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I understand the intuition you use to affirm that argument, I imagine others do too. At t=1 the sequence has ended, and the lamp must be either on or off.
Me and fishfry have insisted that this is a case of missing limit.
There's something going on here about ends and limits. I understood that the issue here is that although the series does have a limit, it doesn't have an end. As an abstract concept, one need not be particularly puzzled by this. But when you locate the series in time, it gets difficult.

It's a dilemma. The definition of an infinite series defines all the members of the series. That takes no time at all - not even an instant. So the time factor is actually irrelevant. But in another sense, each term of the series needs to be worked out, by us, and that is a process. That process must take time; actually, it would take infinite time - i.e. can never be completed.

Yes ok, but that supports the possibility that in the future, our current preoccupation with "mind as computer" will look as dated as "mind as waterworks" of the Romans.
Of course it does. I can't wait to see how it all plays out.
Though there is at least one case where the idea got transformed and returned with a vengeance. I mean the Pythagoras' and Plato's idea that ultimate reality is mathematical, meaning the only reality is the mathematical as opposed to the physical, world, returns as the idea that the physical world is mathematical. Weird.

That it explains nothing? I agree. Like saying "God did it." Or saying the Great Sky Computer (GSC) did it. Except that God is not restricted to being a computation, whereas the GSC is, making God a less unreasonable hypothesis.
My word, there's a discovery! A hypothesis that is more unreasonable than God! This should get a Nobel prize of some sort.

If you allow the transfinite ordinals, the sequence 1, 2, 3, ... has the limit ω. And even if this seems unfamiliar, it's structurally identical to the sequence 1/2, 3/4, 7/8, ... having the limit 1, which is much more familiar.
Yes, I do remember our earlier discussion of this. I don't pretend I understand them, but I do admit they exist - my allowing them or not is irrelevant.

What is the starting point of no axioms? It's like playing chess with no rules.
Did someone mention a starting-point of no axioms? It would be indeed be like playing chess with no rules - or discussing infinity.
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Of course it does. I can't wait to see how it all plays out.
Though there is at least one case where the idea got transformed and returned with a vengeance. I mean the Pythagoras' and Plato's idea that ultimate reality is mathematical, meaning the only reality is the mathematical as opposed to the physical, world, returns as the idea that the physical world is mathematical. Weird.

Tegmark's trolling. And the world is mathematical to us just as it's sound to a bat. The world does whatever it's doing. We do the math. The world is described by the math to a good degree of approximation. It's a metaphysical hypothesis that the world "follows" the math. Clearly the world did not stop following Newtonian gravitation when Einstein came along. Both theories are just approximations to something deeper ... or perhaps nothing at all. Nobody knows.

That it explains nothing? I agree. Like saying "God did it." Or saying the Great Sky Computer (GSC) did it. Except that God is not restricted to being a computation, whereas the GSC is, making God a less unreasonable hypothesis.
— fishfry
My word, there's a discovery! A hypothesis that is more unreasonable than God! This should get a Nobel prize of some sort.

The computational theory of the world requires that the world is a computation. That is indeed more restrictive than the hypothesis that God did it. Computable functions are a tiny subset of all possible functions. There is no reason at all for the world to be computable. I find it unlikely.

That is (to repeat myself): The street corner preacher says that all of us are created in the image of God. The TED talker says that we are all created by the Great Simulator, who operates as a Turing machine. That is a most restrictive stipulation. Far less likely than God. It's ironic that the intellectual hipsters mock God and flock to simulation theory, which is a far less likely hypothesis.

In your opinion, how is simulation theory any less magical and unrealistic than God? And why should God be restricted to be a Turing machine? I never understand this point.

If you allow the transfinite ordinals, the sequence 1, 2, 3, ... has the limit ω. And even if this seems unfamiliar, it's structurally identical to the sequence 1/2, 3/4, 7/8, ... having the limit 1, which is much more familiar.
— fishfry
Yes, I do remember our earlier discussion of this. I don't pretend I understand them, but I do admit they exist - my allowing them or not is irrelevant.

That's why I prefer the 1/2, 3/4, 7/8, ... example. Same structure in more familiar clothing.

What is the starting point of no axioms? It's like playing chess with no rules.
— fishfry
Did someone mention a starting-point of no axioms? It would be indeed be like playing chess with no rules - or discussing infinity.

Mathematicians have incredibly precise rules for infinity. The rules are the axioms of ZF or ZFC set theory.
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No evidence of your interpretation here.

A few quotes with no real context, does little. Anyway, it's off topic, and really sort of pointless to argue a subject like this. You have your opinion based on how you understand Plato, and I have mine. Due to the reality of ambiguity, i don't think there is a correct opinion here.

Fair enough, but to go on, as Plato does, to accuse the sophists of deliberate deception or wilful blindness is completely unjustified (except when, as in the Protagoras,(?) Gorgias (?) someone boasts about doing so – though it doesn’t follow that everyone that Plato accuses of rhetoric and sophistry did so boast.).

The problem is that "sophist" was a word with a very wide range of application at that time. In the most general sense, you'll see Aristotle use it to refer to someone who uses logic to prove the absurd. Zeno might be a sophist in this sense. But also "sophist" referred to people like Protagoras and Gorgias, for their use of rhetoric. And "sophist" also referred to those who had schools and charged money to teach virtue. So there was a range of meaning, but "rhetoric" seems to be the essential aspect, and this is a mode of persuasion which is not necessarily logical. Accordingly, "sophist" has bad connotations, but as Plato demonstrates in "The Sophist", it's very difficult to distinguish a philosopher from a sophist. It appears like either the sophist is a type of philosopher, or a philosopher is a type of sophist.

But accepting that connection is a long way from accepting that he had any doubts about the validity of his conclusions.

The issue is not the validity of the conclusions, it's the soundness. Take the Achilles for example, with two principle premises. First, to overtake the slower, the faster has to get to where the slower was. Second, in that time, the slower will move further ahead. So the faster does not overtake the slower, and this may repeat if the faster is still trying. It appears valid to me, so if we want to refute it we need to look at the premises, as Aristotle did. But when we try to understand how the premises are wrong, then there is disagreement amongst us, because we really can't demonstrate exactly what the premises ought to be replaced with.

,
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It's a metaphysical hypothesis that the world "follows" the math.
Believe it or not, that's an incredibly helpful remark. Not only do I understand and agree with it, but it also enables me to get a handle on what metaphysics is. Sorry, clarification - I am referring to the whole sentence, not just the last five words.

Far less likely than God. It's ironic that the intellectual hipsters mock God and flock to simulation theory, which is a far less likely hypothesis.
I had to look Tegmark up. No disrespect, but he does illustrate the observation that intellectuals are not exempt from normal human desires for fame and fortune, no matter how much they protest the contrary. There's also a normal human pleasure in astonishing and shocking the tediously orthodox Establishment.

That's why I prefer the 1/2, 3/4, 7/8, ... example. Same structure in more familiar clothing.
Yes, we had that discussion as well. You may remember that I had reservations. Same, but not identical, structures, I would say. But I don't expect you to like it. It doesn't matter until it becomes relevant to something.

Mathematicians have incredibly precise rules for infinity. The rules are the axioms of ZF or ZFC set theory.
My apologies. I should have restricted my remark to those who dream up paradoxes. Though perhaps even that is wrong. They may be exploiting the rules themselves, rather than merely breaking them. The mathematical rules for infinity don't seem particularly helpful in resolving these problems.
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i don't think there is a correct opinion here.
Well, I'm almost certain there isn't. But my disagreement with you prompted me to look more closely and acknowledge something that feels like error in one or two respects.

It appears like either the sophist is a type of philosopher, or a philosopher is a type of sophist.
Yes, I understand your account of this. It's important to add that Plato thinks that the sophist mimics the philosopher and what he says is accounted rhetoric because it mimics the speech of the philosopher. (He didn't have a concept of logic as we think of it.) The mimicry is the reason why he condemns both the man and what he says. How does he distinguish mimicry from the real thing? Mimicry seems to be true, but is not. So, in the end, the distinction between the two in his writings is the distinction between those who agree with him and those who do not. I'm not trivializing Plato. It is a universal problem.

Socrates (as presented by Plato) considered himself wiser than anyone else because he knew he didn't know anything, which doesn't seem to leave much room for anyone else (at least in Athens) to be a philosopher. However, his dialogues with sophists do not show Socrates treating them disrespectfully and this is something of a puzzle. The orthodox interpretation regards Socrates' respect as ironic. Maybe it is. But maybe Plato's practice was a bit less dismissive than all this implies.

It is very difficult. If you believe that you have got hold of an absolute guarantee of truth and someone else disagrees with you, the temptation to dismiss them, rather than just their view, is very great. If P implies Q and P is true, but someone rejects your conclusion, what are you to do with them? Have you ever read C. L. Dodgson's article on Achilles and the Tortoise? It faces the problem head-on. I won't spoil the plot. You should be able to get hold of it somewhere on the web. Wittgenstein faces this issue in his discussion of rule-following. I don't know of anyone else who takes the issue seriously.

The issue is not the validity of the conclusions, it's the soundness.
Yes, you are right. I was not accurate. Sorry.

But when we try to understand how the premises are wrong, then there is disagreement amongst us, because we really can't demonstrate exactly what the premises ought to be replaced with.
I had noticed. Which is why I keep trying to suggest other approaches. With little success, I admit.
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It appears like either the sophist is a type of philosopher, or a philosopher is a type of sophist.
In my reply to this quotation, I said
Have you ever read C. L. Dodgson's article on Achilles and the Tortoise? It faces the problem head-on. I won't spoil the plot.
This was a mistake. I intended to spare you unnecessary verbiage in my reply. But what I said was annoying and unnecessary. I'm sorry.
The point of the article is very simple. Achilles and the tortoise are chatting after Zeno's race. Achilles observes:- "I was first past the post, so I won". The tortoise replies:- "I don't accept that." Achilles:- "What do you mean? The first competitor to pass the post is the winner of the race, and I passed the post first, so I won". Tortoise:- "I don't accept that". It continues for some time. There's no resolution - not even Achilles killing the tortoise - not that that would count as a resolution. But we all know what happens in real life when such situations arise.
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Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already.

There isn't one. I've answered this several times already. That's what it means for me to accept P1.

But you need to prove P2. You haven't done so.
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I have agreed repeatedly that we can't "count all the natural numbers backwards" since an infinite sequence has no last element.

So we're back to my post here:

a. I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum

b. I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum

Here is our premise:

P1. In both (a) and (b) there is a bijection between the series of time intervals and the series of natural numbers and the sum of the series of time intervals is 60.

However, the second supertask is metaphysically impossible. It cannot start because there is no largest natural number to start with. Therefore, P1 being true does not entail that the second supertask is metaphysically possible.

Therefore, P1 being true does not entail that the first supertask is metaphysically possible.

You accept that (b) is impossible but you claim that (a) is possible. You have to prove this. P1 doesn't prove it.
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Tegmark's trolling. And the world is mathematical to us just as it's sound to a bat. The world does whatever it's doing. We do the math.

That is the view that mathematical is somewhat of an empirical endeavor. Many disagree however, and think that mathematics is something fixed and representative of the world.
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Tegmark's trolling. And the world is mathematical to us just as it's sound to a bat. The world does whatever it's doing. We do the math.
That is the view that mathematical is somewhat of an empirical endeavor. Many disagree however, and think that mathematics is something fixed and representative of the world.
Certainly mathematics is, in a sense, fixed. But what we are talking about it is applied mathematics. It seems pretty clear that arithmetic and geometry originated in severely practical needs of large empires. But it does seem to have taken off on its own, as it were, as a theoretical enterprise. Here, we are talking about applied mathematics.
I think what @fishfry means to say is that mathematics is the way the world is represented to us. That's the point of the comparison with what sound is to a bat. I would rather say that mathematics is the way we represent our world to ourselves.
It's true that the mathematical techniques we use are fixed - though we also develop new techniques, as in 17th century calculus or non-Euclidean geometries. But we have to work out how they can be applied to specific phenomena.
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So when someone describes the situation in a way that seems to make that fact impossible, why don't we just reject it as inapplicable?
I do, but Zeno's division of the task didn't seem to make anything impossible. To read Aristotle, Zeno seems to believe in the discreetness of anything of magnitude, directly contradicting Aristotle's physics of the day, which were his opinions pretty much by definition. Much of his opinions held for millenia. Some still do.
Also keep in mind that physics was absolute back then, and calculus was unheard of.
So Zeno's premise that in any task of changing location, one must first go to the halfway point. Zeno seems to have put that up there as a ridiculous premise, one in which he didn't believe. It was an attack on Aristotle's position which would find no fault with the statement. Given Zeno's beliefs, the premise above is false, and he attempts to demonstrate why, but of course he can only do so by begging his own opinion, which is the second premise that I've been going on about.

But we allow physical impossibilities into fiction all the time. They even crop up in philosophical examples. "The sun might not rise tomorrow morning"
Not an example of a physical impossibility. Yes, i agree that physical impossibilities can be turned into fiction. Did I say otherwise?

Your point about the final state not being defined is about logic, not physics (despite some people thinking that it is about physics).
The state of Achilles is that he is even with the tortoise. It's admittedly not final because he continues on after the task of overtaking it is complete and takes the lead. There's nothing about that where physics stops being relevant.

The lamp example? That isn't physics. Never was.

In any case, the final state is defined. It must (on or off) or (0 or 1).
A list of valid options is not a definition of a state.

Wouldn't it be more accurate to say that it is undetermined?
Synonym?

But it would be absurd to say that every state in the series is indeterminate.
They are, or at least the existing ones are. None of the ones you listed was an existing step.

The time length is irrelevant.
Says the proponent that time stops.

See my comment above. I suspect Zeno believed his premise to be false, that one must first get halfway before getting to the goal. He was trying to illustrate this belief by driving Aristotle's assertions to absurdity, but he must beg his own beliefs to do this.
If there was a better worded argument provided by Zeno himself, perhaps Aristotle didn't convey the full argument, in the interest of waving away a suggestion that he is wrong. So many modern mathematical tools were not available to them back then.

I gave you Aristotle's wording.
To me it was just another wording, but apparently so since I see it referenced verbatim on so many discussions. Interesting is the total lack of mention of the tortoise.

He rejects most of the arguments because they contradict his own assertions.

The matter of instants appears irrelevant here, and the problem seems to be with the assumed nature of space, rather than time.
The argument is the same with space. He says "time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles". Space qualifies as an 'other magnitude'.

Like above, noAxioms insisted Zeno did not conclude that the faster runner could not overtake the slower,
I said no such thing. Zeno very much is reported to have concluded such things.
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There's something going on here about ends and limits. I understood that the issue here is that although the series does have a limit, it doesn't have an end.

Yes, but the meaning of limit here isn't the same as it is used in Calculus. It is in the sense used here.

His analogy/metaphor implies that mathematics is something that we impose onto the world instead of something that we derive from the world. His position is anti-realist therefore. If he was right, platonism about mathematics wouldn't be such a strong position today.
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Also keep in mind that physics was absolute back then, and calculus was unheard of.
I don't think the calculus is relevant. In any case, I understood that it stated the problem rather than solving it - calculating the result to as close an approximation you aspire to, but never absolutely. I wouldn't be surprised if I got that wrong.

Not an example of a physical impossibility.
If you accept that Twin Earth is not physically possible, there's no need to argue about the sun example. Maybe your imagination is richer than mine.

A list of valid options is not a definition of a state.
Monochrome = (black, white or grey all over)? Red = (indefinite number of shades of red)? Sibling = (brother or sister)? Parent = (Mother or father).

Synonym?
I don't know, what do you think? I had in mind that every step is defined by the formula, which cannot be applied to any step unless it's predecessor is determined (except for the first step.) I wouldn't go to the stake for one or the other.

They are, or at least the existing ones are. None of the ones you listed was an existing step.
Yes. The first step exists if you are looking forward, but if you are looking backward, it doesn't. But in the normal world, the first step is the last step - i.e. exists whichever direction you are looking or even if you are not looking at all. This is Berkeley's world.

His analogy/metaphor implies that mathematics is something that we impose onto the world instead of something that we derive from the world. His position is anti-realist therefore. If he was right, platonism about mathematics wouldn't be such a strong position today.
I don't quite get what "anti-realist" means here. But you are right. I was trying to articulate the idea that counting is not a determinate description, but a system for generating determinate descriptions; we have to apply the system and discover what pieces of the number system apply in each case. Actually, one could see some sense in saying both that the mathematics is derived from the world and that it is imposed on the world.
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I don't quite get what "anti-realist" means here

The idea that mathematical entities aren't real, especially that they aren't abstract objects.
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Socrates (as presented by Plato) considered himself wiser than anyone else because he knew he didn't know anything, which doesn't seem to leave much room for anyone else (at least in Athens) to be a philosopher. However, his dialogues with sophists do not show Socrates treating them disrespectfully and this is something of a puzzle. The orthodox interpretation regards Socrates' respect as ironic. Maybe it is. But maybe Plato's practice was a bit less dismissive than all this implies.

I believe Socrates (as portrayed by Plato) had great respect for the sophists. They displayed power and influence, and this piqued his interest. In Plato's dialogues, Socrates holds lengthy discussions with some sophists, and this would not be possible without the appropriate respect. On the other hand, I also believe that since the sophists presented themselves in a conceited way, as filled with a sort of complete or perfect knowledge, this produced a challenge in Socrates, to demonstrate their faults and weaknesses. Because Socrates had some degree of success in this personal challenge, Plato developed a level of disdain for them.

Prior to Socrates I believe that sophists were generally well respected, and this is evidenced by the power of their rhetoric. Socrates revealed the subjectivity of rhetoric, leaving the character of the sophists who employed it, exposed. The principal sophists who were exposed in this way, were the the politicians of Athens. But Socrates carried on toward exposing those in the even higher level, more exclusive schools of logic (I don't agree with you that there was no concept of "logic" at this time) like the Pythagoreans and Eleatics, and this allowed Plato to class them as sophists. This is where Zeno fits in. And Socrates is portrayed by Plato as having great respect and curiosity for the lofty principles held by these prestigious schools. Nevertheless, despite great respect for the individuals, he sees that there must be flaws in the principles, and therefore proceeds with his personal challenge of engaging the individuals to defend, and ultimately reveal those faults.

I think that the important point is the use of valid reasoning with unsound premises. This is how Aristotle attacked Zeno's paradoxes. But Aristotle didn't have a good understanding of the nature of knowledge, and the effects of faulty premises. He claimed that logic leads us from premises of greater certainty, to conclusions of lesser certainty, when in reality the opposite is true. Uncertainty in the premises is what introduces uncertainty into the conclusions. And the problem is that many premises are intuitive notions simply taken for granted, such as in the Achilles, the premise that the faster must first reach the place where the slower is, prior to passing. In reality, the faster passes the slower without ever sharing the same place.

The Aristotelian view of knowledge is still common place. You'll see that many here at TPF argue that there are fundamental principles, 'bedrock propositions' or something like that, which are beyond doubt, and support the whole structure of knowledge. In reality, those fundamental principles are the least certain because they are taken for granted, lying at the base of conscious thinking, bordering on subconscious knowledge. Those highly fallible intuitions are the ones most needing the skeptic's doubt, but it takes someone like Zeno to demonstrate this.

Says the proponent that time stops.

Huh? I said that time stops? I don't think so. I said that in the scenario of the op, 60 seconds will never pass. But clearly time does not stop. In that scenario, time keeps passing in smaller and smaller increments, such that there is never enough to reach 60 seconds, but time never stops. The claim that 60 seconds must pass or else time will stop, is derived from different premises which are inconsistent with the described scenario.

I suspect Zeno believed his premise to be false...

That's what I was arguing as well, but Ludwig produced references to show that this might not be the case.
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Believe it or not, that's an incredibly helpful remark.

Thanks.

Not only do I understand and agree with it, but it also enables me to get a handle on what metaphysics is. Sorry, clarification - I am referring to the whole sentence, not just the last five words.

Well metaphysics is just "What is reality?" And it can't exactly be our math, because we can see that it wasn't quite what Newton wrote down, and in the end it won't quite be what Einstein wrote down. It's actually kind of strange that math doesn't exactly describe reality, but so well approximates it. Our theories get better and better but never get there. As if reality is the limit of our theories.

Or worse. Our math is like the bat's echoes. Just the only tool we have to understand the world, but greatly limited. And we think we know everything.

I had to look Tegmark up.

Goes a step farther. The universe isn't just described by math, it "is" math. Which is a category error so massive that Tegmark must be trolling. The equations of motion describe the planets, they aren't the planets themselves. The map is not the territory. Just as the source code for a program must be executed on hardware in order to do anything.

Tegmark must be trolling. There is no other explanation. That so many take him seriously is a good reason to be skeptical of experts, celebrity scientists, and "public intellectuals."

No disrespect, but he does illustrate the observation that intellectuals are not exempt from normal human desires for fame and fortune, no matter how much they protest the contrary. There's also a normal human pleasure in astonishing and shocking the tediously orthodox Establishment.

We're in agreement. Bostrom (we're all sims) and Tegmark (we're all mathematical structures) must be enjoying themselves tremendously. Most likely when they write serious stuff, nobody pays attention.

That's why I prefer the 1/2, 3/4, 7/8, ... example. Same structure in more familiar clothing.
— fishfry
Yes, we had that discussion as well. You may remember that I had reservations. Same, but not identical, structures, I would say. But I don't expect you to like it. It doesn't matter until it becomes relevant to something.

Well it's relevant to the Thompson lamp. It's a mathematical model of a sequence with its limit point adjoined. The example is so familiar to me that I thought it would add clarity. To the extent it got in the way, perhaps I should rethink how I present the idea.

My apologies. I should have restricted my remark to those who dream up paradoxes.

Mostly philosophers who prefer to indulge in the vagueness of word games rather than the precision of math. But I concede that many smart people take these puzzles seriously. I respect that, but for some reason the fascination eludes me.

The lamp's defined at each point of the sequence, but it's not defined at the limit. There's no way to make the sequence continuous, se we are free to make the terminating state anything we like. There is no natural continuation. That seems perfectly clear to me. I don't know why it's not perfectly clear to everyone else. I actually have a difficult time seeing the other points of view.

Though perhaps even that is wrong. They may be exploiting the rules themselves, rather than merely breaking them. The mathematical rules for infinity don't seem particularly helpful in resolving these problems.

Maybe we'll get some new infinitary physics some day.
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Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already.
— fishfry
There isn't one. I've answered this several times already. That's what it means for me to accept P1.

But you need to prove P2. You haven't done so.

But you just proved P2 yourself! You agreed that under the hypothesis of being able to recite a number at successively halved intervals of time, there is no number that is the first to not be recited.

This proves that all numbers are recited. This is a standard inductive proof that a high school student should be able to not only understand, but even figure out for themselves. If someone's high school didn't teach them mathematical induction perhaps they picked it up in Discrete Math class; and if not, then the writeup on Wikipedia would suffice.

You have proven P2 yourself simply by agreeing that there is no first number that is not recited.

If no number did not get recited, then they all did.

So we're back to my post here:

a. I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum

[details omitted]

You accept that (b) is impossible but you claim that (a) is possible. You have to prove this. P1 doesn't prove it.

Let's focus on one thing at a time. Regarding your example of counting the natural numbers backward, or letting the sequence get smaller when time goes forward, the 1, 1/2, 1/4, ... idea; I have repeatedly asked you if you understand and agree that any interval of real numbers containing the limit of a sequence, necessarily contains all but finitely members of the sequence.

I need you to understand that in order for me to explain to you how the backwards counting puzzle is resolved.

Since I've asked you several times to just tell me, yes or no, do you understand what I said, and you have repeatedly ignored me, conversational progress can not be made on this point.

So let's stick to the inductive proof, in which you yourself proved P2 is true. Let's get back to the backwards counting example after you tell me, yes or no, do you understand the property of limit points of sequences that I keep asking you about and that you keep not answering.
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Tegmark's trolling. And the world is mathematical to us just as it's sound to a bat. The world does whatever it's doing. We do the math.
— fishfry

That is the view that mathematical is somewhat of an empirical endeavor. Many disagree however, and think that mathematics is something fixed and representative of the world.

Surely few if any people believe math is "fixed." Math is historically contingent and changes all time time, with a massive volume of new papers published every day.

If you are referring to some kind of Platonic math that's already known by God, that we are just discovering, that's an entirely different discussion.

Am I understanding you correctly?

Besides, math can't "represent the world," simply because there are Euclidean and non-Euclidean geometry. They can be used to represent the world; but they can't both be true, hence they can't both "represent the world." They can only be used to represent the world.

Math can not tell you what's true about the world. It can only be used to model various aspects of the world. That's different.
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Certainly mathematics is, in a sense, fixed.

I'm asking, in what sense? Surely math has never been fixed. It's always changing. It's a human activity.

Do you believe that God's math book is already written? Is that how you are defining math, as a Platonic ideal "out there" that exists even before we've discovered it?

But what we are talking about it is applied mathematics. It seems pretty clear that arithmetic and geometry originated in severely practical needs of large empires.

Originated as, yes. But that doesn't restrict how math is seen today.

But it does seem to have taken off on its own, as it were, as a theoretical enterprise. Here, we are talking about applied mathematics.

I don't recall stipulating to any such restriction.

In any event, what is the difference between abstract and applied math? Only time and historical contingency. Non-Euclidean geometry was abstract and useless in 1840, and it became applied (to general relativity) in 1915. [Someone complained a while back about my phrasing. I could say Riemannian geometry, since there are other flavors of non-Euclidean geometry. I'm making a different point which I hope is clear].

Likewise number theory, regarded as supremely useless since the time of Diophantus; and now the basis of public key cryptography, the basis of modern digital commerce, since as recently as the 1980s. That's 2200 years of uselessness, only to become supremely useful.

The difference between abstract and applied math is time and history.

I think what fishfry means to say is that mathematics is the way the world is represented to us. That's the point of the comparison with what sound is to a bat. I would rather say that mathematics is the way we represent our world to ourselves.

I could have said that. I could have said your formulation too. My only point is that how humans model the world and the world itself are two different things. Aren't those Kant's phenomena and noumena? I'm not a philosopher but that's what he meant, right? Humans try to explain the phenomena. We can't know the noumena. That's Plato's cave analogy again.

It's true that the mathematical techniques we use are fixed - though we also develop new techniques, as in 17th century calculus or non-Euclidean geometries. But we have to work out how they can be applied to specific phenomena.

I think you are agreeing with me. Abstract today, applied tomorrow. Or often the reverse. We invent new abstract math to help us understand some real world application. It goes back and forth.
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But you just proved P2 yourself! You agreed that under the hypothesis of being able to recite a number at successively halved intervals of time, there is no number that is the first to not be recited.

I agreed that if P2 is true then C1 is true, as I have agreed from the beginning.

This doesn't prove that P2 is true.
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As if reality is the limit of our theories.
Since I don't know what "reality" means in its philosophical sense (which I designate by "Reality", but I do know, roughly, what you mean by "the limit of our theories", I would prefer to say "The limit of our theories is Reality". I'm of the school that teaches that the philosophical sense is metaphysics, and nonsense. But, since I arrived on these forums, I've had to recognize that, in philosophical discourse, "Reality" is a term in regular use and with some level of common understanding.
It's still a bit broad brush. I can understand it in the context of the inescapable inaccuracy of measurement in physics, etc, contrasted with the preternatural accuracy of (many, but not all) mathematical calculations. It's a version of Kant's regulative ideals and gives some content to phenomena/noumena and an explanation how they might be related.
Or worse. Our math is like the bat's echoes. Just the only tool we have to understand the world, but greatly limited. And we think we know everything.
Not the only tool. We have sound as well. Not that we know everything, thank God.

The universe isn't just described by math, it "is" math. Which is a category error so massive that Tegmark must be trolling. The equations of motion describe the planets, they aren't the planets themselves. The map is not the territory. Just as the source code for a program must be executed on hardware in order to do anything.
Tegmark must be trolling. There is no other explanation. That so many take him seriously is a good reason to be skeptical of experts, celebrity scientists, and "public intellectuals."
Well, I would certainly want to get him to explain what he means by "is". That might slow him down a bit.
Intellectuals have human motivations and follies just like everyone else - and some of them would do well to acknowledge that. I understand also that it is irresistibly tempting to explain people's failures to recognize conclusive rational arguments in ways that they will not like. But one needs also to understand that can be a trap. Hence Plato turned a classification of the philosophers he disagreed with into a term of abuse - "sophist", "rhetoric". You may have noticed that I'm engaged in some discussion with @Metaphysician Undercover about this issue in relation to Zeno. They, and, apparently @noAxioms cannot believe that Zeno believed his own arguments - and that's not an irrational response because they are incredible. Nevertheless, I can't believe that they believe that. It's not easy. But I think it is important not to follow Plato's example in this respect.

The example is so familiar to me that I thought it would add clarity. To the extent it got in the way, perhaps I should rethink how I present the idea.
I don't think there was anything wrong with your explanation. There's no such thing as the bullet-proof, instantly comprehensible, explanation. On the contrary, it helps to allow people space to turn what you say round and poke it and prod it. It's part of the process of coming to understand a new idea.

The lamp's defined at each point of the sequence, but it's not defined at the limit.
Quite so. It's a sequence, but also a chain, because each point of the sequence depends on its predecessor. The reason it's not defined at the limit is that we can never follow the chain to its' conclusion - even thought the conclusion, the end, the limit, is defined.
It seems paradoxical, because the limit is established before the chain can begin. The first step is to define the limit and the origin; that gives us something we can divide by 2 - and off we go.
This may not be mathematics. But I do maintain it is philosophy.
The consequence is that the series "vanishes" if we try to look back from the "end". It's existence depends on our point of view. I don't suppose that any mathematician would be comfortable with that, but I plead that we are talking about infinity and standard rules don't apply.

I'm asking, in what sense? Surely math has never been fixed. It's always changing. It's a human activity.
Originated as, yes. But that doesn't restrict how math is seen today.
I think you are agreeing with me. Abstract today, applied tomorrow. Or often the reverse. We invent new abstract math to help us understand some real world application. It goes back and forth.
I agree with all of that. But I think it is very, even hideously, complicated.
It seems to me that we should always be specific about what is fixed and what is not. There may be disagreement about what goes in to which classification or what "fixed" means. But to say "math" without specifying further leads to confusion.
Arithmetic, for example, is (relatively) fixed, though it may be modified from time to time. The inclusion of 0 and 1 as numbers is an example. Number theory might count as another example - I'm not sure about that. But once the methods of calculation are defined, they are fixed and the results from them are fixed as well. One could say, however, that both methods and results are discovered rather than defined, because there are ways of demonstrating whether a particular procedure gives the right result or not - through the application of the results or through the application of criteria like the consistency and completeness of the system. Euclidean geometry is similar, so far as I'm aware.
Algebra, calculus, non-Euclidean geometry, infinity theory are all additions to mathematics, rather than replacements of anything. It is almost irresistible to speak of them as developed or created rather than discovered, but since they share something with arithmetic and geometry, there are some grounds for speaking of them as discovered, because they were always possibilities, in some sense. What is it that is shared? The best I can do is to say something like logic - a sense of what is possible, or permitted.

This is not irrelevant to this thread. Once we have realized that "+1" can be applied to the result, it would not be wrong to say that the result of every step is fixed, whether or not we actually do add 1 to the 3,056th step. The result of each step is "always already" whatever it is. (I think it derives from Heidegger, but that doesn't prevent it from being helpful.) It captures the ambiguity between "+1" as something that we do and something that is done as soon as it is defined, or even before that.
As a result of the simple recognition of a possibility, we find ourselves plunged into a new and paradoxical world. I mean that it is simply not clear how the familiar rules are to be applied. Which makes it clear that we have to invent new ones - or are we discovering how the familiar rules apply or don't? I don't think there is a determinate answer and "always already" recognizes the ambiguity without resolving it.
When we refer to a step in the series, are we talking about something that we do (and may not do) and which actually takes time or something that is "always already" done, whether we actually ever do it or not?
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If you are referring to some kind of Platonic math that's already known by God, that we are just discovering, that's an entirely different discussion.

Am I understanding you correctly?

Yes.

Besides, math can't "represent the world," simply because there are Euclidean and non-Euclidean geometry. They can be used to represent the world; but they can't both be true, hence they can't both "represent the world." They can only be used to represent the world.

Math can not tell you what's true about the world. It can only be used to model various aspects of the world. That's different.

You are assuming a non-realist view of mathematical entities again. You can still have Euclidean and non-Euclidean facts in the world as different facts just like algebra and calculus are different facts. Many philosophers think mathematical objects are real objects that exist outside of space and time.
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The idea that mathematical entities aren't real, especially that they aren't abstract objects.
Many philosophers think mathematical objects are real objects that exist outside of space and time.
It would be merely picky to ask whether "+" and "-" are objects, because it is obvious that they are operations to be carried out on objects. Still, there is a question what that means. But it would a distraction from the main event.

Take numbers. Does anyone deny that numbers exist? Does anyone claim that they are concrete in the way that bricks and timbers are? They are like objects in some ways, in that they can be distinguished one from another and counted. But what kind of objects are they?

Geometrical shapes like triangle and circles seem to be different. In one way, physical objects are called triangle and circles, but are acknowledged to be approximations to the ideals of geometry. Ideals are not physical objects and do not exist "in space and time". They certainly exist and are real in that sense.
But what does "outside space and time - beyond the fact that there is no possible answer to the question "Where are they now?" What does that mean - I mean what do "exist" and "real" mean here? True, we can say that they are abstract, but what does that mean - apart from "not physical" or "not concrete".
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It would be merely picky to ask whether "+" and "-" are objects, because it is obvious that they are operations to be carried out on objects.

I don't know what platonists say about mathematical operations. Perhaps they would say they are relations among numbers (which for them are real things). Most kinds of dualism for me are overly problematic and I think only nominalism is really sensible when we get to the bottom of things.

Does anyone deny that numbers exist?

Putting it bluntly, nominalists and conceptualists and every kind of anti-realist strictly defined.

Does anyone claim that they are concrete in the way that bricks and timbers are?

In a way, immanent realists do.
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I think only nominalism is really sensible when we get to the bottom of things.
Oh, well. That changes everything. I thought I was talking to a platonist and trying to get him to face up to some of his problems. But that's a bit futile.

Putting it bluntly, nominalists and conceptualists and every kind of anti-realist strictly defined.
So you deny that numbers exist? Really?

Why do you keep reminding us that platonists exist? Is it perhaps because you think people should not say mathematics is thus and so, but be more specific? Or because people so often say that mathematicians think this and that when it is plain that only some mathematicians think those things? Those are tendencies that annoy me.
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I don't think the calculus is relevant.
Well, when was the notion of limits of a series introduced? Not back then I think. I'm not an expert in the history, but Zeno was definitely using techniques beyond the state of the art at the time. Good for him.
There are people today that say that there are no real infinities, whatever that means. I think this might be one example of such an assertion.

If you accept that Twin Earth is not physically possible
Where there's not-water? I accept that as a physical impossibility, yes, but overtaking a tortoise is not.

there's no need to argue about the sun example. Maybe your imagination is richer than mine.

A list of valid options is not a definition of a state.
— noAxioms
Parent = (Mother or father).
Well illustrate. A list is not a parent, so I disagree with the '=' you put there. I'm sure there is a correct symbol to express that any member of that list satisfies the definition of parent.

Tegmark must be trolling.
I replied to this in the simulation topic since discussion of it seems to be of little relevance to this topic.

But you [Michae] just proved P2 yourself! You agreed that under the hypothesis of being able to recite a number at successively halved intervals of time, there is no number that is the first to not be recited.

This proves that all numbers are recited.
The two of us also seem to be on the same page.

I said that time stops?
Not in those words. "Does not allow for a minute to pass", like somehow the way a thing is described has any effect at all on the actual thing.
The specifications do not allow for a minute to pass, that's the problem.
Anyway, I see nothing in any of the supertask descriptions that in any way inhibits the passage of time (all assuming that time is something that passes of course).

I don't think so. I said that in the scenario of the op, 60 seconds will never pass.
The OP scenario is pure abstract, and it directly describes a state beyond the passage of a minute.

But clearly time does not stop. In that scenario, time keeps passing in smaller and smaller increments, such that there is never enough to reach 60 seconds, but time never stops.
Ah, it slows, but never to zero. That's the difference between my wording and yours. Equally bunk of course. It isn't even meaningful to talk about the rate of time flow since there are no units for it. The OP makes zero mention of any alteration of the rate of flow of time.

So you deny that numbers exist? Really?
Not to put words in anybody's mouth, but such a statement depends heavily on the definition of 'exists'. For instance, does the number 37 have a location somewhere in our universe? When was it created? That references a definition of "is a object in our universe". If you define 'exists' as 'is an abstract concept in some mind somewhere', then 37 exists as long as somebody is thus abstracting. It's still a version of 'is part of the universe'.
I am not really sure of the definition Lionino is using. I didn't get it from the brief context.

Look at Tegmark's view mentioned above. He is definitely using a definition of 'exists' that doesn't supervene on our universe, and suggests that the reverse is the case.
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So you deny that numbers exist? Really?

I don't have a very strong stance on this debate exactly, I am aversed to dualism in objects, so positions like platonism irk me immediately. For me, numbers exist more like Superman exists or an equation exists rather than how my hand exists.

Is it perhaps because you think people should not say mathematics is thus and so, but be more specific?

Because I think people should not claim X when whether X is far from being settled by specialists. Not exactly the same but close to how you put it:

Or because people so often say that mathematicians think this and that when it is plain that only some mathematicians think those things?
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In any case, I did a short breakdown of the topic here: https://thephilosophyforum.com/discussion/15080/grundlagenkrise-and-metaphysics-of-mathematics
Tones, who knows about the topic much more than me, had a few corrections to make about it, some of which I have implemented already.
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Because I think people should not claim X when whether X is far from being settled by specialists. Not exactly the same but close to how you put it:
Yes. That annoys me as well. Though there has to be a little wriggle room, doesn't there? Philosophers, in particular, would be very constricted if such a rule were strictly enforced. Though I do agree that some philosophers would do well to be much more cautious than they are. For example, it is clearly wrong to treat the latest speculations from speculative cosmology as established fact.

Zeno was definitely using techniques beyond the state of the art at the time.
Yes. And as you say, they were beyond the state of the art at the time, so what he was doing needs to be rather carefully described (unless you are going to propose time travel.) It is very difficult to handle anticipations of later developments in historical texts. Some people have seen anticipations of Einstein in Berkeley. In a sense, they may be there. But I think that's merely a similarity rather than an anticipation. I don't know how to represent this case properly.

There are people today that say that there are no real infinities, whatever that means.
Yes, and I think that @Lionino may have been protesting at such ways of talking. If one is not a platonist, the way to say what you want to say is to conceptualise "real" in a non-platonic way. To outright deny that infinities exist is just attention-seeking. Though perhaps philosophers are not exempt from such a very human temptations.

A list is not a parent, so I disagree with the '=' you put there. I'm sure there is a correct symbol to express that any member of that list satisfies the definition of parent.
I've noticed a variety of extensions of the use of "=" lately, so I'm sorry if I misused it. I'm glad you recognized what I was trying to say.
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For me, numbers exist more like Superman exists or an equation exists rather than how my hand exists.
Yes, for me, that is the most helpful approach. Different kinds of object - different modes of existence. If you haven't come across it before, you might find this reference useful.
Fictionalism is an approach to theoretical matters in a given area which treats the claims in that area as being in some sense analogous to fictional claims: claims we do not literally accept at face value, but which we nevertheless think serve some useful function.
The only downside I can think of is that it might lead to us conceding that God exists just because so many people believe that he/it does. But then, the same would apply to Zeus, Apollo, Thor, Loki, Horus, Ptah etc. So no-one could draw the conclusion that one is a believer in any of them.
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