Also keep in mind that physics was absolute back then, and calculus was unheard of. — noAxioms

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. — noAxioms

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. — noAxioms

Monochrome = (black, white or grey all over)? Red = (indefinite number of shades of red)? Sibling = (brother or sister)? Parent = (Mother or father).

Synonym? — noAxioms

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. — noAxioms

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. — Lionino

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.

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. — Lionino

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.

It appears like either the sophist is a type of philosopher, or a philosopher is a type of sophist. — Metaphysician Undercover

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. — Ludwig V

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.

i don't think there is a correct opinion here. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

I had noticed. Which is why I keep trying to suggest other approaches. With little success, I admit.

It's a metaphysical hypothesis that the world "follows" the math. — fishfry

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. — fishfry

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. — 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.

Mathematicians have incredibly precise rules for infinity. The rules are the axioms of ZF or ZFC set theory. — fishfry

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.

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. — Lionino

Me and fishfry have insisted that this is a case of missing limit. — Lionino

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. — fishfry

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. — fishfry

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. — 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.

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.

If jorndoe is representing the view well, I am confident both have good reasons to make such equations; I was exploring ways to make the semantics of "metaphysical" not fully overlap with "logical" or "physical". — Lionino

Well, whatever prompted you, the project makes sense to me and I agree with Toulmin. I'm not convinced about the relationship of those propositions with metaphysics or their classification in the analytic/necessary/a priori constellation. However, preserving those concepts doesn't seem to me particularly important. I would be quite happy to abandon all of them.

It's simple, talk to people, ask them. — Metaphysician Undercover

If I knew how to ask without leading them into philosophy, I would.

Well, there is a lot of information available from Plato. — Metaphysician Undercover

The Stanford Encyclopedia is the best quick reference that I know of for something like this.

Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato’s Parmenides. There we learn that Zeno was nearly 40 years old when Socrates was a young man, say 20. Since Socrates was born in 469 BC we can estimate a birth date for Zeno around 490 BC. Beyond this, really all we know is that he was close to Parmenides (Plato reports the gossip that they had a sexual relationship when Zeno was young), and that he wrote a book of paradoxes defending Parmenides’ philosophy. Sadly this book has not survived, and what we know of his arguments is second-hand, principally through Aristotle and his commentators

SEP - Zeno's paradoxes
From what I could find, Aristotle has very little about Zeno and nothing about his motives. But what he does summarize (some of) the arguments, which Plato doesn't.
I hadn’t realized quite how close in time they were. It seems that the scenario in the Parmenides, which seems to be far and away the best source we have, could have taken place. Not that we know that it did. On the face of it Plato is not an implausible source – if only separating out the history in Plato was not so complicated.

The evidence surveyed here suggests that Zeno’s paradoxes were designed as provocative challenges to the common-sense view that our world is populated by numerous things that move from place to place.

No evidence of your interpretation here.

Thus, while Zeno accepts Socrates’ point that his own arguments aim to show that there are not many things, he corrects Socrates’ impression that, in arguing this point, he was just saying the same thing as Parmenides in a different form.

Or here.

Plato’s references thus consistently connect Zeno with the rise of eristic disputation, and it is perfectly plausible that his arguments against plurality and motion would have been well-known examples of making the weaker case seem the stronger.

Now, this is another example of what I was talking about. Plato (and others) were confident that Zeno’s case was weak. 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.). I have seen it often before, particularly in the last year on these forums. But it is most disheartening.

Zeno’s influence, however, on the great sophists who were his contemporaries and, more generally, on the techniques of argumentation promulgated among the sophists seems undeniable.

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

Zeno was not a systematic Eleatic solemnly defending Parmenides against philosophical attack by a profound and interconnected set of reductive argumentations. Many men had mocked Parmenides: Zeno mocked the mockers. His logoi were designed to reveal the inanities and ineptitudes inherent in the ordinary belief in a plural world; he wanted to startle, to amaze, to disconcert. He did not have the serious metaphysical purpose of supporting an Eleatic monism” (Barnes 1982, 236).

I was wrong about that. I elided Parmenides with the Eleatics, though the difference is, perhaps, somewhat metaphysical (!). However, the difference matters when it comes to Zeno, so now I can get it right. It does not follow that Zeno did not believe that his conclusions were not true.

All the quotations above are from SEP - Zeno of Elea

By Chalmers, logical = metaphysical; by Shoemaker, metaphysical = physical. — jorndoe

It would be a mistake to apply (((P = Q) & (Q = R)) implies (P = R)) without checking very carefully whether "Q" means the same for both of them. It is not something one could take for granted. I wouldn't take that thesis seriously without cross-questioning the author very carefully.

It was more of taking the phrase "metaphysically (im)possible" to mean "there is (not) a possible world where" and seeing where that leads. And if it leads anywhere is that maybe the definition of metaphysically possible is «that which follows the rules of the game». That seems abusive of the meaning of the words, or the words are not well-defined (many would say so for "metaphysics"). — Lionino

I doubt if it is possible to abuse the word "metaphysics". It has been abused so often in the past. In fact, it is so ill defined that I'm not sure what would count as abuse.
Three points:-
I have problems with the term "synthetic necessity" because I don't understand what that does to the meaning of "contingent". (I'm taking the Kripke-style interpretation that it means "In any world in which ...., this must be the case." - and in "in any world in which knock-out tournaments are played, it cannot be the case that two opponents in round 1 can meet each other again in round 2".) Tempting as it is, since logic is also a (language) game, or at least has rules, if metaphysics is that which follows the rules of the game", it aligns metaphysics with logic. But I do admire Toulmin's argument and recognize that he identifies an important class of propositions that have not figured much in philosophy.
I'm afraid I understand the possible worlds interpretation of possibility as simply possibility. Either way, of course, that aligns metaphysics with logic.
Many of the uses of apparently metaphysical language seem to me to be a simple matter of using what logic describes as "de dicto" as "de re" - possibly without being aware of what they are doing.

I must disagree there. If there are two different descriptions of a fictional situation, and the description affects the thing being described differently, then they're describing two different things, not the same thing in two different ways. — noAxioms

I see your point. But you must know that there is a great deal of philosophy around your view of this. But I won't try to drag you through it, is because I'm not sure how relevant it is. Yet.

The tortoise being overtaken is fiction, but mirrors real physical situations, unlike almost all the other examples in this topic. Describing the motion of Achilles as normal or as a supertask has zero effect on the ability of Achilles to overtake the tortoise. — noAxioms

I agree with that. 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 must clarify that the lamp itself is physically impossible, making it fiction. I said 'faulty', which it is not. It measures something undefined, so it isn't a contradiction (a fault) that the final state isn't defined. — noAxioms

But we allow physical impossibilities into fiction all the time. They even crop up in philosophical examples. "The sun might not rise tomorrow morning". "Twin Earth has water that is not H2O". I won't even mention philosophical zombies, brains in vats or simulations.
Your point about the final state not being defined is about logic, not physics (despite some people thinking that it is about physics).
In any case, the final state is defined. It must (on or off) or (0 or 1). Wouldn't it be more accurate to say that it is undetermined? Or is the final state the one immediately preceding the limit; in any case, it is not determined. So is the one before that.... But it would be absurd to say that every state in the series is indeterminate. It seems that whether anything here is determined is a question of how you look at it - from the beginning or from the end.

The difficult thing is that many human beings .... think that our sense perceptions of "the everyday world" are a direct copy of the way an independent world would be. — Metaphysician Undercover

No, I don't think that they think that. It is a philosophical thesis. I'm not sure it is possible to articulate what people who have not thought about the question think the answer to it is.

But I do not think that this was what he was sincerely trying to prove. — Metaphysician Undercover

So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it. — Metaphysician Undercover

I don't think we have anything near the evidence required to divine Zeno's motives. We don't even have his articulation of the argument.

Clearly he could observe motion, and he would know that this would be considered a ridiculous proof. — Metaphysician Undercover

Zeno came from the Eleatic school, so the first principle was "being", stasis, but what he was demonstrating was that this principle was insufficient to understand reality. — Metaphysician Undercover

So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it. — Metaphysician Undercover

But you don't know that he recognised what is so very clear to you, that the argument was ridiculous, or that he had "apprehended the faults in that ontology", though I admit that if he had understood what you understand, he might well have been poking fun at it. Still, other people since then have poked plenty of fun at it. But that's not a substitute for understanding the argument.

hat's why Socrates and Plato took interest in the sophistry of the Eleatics. The Eleatics could employ logic to prove absurd things, and this showed the gap between the "becoming" of the physical world, and the "being" of the Eleatics and Pythagorean idealism. — Metaphysician Undercover

I agree that it is very likely that Plato/Socrates was addressing the apparent incompatibility of the perceived reality of change and the Eleatic rejection of that perception as illusory. The "two worlds" solution has its problems and, for my money, Aristotle's solution was much better.

The time length is irrelevant. — Metaphysician Undercover

The exact length is indeed irrelevant. But the dimension of time is not. On the contrary, it is embedded in the argument.

They're clearly being confused (b)y maths. — Michael

.... and, as I think you must know, they think you are being wilfully dogmatic. That disagreement is what needs to be understood.

If you're trying to find a "solution" you won't find one. — Michael

I'm not trying to find a solution, just to understand what's going on. Not so much why it's wrong, but why anyone would think it was right. Where does the illusion come from?

Given that in C2 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then C1 is necessarily false. The supertask described in C1 is impossible. — Michael

I think I've just understood the significance of your A and B propositions. They are what justifies your formulation of the problem as a contradiction.

There is no last step before t1, hence no coherent definition of X at t1. But also at no point between t0 and t1 is there a step where X goes from being defined (as either "0" or "1") to being undefined, and the definition of X is always retained until redefined to something else. It's a simple contradiction. — Michael

If there is no last step before t1, there is no last-but-one step before the last step and no last-but-two step before that. And so on. The entire sequence unravels.
If you look at the series one way, it looks perfectly in order. If you look at it another way, it collapses entirely - it's not just a problem about defining the state of X at t1, but about defining the entire sequence.

A1. At t0 X = 0
A2. Therefore, at t1 X = 0

B1. At t0 X = 0 and then at t1/2 X = 1
B2. Therefore, at t1 X = 1

C1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on ad infinitum
C2. Therefore, at t1 X = ? — Michael

Going back to your propositions A, B, C, it seems a fair guess that the problem is the insertion of "ad infinitum". That's the difference that causes X to become undefined. Our instinct that it should work derives from the fact that the series works perfectly well even if we do not insert any definite number of steps:-

D1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on for n further steps where n is an even number.
D2. Therefore, at t1 X = 0

E1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on for n further steps where n is an odd number.
E2. Therefore, at t1 X = 1

I think that's more or less what I was looking for.

No, it was three separate situations. Sorry if that wasn’t clear. — Michael

Oh, I see now. You did explain, but I didn't pay enough attention.
Though I don't quite see how your B2 follows from your B1. But I don't think it is important.

It is applicable to t1, but given the supertask described in P3 there’s no coherent answer to the definition of X at t1 (no final redefinition before t1) proving P3 to be impossible. — Michael

You mean that we don't know the state of X at the last step before t(1), even though X must have been in one state or the other? (We don't have to work laboriously through each step. We just have to know how many there are steps there are between t(1/2) and the last step - we could work it out from that.)

It seems to me that we can work out the value of X for each and every step between t(0) and 1 if we work forward from t(0) but not if we try to work backward from t(1). In other words, whether X has a value at any stage depends on whether we define that stage in relation to the beginning or the end of the series. That seems very odd to me. But perhaps I've misunderstood. But I would be inclined to call a definition like that somewhat ill-formed.

Given that in C3 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then P3 is necessarily false. The supertask described in P3 is impossible. — Michael

That's clear as crystal. Your conclusion coincides with mine, so I'm perfectly happy with the argument.

P1. At t0 X = 0
C1. Therefore, at t1 X = 0 — Michael

This puzzles me. Is this t(1) the same t as the t(1) in C3? It can't be. There must be a typo there somewhere.

One question, then - The state of X at any t(n), depends on its predecessor state at t(n-1), doesn't it? Isn't that a definition? Why is it inapplicable to t(1)?

I meant, that they can mislead us when we apply the principles to the activities of the physical world. — Metaphysician Undercover

I think that's perfect. It's the conjunction of mathematics and - what can I say? - the everyday world.
What's difficult is the decision which is to give way - mathematics or the everyday world. Zeno was perfectly clear, but some people seem to disagree with him.

What is evident, is that we do not know how things move, and the exact "path" through space, that things take, whether they are big planets, stars and galaxies, small fundamental particles, or anything in between. — Metaphysician Undercover

That suggests that we do know roughly how things move. I don't think that's what at stake in Zeno's thinking. His conclusion was that all motion is illusory. The only alternative for him was stasis. But I guess we can do better now.

Yes, it affects how we think of them. It doesn't effect the situation, despite all the assertions to the contrary by several members. — noAxioms

Yes - unless it is a fictional situation - whether in the philosophical or the literary sense.

A thought experiment is a valid method of deriving conclusions from premises. They only get fictional if the premises are faulty, such as the lamp, a device which cannot physically operate as described. — noAxioms

That may explain why I have been confusing them. Thanks for that.
I have wondered whether one could replace the Thompson lamp with a question, such as whether the final number was odd or even. That would work if you start with an odd divisor and don't express everything in decimals. Perhaps it would work for all examples if you ask whether the number of steps taken is odd or even when the minute is up. I think.

but only because the axioms of continuity and infinite divisibility are themselves misleading. — Metaphysician Undercover

You mean because they allow the convergent infinite series?
Mathematically? Physically? (I'm inclined to think you mean physically, because of your reference to fundamental particles.)

So Zeno simply demonstrates how standard conventions are actually misleading us. — Metaphysician Undercover

Well, we've caught them out misleading us before, so I suppose they may be doing it again.

And so, fundamental particles take every possible path when they move from A to B, because the direct spatial route does not allow them to get ahead of the tortoise. — Metaphysician Undercover

Is the direct spatial route not available because it contains a convergent regress?
What path does Achilles take? (I assume he is not a fundamental particle.)

Some of both, I'm sure. The impetus thing had to go (survived until Newton, not bad...), but one could argue that it is a poor description of inertia. — noAxioms

Well, one could argue that it isn't a description of inertia, but of certain phenomena which are better described by inertia. Either way, impetus proved unhelpful and alternative conceptualizations proved more helpful.

That's my point. The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers. The historical contingency is an argument against the theory, not for it. — fishfry

You are right that the historical contingency should make us suspicious. (Descartes, by the way, has a description of statues "animated" by a hidden hydraulic system - I think in Versailles). But I don't think the process is simply over-enthusiastic. It seems reasonable to try to apply a new discovery as widely as possible. That way, one discovers its limitations.

The smallest thing still is. Unfortunately the word got applied to something that was a composite object, and they kept that instead of renaming the assembly and keeping 'atom' for anything fundamental. — noAxioms

Oh, I don't know. Given the conceptual revolution that happened when sub-atomic physics arrived, it's not a bad idea to signal the change by leaving atoms where they were.

So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is. — fishfry

That's more or less one Ryle's favourite arguments against dualism.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards. — fishfry

Clearly "<divide by> 2" is not applicable at 0. Would it be right to say that "+1" begins at 0 and has no bound and no limit, and that "<divide by> 2" begins at 1 and has no bound, but does have a limit? But they both they have a defined start and no defined end.

Not always. Just a minute. I know, Zeno doesn't give the time, but we've been using a minute. The way the scenario is described has no effect on the situation compared to a different way of describing it. — noAxioms

That's true. But different descriptions of the same situation can affect how we think about that situation. An additional difficulty, I suspect, is that our descriptions are fictional (sorry, thought-experiments), which means that the context is limited and evaluations of descriptions much more difficult. They need to be assessed in a different way - as useful or not.

I think @noAxioms looks at Zeno in a different way. — Metaphysician Undercover

Yes. You cannot necessarily decide that just one way of looking at things is true and all others false. They are better evaluated as useful or not. I think that applies here.

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right? — fishfry

Yes. The difficulty is how to evaluate a starting-point. True or false isn't always relevant. Which means that it can be difficult to decide between lines of reasoning that have different starting-points.

The paradox is like this. Both Achilles and the tortoise are moving, but the tortoise has a head start. So at t1 Achilles is at location A and the tortoise is at location B. At t2, Achilles reaches location B, but the tortoise has moved to location C. At t3, Achilles reaches location C, but the tortoise has moved to location D. As this procedure will carry on without end, Zeno concludes that the faster runner cannot overtake the slower. — Metaphysician Undercover

So are you going to conclude, with Zeno, that motion is impossible? or that Zeno is analyzing the situation in a misleading way?

We are already able to create systems that appear like a conscious subject on a passing glance (though humans also occasionally ascribe consciousness to anything from cats to rocks, so perhaps that's not surprising). It seems likely that we'll be able to create artificial systems which are indistinguishable from conscious subjects in a number of circumstances in the near future. — Echarmion

What it shows is that being a person is not simply a matter of fact, like weighing 15 stone or being 6 ft tall. It is a whole network of concepts (language game) which define, not only the properties a person has but their abilities and responses and, most important, the relationships we can have with them. So we can decide to treat as persons things that we know aren't "really" persons. Wasn't there a film in which someone fell in love with one of the voices that they give to machines these days?

It is also possible to treat cats (and people) as physical objects. Sometimes this is "dehumanizing" and morally objectionable. But analysing people as machines has also been incredibly productive. So it's not simply false (or true).

It seems likely that we'll be able to create artificial systems which are indistinguishable from conscious subjects in a number of circumstances in the near future. — Echarmion

There is a story that Hitler was able to throw a tantrum whenever it suited him. He may have been faking it at the beginning, but people around him had to treat it as genuine. They ended up not being able to tell the difference, but then having to respond on the basis it was genuine. The question whether it was genuine or merely indistinguishable was impossible to answer. But it wasn't just about some fact about Hitler; it was also about their decision how to respond.

Sure, but those mathematical principles are not the premises described by Zeno. — Metaphysician Undercover

Achilles is moving, and described as doing this in a way in which he will always have to move further before he can overtake the tortoise. Since he will always have to move further before he will overtake the tortoise, we can conclude logically that he will never overtake the tortoise in that described activity. — Metaphysician Undercover

Case closed, then.

(A) more complex model for the universe does not effect a simple geometric model at all, no. The simple model simply isn't fully applicable to the reality it is supposed to describe, just like Newtonian physics isn't fully applicable to the same reality, despite the fact that they'll continue to teach it in schools. — noAxioms

. That's a relief. I suspect that there are still people around who have difficulty with the difference between "not fully applicable" and "false". I still wonder (when I haven't anything more important to wonder about) whether Aristotelian physics is not fully applicable or not physics or false. I don't think anything important hangs on the answer, but still, that doesn't usually bother philosophers much.

Somebody still suggests that matter is continuous? I mean, that sort of went out the window a couple centuries ago. — noAxioms

All I was pointing to was the conceptual explosion that happened when we finally split the atom. (Which, you will remember, was by definition unsplittable).

In the sense that there's a self consistent narrative going through those works of fiction whose behaviour is impossible to translate to our universe, those universes would be metaphysically but not physically possible. — fdrake

This is a fascinating issue, mostly swept under the carpet in philosophy. I don't say that you are wrong.
I think it was Aristotle who first articulated the idea that a fictional story must be at least plausible. (Does that mean "possible"? - possibly). The idea that it requires "suspension of disbelief" was, apparently first articulated by Coleridge in 1817. There's a distinct tension between these two requirements. Both high-light that the audience/reader needs to collaborate with the author. It seems to me that the collaboration is at least sometimes secured by "arm-waving" by the author at awkward moments to distract the audience's attention and the audience not pressing questions that would be irresistible in other circumstances. The concept of magic is a good example. Science fiction stories usually put up a better front than that, but nonetheless... The issue comes home to bite philosophers when we offer examples - thumbnail stories. (I won't give examples for fear of setting off a hare and distracting us all). The difficulty for us is to distinguish arm-waving from actual possibilities (!).

The conclusion that Achilles cannot overtake the tortoise does contradict empirical evidence, that's the reason it's called a paradox. — Metaphysician Undercover

Yes. Disagreements between logic and experience are not unfamiliar. Experience usually wins, because logic is more adaptable than it seems. (I realize that may seem like heresy in a philosophical concept, but doesn't experience support it?)

I think that's a species of metaphysical possibility - a different physics. What would distinguish that from logical possibility, in my book, is that there are simply more ways of being noncontradictory than being unable to exist in our universe. Like flibbertygibbets. And nonmeasurable sets. And, maybe, abstract categories. — fdrake

Surely a different physics will have to be consistent and complete - when it is finished. That looks very like "logically possible", doesn't it?
As for the rest, you seem to understand existence as a single category. Perhaps you believe the same of reality. That is not how I understand either term. Existence has many different modes? categories? which are defined contextually. Ditto reality. I understood a flibbertygibbet to be a silly person who talks too much, so they very much do exist in my universe and I envy you yours. Things like non-measurable sets and abstract categories exist all right, but not in the same way/mode/category as tables and chairs. So do fictional things like Achilles and his tortoise and the Gorgon's mirror.

If no particular step can overtake the tortoise, then the tortoise, by the described motion cannot be overtaken. Where's the need for another premise? — Metaphysician Undercover

I'm afraid that if you condescend to use ordinary arithmetic, one can predict exactly when Achilles will overtake the tortoise, given data about how fast each contestant moves and the size of the handicap.
Not being a fully trained mathematician, I'm not sure about it, but I suggested this earlier and no-one has contradicted me - yet. Perhaps it is just too boring.

Neglecting acceleration, let's say Achilles gives the tortoise a head start of 100 units of length and that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds. — Ludwig V

That's what "first cause" arguments attempt to do. They describe the temporal aspect of "a process", "a thing", or similar term, in such a way that it necessarily has a beginning and an end in time, then they produce a logical argument from that description. It's an attempt to bring the realm of material (physical, or temporal) reality to bear on the realm of logical possibility, by stating premises which are supposed to represent the essence of material (physical) reality, and restricting logic with them. Another example of a similar restriction is the law of identity, and the other two fundamental principles. — Metaphysician Undercover

That is a very interesting take on the argument, though I don't understand how this applies to the law or identity. But then, I don't understand the law of identity, either. What are the other two principles?

I would feel very ashamed of myself if ChatGPT had access to read my posts with poor grammatical skills. — javi2541997

It has occurred to me to wonder why anyone trains these machines on text that has not been vetted for grammatical and other errors. (My bugbear is typos). The results could be, let's say, interesting.

Just another reason for treating their output with immense caution.

The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too. — fishfry

I am so sorry. I started a hare by mistake. The horse first appeared in this comment

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said. — Ludwig V

So a horse here is shorthand for whatever physical object one is trying to put into mathematical harness. Zeno's horse is the tortoise, or Achilles, or both.

I was wondering about what is actually meant by 'metaphysically possible' or 'logically possible'. The latter is probably the same as 'mathematically possible', but I'm wondering how the former is distinct. — noAxioms

I asked about this earlier in this thread. You can find what I got from it here. I'm not at all clear what people who use the term metaphysics mean by it. For the time being I'm treating the "metaphysics" and "logic" as co-terminous, if not synonymous.

Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point. But I gave some QM examples that suggest a non-continuous model of reality. — noAxioms

I have been wondering about exactly that point, and trying to work up the courage to articulate in this context. Thanks. If physics requires a non-continuous model of reality, then so be it, but then it would be empirical (physical) and wouldn't affect the geometrical concepts, would it? If what happened to the question whether matter was continuous or not is anything to go by, I think that a third alternative (not yet available) is most likely.

Yes. I've heard stories. I was very lucky to be able to work for the same institution for forty years. But I managed that by turning my hand to whatever the institution needed. Few philosophers have taught as wide a range of philosophy as I have and I always had administration on my work programme as well. The best career advice is probably flexibility - even if you have a specialism. But that's wise after the event.

So you'll probably get your wish: no matter how poor they are, educated people will be crippled with debt before they even get started. — Vera Mont

In the UK, the student loan repayment scheme was predicated on the "graduate premium" - that is, the idea that students would earn more money with the degree than they would have done without it. That's what was supposed to fund the repayments. At the time (in the nineties) this idea had something to be said for it - though it was always clear that some students, for whatever reason, would not earn much, if any, premium. Now, graduates are expected to repay their student loans, and a mortgage and repayments for car, white goods, furniture and fittings and save for their pension, and the student premium has largely disappeared (partly because of the increase in the supply of graduates.) The company store seems almost benign by comparison.

Look at the context to which my "Zeno's horse" was a reply. — noAxioms

Yes. I realized soon after I had logged off what you were talking about, went back in and edited my response. Too late to avoid revealing how dumb I had been. Never mind, it happens.

At best he showed that one example is undefined......To prove something impossible it must be shown that there is not a single valid one. — noAxioms

That seems to me a good response, though not quite the knock-out blow one would hope for. But it seems to me also a perfectly good reply to a purely mathematical version whether last number is odd or even.
So supertasks are like Gettier problems. Whack one on the head, and another pops up, specifically designed to avoid your refutation. It gets really wearisome, but no-one seems able to find a general refutation. One just gets bored in the end.

It is very valid to apply mathematics to physics, but it really helps then if that to which it is being applied is actual physics. — noAxioms

Very true. I'm afraid what I wrote is a rather embarrassing case of tunnel vision. But it rather matters what mathematics you are trying to apply to what physics. Sometimes it's a case of finding the right mathematics to apply. Which means that it is the physics that's in charge, so to speak.
But that doesn't apply here. Indeed, there's a question whether this branch of mathematics applies to any physics - not that that's an objection to the mathematics itself - just that this isn't the right application of it. That would be a solution, though. (I won't mention the issue of possible future physics'. I don't say it's just arm-waving to discuss it, but it is pretty close.)

Zeno's horse is quite real. Almost none of the others are. — noAxioms

In that case, it is clear what the right mathematics is. (IMO) One of the ways in which Zeno is a better paradox-maker than the others.

Creation of a device to measure a non-existing thing is not actual physics. — noAxioms

If you mean Thompson's lamp, quite so. (Do I understand correctly that Thompson actually argued that supertasks are impossible?) It is a fairy tale which seduces us to look at it wrongly.

the OP involving many non-relevant fairy tale elements and probably don't even understand what the staircase question is. — fishfry

There is no bottom, and the OP did not suggest a bottom step. He is done, and no stairs are observable. It's mathematical only, but framed with a physical sounding analogy, which makes it fall apart. — noAxioms

You seem incapable of moving beyond the maths and looking at how you're trying to apply the maths to some proposed real world activity. — Michael

So there is a common understanding of what the issue is. Your disagreement is about different ways of responding to it. Don't you think?

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said.

But this problem makes me think that they were wrong. One issue that comes to mind is the issue of making a 2-dimensional map of a 3-dimensional sphere. Euclid doesn't work (accurately). But the problem is resolved by developing a different geometry, which breaks some of Euclid's rules. (I realize I'm oversimplifying here, but I hope I'm not hopelessly mistaken.)

One point to take into account here. This is a thought experiment, so, while the mathematics is real, the horse is not physical, but imaginary, and the difficulty is to work out what rules apply to that in-between context.


I followed the link you gave me and found another link -

Update: https://thephilosophyforum.com/discussion/comment/866916 — Lionino

. This is your exposition of Toulmin's argument about synthetic necessities. Toulmin (for whom I have a lot of time) clearly identifies a class of propositions which orthodox philosophy has not recognized. But he is right.

In order to provide a nutshell explanation, I would say that the point is that the rules of a game can rule out possibilities which are physically possible, but violate the rules of the game. So in a way, they seem to be ruled out a priori or analytically, yet they are physical possibilities. Hence he classifies them as synthetic necessities.

You are interested in exploiting that to define metaphysics. Perhaps that works, perhaps it doesn't. (It's not as if there is any interesting alternative.) But the problem at hand is whether this helps with our problem. I think it does, because it suggests that it is not a matter of discovering what the rules are or what they imply. There is no truth of the matter, because it is a matter of deciding how to apply the rules to a situation which they were not designed to cater for. That's not the same as saying that it is an arbitrary decision, since decisions here may well have consequences elsewhere.

In the covid period, massive government spending went to the top tier of the economy, while main street got crushed. The $600 stimmy checks were all the middle class got. Was this massive transfer of wealth upward from the middle class to the elite just an accident? Or was it all a plan? A crisis that the big players didn't let go to waste. — fishfry

Yes. I've seen some analysis of this. The media told us it was about supporting the workers, but it wasn't. It was about supporting the economy. Actually, there was a real problem about that. In lockdown without support, businesses would have gone bankrupt. A difficult problem. But the solution didn't have to be so skewed.

The Top 1% of Americans Have Taken $50 Trillion From the Bottom 90%—And That’s Made the U.S. Less Secure — fishfry

Yes, I've seen the reports about that. It's much the same picture in the UK and I'm sure elsewhere.

It seems to me that there are three aspects to all this - each interacting with both the others. There's power - physical (The military and enforcement of the law) and social - conditional on social structures. There's psychology - mass and individual. There's ideology. The interactions are conditioned by two opposing tendencies - competition and co-operation.

Perhaps I'm writing the beginnings of another thread. How far it would be philosophical is a question.

And without a culture to enjoy, how will they live? — finarfin

The ancient Romans had it right. Bread and circuses. People do not live by bread alone.

An occupation's value to society is roughly related to its economic price, and the number of workers in that field. — finarfin

Yes, the labour market is a market. But like many others, it isn't a free market - meaning a willing buyer and a willing seller - meaning that both sides can walk away without a deal. Work is like fresh food - it can't be stored when it isn't needed. Roughly, if work means food and shelter, everyone needs work to-day for to-day. The other is social expectations. You don't find out the economic value of dust-
men until they aren't working. Then, all sorts of nasty stuff hits the fan. Dustmen and doctors are both essential to health - and how do you put a price on that? The actual differential between the two is heavily influenced by social expectations.

Does that mean I shouldn't be on the workers' side after all? — Vera Mont

It's good to speak up for those who don't have a voice. But it is better if those who don't have a voice can have their own. But somehow, the system needs a balancing factor - a referee or arbiter, who is neutral. That's a valid position as well. Workers can be greedy, competitive, and self-interested just as much as capitalists - indeed, arguably, capitalism expects that.

A neocon/neoliberal/CIA plot all the way. — fishfry

I'm not going to disagree with you. But I think regarding it as a plot in the standard sense is not the best way to think about it. I think it was the result of a consensus or "group think" - everybody agreed about the basics and so acted in concert without needing to deliberately plan or co-ordinate anything. Another factor that contributed was more complicated. The distinction between communists and Russians was blurred, that it was easy to continue the suspicion and hostility even when the ideological cause of it was removed. Russians were "othered" during the communist years and remained under suspicion even after communism fell.

Other way 'round I think. Clinton and the neoliberals did spread prosperity around the world, at the expense of the manufacturing base of America. — fishfry

They did so in the wrong way. The banner of free trade was pinned to the eternal search by capital for cheap labour. The irony of it is that the recipient countries didn't benefit all that much. In general, much of the wealth went to a minority of people who formed a new capitalist class in the recipient countries. It was actually a continuation of colonialism in a slightly different format.

They just wanted to be friends, but the neocons only want war. — fishfry

They seem to lack a sense of bargaining and deal-making. If you regard it as a competition with winners and losers, you have missed the point. It is of the essence that you allow the other side to make its profit.

Not sure I share your trust in the ability of our leaders to "spread the wealth around," as Obama put it. — fishfry

Yes, "share their wealth" is a lazy way to put it. It already implies taking something away. But see last comment. But my point was not that I expected them to be overcome with generosity, more that it is not in the long-term interest of the wealthy (even of the moderately wealthy) to prevent others from becoming prosperous. It might mean somewhat lower profit margins, but it doesn't necessarily mean actually taking anything away that they already possess. Its like the argument that it doesn't pay to rip off your customers too much, because they won't come back if you do.

Very interesting. If only I knew what "metaphysically possible" means? Can you help?

(I do know what "I said <x>" meant and what seconds are)

Oh, well, I can't make you do anything you don't want to do.

As we see from the graph, years of education is just as or less important than intelligence. If intelligence wasn't that important, we would see much higher variation in those less privileged occupations. But it isn't so, most fall under 95, the variation is small.
Such is the reality of genetic determinism, life sucks. — Lionino

Life certainly does suck. But I'm not at all sure that genetic determinism is the explanation and even less sure that IQ tests measure it. The most important point is that the validity of IQ tests is controversial and so is the very concept of intelligence or general cognitive ability.

For more details, see Wikipedia - Intelligence Quotient

I'll grant you that Marx's predictions about late-stage capitalism seem to be coming true. We don't actually have much capitalism anymore, we have an oligarchy causing unsustainable inequality leading to a revolution or a cyber totalitarian nightmare. The system's broken. In fact the economy is only being held up by government borrowing and printing at this point. You and I may be in agreement on some things. — fishfry

I can sign up to that. It all went wrong in the 1990's, when the West and capitalism indulged in triumphalism instead of recognizing the need to spread prosperity around the world. (WTO is supposed to help with this, but does not work - at least, not anything like enough.) They should have started with a Marshall Plan for Russia and then similar plans for all the other underdeveloped areas of the world. Very expensive, but cheaper than yet another world war.
I mention this because it is a case of the general problem posed for this thread and to have an excuse for promoting the argument for enlightened self-interest as a way of breaking through the reluctance of the wealthy to share their wealth (beyond charity, which they remain in control of).

I love when people agree with me. It happens so seldom around here — fishfry

So do I. There's a paradox about agreement, that it is the purpose, but also the end, of the discussion. So people tend to focus on disagreements.

And in fact we have a name for that. In ortdinal theory, an ordinal with a predecessor is a successor ordinal. And an ordinal without a predecessor is a limit ordinal. So your intuitions are spot on. — fishfry

I found that discussion very helpful.

But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence. — fishfry

Have I understood right, that 0 means "walker is not on the step", and that "the step" means "the step that is relevant at this point" - which could be 10, or 2,436? So 0 would be appropriate if the walker is on the floor from which the staircase starts (up or down)
My instinct would have been to assign 0 also to being on the floor at which the staircase finishes (up or down). It makes the whole thing symmetrical and so more satisfying.

That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is. — fishfry

I don't like that way of putting it, at least in the paradoxes. Doesn't the arrow paradox kick in when you set off in the.reverse direction? Or perhaps you are just thinking of the numbers as members of a set, not of what the number might be measuring. I suppose that's what "ordinal" means?

I confess to not knowing the answer to Zeno. It's a clever argument. Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But Michael objects to that, for reasons I don't yet understand. — fishfry

Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask? I don't understand this point. — fishfry

Michael's way of putting the point is, IMO, a bit dramatic. The boring truth for me, is that the supertask exists as a result of the way that you think of the task. If you think of it differently, it isn't a supertask. It's not about reality, but about how you apply mathematics to reality.

Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery. — fishfry

Well, if you insist on describing things in that way .... I'm not sure what you mean by "model". I think of what we are doing as applying a process of measuring and counting to space - or not actually to space itself, but to objects in space. A geometrical point has no dimensions at all. So it is easy to see how we can pass infinitely many points in a finite time. (I'm not quite sure how this would apply to numbers, but they do not have any dimensions either.) This doesn't apply to the paradoxes we are considering, which involve measurable lengths, but it may help to think of them differently.

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael

That's all very well. But it also takes us back to the question what this "operation" actually is. If you think of it as an action that takes a measurable amount of time, you can't, by definition. When we perform a calculation, that is an action in physical time. But a mathematical operation isn't quite like that, and somewhere in that is the answer (possibly).

Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.
I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory. — fishfry

Yes, but you didn't speak all the natural numbers, and indeed, if induction means what I think it means, your argument avoids the need to deal with each natural number in turn and sequence.

I'm sorry this is a bit scrappy, but there are lot of issues going on at the same time here. Great fun!

Why should we limit ourselves to computer simulations? Our world could be simulated inside of a cosmic brain. — Scarecow

Interesting.

A cosmic brain would be at least very like a god and there are plenty of ideas along those lines - and plenty of people believe them.

So the simulation hypothesis could be seen as a new version of an old idea, perhaps more suitable for our materialistic culture.

The question then arises why people actually believe them?

Well, there are different ideas of what constitutes a waste of time.

I do think that Descartes' exercise is a waste of time. It's just that I don't equate all theoretical work with wasting time.

Mind you, the second phase of Descartes' project is to find one's way out of the scepticism of the first phase, so perhaps the waste of time is allowing oneself to become stuck in the first phase of it.

That's not what the quotation says - unless you take "not involving practical action" to mean "waste of time".

Similarly, why don't we sometimes notice violations of the laws of physics? — jasonm

Well, perhaps we do. But when we do, we don't immediately assume that they are violations of anything. The most reasonable assumption is that we don't understand what is going on. Sometimes, it turns out that what we've noticed doesn't violate our laws of physics. Somtimes we decide that our laws need to be revised. It would take an awful lot to conclude that the phenomena betray the hidden machinery of a simulation. To conclude that would be no more reasonable than concluding that God had performed a miracle.

Fair enough. How do we know what the locale is really like, so that we can evaluate the simulation as accurate or not?

Such things are good only for having fun and creating sci-fi stories. — Alkis Piskas

Yes. But some people have peculiar ideas of fun. Other people get annoyed and engage in the forlorn hope of persuading them to stop being so silly.

It reminds of Descartes, but it is not strictly the same. — Lionino

True. The point of the comparison is to introduce some perspective and suggest that these thought-experiments are subject to similar criticisms.

Light reflecting off of objects and producing color and form in mind is a kind of simulation. — Barkon

Really? What is it a simulation of?