sense". I would rather not have to try to find another definition. "Cogent and useful" can mean different things in different contexts.

you only need to establish a procedure of how you would do it, or even just prove that such a procedure exists. — SophistiCat

I don't disagree. But half the problem, for us ordinary folk, is understanding that procedure, especially if, as in this case, it can't actually be carried out. The difficulty is understanding the difference between "and so on" as laziness, when it could be carried out, but one is too lazy or busy to actually do so, and "and so on" in the context of a mathematical induction, when it can't. In the background, I understand, there are people who have doubts about the validity of mathematical induction.

Locations are in physical space. This isn’t a math problem yet. — Fire Ologist

I think that's too simple. It's about the applied math. The issue is about applying the math to physical space (and time). After all, there is no problem about applying ordinary arithmetic to these situations.

When we reach the 200m finish line, what distance and what time is displayed on the screen? — Michael

Sometimes, I am so slow I cannot believe it. The answer to the question is available, if only you would apply ordinary arithmetic to the problem. All the paradox proves is that an analysis in terms of a convergent series does not apply to the question.

The thought experiment is only to examine the internal consistency of continuous space and time, not the practicality of carrying out the experiment. — Michael

It is as well not to confuse the conclusion you want to draw from the analysis with the point of the thought experiment. After all, Zeno did not draw your conclusion from it. Nor do I.

In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. That’s a square circle. We don’t get out of the gate. The infinite is by definition uncountable, — Fire Ologist

When I first saw the phrase "countably infinite", I thought that was absurd, and I still think it is an unfortunately ambiguous description of what it means. I would put it this way - any (finite) part of the infinite set can be counted, even though the whole of the set cannot be counted in one go. But I think that Wikipedia also puts it in a reasonably clear fashion.

Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements — Wikipedia

Though I would have said "even though the counting may never finish due to an infinite number of elements."

If this is a mathematics conversation then why are we ever referring to stairs, lamps, hotels, switches, starting lines at races?? — Fire Ologist

I read somewhere that Hilbert never discussed his hotel after the casual mention of it in a paper, even though it provoked enormous discussion. I'm pretty sure he invented it only to help people realize what infinity means. All these cases play in the border country between the mathematical and the physical; they are entirely imaginary (not in the sense that they are possible, but only in the sense we can imagine impossibilities). Consequently, the normal rules of possibility and impossibility are suspended and people think the fact that they can in some sense imagine them means that they are, in some sense, possible.

when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. — Fire Ologist

I agree with that and for that reason think that to say that the conclusion or limit of the set can be anything at all is misleading. In a convergent series, specifying the limit is essential to defining the series. But that doesn't mean that the function that generates the set can generate it's own limit. In fact, if it could, it wouldn't be an infinite set.
The other problem is that Zeno, and most people since him, lose track of the difference between an analysis and a dissection. If I measure the length of the race-track as being 10 units long, the race track is completely unaffected; my measurements are an analysis. If I then cut the race-track into 1-unit lengths, the race track is affected, and what I have done is to dissect it. One can analyze the race track in terms of a given convergent series. But that does not exclude other analyses, including my analysis of it as 10 units long.
My final issue with Zeno is that he forgets that if Achilles is travelling at a constant speed, he will take less and less time to travel each segment of the series, approaching an infinitesimally small time as the segments become smaller and smaller. And, of course, since he can cover a segment in an infinitesimal amount of time, covering an infinite number of them in a finite time becomes less of a problem.

Sure, they can be realized in actions, but they are not necessarily. — Janus

Surely you don't mean that love or concern may never shows themselves in any actions at all? The moral worth of that is, let us say, debatable.

I think there are higher and lower states of consciousness. — Janus

There's that higher/lower metaphor again. But I can't see just what you mean without examples.

I'm not advocating removing the anyone's rights. That would be in the domain of legal policy and that is not what I am addressing. — Janus

Those laws have been developed from what many people think are moral imperatives. Think of Kant's categorical imperative.

what I've been saying is that one who is concerned only with their own interests is morally lower than one who is concerned with their own interests and the interests of others. — Janus

I'll just substitute "worse" for "lower". OK? Certainly most selfish people are hypocritical at some level, since their personal interests depend on mutual recognition of other people. My property is my own, but only because other people have the same rights as I do.

Others do, like Zeno's and Bernadete's. — Michael

I don't think that either of them suggests that space or time may be discrete. In any case, you seem to accept that that's a different topic.

Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off. — Michael

That seems to me to be a good diagnosis of the issue with supertasks. All that is then needed to free people from the illusion that walking to the fridge can be mathematically analysed in many ways, none of which affect physical reality.

I think there are abstract things and concrete things. But physics these days pushes hard on the nature of physical things. Is there a philosopher in the house? — fishfry

The trouble is that many philosophers seem to be hypnotized by physics, and seem to forget that physicists develop their theories and conduct their experiments in ordinary human reality.

At the very least you've got work to do in "dephysicalising" or "physicalising" the intuitions regarding number and processes you have. — fdrake

Spot on. The difference between analysis (in the head) and dissection (on the bench) seems obvious, but turns out to be quite difficult to trace in certain situations.

There's no miracle. Motion isn't continuous; it's discrete. — Michael

Your argument doesn't prove that.