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?

A technologically mature “posthuman” civilization would have enormous computing power. Based on this empirical fact, the simulation argument shows that at least one of the following propositions is true: (1) The fraction of human‐level civilizations that reach a posthuman stage is very close to zero; (2) The fraction of posthuman civilizations that are interested in running ancestor‐simulations is very close to zero; (3) The fraction of all people with our kind of experiences that are living in a simulation is very close to one.

If (1) is true, then we will almost certainly go extinct before reaching posthumanity. If (2) is true, then there must be a strong convergence among the courses of advanced civilizations so that virtually none contains any relatively wealthy individuals who desire to run ancestor‐simulations and are free to do so. If (3) is true, then we almost certainly live in a simulation. In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2), and (3).

I suppose it is not unreasonable to apply a judicious, but not radical, scepticism to this argument or at least some of it. Long posts risk not being read, so I shall focus:-

(3) The fraction of all people with our kind of experiences that are living in a simulation is very close to one.

I charitably assume that "all people" means all people past, present and future, including artificial people developed as part of a holistic simulation - if there be any such. What is the evidence that there are any people with our kind of experiences living in a simulation - apart from NASA experiments, archaeological research and trials for deep-sea mining? None that I know of.

But the Doomsday Clock is at 90 seconds. Extinction without any evolutionary descendants is a real possibility and should have been included as a fourth possibility - amongst many others.

If (3) is true, then we almost certainly live in a simulation.

That seems to follow. However, let us note that "we almost certainly live in a simulation" assigns a probability of, say, 0.99 to "we live in a simulation".

In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2), and (3).

In probability theory, it is indeed regarded as sensible to do that if we have listed all the possibilities. But the rule only really applies in mathematical probability, which this exercise is certainly not. The Bayesian notion of credence is based on an admittedly subjective evaluation of the evidence for each outcome. But there is no real evidence for anything here, so the assignment is arbitrary, rather than sensible.

In any case, what we actually have is a probability of 0.33 that the probability we are living in a simulation is 0.99. That seems a good deal less than almost certain and almost certainly less that 0.5.

Bostrom is saying that one of these is almost certainly true: — Michael

Yes he does, and, as you say, he doesn't say which of them he thinks most likely - though many people seem to have decided that 3) is the best bet. I've no idea why.
But I couldn't see why Bostrom thought that one of those three must be true.

the physics would likely have developed more or less at pace with our own history. — noAxioms

I don't see why you say that. I think you are assuming at least a soft determinism? But given that the starting-point of the history in the simulation is not more than roughly the same, I don't think you have any real basis for that assumption. I grow even more sceptical when I remember the argument that small differences, over time, can result in big differences. Remember, there were times during the Cold War when nuclear holocaust hung by a thread.
You say that you wouldn't necessarily run detailed simulations of everything at the same time, but switch to closer simulations when necessary to maintain the illusion. That's all very well, though it imposes an extra burden on the machinery because it will have to be aware of what people are attending to at all times.
But what I'm thinking about is what would happen if the people inside the simulation decided to do some physics, including, of course, experiments. That would require interaction between the person and the machinery even when they are carrying out experiments designed to reveal the physics at work. If the progress of science is anything to go by, it wouldn't be easy to fool them all the time.

Don't know what you mean by QM and GR being wrong. They're not, but they're not necessarily the physics of whatever is simulating us. — noAxioms

Didn't you say something to the effect that quantum mechanics and general relativity couldn't be simulated? Perhaps I misunderstood.
There are two physics involved. One is the physics of the simulated world, which would need to be quite like ours. The other is the physics of the world in which the simulators exist.

How lucky are we to have survived to the point of being able to put together these simulations? — noAxioms

Even luckier to have the resources to waste on such a project.

From the viewpoint of fish in an aquarium, is their existence a simulated life, in that the aquarium simulates the ocean, or is it a real life, in that their environment is all they know. — RussellA

Both make some sort of sense. It would also be possible to say that the aquarium is a little bit of the ocean. It would also be possible to say that, since the water they swim in and the food they eat are both real that their world, though small, is not simulated.

Takeaway pizzas simulate real food, social media simulates real life, sports events simulate medieval battles, surveillance cameras simulate the nosy neighbour in a small village, modern government tries to simulate the parents in a traditional family (whether father of the nation or mamala to the people) and corporate employers simulate the process of having to work for essential food and shelter. — RussellA

You are confusing "simulate" with "is like".

As regards argument 1), the designers do pop out at times, but the life-form is not aware because of their necessarily limited intelligence.
As regards 2), similarly, the life-form doesn't notice violations in the laws of physics also because of their necessarily limited intelligence
As regards 3), the same amount of computing power would be required to house a "simulated" world as a "real" world.
As regards 4), the simplest explanation for fish in an aquarium is that they exist in a simulated world — RussellA

1) Do you have any evidence for that?
2) Do you have any evidence for that?
3) The real world does not need to be housed and therefore does not require any computing power.
4) What is the question the fish are asking? If there's no question, there's no need for an explanation.

You may respond that humans are the supreme intelligence in the universe and are all knowing, yet humans have only been around for 2.8 million years whilst fish have managed to successfully survive in a hostile world for more than 500 million years, regardless of whether this world happens to be "simulated" or "real". — RussellA

What does that have to do with anything?

If we are a simulation and there is a world outside ours, how would we know what is possible? Since we know nothing of the world outside the simulation, we don't even know if it is done via computers. Would it not be a mistake to assume that what applies in our world applies outside it? This seems an odd position to take. — Tom Storm

It rather depends on what your project is. If the project is to make a space for fantasies, then the fact that we don't know is an opportunity, not a problem. Myths have always been stories told where knowledge was not possible. This is a myth.

Also, I think many do not realize that the “God hypothesis” has come back in a stealthy sort a way. Instead of the watch needing a designer, the simulation needs a simulator. — Richard B

Good point.
Come to think of it, the argument could be seen as a version of Berkeley's "proof" of God.

If we are a simulation and there is a world outside ours, how would we know what is possible? Since we know nothing of the world outside the simulation, we don't even know if it is done via computers. Would it not be a mistake to assume that what applies in our world applies outside it? This seems an odd position to take. — Tom Storm

It rather depends on what your project is. If the project is to make a space for fantasies, then the fact that we don't know is an opportunity, not a problem. The point where knowledge runs out is always an opportunity for myth to fill the gap. This is a myth.

That's one of the cognitive traps of the lamp problem. IMO the final state is simply not defined by the premises of the problem, AND there is no solution that makes the sequence continuous, therefore spaghetti is as sensible as anything else. And I've convinced myself that this is the solution to the problem. — fishfry

I agree with you. It suits my approach well, in that the existence of the problem is a result of the way it is defined, or not defined.

The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous. — fishfry

That's the way ω is defined, isn't it? Although I'm not sure what you mean by "continuous" there.
I still feel uncomfortable, because it does get to the bottom of the stairs by placing a foot on each of the stairs, in sequence. But that's exactly the hypnotism of the way the problem is defined. And if an infinite physical staircase is the scenario, then anything goes..

I have been known to take my bat home when I think that a discussion has become fruitless, even before it gets bad-tempered.

Yes ok, so the coach can turn into a pumpkin and the lamp can turn into a plate of spaghetti. Are you agreeing with me on that point? — fishfry

Yes. But I have an obstinate feeling that that fact is a reductio of the process that generated it. So I'm not questioning what you say, but rather what we make of it.

I was making my point about mathematical convergent sequences. Don't know whether it strictly applies to walking. — fishfry

It may be a bad habit to think of applications of a mathematical process. But that's what's going on with the infinite staircase. So it might be relevant to that.

I understand. It's probably best not to comment any further.

Your are being too kind to call this even “highly improbable”. Just because we can imagine such fanciful scenarios does not mean they are possible. — Richard B

I agree. I understand the argument as being a version of Cartesian scepticism. The possibility that God, or an evil demon is feeding us false information is also a fanciful scenario. The paradox of the situation is that believers in it have to put more faith in their fancies than in their experience.

Descartes did at least try to insulate his reflections from being mistaken for real, by proposing them as a thought experiment for academic purposes. I say "try" because so many people have treated his conclusions as objectively true and struggled to escape from the conclusion, when, on Descartes' own presentation, all they have to do is get up from their warm oven and go for a brisk walk in the fresh air.

So you cut corners and don't simulate at that level unless something intentionally is paying attention to that level. — noAxioms

So there are two ways that a simulation of our world would differ from the real world - sorry, the world as we know it.
The first is that the whole of our world could not be simulated, because the hardware would have to be bigger than the whole (real) world.
The second is that exact simulation of even a small part of the real world, down to sub-atomic and near-light-speed events could not be constructed, for the same reason.
So it would not be possible to simulate the progress of research in physics over the last 100 years or so?

I think you'll have to say that the hardware of this simulation we live in must be much, much more powerful than anything we can conceive of and that QM and GR are false. No?

We might be 27 levels down, but there's a base reality up there (as is typically presumed), and that one is 'the reality' by the definition implied by a topic like this. — noAxioms

Surely, we have to presume there is a base reality, or face an infinite regress.

Surely the programmers can create whatever physics, chemistry and phenomena they like. — Benj96

Not quite. They can simulate them, but that just means they can create an illusion of them. They can't create them for real.

Ofc I'm on some fantasy rant here. But I enjoy dabbling in wild metaphysical speculation — Benj96

Yes. It's a curious game. I've never understood the rules.

It might not be a 3 dimensional space world with physics as we know it. That's kind of likely actually since our physics cannot be self-simulated. — noAxioms

Do you mean that no-one living in our world could create a simulation of our world? !!!

The argument still is valid that if we're 'probably' simulated, and if the simulating world is similar to ours, then they're also 'probably simulated'. But that's a lot of 'if's. — noAxioms

Yes, it is. An infinite number, to be exact.

The battle is real to those in the simulation, but not real to those running the simulation. — noAxioms

That's just a posh way of saying that the battle seems real to those in the simulation. Reality, by definition, is not "in" the simulation, but outside it.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end. — fishfry

I think that's all right. When I walk a mile, I start a potentially infinite series of paces. When I have done (approximately) 1,760 of them, I stop. The fact that the 1,760th of them is the last one is, from the point of the view of the sequence, arbitrary, not included in the sequence . The sequence itself could continue, but doesn't.

There's no part of the segment that isn't a point (or a set of them), and yet points have no size, so no finite number of them can actually fill a nonzero length of that segment. — noAxioms

OK. Is that because they have no dimension - are not a part of the line?

The problem I was trying to point out that is that, if we admit a ∞-th step, this step should be associated with a state in one of those mechanisms Michael made up. — Lionino

Because <the infinity symbol> can't be associated with any natural number?

I agree with that too. In the end, I don't think reasoning about infinity gets us anywhere. — Lionino

Then the ultimate paradox is that there seems to be no end to the reasoning.

You complete all the steps by time 1, so the task is then complete. No contradiction so long as we don't reference 'the highest natural number' which doesn't exist. — noAxioms

I don't quite understand. Is the point that the simple arithmetic analysis doesn't reference the highest natural number, so that way of reaching it is OK. It doesn't look like completing all the steps to me - it looks more like jumping over them. But I have travelled over all the spaces.