it's not necessarily that well suited to philosophy. It's more well suited to disciplines that have explicit agreed upon correct answers, and philosophy seems to have remarkably few of those. — flannel jesus

You are too kind. The model for the site is clearly (not "not necessarily") "those disciplines that have explicit agreed upon correct answers". That model is not at all well suited to philosophy, which lives and breathes on disagreements. If the model was strictly followed, it would be an extremely boring place.

There's actually a (somewhat) hidden tension between the model and the democratic structures (modified by the existence of the moderators, who do a reasonably good job) that are also built in to the system. The result is a bit chaotic and inconsistent, and, of course, there is a good deal of using the rules to try to enforce specific philosophical views.

In my opinion, the system of building a reputation by scoring points and collecting badges is much more problematic - and there is a similar system on Reddit, if I remember right. I found that it is quite hard to resist the temptation to score points rather than doing some philosophy.

It's basically, "If someone can disagree with your answer, then it isn't a good answer." To the extent that an answer required thought or creativity or any substantial form of agency, it isn't a good answer. — Leontiskos

You are quite right. But the expectation that answers (and questions) should have some basis in existing philosophy is not altogether unwelcome. But perhaps that's just my personal taste.

There is scope for real philosophical work in the chat rooms, which are not subject to the same restrictions. I found those the most fruitful part of the site.

Don't you think it is ironic that the critique of PSE was posted on Reddit? Talk about pots and kettles. I thought that the response to this question when it was posted on PSE was, overall, quite well balanced and sensible.

PSE echoes Wikipedia, which has similar problems. The place of an expert in the democratic and inclusive times that we live in is difficult and needs constant negotiation.

I have used PSE but found it difficult to negotiate. I find myself using TPF more and more and PSE less and less. I have looked at Reddit, but never signed up - not my cup of tea at all.

There are four words we can use to adequately, discreetly and clearly delineate the four positions of relevance — AmadeusD

I remember that discussion. Thanks for the reminder.

In the context of the assumption that it is an empirical debate, I'm content with C. In favour of it is the idea that existence claims are always empirical. That's more complicated than it appears.

But in any case, the fact (and I do think it is a fact) that the empirical debate cannot be resolved suggests that it is not simply an empirical debate. The heart of the problem is that the debate is not about the evidence, but how the evidence is interpreted. That means that the proposition that God exists is not empirical, but is a principle of interpretation.

Short story, it is an extension of the concept of a person and the associated language-games. The gods of animism are much more like a personification of their various powers than anything else and monotheism is an extension of that.

So that brings into question whether there is a coherent minimalist (or maximalist) concept of God. I think that there is not. D may be more appropriate to that.

But if I think that theism is irrational, I must think that anti-theism is also irrational, which suggest that D may be more appropriate.

Do you agree that the philosopher must uphold, almost, a fiduciary duty towards the public, in terms of living a certain life? — Shawn

The public certainly seem to think that everyone in the public eye is expected to do that. Yet it is not clear to me that the public think that they should uphold the same standards. One might argue that people in the public eye are often role models for others and so have an additional responsibility to conform to a higher standard. But if that's so, everyone is in the eye of some of the public and is likely to be a role model for some people, so everyone has a similar responsibility.

I assume most people (philosophers or not) are flawed and limited beings — Tom Storm

And one would have thought that a certain level of tolerance and even forgiveness might be expected of the public - unless the public never sins.

All we have is a text and the text is a fecund vehicle for alternative interpretations. — Tom Storm

It certainly is. A biography is also subject to interpretation. It is probably a good idea to wait until it is over, but even then a final judgement is difficult to arrive at.

To give an example, take Socrates. His life and philosophy seemed inseparable. — Shawn

It's a good idea to remember always that Plato's account was more hagiography than biography.

Whether Heidegger was a Nazi or not (for me) may well taint our experience of his work, but it says little or nothing about whether the work is any good. — Tom Storm

The interesting question is what basis, if any, there is for Nazism in his philosophy. I don't think there is a determinate answer, but it is worrying.

I don't see why you put Witty on a pedestal. It's well known that he was an awful person. — Heracloitus

I agree. Yet he had friends. I don't think I could have been friends with him, and I don't suppose he would have wanted to be friends with me. I don't care either way, he is an amazing philosopher.

There are questions to be asked about the involvement of both Berkeley and Locke in the slave trade as well. In their case, there is a tricky issue about how far we pay attention to the context that they lived in. I don't suppose anyone is much bothered about whether Plato and Aristotle (or Socrates) owned slaves. Rousseau's life presents other issues.

Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent. — Metaphysician Undercover

There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why.

Jgill talked about how the lamp would "appear", and this implies a sense observation, and empirical judgement. The point I made is that the description describes something far beyond our capacity to sense, so it is incoherent to talk about how this described thing would "appear". — Metaphysician Undercover

I agree that this isn't really about anything empirical, but it sort of seems to be.

OK. You and @fishfry both believe that the supertask is impossible. But you believe that is because it is contradictory and fishfry believes that it is because the last step is not defined. Am I right about that?

I see no contradiction in Thompson's lamp, only a failure to define the terminal state.
— fishfry
See here. — Michael

I followed your link and found this quotation from Benacerraf's Tasks, Super-Tasks, and the Modern Eleatics. I've put the passages of interest in bold and italicized the passage quoted from Thompson for clarity.

Thomson's first argument, concerning the lamp, is short, imaginative, and compelling. It appears to demonstrate that "completing a super-task" is a self-contradictory concept. Let me reproduce it here:

There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button, the lamp goes off. So if the lamp was originally off and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on. ... After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? ... It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.

Pause here. I think Thompson means that

It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on.

contradicts

But the lamp must be either on or off.

That seems to be true.

But the passage continues (Benacerraf's words): -

Rarely are we presented with an argument so neat and convincing. This one has only one flaw. It is invalid. Let us see why. Consider the following two descriptions:
A. Aladdin starts at t0 and performs the super-task in question just as Thomson does. Let t1 be the first instant after he has completed the whole infinite sequence of jabs – the instant about which Thomson asks "Is the lamp on or off?" – and let the lamp be on at t1.
B. Bernard starts at t0 and performs the super-task in question (on another lamp) just as Aladdin does, and let Bernard's lamp be off at t1.
I submit that neither description is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).

That also seems to be true. The three sentences in bold in the first passage are not individually self-contradictory, but the conjunction of the three (the concept of a supertask) could be described as self-contradictory. Nor are Benacerraf's A or B self-contradictory. They could be both true, if a third state that is neither on nor off were possible. Perhaps Benacerraf was assuming that there isn't.

But @fishfry says that the final state is not defined. That would indeed be a third state which is neither on nor off. The idea that this is the case, is supported by the fact that both Thompson and Benacerraf feel the need to consider both alternatives. The point is simple enough - the definition of the infinite set is such that there can be no last step, in virtue of the definition, every step has a successor. So the last step is not defined. You may be thinking that there must be a last step in a convergent series in the series that we've been considering here, it is 1, or 0. But those are limits, not last steps. The series itself by definition cannot reach that limit, so 1 (or 0) cannot be a step in the series. One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series.

I think I'll leave it there.

PS Since I started writing this, the link to the post that I copied this quotation from seems to have become non-functional. Very odd.

Ok, so we don't know anything for sure, not just the matter of whether there is a God. — Lionino

Well, the first half of that is debatable, but let's save that for another time. You seem to have agreed on an agnostic position.
From observation of philosophical debate it is clear that both (some?) theists and (some?) atheists agree (though I've not seen either side explicitly acknowledge the fact) that the question whether there is a God is empirical. Let's have a closer look at it.

First, it implies that there is a concept of God - in fact several concepts, but let's take the minimalist one. Let's treat it as a hypothesis. Both sides, presumably have an answer to the question what is to count as evidence, either for or against. It comes down to experiences. Theists will cite certain experiences, which are not universal, but are not uncommon, and the various mysteries that exist in the sciences, and possibly the idea of an experiment, in the form of prayer. Atheists will cite the lack of any experiences that specifically prove God exists (and discount all the evidence given by the theist). On top of that, I think that they will not be able to explain what experiences might convince them. Certainly, I can't and I've never seen anyone try.
There has been considerable debate about where the burden of proof lies. Each side makes a case that the burden of proof is with the other side. So no agreement there.

Transcendent experiences may well convince the person who has these experiences, but are of little or no value to the atheist, because they are so subjective. Experimental prayer doesn't stand up to scrutiny by the standards commonly applied in science. A God of the gaps certainly won't convince anyone who isn't already convinced - it comes down to interpretation of the evidence.

Atheists seem to be in a stronger position if the burden of proof lies with the theist. However, proving that unicorns, for example, don't exist is at least very hard. Arguably the only reliable case is a proof that they could not possibly exist - i.e. that the concept of a unicorn is incoherent in some way. The same applies to the concept of God. But that's not going to convince a theist.

I don't see what might happen to resolve this. Suggestions welcome.
For me, both theism and atheism are irrational, even if they are empirical claims. Which leaves agnosticism as the only rational position.

Sorry about that. I typically select the entire post and hit Quote, and it seems to lose a lot of the attribution. — fishfry

Yes, I find that as well. I work round it by selecting only the quoted text, not including the link that gives the attribution. Then, you can hit "quote" and the system does pick up the attribution. Then, if you separately select the response, it is copied and attributed in the normal way.

The limit is not part of the sequence. so that doesn't run the sequence backward. I am not sure what point you are making about the sequence. The dots merely indicate that the sequence progresses indefinitely. — fishfry

Neither am I, on reflection. I was trying to articulate the point that one can count forward, but not backward, so I don't think anything is at stake.

"If you know what you're doing you're not learning anything." Think I read that somewhere. — fishfry

Yes, I like that. I'm a bit of a contrarian, so I'm tempted to reply that I don't need my surgeon to learn anything while he's cutting me open. Indeed, I would be rather concerned if I thought he was. It applies better to artistic, experimental, open-ended activities - like philosophy and maybe mathematics, at least sometimes.

I'm out of my depth on that. Don't understand what's meant by realism or anti-realism. Simply don't believe that 2 + 2 = 4 has a truth value before some intelligent entity shows up to pass judgment. — fishfry

If you don't understand what realism vs anti-realism means, you have understood correctly - as I see it. Some people would argue that the proposition that "2+2 = 4" does indeed only have a truth-value only when someone passes judgement on it but that 2+2 = 4 independently of anyone doing that i.e. is objectively true. There's a temptation to think that mathematical truth is eternal, i.e. always has been true, always will be true, whatever happens. But that's a mistake. It makes no sense to assign a place in the time series to 2+2 = 4; there is no meaningful way of doing that. (Grammarians recognize a tense that is called the timeless present which is exemplified in propositions like this.)

Ok. Don't think I disagreed with anything you said. — fishfry

I'm glad it made sense.

If one watches the lamp in a dark room, at some point it will appear to be on continuously. — jgill

Quite so. I've been thinking that the rules of the game require one to classify that as a purely physical phenomenon. But I prefer versions of this problem that define a sequence (0,1,0,1,...) and align that with the lamp. Even better, I think, we can count the steps in the convergent series and not that odd and even numbers alternate and ask whether the last count when 60 seconds are up is odd or even. Nothing physical intervenes.

I think that if the lamp is going on and off at an infinite rate, then it's not correct to say that it would be on at any particular time, or off at any particular time, because it is going on and off at a rate faster than our ability to determine a particular time. — Metaphysician Undercover

That's true, but seems to be a purely physical limitation. It raises the question whether that means it is really on or off, or a some sort of in-between state. Fluorescent lights flicker on and off all the time (at least if they are running on AC, and we just say they are on. And it is true that for practical purposes there is no relevant difference between that light and sunlight or candle-light.

Either way, the point is that it's special pleading to argue that it's possible to have recited the natural numbers in ascending order but not possible to have recited them in descending order. It's either both or neither, and it can't be both, therefore it's neither. — Michael

I'm not clear whether you are thinking of reciting as a human action that takes time. In which case, there will come a point in your recitation when you physically have to stop, but have not run out of natural numbers. (If we are talking about a series that is convergent in time, it will take longer to utter the word(s) than the time available.)
To put it another way, to say that the series of natural numbers is countable, doesn't mean that there is anyone (apart, perhaps, from God) who could actually, physically, count them. In my book, it just means that they are defined in such a way that they could be counted and, indeed can be counted, so long as one chooses to count only a part of the series. It doesn't follow from the definition of the natural numbers that anyone could count all of them.

I have to disagree. What you describe is a rate of acceleration which would produce an infinite speed. The rate at which you recite the numbers becomes infinite before 60 seconds passes. And, despite the fact that infinite speed is in some sense unintelligible, it is clearly not at all the same as being stopped. — Metaphysician Undercover

No, it isn't the same as being stopped. Being stopped is an everyday occurrence. Infinite speed, is, as you say, unintelligible. If that's what underpins the supertasks, it makes sense of the narratives - apart from the fact that it doesn't answer the question whether the lamp is on or off.

The rules of this (language-game) still make no sense to me.

It's not so clear to me, many people treat God as if it were something explanatory, sometimes even empirical, in the broad meaning of the term (which includes personal experience). — Manuel

Yes. People may differ, of course. The view I expressed is unlikely to be acceptable to many believers - though there may be some, with philosophical inclinations who could accept it. There are theologians who would be able to recognize a view like mine.

Why did I get a bonus at work? God is gracious. What caused my existence? God. Etc. — Manuel

A nice simple example. But if you look a bit closer, you may think that what is on the surface is not the whole story. When you don't get a bonus, even though you worked just as hard, with the same good results, you don't think maybe it isn't God who gives you the reward, but your employer. You think that God must be angry with you and search for reasons why that might be so. You don't think maybe God is a bit strapped for cash this year so is having to cut back. The idea that it is God who dishes out rewards is protected against refutation. That's important. (I'm sketching here to avoid reams of writing and reading.)

But I do not think that asking for some properties or attributes or facets of God is asking for too much. The more which can be given, the better we can proceed. If it is limited to a Great Being, or a supreme force, then I do not know what this means, or at least, it is very nebulous. — Manuel

Yes, that's a fair demand. Too many "proofs" of God don't explain what that means. (Hence, we find that the God of the philosophers bears little resemblance to the God of the believers, and that's a problem.)

And there's no arguing with axioms, except by their results. In this case, the argument has to be about what life the believer leads — Ludwig V

In many ways, I'm not happy to be dealing with a God about whom there can be no argumentation. Hence belief in God as a matter of faith, not subject to rational comment, is far too comfortable a retreat for believers. That's why I suggested how the argument might go.

So I think we can have arguments about God, even if there may be no chance of getting each other to agree. — Manuel

You are maybe a little too pessimistic. People do sometimes abandon their faith. But it's a complex process that may include rational arguments, but religious belief involves more than that, so they are only one factor.

I think agnosticism is better, with atheism being applied in specific instances. — Manuel

That's a very reasonable position.

"Erm, I can't say either way", even though there is nothing logically contradictory about a green floating donkey tidally locked behind Jupiter in respect to the Earth. — Lionino

On that basis, agnosticism is the only rational response. (It is my preferred response if people ever ask me.) But there are a number of physical impossibilities, not to mention improbabilities, about that the green donkey hypothesis that make it, in my view, unreasonable to be agnostic about it. I assume that you focus on logical possibilities because that's the tradition of our philosophy. But we have to live with physical impossibilities as well, so it seems a bit peculiar to ignore them, if what you want to understand is human beings.
Wittgenstein imagines himself in conversation with a philosopher about the question whether the tree they are sitting under really exists, and then realizes that he has to turn to anyone nearby who's listening and explain "It's all right, we're only doing philosophy". If it's only philosophy how can it matter to actual human beings?

First, define what God is, then we can say if we know enough to say, with certainty, that such a thing exists or does not. Maybe we can't reach certainty, in that case we shift to probabilities. — Manuel

That's the logical procedure, and some theists do like to try to follow it. But God isn't an empirical hypothesis. It is how you frame your life. What God means, according to the religions, is how one should (try to) live one's life. (What science means is not just the philosophy of science, but how you do it in practice.) Admittedly, how that works out in practice can be a bit puzzling to outsiders, but that's how the ideas work. (The same is true of science) To put it another way, if you start by defining God, that may turn out not to be a hypothesis, but an axiom. And there's no arguing with axioms, except by their results. In this case, the argument has to be about what life the believer leads.

“If you crush a cockroach, you're a hero. If you crush a beautiful butterfly, you're a villain. Morals have aesthetic criteria.” - Nietzsche — BitconnectCarlos

He's right, of course, in his annoying way. Either there's a justification for that difference or there isn't. If there isn't, then morality is deficient. But I think there is. Cockroaches are annoying and dangerous. Butterflies mostly are not, but they are beautiful - except perhaps when they are caterpillars. (That's awkward, I admit) I don't see anything dubious about not destroying beautiful things that do no harm and something very dubious about not destroying dangerous things that are harmful.

If something is not necessarily right then it could possibly be wrong. Evolution helps us survive, not necessarily thrive or self-actualize. — BitconnectCarlos

Do you not think that the values that we define as necessary for those two are given (majorly) by evolution? — Lionino

Evolution doesn't give a toss whether individuals or a given species survive or not. It doesn't even care much if a species survives. It is a consequence of the genetic variation of individuals within a species and the random effects of that variation on the survival and reproduction of traits amongst those individuals. Morality has nothing to do with it.
Homo sapiens is a social animal. So are many other species. It is curious that we so often see ourselves as individuals and society as an optional extra and a problem. But surely that fact sociality is so common should lead us to conclude that social living enables individuals to survive and reproduce better than competitors. I would agree that this may well have something to do with morality, insofar as morality is about social living. Evolutionary biologists regard this as "kin selection", based on preserving the genome and nothing at all to do with morality, so there is more to be said here.
I do agree that evolution doesn't have much to do with thriving or self-actualizing, as we understand it. Though it does seem very plausible that if morality interfered with the ability to survive at least until reproduction, it would surely die out. (Can you imagine a society in which everyone was celibate? Not for long.) So evolution must influence morality at least in that negative way.
The idea that ethics and morality are not merely about how to live in a society, but also about how to live well as an individual (that is, as an individual in society). Answers to that must be based on ideas about what human beings are and what they can be. But evolution, though it has an impact on everything, does not dictate everything, (though evolutionary biologists seems to forget that), so it is not impossible to choose different ways of living with the constraints of survival and reproduction. If our lives are really limited to survival and reproduction then they are grim indeed. It is better to regard them as the preliminaries to living well but not the whole story. There's more to be said, of course, but I'll leave it there.

I think that (1) is a tautology — Michael

I agree. By "we" do you mean us human beings? You and I? If so, we will necessarily stop, if only when we die.

whereas no evidence has been offered in support of (2). — Michael

Assuming that there are people who believe this, it is reasonable to assume that they can offer what they think is evidence. So it's truth depends on what you mean by "evidence".
By "recite", do you mean some event that occupies a finite amount of time (larger than 0)? In that case, assuming you mean "all the natural numbers", 2 is false, or at least logically impossible.

I hope not, my sources are academic.
— Ludwig V
I have no doubt, and I hope I am sufficiently conveying the humble limits of my knowledge in this area. — fishfry

There's a confusion here. The remark you quoted, which the system attributed to me, is actually @Lionino. I could claim academic sources from what I'm saying, but I read them a long time ago, and if you asked my for attributions, I would have to spend a long time looking them up.

By definition, an infinite sequence is a1,a2,a2,… It only goes forward. Though if the elements are decreasing (as 1, 1/2, 1/4, ...) the points go from right to left. — fishfry

I take your point. So the dots reflect the lack of definition and trying to run it backward finds the dots at the "beginning", so the "beginning" is not defined. But one could define a similar sequence that runs (0, 1/2,1/4.... 1), couldn't one? That would not be the same sequence backwards, of course.

I freely admit to my philosophical ignorance, so I am out of my depth in these matters. — fishfry

But no, that is not about the world. The world is what's real, what's physical. — fishfry

Welcome to my world. Being out of one's depth in it is almost a prerequisite of inhabiting it, so that's not a problem. It would probably unfair to say that people who think they are not out of their depth are always wrong (compare relativity and QM). But it is certainly true that you need to be a bit out of your depth to be doing any serious work. If you have everything sorted out and pinned down, you've lost your grip on the problem. (Wittgenstein again)

Unfortunately "The world is what's real, what's physical" is a metaphysical remark (at least, it is if there are any philosophers around), so you've jumped into the water without, perhaps, intending to. The question is whether numbers, etc. are real things that are not physical; platonist-type theories see numbers as real things that "transcend" the physical world. Don't ask me what "transcend" means - or "thing", "entity", "object". They would probably prefer to tell you what transcendence etc. are. But that's the same question in a different mode. Their mode is metaphysics. Mine is linguistic.

What I was doing, in response to what Lionino was saying, was putting realism and anti-realism together - since they are defined in opposition to each other - and then asking what they disagree about. (There are many varieties of both sides of this coin, so I'm simplifying, and arguably distorting.) In particular, I'm trying to show that "real" is not 'really' in contention, since no-one could deny that numbers are real - what is at stake is different conceptions of reality. And you see how slippery this is because in mathematics, not only are some numbers real and some imaginary, other numbers (like transfinite ones) are neither. Worse still, the imaginary numbers are numbers and exist, so must be real - in the philosophical sense. (At least, you can put me right if I'm wrong here.)

What "real" means depends on the context in which you are using it. Some philosophers want to use "real" in a context-free sense. But that generates huge complications and confusion. Better to stick to contexts. (The same applies to "exists") That's why I try to avoid metaphysics and metaphysicians will classify me as a linguistic philosopher - and that is indeed where I learned philosophy.

Let me ask you a different question. Before chess was invented, did all the games of the grandmasters exist "out there" in Platonic space? Did the collected games of Bobby Fischer exist before he played them? After all, each game could be encoded as a number, and the Platonists believe numbers exist independently of minds. I find that difficult to believe, that all the symbolic works of humanity exi(s)ted before they were created. — fishfry

All right. Those are good questions. They lead one in a certain direction. I am very sympathetic, so it would be better to let a platonist answer them directly. But I don't think that platonism needs to rule out the possibility that humans might be able to create some things, such as fictional stories - (although Plato was very scornful about such things on moral grounds, though he made liberal use of them himself.) - and games.
But in this field, it is as well to understand your opponent's (colleague, hopefully, in a joint attempt to discover truth) position. So consider. Games like chess are unlike games like football. Once they are defined, all the possible games are defined (so long as you limit the number of moves). So you could argue that the Sicilian defence, for example, was not created, but discovered. That's the germ of platonism.
In the end, I think, one has to see these arguments, not as simple question of truth and falsity, but of how you think about things. The answers, then, are quite likely to be pragmatic or even moral.

Humans create. That's what we do. Humans are, if you like, the very mechanism by which the universe figures out if 2 + 2 = 4. — fishfry

Yes, that's fine. There is an approach that sees humans (and perhaps some animals) as the means by which the universe becomes self-conscious. I think that's going a bit too far, but I can see the attraction.
My enemy in this field is dogmatism.

Remove God and life can lose its sanctity quickly. — BitconnectCarlos

That right? All the time the majority of the people believed in God, none of them killed any other? — Vera Mont

The truth is, you are both right.
What religions don't often face up to is that brotherly love and sanctity are actually applied only to believers. When it comes to unbelievers, all too often it's a different story. (Unbelievers includes those of a different sect.)
It's difficult to state this accurately. Not all religious people all the time regard unbelievers beyond the pale of sanctity, but it frequently goes that way.
But I don't think history shows religious people any worse than irreligious or atheistic people. (Though the majority of people through the majority of history have been religious, so the comparison is a bit flaky.)

Maybe you are misunderstanding what "abstract" means in those quotations. It doesn't mean something that we conceive in our minds, but a real object that exists independently of any conscious being, but that is outside space and time. — Lionino

But one of the minimal characteristics of mathematical realism is that things such as "2+2=4" are true and they are true even if we are all dead — in other words, it is about the world. — Lionino

If both of these are true, then we need to be very careful about what we mean by "the world". There is an application that takes "the world" to exist in space and time. Note, however, that the space-time world continues to exist even if we are all dead, even if we never existed at all. If "the world" includes everything that exists, then it can, of course, include things that exist "outside" of space and time - provided that we understand how anything can exist "outside" space, which seems to indicate a location, but does not.

Is not existing at any particular location in space the same as existing outside space? Where does platonism or Darwinism exist? or football or judo? Or the possibility of rain where I live tomorrow? Or the English language? Or the recipe for doughnuts?

Sorry - rhetorical questions. I realize that you are reporting the things that platonists might say.

Reality is what's really going on in the world. Not sure why you regard that as problematic.
A bat has a particular view of the world, as does an ant, as does a sea slug. None of them, and that includes us, know what ultimate reality is. Not sure what your objection or concern is with this idea. — fishfry

I agree with "a bat has.... what ultimate reality is" But then, I wonder what the status of "what's really going on in the world". Is that ultimate reality? From what you say, the answer is not clear. My concern is that both "ultimate reality" and "what's really going on in the world" are not defined in a way that reminds me of the way that the last step in a converging series is not defined - and cannot be defined. Yet, the sun is really shining at the moment and there really is a war in Ukraine - in short, we all (including bats and ants and slugs) live in the same world and interact in it.

Physics is inaccurate, but what if it's wildly inaccurate, as inaccurate as an ant's view of the world relative to the real world? — fishfry

But how can you say that an ant's view of the world is inaccurate? I think I can grasp what you are getting at when you say that physics is inaccurate. It reflects the fact that physics is an on-going enterprise. "What if it's wildly inaccurate.." is a style of question that I'm very sceptical of. It reminds me of "what if everything's a simulation?" I classify it as a speculation and not capable of a meaningful answer.

As I understand it, Tegmark believes the world is a mathematical structure, like a group o a topological space. — fishfry

One might interpret that belief as a dramatic way of putting the point that we can find a mathematical structure that applies to the world. If he doesn't mean that, I want to know what he means by "is".

I have the worst habit lately of only responding to my mentions and not reading the rest of these threads. — fishfry

A very sensible policy. It is easy to drive oneself crazy by trying to respond to everything. But sometimes I can't resist intervening in discussions that haven't mentioned me. It doesn't always work, in the sense of developing into something interesting, but some times it does.

I ended up spending all my time explaining the ordinals and that detracted from my resolution of the lamp. — fishfry

That's my fault. Sorry. I did benefit very much.

But the limit isn't defined in the lamp problem. — fishfry

Yes, I understand that now. I was talking about the limit of the convergent series. The series "0,1,..." has no inherent limit. If it ever is limited, it is by some event "outside" the series. That's badly put. I just mean that I can stop following the instruction for any reason that seems good to me or even none at all. The series as defined is infinite.

I'd say that the standard mathematical rules for dealing with infinity are perfectly clear, and do apply. — fishfry

I didn't mean to suggest that wasn't the case. Thinking of the series backwards is a vague handwavy imagining. That's all. I intended to contrast that with a series that can be defined forwards or backwards. It's odd, that's all.

Yes, sure, a fixed body of knowledge evolves. But that body of knowledge is added to every day by every math journal and university colloquium. — fishfry

Both sentences are true - the first sentence does not imply anything platonic, in my view. I think the difference between us is a question of emphasis rather than an actual disagreement.

I believe I lost track of what this paragraph referred to, sorry. — fishfry

Yes, that was a step too far, and it is very speculative, more a musing than a thought. I should not have pursued it. Let's just let it go.

I read your post. It is really helpful. I don't know enough to respond meaningfully, but I have a feeling I shall find my way back to it from time to time.

It did provoke the heretical speculation that it is an assumption that just one of these accounts applies to the whole of mathematics. Perhaps mathematics is not just one language-game, but a family of them.

For me, numbers exist more like Superman exists or an equation exists rather than how my hand exists. — Lionino

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. — Stanford Encyclopedia - Modal Fictions

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.

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: — Lionino

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. — noAxioms

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. — noAxioms

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. — noAxioms

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.

I think only nominalism is really sensible when we get to the bottom of things. — Lionino

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. — Lionino

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.

The idea that mathematical entities aren't real, especially that they aren't abstract objects. — Lionino

Many philosophers think mathematical objects are real objects that exist outside of space and time. — Lionino

So many questions about this.
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".

As if reality is the limit of our theories. — fishfry

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. — fishfry

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." — fishfry

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. — fishfry

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. — fishfry

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. — fishfry

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

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. — fishfry

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?

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.