I'd agree except that the law of identity was first, set theory came along after. — Metaphysician Undercover

Temporal priority is not logical priority.
Whether a proposition is obvious (self-evident) or (self-evidently) absurd are subjective.

Well, if the law of identity is an obvious self-evident tautology, then it appears like there must be something wrong with set theory if it's in contradiction with what is obvious. — Metaphysician Undercover

There are plenty of ways to formulate that law without using the word "same". In any case, "same" in that context just means "same object", so it isn't absolute. moreover, If you drive my car, you don't drive it at the same time.

The thing is that everything about it is not the same, only those named qualities are the same, and that's why it's incorrect to say that it is "the same" in that unqualified sense. — Metaphysician Undercover

There is no unqualified sense of "same".

Not everything about the two is necessarily the same, only the stipulated required qualities. — Metaphysician Undercover

We agree!

So it is incorrect to say that the two sets are the same, in the unqualified sense, — Metaphysician Undercover

There is no unqualified sense of "same".

He (sc. Aristotle) claimed that the law of identity was necessary to battle against sophists who could logically demonstrate absurdities. — Metaphysician Undercover

I'm sure that Aristotle would not object to my regarding that as not a logical argument.

That's precisely why supertasks are impossible. — Michael

My point is that I think that the disagreement between you and @fishfry is about different ways to make the same point.

There's nothing wrong with defining, or performing, a recursive function. There is a problem with claiming that it is possible to have completed a recursive function. — Michael

Quite so. Wittgenstein made much of the endlessness of infinity and asked how it was possible. You may know what his answer is. If you don't, it is easy to look it up. (It would be far too long to try to outline it in this context and you likely know anyway.
My question is, how do we know that it is not possible to complete a recursive function (in the sense of writing or speaking each step that the function defines)? BTW, I don't think there is any particular problem about defining such a function, though you could argue that it is the result of misapplying an operation that is perfectly harmless in other contexts, like √2 or a self-referential pronoun like "I" or "this".

For example, proving that √2 is irrational doesn't rest on trying and failing to write it down, but on showing that the assumption that √2 is rational leads to a contradiction - reductio ad absurdum.
In the case of infinity the argument rests on mathematical induction. I understand that not everyone is happy with that argument but it seems OK to me. A recursive function is defined, but in such a way that its end is not defined.
So even if a sequence cannot be completed empirically, so to speak, there are ways of proving that it is endless with attempting the impossible refutation of writing down each step.

The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times. — Michael

The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined.

You have not defined the terminal state. So why do you think there should be a sensible answer for what it is? — fishfry

Surely, the contradiction is the result of the lack of any definition of the terminal state. If the terminal state could be a plate of spaghetti, why couldn't be a lamp that is neither on nor off?

I really cannot see what you two are arguing about. Why does the difference matter?

You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. It's not a real light bulb, it's not driven by a real circuit. — fishfry

The plate of spaghetti is a great dramatic way of making the point that there is no definition. But the series is defined on the basis that its limit is 1. You can't derive 1/2 from a plate of spaghetti.

Conclusion: set theory is in violation of the law of identity. I've explained to you why this is the case. Do you agree with me? — Metaphysician Undercover

It seems pretty clear that "same" does not have the same (!) meaning in the context of set theory that it has in the context of the law of identity. How could it? The definition that applies in the context of the law of identity is inapplicable to the context of set theory, and vice versa. So why don't you conclude that the use in the context of the law of identity violates the use in the context of set theory? It seems to be an arbitrary choice.
The meaning of "same" depends on its context. There are many other concepts of which that is true.

But this is the real hard question. Being in a prison implies one is not free, so the question then is, what is the nature of freedom? — Constance

So freedom is always there as it is our nature, our existence, to stand in this openness of possibilities, but this is forgotten. — Constance

Yes, freedom is about possibilities. Prison means that certain possibilities are denied. All of that is true if I am in prison. But what freedom means in that context is perfectly clear, both in respect of the possibilities that are denied to me and in respect of the possibilities that are open to me. Your question implies that something is not clear. For me, the question of the nature of freedom seems to be posed in a vacuum, without context. Some would call this the quest for absolute freedom, but trip up because without context there is neither freedom nor constraint.

Just going along, day by day, is a bit like being a thing, for a thing doesn't have choices. — Constance

It can be a bit like being a thing, but it is also being free. It depends how you look at it. Either way, it is where we live.

Pain is OF the world, not of our laws that deal with pain. Pain is this primordiality, a givenness of our existence, and will not be spoken. — Constance

It certainly is a given. I'm not sure what you mean by speaking of pain. We can certainly talk about it, and we do - especially when we are suffering from it.

so the question then is, what is the nature of freedom? — Constance

That's (one version of) the question that philosophers ask. But they don't think through what that question means and so end up is quagmire.

And you likely know that Heidegger, Sartre, Kierkegaard, and so on, including Kant and his rationalism, all have something to say about freedom. — Constance

Quite so. But I'm intrigued that you go through a huge process and end up in the same place that I'm in. Pain is part of life. So what is at stake here?

Forgive me - what the "GS"?

I'm sorry if I confused things. I'm happy to take answers from anyone who is moved to provide one.

One need not think of specifically the physical realm as a closed system, one can instead imagine (physical realm plus mind realm) as a combined closed system. And an LD that's fully aware of what's going on in all the relevant realms of the combined closed systems is still conceivable. — flannel jesus

All I was saying is that it makes no difference whether one thinks of the universe as a closed system or as a combined closed system. It is just one way of thinking about the universe. It may be useful, but is it true? What is the evidence one way or the other?

It's Laplace's premise. It's not mine. I don't believe it to be the case. — Patterner

That's fair enough. I'm just trying to say that it isn't an empirical idea - no amount of empirical evidence will confirm it, or refute it.

There's nothing wrong with defining, or performing, a recursive function. There is a problem with claiming that it is possible to have completed a recursive function. — Michael

Quite so.

Can you prove that it is impossible to complete a given recursive function?

For reference:-

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. — Laplace

It's just a day-dream.
You can summarize this as "If determinism is true, it would be possible to predict everything". Not very exciting, is it?

One need not think of specifically the physical realm as a closed system, one can instead imagine (physical realm plus mind realm) as a combined closed system. And an LD that's fully aware of what's going on in all the relevant realms of the combined closed systems is still conceivable. — flannel jesus

"IF the universe is a closed system.." we can make all sorts of deductions and predictions. But is it? What's your evidence?

Determinism rules all things, and LD has the perception and intellect to figure everything out. — Patterner

"IF determinism rules all things..." but does it? What's your evidence? Laplace is perfectly clear that "we may regard the present state of the universe..." He doesn't pretend that this is any more than a possible way of looking at things.
Similiarly, he says clearly " An intellect which at a certain moment would know all forces ... and all positions of all items..., if this intellect were also vast enough...". He doesn't even suggest that this is possible. (Interesting that he doesn't mention that God would be such an intellect.)

I answer Yes to both. Why not? That's the premise. — Patterner

You need to establish the premises in order to assert the conclusion.
Still, consider your answers:-
1 If LD cannot figure some things out, what follows? Does it follow that determinism is false? Yes.
The catch is that you have to wait until LD has figured everything out before you know whether there are some things it cannot figure out, and even longer before you know that it has not just made a mistake.
2 If LD can predict everything accurately for the next n years where n is any number you like. Does it follow that determinism is true? Yes
No, it does not follow that determinism is true, even if you can predict for any finite number of years ahead. Even if the universe is finite and time will run out, and LD predicts that, it will not follow that it got things right because it's calculations were correct.

No, the mathematical operation of division cannot be applied to an infinite number of times, for the reason explained above. — Metaphysician Undercover

Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. — Wikipeida

I know it is only Wikipedia, but I'm sure that more authoritative references could be found.

What is wrong with that?

LD is also aware of where every particle in the universe outside of our solar system is, which way each is going, and can calculate which will interact with our SS, and when. Even if two hunks of rock a thousand light-years away that are not heading this way are going to collide, and some debris from that collision will then head this way. — Patterner

Nonetheless, it is treating the universe as a closed system.

We seem to be unable to communicate. — noAxioms

Oh, I don't think it's as bad as that.

But there very much is storms and rain in the world simulated. It wouldn't be a weather simulation without such things. — noAxioms

No, there are only simulated storms and rain in the simulated world.

The point of the simulations would be lost if real people capable in their own right of acting and reacting in their world.
That sentence lacks a verb, and you lost me. ....
Your wording in the verb-less sentence suggests that simulated people would perhaps need to exert some sort of free will over the physics of the simulation. That model isn't compatible with Bostrom's view. — noAxioms

I'm sorry. I'm afraid I can't re-construct what that sentence was supposed to be. But your version of it is what I was trying to say. I can believe that it is not compatible with Bostrom's view. The question is whether Bostrom's view is coherent.

Once you suppose that the simulations are conscious

I don't think anybody is supposing that. See the above. — noAxioms

Similarly, a simulation of a conscious being would not make a computer conscious, but that doesn't mean that simulated person is not conscious. Bostrom suggests that is exactly what's going on. — noAxioms

So Bostrom does suggest that the simulations of people "inside" the (non-conscious) computer are conscious.

I am not an avatar in a video game, for the usual Cartesian reason. There's a "me" in here having subjective experiences. — fishfry

I'm agree with fishfry here, but adding that if the "me" in here is having subjective experience, then I must be able to interact with the presented illusory environment, that is, I can cause things to happen in the environment and get appropriate feed-back from the environment. But that would make me a real person, not a simulation (though I might be a clone.)

Yes, a simulated person would behave differently than 'their originals', which I put in quotes because there are no originals in the scenario in question, — noAxioms

There's an ambiguity here. There could be simulations of people that are like fictional people. Their originals would be people in general, not people in particular (though an ancestral simulation suggests that they would need to be people in particular - if they aren't, then what makes it an "ancestral" simulation.)

So one is always already IN some historical framework (this for Heidegger was the essential ontology for dasein), bound to a particular finitude. — Astrophel

There are two issues with this. First, the framework that I have learnt is not bounded, in the sense that it has infinite possiblities within it. Second, it is not a fixed framework, but is subject to change and development - Derrida is acutely aware of this, isn't he? So I ask the question, what tells us that we are "bound" to a particular framework? Awareness of history, perhaps, and/or awareness of change. Perhaps we should think of our historical framework as a starting-point, rather than a prison.

But it is a very sticky matter simply because one has to bite this absurd bullet that says as I acknowledge my cat on the sofa, it is somehow existentially remote from possible understanding. There is this impossible distance between me and the cat that says I know, but I really don't know in the deeper ontology. This distance is about language and the world. — Astrophel

I can, and do, acknowledge my cat on the sofa and acknowledge also that I do not know - am not aware of - everything that the cat is. Some things may be beyond any possibility of knowing, such as knowing (i.e. experiencing) the lived world of the cat (because I could not be the cat without ceasing to be me, a human being). There is surely, no harm, in admitting my limitations while at the same time acknowledging the cat is "really" there, and on the sofa.

I'm not sure that there are not some typos in this - I certainly can't construe it:-

the pain from this broken knee cap is does not issue from a construction of beliefs about pain, and the prohibition against bringing this into the world some from the pain itself, not as the pain is construed, interpreted. Pain qua pain makes sense even though the language that speaks it cannot speak the world, so to speak. — Astrophel

But, yes, the world resists us and obtrudes on us - however much we may try to control it or ignore it. That's how reality becomes real for us as we exist in our framework - and, of course, how our framework has to stretch and adapt to accommodate it. The limitations we posited at the beginning do not exist.

I believe this is saying that LD knows everything about everything IF everything about everything is deterministic. That, I believe, is the point of Laplace's thought experiment.
But if all of reality is not deterministic, LD's calculations would not be able to figure everything out. — Patterner

If you think of some restricted problem, such as the movements of the planets in our solar system, this seems to work. But it treats the solar system as a closed system and restricts the predictions that are made about it. Laplace is generalizing from that to everything. That's not a defined system and it posits a range of predictions restricted to those that physics can make or a final and complete physics of the future. Don't you think that is a rather generous assumption?

Even if you swallow that assumption, consider:-
1 If LD cannot figure some things out, what follows? Does it follow that determinism is false? No.
2 If LD can predict everything accurately for the next nyears where n is any number you like. Does it follow that determinism is true? No.

Laplace's demon proves nothing.

Worse than that, if LD can accurately predict some things, does it follow that they happened because of LD's prediction? No.

Laplace's demon is irrelevant.

By most definitions of 'reality', yes, a simulated world would be a reality of its own, but it being called a simulation is an explicit admission of it being dependent on the deeper reality running the simulation, just like saying 'God created the universe' makes the explicit relation of the universe being dependent on the god. — noAxioms

The meaning of "dependent" is context-dependent. The dependence of a simulation on its deeper reality is quite different from the dependence of a created object on it creator. If one thinks of some entity having created a universe, the implication is that the creation exists in its own right. Insofar as a simulation is a reality of its own in the way that a story is a reality of its own, it will not exist in its own right and remains under the control of the story-teller, even though it may have an internal logic that is not the same as the logic of reality.

I'm a sim fishfry and there's a "real" entity fishfry who's being simulated, but who isn't really there. — fishfry

..... unless you think of fishfry as an avatar. On the other hand, if I am a simulation that is not aware of the fact, I must be able to act and react in my world. In that case, I am not a simulation of anything.

Bostrom suggests a sim of ancestral history, which means that random new people get born, and these people do not in any way correspond to actual people that might have existed in the history of the GS. — noAxioms

I can think of models of the weather system that are used to predict the weather. They can be called simulations. They remain quite distinct from the actual weather. There are neither storms, nor rain, nor sunshine inside the computer. Yet the point of the exercise is that it remain as close as possible to what actually happens/-ed. (I can't imagine what the point of ancestral simulations would be, if not that.)
Once you suppose that the simulations are conscious (perhaps per impossibile), those people are not simulations. They might be clones (even if they are not clones in the classical sense). But they would be actual people, perfectly capable of behaving differently from their "originals".

The history being simulated is quite different than the one that actually happened way in the past history of the GS world, although the initial state of the simulation presumably had similarities to some actual past state of the GS history. — noAxioms

The point of the simulations would be lost if real people capable in their own right of acting and reacting in their world. It wouldn't even be a way of running an alternative history. Or is there some other point at stake here, that I've failed to grasp?

I can't respond to all of this. I'll just pick out some remarks for comment. I'll try to respect their context.

Phenomenology asks the right questions. — Astrophel

How do you know they are the right questions?

Also, he never read phenomenology beyond Kierkegaard. — Astrophel

I expect that's true. On another thread recently, someone remarked that he never read Aristotle; from the context, it seemed natural to infer that this was a deficiency. I thought it remarkable that someone would think that any philosopher who had not read Aristotle was deficient in some way.

Wittgenstein was not aligned with the positivism that so emphatically rejected metaphysics. He was different. A great admirer of Kierkegaard, he insisted that meaningful talk had no place in metaphysics because it would offend the most important part of our existence. He writes in Value and Culture: Divinity is what I call the Good. And would go no further — Astrophel

Yes, his position was much more nuanced than many of his contemporaries. But he had very little, if anything, to say about it. We are left with the business about speech and silence, which is a blank sheet of paper on which we can write more or less what we wish to - and people do.

Heidegger, the greatest philosopher of the 20th century, perhaps ever. — Astrophel

He's certainly an impressive figure. But those accolades come and go. They said that about Russell at one time, and Wittgenstein. I'm not good at hero-worship.

Yes, as I said. One cannot doubt the apodicticity, but one can doubt the way language takes up the world. — Astrophel

I'm not sure about apodicticity, so if you don't mind, I'll just talk about certainty.
That doubt is unresolvable, because it frames the issue in the wrong way. In the first place, as Wittgenstein argues (mostly in the early period) just as one cannot draw a picture of a picture, one cannot expect to explain in language what the relationship is between language and the world. As he would say, it "shows itself", just as a picture (once we have learned to interpret it) shows what it is a picture of.
But the big mistake is to think that the problem is about the relationship between language (as given, and our starting-point) and the world. Language arose in the world, from the world, to be of use in living in the world. Hence the only question is about the relationship between the world (as given, and our starting-point) and language, just as we assess a picture by comparing it to the world and not the world by comparing it to a picture.

Mostly, as I pointed out, it is framed in language and analyticity itself is a language construction, and so one would have first to establish that language itself is apodictically certain. — Astrophel

Language is a construction, in a sense, yes - in the sense that a game is a construction. Actually, it is a set of rules (or several sets of rules). There was not law-giver who laid them down - they evolved in the interchange of our social lives in the world - the useful rules stayed, the useless ones disappeared without trace. What makes those rules certain is that we keep them - nothing else. (Actually, we don't keep them - we mess with them all the time, as Derrida realized, but set that aside for the moment) In itself, however, language is neither true not false. It is the means by which we assert and ascertain what it true and what is false. The certainty that Descartes was after was to be found or lost in the use of language, not in language.

Thus should not the true object of an inaugural inquiry be the Being of the ego rather than the ego itself, or more precisely, the Being in and by which the ego can rise to existence and acquire its own Being? — Astrophel

I'm not not particular about where I find good philosophical ideas and I'm quite pragmatic about which school or tradition the ideas originate from. Heidegger and the others have some good ideas from time to time. But I think I can detect eyewash as well. Unless I think of it as a sort of (not very good) poetry.

In any case, it is far too late for an inaugural enquiry. The horse has left the stable and is busy ploughing the fields etc. We arrive or are thrown into a world that includes language and we gradually learn to participate. There are no beginnings or foundations to be found apart from that.

What I've explained though, is that infinite divisibility is really incoherent due to self-contradiction. So the supertask is not even logically possible. It just appears to be, when not subjected to critical analysis. — Metaphysician Undercover

.... and when one analyses it, it is a confused mixture of physical possibility and logical possibility, each of which are coherent on their own.

It's only the contradictory notion, that a finite thing can be divided an infinite number of times, which produces the paradox. — Metaphysician Undercover

A finite thing certainly cannot be divided an infinite number of times, if by "divided" you mean "physically divided", subject to clarification of what you mean by a finite thing.
At the same time, it is possible to divide it into halves, quarters, etc. (how many fractions are there?) and into feet, inches, etc. and into metres, etc, and according to an indefinite number of other units of measurement. To physically divide in one of those ways excludes dividing it in any other way, so you can't divide it by all those things at the same time. But those possibilities do all exist, all at the same time.
On the other hand, the number 2 is finite, in one sense. Yet it cannot be physically divided at all (because it is an abstract thing), yet it can be divided by a familiar mathematical operation, and that operation can be applied to it an infinite number of times. (No, I'm not talking about space or time.)

From past experience I understand that fishfry is very slow to accept the reality that some principles employed by mathematicians are incoherent. — Metaphysician Undercover

I wonder if that's because the principles that you are applying to mathematics do not apply to mathematics? For example, numbers are abstract objects; they do not exist in space and time. Geometry is not about physical objects, but about ideal objects, which do not exist in space and time. Abstract entities that do not exist in space or time are not subject to the restrictions you wish to impose on space and time - obviously. You may or may not regard such entities as not true (or not real) objects, but that's neither here nor there.

I agree. Michael keeps saying supertasks are metaphysically impossible, and I think they're metaphysically possible. — fishfry

Now I'm confused. I thought you didn't know what "metaphysics" means - or what metaphysics is.

No. I'm saying that there's no natural way to define the terminal state. There are lots of ways to defined it. I define it as a plate of spaghetti. — fishfry

I'm puzzled now about "natural". If the terminal state of the lamp is not defined, there is no way to define it - natural or otherwise. Or, possibly better, any arbitrary state will do. Hence the plate of spaghetti.

the lamp problem, which only defines the state of the lamp at the points of the sequence, and not at the limit; — fishfry

Yes, of course - and since it is not defined, Michael can derive a contradiction - two equally possible or impossible states.

"Complete" is not an applicable mathematical term. Unless you want to say that sqrt(2) completes the sequence 1, 1.4, 1.41, 1.414, ... That's an acceptable usage. But it doesn't mean there is any kind of magic jump at the end. It just means the terms of the sequence are arbitrarily close to the limit. — fishfry

H'm. I would be quite happy with that acceptable usage. But it suggests that 1,1.4,1.41, 1.414... is incomplete, and we are back with the temptation to think that series can somehow be completed. It's probably better to stick with "not applicable".

Under the thought process experiment of "adding the next term" at successively halved time intervals, I'd say it completes in finite time. But that confuses people because we're conflating math and physics. — fishfry

I think that's the heart of the problem. My only hesitation is that the lamp is imaginary, so it sits on an ill-defined boundary between the two. I'm very suspicious of the idea that anything anyone can imagine is (logically) possible. Twin Earth is a good example. But there's a raft of others.

possible worlds exist as concrete entities in logical space, — fishfry

I don't know what to say. Ryle would go on about category mistakes. In poetry (or politics) people sometimes talk of a "tin ear". That's exactly what this is - a rhetorical gesture that confuses "concrete" with "well defined" and with - well - concrete. It's protesting too much. There must be some repressed doubt going on there.

Well it all went over my head when I took a MOOC on the subject. — fishfry

You are lucky. It will spare you a world of grief and confusion. Modal logic can look after itself.

The nested interval construction can be explicitly written down. I perhaps am not sharing your vision here. — fishfry

The system is telling me that you mentioned me in the context of this comment in the thread on the Fall of Man paradox, but I can't find any mention of me. But the system is doing some weird things anyway, so I'm not going to worry about it. I do regret not having been aware of the thread sooner. I thought it had something to do with theology.

I can't contribute to the discussion you are involved in there, but this quotation:

I concur that this narrative couldn't unfold in our physical reality, but your argument doesn’t address the core of the paradox. The inclusion of God and the Garden of Eden in the story was specifically to lift us beyond our finite limitations. — keystone

does make me think that the same problem, of the interface between mathematics and empirical reality, is at the heart of this paradox as well.

If this is familiar territory, then I can push just a bit. — Astrophel

You can. But it is the first step into a swamp that sucks you in... But then, you are mired in it anyway, so perhaps it will help to point out that there are ladders that can get you out. You just need to ask the right questions.

But it is not an interpretation, just a term that designates the cogito's objects. — Astrophel

I deduced that. But it already palms off on me a model of thinking about thinking.

And then "doubt would be meaningless without certainty": depends on what is meant by certainty. — Astrophel

"Parent" and "child" are interdependent. Both are defined at the same time. This may be somewhat hidden here because of an accident of our language. "Certain" has two meanings, one psychological and one objective. The opposite of "certain" in the objective sense is "uncertain", which seems to have no psychological correlative; but it does exist, since we have "doubt".

But while certainty implies doubt, for all things can be doubted, even logic, you would have to clarify how all doubt implies certainty. — Astrophel

"I doubt whether p" means "I don't know whether p is true or false", which implies "I know that p might be true or might be false", which implies "I know that p might be true".

Well, it has already been done, but this, of course, has not reached the ears of "people". — Astrophel

The message must be getting drowned out. But you are missing out all the others who have tried. Hume, Russell, Husserl, Wittgenstein, and maybe others.

Perhaps after AI has delivered us from drudgery, the world will see that phenomenology is the one true view. — Astrophel

Perhaps. But I think it more likely that most of it will immerse itself in games and simulations and internet fora - moderated by AI, of course.

As to "all things can be doubted", do you include "If P implies Q, and P, then Q"?

Therefore, the god of the gaps is immaterial in every sense. — Vera Mont

I don't think Berkeley would be pleased. But perhaps that's immaterial.

the apprehended world is just as indubitable as the conscious perceiving agent that affirms it. In an important way, there simply is no such thing as Cartesian skepticism, that is, until one makes the move toward interpretation. One does doubt in ordinary ways, and certainly one can doubt the science and everydayness that is constructed out of the cogitata, things present before us. — Astrophel

That could be the beginning of a long argument, which, I guess, would be a trip through very familiar territory. For me, "Apprehended world" and "cogitata" are the dubious interpretations, not the everyday world. In my view, what Descartes missed was the elementary point that doubt implies the possibility of certainty; doubt would be meaningless without it.

like all of those omni's, and notions of the creator and the source of judgment, and so on. — Astrophel

Yes, those are the reasons I think that the concept is incoherent. Getting rid of traditional metaphysics is a lot harder than many people thought in the mid-20th century (and, indeed, earlier, back to the 17th century). I am sceptical about whether it is going to happen.

As soon as humans are eradicated. — Fire Ologist

It isn't necessary to wait that long. There are non-theistic ways of life. Confucius (?), Buddha, Stoicism, Epicureanism.

What difference does it make whether something you might choose to call God exists in a non-empirical dimension? — Vera Mont

Is there a non-empirical dimension?

So the fact that the status of the lamp at t1 is "undefined" given A is the very proof that the supertask described in A is metaphysically impossible. — Michael

This is a contradiction, therefore Thomson's lamp shows that it is logically impossible to have pushed a button an infinite number of times. — Michael

That's right. But there's nothing special about the lamp. It is impossible to complete any action an infinite number of times.
I wasn't accurate enough in my last post. The representation of the series is misleading. (1/2,1/4,1/8...) invites one to think of the dots as an abbreviation for a longer sequence, which could be written out in full. But there is no possibility that it could be written out in full. The notation does not define an end,

In the message you linked to, you concluded:-

So the fact that the status of the lamp at t1 is "undefined" given A is the very proof that the supertask described in A is metaphysically impossible. — Michael

I think that this is what @fishfry was saying. (Substituting "logically impossible" for "metaphysically impossible".)

So what is the argument about?

It is necessary that the lamp is either on or off after 60 seconds, and for it to be either on or off after 60 seconds it is necessary that the button can only been pressed a finite number of times before then. — Michael

That looks to me like a valid argument to the conclusion that the Thompson lamp is a physical impossibility, because switches and electrical currents are not infinitely fast. What's wrong with that?
My problem is that I don't understand what metaphysics is, if it is not logic.
I may be wrong, but it seems to me that those who believe in supertasks think that by stipulating that the lamp can be switched infinitely many times in a limited time, they can make it so. But they can't. Supertasks play on the difference between the physically possible and the logically possible to create an illusion.

Here is a hard question, the second hardest question I can think of: how is knowledge possible? — Astrophel

I hate to say it, but I would not be able to reject an accusation of "whataboutery" if I tried to change the subject to a general philosophical discussion about knowledge. My reaction may be conditioned by my view that much of epistemology has been thoroughly distorted by Cartesian scepticism and the belief that the only certainty is logical certainty; the latter of course, rules out all empirical knowledge out of hand. There is also a danger that if your interlocutor is not convinced by Descartes, your opportunity to persuade them on this specific issue will be lost. Faced with an argument about the existence of God, you try to prove that we don't know anything anyway. No, I don't think so.
Mind you, with a suitable interlocutor, I would be inclined to try to persuade them that the question of God's existence cannot be answered by purely empirical evidence.

What I reject out of hand is that lukewarm admission that there may be some kind of supernatural something behind or underneath of the universe, and that something could be called God - because we can't prove it ain't so. Why should we bother with such a fruitless conjecture? Just not to be called atheist? — Vera Mont

We should wear our badge with pride and not let the opposition use it as a term of abuse.
I it is a mistake to allow supernaturalists to pretend that the question is an empirical one. That is a wedge argument, designed by believers to open the mind of unbelievers. But stick to the issues.
If someone claims to have seen the Virgin Mary, some will propose an explanation such as a hallucination, or down-right lying, and some will propose an explanation such as a vision granted by Heaven. Which is the most likely (plausible)? Even the RCC adopts the former. (This is a version of Hume's argument against miracles.) The RCC is prepared to consider further evidence, and may change its mind, and adopts the latter. Now, either that evidence is of a kind that anyone can accept it, or it is not. What the RCC considers to be evidence is, I understand, evidence that the visionary has caused further miracles, which is clearly circular. No need to fall for blandishments.
If you really need a bolt-hole, you can insist that, where there is insufficient non-circular evidence, the answer is to classify the event as unexplained.
If you get really desperate, you can point out that if a supernatural explanation were to be established (perhaps amongst the paranormal phenomena) on the basis of normal evidence, (not that I can imagine that happening) the explanation would then become a natural explanation.

The fine-grained nature of the world we live it might just be a function of adaptive creative algorithms which feed off of past events, in the simulation. — AmadeusD

I guess that's so. But that would mean that the simulation is a reality of its own, independently of the "real" reality. (As a story has its own logic, even though it is just a story) Still, the algorithms are part of reality - they are not simulated, are they? - they wouldn't really be algorithms if they were simulated. So the simulaton may be different from the real world in all sorts of ways, but it needs to be built from and in the real world.

Much more important than stories about the elements are stories about dead people. — Vera Mont

I don't know about "more important", but I agree that this is something that is addressed in all religions, and it is important to people.

Like screaming children in burning cars. Suffering, that is. That is not a story. — Astrophel

Yes, of course this is also part of the mix. The book of Job comes to mind.

Without intelligent makers, there would be no couches or shoes. — Vera Mont

Of course, and so it is easy to see why creation stories are included in so many mythic cycles.

Before there was worshipping, Gods, and all the trappings of these churchy fetishes (I like to call them), there was a basic problematic built into existence that gave rise to the worshipping and the rest. — Astrophel

there is something empirical behind that swathe of (potential) nonsense. Thunder/rain Gods are one. — AmadeusD

It is not enough, it seems to me, to dismiss the whole business as superstition. We can't pretend that it isn't still important to human beings. It would be reasonable to suppose, wouldn't it, that religion addresses issues that are still important to us? The question of it's historical origin is one way, though it is unlikely that we'll get more than plausibility this far from the events.

I think explanation of cause only gains importance after the concentration of humans in walled cities - after we cut ourselves off from nature and felt we had to master or conquer nature. — Vera Mont

Causal explanation in our sense is a more recent development. It is part of the scientific revolution in the 16th and 17th centuries. But walled cities, agricultural technologies and religion (in our sense) all seem to have arisen at, very roughly, the same time. (Some people talk of an Age of Wisdom.) It makes a lot of sense to see them as interlinked and interdependent. There are many themes built in to religion. It addresses human concerns, but also, as Nietzsche so emphatically pointed out, is involved in the power struggles in the new, complex human societies in the new cities. I think he missed a trick, in fact. Religion gave power to a new version of the shaman - the priest - and supported or enabled much larger human societies. But it also gave a voice to people who are oppressed in those societies.

My best guess is something like: delving into the human psyche reveals that it differs from inanimate objects. That much, I have already stipulated as self-evident. — Vera Mont

It seems more plausible to me to see the sharp distinction between animate and inanimate - and conscious and not conscious - is a product of our times, specifically of - again - the scientific revolution. In natural language, there is no sharp distinction between action in the sense of what human beings do and action in a broader sense. I mean, quite simply, that we talk of, for example the wind blowing the door shut, the lightning striking a tree, the sun drying the washing without batting an eyelid. We all know the difference, but that's because of our intellectual training. Personification of the inanimate in that way is built in to our language.

After all, if one is going to dismiss spirit, it has to be made clear what the term even means apart from the mundane casual (causal?) thinking. — Astrophel

Yes. I do accept that it means something to those who talk about it. My problem is that I don't really understand what that meaning is. Too often, it seems like a way of escaping awkward questions.

Obviously, the lure of magic, wish-fulfillment, personification of natural phenomena and all those impulses that begin with ritual and eventually culminate in huge international institutions like the RCC, is very much a part of that interest. — Vera Mont

RCC = Roman Catholic Church?

But isn't there something "behind" the stories that a person cannot wimp out on even if she tried? — Astrophel

For me, phenomena like personification and our ambivalent (or complicated?) attitude to animals is a clue. The concept of a person can be applied to things that are like people in some ways, but not others and it is particularly tempting for societies that don't have the benefit of modern science. If you think that some sort of super-human being is throwing the furniture around in heaven it is less alarming than not knowing. You can take steps to appease its wrath, which is comforting even if ineffective.
I have a feeling that some conspiracy theories have the same origin.

Perfectly reasonable, IMO. It does seem to exist by definition, rather than anything else (conceptually). — AmadeusD

Yes. The trouble is that believers wouldn't buy that. They think that God is real, so the problem is to discover and describe him/her/it.

I don't quite understand why this would be the case? — AmadeusD

I was thinking what might persuade me to think that a religion was rational. If someone posited God as an axiom, and thought through the consequences for their life and lived accordingly, that would be rational, wouldn't it? Then, if religion was rational, atheism could posit (or just not posit) the axiom but still be rational, if they thought through the consequences and live accordingly. However, if both ended up living the same sort of life, it would follow that the axiom was unnecessary and could be abandoned. That would be a rational approach to religion. Not altogether implausible.
There is a school of theology, known as presuppositionalists, (look up a theologian called Van Til) who do adopt this approach. However, they posit that the Bible is true, which is a whole different ball game. I've never seen a philosophical discussion of this, but that's understandable. If you presuppose that the Bible is true - well, other than as a historical document and evidence for history - there's very little to argue rationally about.

think this the case for a lot of agnostics - they can just leave off the issue entirely by claiming that looking for the evidence is a fools errand. — AmadeusD

"Can't be bothered" as opposed to "Don't know". I'm sure there are people, perhaps many, who are like that. They'll go with the crowd in the end.

it boiled down to just not liking uncertainty. I think this the case for a lot of agnostics - they can just leave off the issue entirely by claiming that looking for the evidence is a fools errand. — AmadeusD

Yes. A lot of philosophers are very bothered by that, as well.

I don't care if there are supertasks or not, but I am driven to straighten out the bad thinking around limits (or die trying, is more like it). — fishfry

I'm entirely in favour of the project, but, to be honest, I don't think it is worth dying for.

In math, the notation 1/2 + 1/4 + 1/8 + ... does NOT denote a process or a sequence of discrete steps. — fishfry

I think that's the first time I've encountered anyone on these sites who understands the difference between "discrete" and "discreet". Not patronizing, just saying.

Likewise 1/2 + 1/4 + 1/8 + ... and 1 are two text string expressions for the same abstract object, namely the number we call 1. — fishfry

Now you have me a bit puzzled. In my book, that means that the equation is about the complete series, which seems at odds with the idea that it can't be completed. What does "complete" mean? Or does it mean the sense in which it is "always already" complete? (see below)

Some people regard all possible worlds as equally true. That viewpoint doesn't resonate with me. — fishfry

It might be easier to understand if you thought of them as regarding all possible worlds as equally possible. I could understand that. I hope they don't mean that all possible worlds are equally actual....
But some people tend to think only of one kind of possibility - logical possibility. But there many other sorts - physically possible, legally possible, practically possible, etc. etc. I say that possible means different things in different contexts, but it may be that I should be saying there is cloud of possible worlds for each kind. Or maybe physically possible worlds are a subset of logically possible worlds. It's all very confusing. But I shouldn't get too snooty. There is, apparently, a need to this concept in modal logic, but I don't understand what it is.

You lost me here. I believe I was arguing to Michael that it's at least conceivable that we execute a Zeno walk on the way to the kitchen for a snack; and that therefore, the idea is at least metaphysically possible. That's all I'm saying. — fishfry

Oh maybe I understand ... you're saying that just because the path can be infinitely subdivided, does not mean that I'm actually executing that sequence. I think I disagree. I have to traverse each of the segments to get to the kitchen. — fishfry

Well, in that case, you are also traversing the infinitely many possible points along the way, as well as the convergent series based on "<divide by> 3" and all the other series based on all the other numbers, plus all the regular divisions by feet or metres. Or maybe you could decide that all these ways of dividing up your journey are in your head, not in the world. Think of them as possible segments rather than chunks of matter or space.

But in math, 1/2 + 1/4 + ... is added together all at once. And the sum is exactly 1, right now, right this moment. — fishfry

Yes. Thanks for clarifying that for me. That's what I was trying to express when I started babbling on about "always already" in that post that you couldn't get your head around. The comparison with Loop program captures what I've been wrestling with trying to clarify. All that business about getting (or not getting) to the end... It's important though that it's a physical process which takes time. You can switch it off at the end of 60 seconds, and see how far it got, but it won't have completed anything, will it?

But all I'm saying is that it's at least conceivable; in which case it's not metaphysically impossible. I don't have to argue strongly that it's true; only that it's at least barely conceivable. — fishfry

There's no clear criterion for what is conceivable and what is not, in spite of generations of logicians. It seems pretty clear that some people have a much more generous concept of that than I do. There are famous philosophical issues around that many people seem able to conceive of, but I can't. I don't know what's wrong with me.

Right. Aleph-null bottles of beer on the wall, aleph-null bottles of beer. You take one down, pass it around, aleph-null bottles of beer on the wall ... :-) — fishfry

Make sure you get one of the ones that you can't finish drinking. You would not be popular if you passed round one of the ones that you can't start drinking.

Did my quoting get messed up? Michael keeps saying supertasks are metaphysically impossible, and I want to make sure I understand what he means by that. — fishfry

You won't have bothered with this exchange - his comment, my reply:-

But I would even go so far as to say that supertasks are logically impossible (as shown by the above argument and Thomson's lamp). I simply went for the phrase "metaphysical impossibility" because it's the weaker claim. — Michael

I think it would be better to stick with the strong claim. At least it is more comprehensible. — Ludwig V

The point is simply this: at the point we make a decision, there is a set of determining factors: beliefs, genetic dispositions, environmentally introduced dispositions, one's desires and aversions, the presence or absence of empathy, jealousy, anger, passion, love, and hatred. — Relativist

Surely, there's no problem about one acting in accordance with one's beliefs and desires (and emotions are an aspect of one's desires). That's what freedom means. What would be it be like not to act in accordance with them? So even if you say these are determining factors, they are not factors that threaten free will.

Yes, I realize that sometimes we feel that our emotions have "carried us away", but other times our emotions are exactly what we want to do. The interesting question is to understand which emotions on which occasions threaten our freedom and which don't. In the same way, we sometimes talk of being in the grip of a desire - addiction and habit are the greatest threats with social expectations and manipulation a close second - and sometimes doing what we want to do just is freedom. Again, the interesting question is to distinguish the two. Beliefs also sometimes mislead us - it is better to act on what we know, but hard to be sure which beliefs are knowledge and which not, but that's the interesting issue. No-one wants to act on false beliefs. Everyone wants to act on knowledge. Sweeping generalizations just create fake philosophical problems.

In my opinion.

But I would even go so far as to say that supertasks are logically impossible (as shown by the above argument and Thomson's lamp). I simply went for the phrase "metaphysical impossibility" because it's the weaker claim. — Michael

I think it would be better to stick with the strong claim. At least it is more comprehensible.

Metaphysical impossibilities are things which are necessarily false; e.g. see Kripke's Naming and Necessity in which he argues that "water is H2O" is necessarily true even though not a priori (i.e. logically necessary). — Michael

Yes Kripke does claim that. But he waters down the meaning of "necessarily". For him, it no longer means "in all possible worlds", but "in all possible worlds in which certain conditions hold". But contingent means, or used to mean, "true or false depending on certain conditions". So, on this account "necessarily" means what "contingent" used to mean. Talk about having your cake and eating it!

Where do i get one of these metaphysical beers? — Metaphysician Undercover

At your local metaphysical beer shop, of course. I'm sure Google knows its address and will give you directions. (Shops never stock both metaphysical and mathematical beers at the same time. They fight, you know - very messy!)

I clearly explained though, it isn't "infinite" which is incoherent, it is "infinite divisibility" which is. "Infinite divisibility" is a specific application of the term "infinite" which is incoherent. It is incoherent because the concept of "infinite" is incompatible with, inconsistent with, or contradicts, what is implied by the concept "divisible". Therefore the two together as "infinite divisibility" is self-contradicting. — Metaphysician Undercover

Of course you did. I'm sorry. But in any case you've just accepted that mathematical objects aren't true objects. So what's the problem?

Mathematicians have made "infinite" into a new term, which really has very little resemblance to its metaphysical roots. — Metaphysician Undercover

I would be happy to accept that there are two concepts of infinity here. I think that may be because their concept has its roots in mathematics, whereas the metaphysical concept has roots elsewhere..

This leaves mathematics, and mathematicians in general, as fundamentally incapable of dealing with the metaphysical problems involved with the concept "infinite". — Metaphysician Undercover

So we just have a case of Domains of Magisterial Authority, and no need to fight about it.
Our only remaining issue is whether the problem of Achilles and the tortoise and Thompson's lamp is a mathematical problem or a metaphysical problem. Or maybe it's just a question of understanding two solutions to the same problem. They clearly won't be incompatible.

Again, I do not follow. Metres can be divided. We have centimetres and millimetres. But when we measure, at some point an approximation is made, a rounding off. — Metaphysician Undercover

Quite so. And we know that it is an approximation because we know what more and less accurate or precise measurement is. The exact measure, in the physical world, is the limit that empirical measurements can approach and never reach. That's mathematics and logic.

I don't think that this is relevant. I believe the analysis applies to all objects. But there is a problem with supposed "mathematical objects", and this is that we assume them to be infinitely divisible. And this assumption creates incoherency. This incoherency renders the supposed objects as not true objects. — Metaphysician Undercover

I'll take that. I wouldn't put it the same way, but it's near enough. I think, by the way, that you would have a tough job to convince mathematicians that there is an incoherency in the concept of the infinite. But that's not my problem.
Irrational numbers are built in to the system, so are recurring numbers, and so is infinity. You can't have one without the other. Sad, but true.

Lost in the ether, forever. — fishfry

I expect we'll survive.

I have not said that. I have said that I have no strong opinion about supertasks and am entirely comfortable arguing either side. — fishfry

Thanks for clarifying that. I find it quite hard to remember what everyone's position actually is. It gets lost in all the detail.

One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series. — Ludwig V

I would be very grateful if you could help me clarify this. When you say:-

When a mathematician says that 1/2 + 1/4 + 1/8 + ... = 1, they don't mean that you can perform this calculation with pencil and paper before lunchtime. They mean that the two expressions on either side of the equal sign denote the same real number. — fishfry

That's not quite as simple as it looks. The left-hand side will never equal the right-hand side as long as I try to make them equal by adding further steps in accordance with the same rule (...1/16, 1/32...). That's what it means to say that 1 is the limit, not the last step. But if I add 1/8 again, the two sides will be equal. Does that count as completing the sequence?

Ok. Possible worlds. I actually took a class where we talked about that, but I have a hard time understanding the concept. There are people who think possible worlds are real. I'm not one of them. And the whole metaphor is lost on me. — fishfry

Whether possible worlds count as real depends entirely on what you mean by "real". For some people, "real" comes down to true. If it is possible that it will rain tomorrow then possible worlds are real because it is true that it will rain tomorrow. For others, a possibility is not actual, so cannot be real.

And even then I reject the claim on its own merits. I could argue (not that I do, but that I could -- hope that's clear) that if time is modeled by the real numbers (agreed, that is a dubious assumption) then I perform a supertask every time I get up to go to the kitchen for a snack. I named my refrigerator Zeno. — fishfry

Quite so. But I think there is a confusion going on here. If you'll allow a temporary and artificial distinction... Roughly, it's the difference between an analysis, which doesn't change or affect its object, and a division or separation which does. That's the difference between measuring a plank of wood as 10 cm long and cutting it into 1cm lengths. The first is an analysis, the second is a division.
There are infinite ways in which I can mark out the plank, and they are all true at the same time and the physical object that is the plank is unaffected by any of them. True, the marks will be physical objects, so there will be limits to what I can do. But the system allows me infinite possibilities, including a convergent series. None of these makes the slightest difference to the plank. So when you visit Zeno for a beer, the fact that there are infinitely many analyses of your journey does not make the slightest difference. It's all in your head.

(Here's a thought. When you drink your beer, you have to drink 1/2 of it and then 1/4 of it and then... Your beer will never be finished. :smile: But then, a similar argument would show that you can't even start drinking it. :sad: )

You could probably help me out by clearly defining metaphysically impossible. — fishfry

It simply isn't clear. "Metaphysics" is a word looking for a meaning. There is some connection with logic, but what differentiates the two is a mystery.

Was this from you to me? That post of Michael disappeared for me as well. — fishfry

I still can't find it. I copied the quoted passage into my message, but not the commentary. Which is a pity.

c) it is metaphysically possible to recite the natural numbers at successively halved intervals of time — Michael

There's another strictly philosophical issue. I know that metaphysics overlaps with logic. I'm still trying to work out whether it is identical with logic.

Supertasks cannot be performed in any possible world. — Michael

suggests to me that it is a question of logic.
Subject to that, I do agree that c is false. I think that those who disagree with you think that they can stipulate by definition some sense in which it is true.

That there really is possible 'evidence' for God which is 'true' regardless of how any particular human sees it? — AmadeusD

Well, I was accepting the widespread belief that the issue is empirical and trying to think through the consequences. I hope I demonstrated that, as at present conducted, the debate will not be resolved, because the two sides talk past each other. On that assumption, agnosticism is the only rational possibility.

I don't rule out the remote possibility that something might turn up that would work as empirical evidence for God. But I can't imagine what that might be. If God did turn up in some way, I would have a great many unanswerable questions to discuss with them.

My actual position is that the concept of God is incoherent, which means that I can neither assert not deny that such a person exists - the concept would have to be coherent for either assertion to be meaningful. So I can't classify myself under any of the four propositions you listed. (I suggested D earlier, but I've changed my mind.)

However, on my understanding of what a religion is - a way of life and a collection of practices and attitudes, if the existence of God is treated as an axiom, it may well be rational. Atheism, then, would be the adoption of the non-existence of God as an axiom, and could well also be rational. (Geometries based on different axioms are all rational - consistent and complete.) Axioms are not adopted on rational grounds, they can only be assessed by their results.

I don't know why we would somehow attribute an ontological free lunch to the concept of God simply to avoid having to resolve the issue. — AmadeusD

I don't quite understand what you mean. What could I do to bring matters to a head?

There is nothing that is divisible infinitely, therefore this ideal needs to be excluded as necessarily an attempt to do the impossible. — Metaphysician Undercover

.... apart from a geometrical straight or curved line. I grant you that that is a concept of an abstract, ideal object. I grant you also that such division does not necessarily affect the unity of the object in any way.

It is a collection of distinct wavelengths, and I believe it is divided by the harmonic principles of the Fourier transform. — Metaphysician Undercover

That is the best representation of colour that physics can manage. But most people do not know about wave-lengths or Fourier transforms. So when I choose a red coat to wear to-day, how do I manage that? The colour that I am aware of is divisible in the sense that there are many colours and shades of colours. These correspond only roughly to the wavelengths of light.

No we don't need infinite divisibility, for the same sort of reason that we need infinite numbers, for the reasons I described. — Metaphysician Undercover

So how can we be sure that anything can be measured in terms of metres, if metres cannot be divided so that they exactly measure the length we are measuring?

I need to think about how to tackle your metaphysics.

An afterthought. Do I understand rightly that your analysis of wholes and parts applies to physical objects, and not to mathematical ones?

In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. — jgill

It's a bit more complicated than that. Bulbs like fluorescent ones flicker, but the light really is constant. It's like what is called "motion illusion" or the φ phenomenon. Film and television both rely on it. In a sense, the motion is an illusion, but in another sense, it isn't. The illusion of constant light, paradoxically, is real.

The problem though, is that in the prescribed scenario there is no such thing as "a certain rate". The rate is not constant, but rapidly increasing. The only constant is the rate of increase. That rate of increase is what I say is incomprehensible and incoherent. — Metaphysician Undercover

Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems.

One is to align the infinite sequence "0,1,0,1,0,1...." with the steps in the regress and to ask whether 0 or 1 can be assigned to the last step. An even simpler version is to align the natural numbers with the steps in the regress and ask whether the final step is an odd or an even number.

Those are clearly abstract sequences, and don't have the complications of trying to align the steps of a physical sequence to the steps of the regress. I agree that trying to do that is very problematic and almost certainly incoherent.

This is the trick of the whole thing. It really is about empirical things. These empirical things are space and time, each of these is known through experience. Then we take these empirical things and pretend that they are absolutely abstract, purely ideal, and stipulate ideal principles like infinite divisibility. — Metaphysician Undercover

I don't have a problem with ideal principles. They are very useful. We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical.

In the case of division though, we may assume that infinite divisibility would allow us to divide anything anyway, but this is really incoherent. That is because division implies, or requires logically, that there is something, an object of some sort, to be divided, and its divisibility will always be dependent on the sort of thing that it is. An object, or thing is a unity of some type, and as such there is always limits to its divisibility, whatever unifies also determines divisibility. — Metaphysician Undercover

I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility. But even that depends on the level of description you are applying, or, if you prefer, the level of analysis you are using.
The weight of a medium sized dry good is infinitely divisible. We couldn't measure its weight or calculate its centre of gravity it if it wasn't.
The colour of something isn't divisible at all. (That's why colour is disregarded in physics and treated as a mental phenomenon, whatever that is.)
You may say that these are not "objects", but that reinforces my impression that you are unduly focused on just one phenomenon in the physical world.

Then, someone creates a scenario, like the lamp or the op, which utilizes this purely ideal feature of infinite divisibility. Now we do not properly separate the purely ideal from the empirical, in our minds, so that "empirical time" interferes, and we say that 60 seconds must pass, it has to because experience tells us that it will. But that is allowing "time" to be an empirical thing. — Metaphysician Undercover

You may be right. I'm afraid that I'm like Augustine. I don't know what time is, though I do know, of course, what time it is right now and what time I woke up.
The great trap with infinity is the temptation to try to get round the endlessness of the sequence by hacking through each step in it.

There's little accounting for what students will do. More interesting is what you do. Assuming you're teaching philosophy and that you have some legitimate pedantic purpose in allowing them to use sources like PSE and TPF, what is your intention/goal in so doing? — tim wood

It isn't a question of allowing them to use PSE or any other site. They will use it or not, as they see fit.

When I was teaching, the most difficult thing was to persuade students to debate philosophy with each other, outside the class-room. I felt then that anything that encouraged them to participate was a good thing. True, they might pick up wrong-headed views, but they are liable to do that all by themselves.

So if I was teaching now, I would encourage them to use these sites and to bring back to class what they found there (within reason). It could be a valuable stimulus to discussion in the class-room or tutorial session. Of course, there would have to be health warnings, but I'm pretty sure the benefits would outweigh the risks. And I would have to insist that the most important thing is to READ THE BOOK.