If there was an easy knock-out blow to it, it wouldn't be a topic on philosophy/mathematical discussions.That seems to me a good response, though not quite the knock-out blow one would hope for. — Ludwig V
I see that you have an opinion, and that you are attempting to rationalize this opinion. But you leave some pretty low hanging fruit in this post, and rather than have me point them out and you denying whatever it is I post, I invite you to step into my shoes and critique the above. If your opinion was the opposite, what portions of the above argument would you put in bold and say is wrong?This logical consequence can be shown when the experiment is explained more clearly:
A1. At t0 the lamp is off
A2. The button is pressed only as described by this sequence of operations: at t1/2 I press the button, at t3/4 I press the button, at t7/8 I press the button, and so on ad infinitum
Compare with:
B1. At t0 the lamp is off
B2. The button is pressed only as described by this sequence of operations: at t1/2 I press the button
The status of the lamp at t1 must be a logical consequence of the status of the lamp at t0 and the button-pressing procedure that occurs between t0 and t1 because nothing else controls the behaviour of the lamp.
If no consistent conclusion can be deduced about the lamp at t1 then there’s something wrong with your button-pressing procedure. — Michael
OK, the bold line is telling. There is something wrong with the procedure. I've pointed it out in several posts. The lamp isn't broken. That violates the mathematical definition of how the thing works. There is no physical lamp since physics cannot do what is described.The important part is in bold. If there is a problem with the button-pressing procedure, which there is in the case of A2, then this problem remains even if the button is broken and doesn't actually turn the lamp on — Michael
This is not a supertask, not even as the tick rate increases arbitrarily high, because the cake (if it is continuous, which a physical one isn't) is going to take forever to consume at any clock rate.A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation. — fdrake
This is not a supertask, not even as the tick rate increases arbitrarily high, because the cake (if it is continuous, which a physical one isn't) is going to take forever to consume at any clock rate. — noAxioms
The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency. — Michael
By providing a standard mathematical object which is infinite, has no final element, tends to an end state, and has an infinite number of occurrences ("steps"), but occurs in finite time. — fdrake
The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible: — Michael
The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency. — Michael
The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:
I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.
How does one start such a supertask? — Michael
From Tasks, Super-Tasks, and the Modern Eleatics:
What conclusions are we to draw from this rather heady mixture of genies, machines, lamps, and fair and foul numbers? In particular, has it been shown that super-tasks are really possible – that, in Russell's words, they are at most medically and not logically impossible? Of course not. In a part of his paper that I did not discuss, Thomson does a nice job of destroying the arguments of those who claim to prove that super-tasks are logically possible; had there been time I should have examined them. In the preceding section I tried to do the same for Thomson's own neo-Eleatic arguments. I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his (supposing that I did destroy them). — 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? — Ludwig V
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
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.) — Ludwig V
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. — Ludwig V
Not sure what you mean by potential cardinality.
— fishfry
Pick a number, say 27. I believe it has been shown that there exists a set the cardinality of which is 27, if that's valid terminology. — noAxioms
One could also reference aleph-26, — noAxioms
but I'm not sure that one can prove that no sets exist with cardinalities between the ones labeled 1 through 27. — noAxioms
Point being that you get no increase in computational power from parallelization.
I beg to differ. A 16 processor machine can sustain a far greater work load than a single-processor machine. The Cray machines were highly parallelized (SIMD architecture) in which thousands of floating point operations were performed by every instruction. These machines were great for stuff like weather simulation. — noAxioms
No function is computable by a parallel process that's not already computable by a linear process.
With that I agree. But that same function can also be done by paper & pencil. You said 'powerful', a reference to how fast the work is completed, and more processors helps with that. — noAxioms
Coloring the steps reduces to the lamp.
I notice that any scenario with a contradiction involves invoking magic. Suppose this physically impossible thing (infinite gods, stairs requiring faster-than-light speed, lamp switches that operate without delay. No magical measurement of something nonexistent. Zeno doesn't do that. No magic invoked, and the first premise thus produces no paradox. — noAxioms
My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.
Oh it serves its purpose, but correct answers are not promoted above the others, and apparently a great deal of their posters don't know what they're talking about when it comes to stuff like this. — noAxioms
Zeno's horse is quite real. Almost none of the others are. — noAxioms
Exactly so. I have correct my post. I meant valid and wrote 'sound' in haste. A simple application of modus ponens shows the lack of soundness of Zeno's conclusion iff empirical knowledge is given any weight.
The conflicting premise which would be used to disprove this, the limitations of divisibility
The conflicting premise seemed to be a denial of the completability of a supertask. He never suggests a limit to divisibility. — noAxioms
The sum of an infinite set of identical finite numbers is not finite, no matter how small the number being summed. It needs to complete in finite time to be a supertask.Why? The ticks per second is also going to infinity. — fdrake
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.does not prove that the following supertask is metaphysically possible: — Michael
Gotcha. No argument then. As I already pointed out, you had referenced power instead of computability: "there's no difference in computational power between parallel and serial processing." and I took it as a statement of work over time.No. I'm talking about computability theory. — fishfry
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.Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.
Yes. Search for 'horse' in the last 20 posts or so.The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too.
Because of this, empirical knowledge doesn't prove pretty much anything to be possible or impossible. That's why science theories are supported by evidence and not by proofs. They'd be theorems, not theories, if they were provable.But empirical knowledge has problems like what Hume showed with the problem of induction. Because of this, empirical knowledge does not prove the supertask to be impossible. — Metaphysician Undercover
I beg to differ. That simply does not follow from the description. Zeno describes a physical completable supertask, which is only as possible as the soundness of his first premise.That the supertask is not completable is not denied, that it is not completable is what actually leads to the problem. In Zeno' paradox Achilles never catches the tortoise because the supertask is never completed.
Again I differ. The supertask (if that premise is true) is not fiction. I mean, my opinion is that there isn't a physical supertask, but opinion isn't evidence, and I have no evidence (let alone proof) that it isn't a supertask.Achilles will pass the tortoise, and in the op 60 seconds will pass. This shows that the supertask as a fiction.
No. I'm talking about computability theory.
— fishfry
Gotcha. No argument then. As I already pointed out, you had referenced power instead of computability: "there's no difference in computational power between parallel and serial processing." and I took it as a statement of work over time. — noAxioms
I brought this up in my simulation-theory topic. A simulation of Earth to a precision sufficient for consciousness can be done by pencil and paper, or by dominos falling, — noAxioms
The latter is really interesting: set up dominos so that you get the function of a Turning machine. Not easy, but it seem that it can be done. — noAxioms
Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.
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
The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too.
Yes. Search for 'horse' in the last 20 posts or so. — noAxioms
The sum of an infinite set of identical finite numbers is not finite, no matter how small the number being summed. It needs to complete in finite time to be a supertask. — noAxioms
Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. — fishfry
Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? It's not something that we can just assume unless proven otherwise. Even Benacerraf in his criticism of Thomson accepted this. — Michael
Feel free to give a reference, else I can't respond. — fishfry
I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his.
I beg to differ. That simply does not follow from the description. Zeno describes a physical completable supertask, which is only as possible as the soundness of his first premise. — noAxioms
Declaring something to be impossible is a strong claim and requires strong evidence. — noAxioms
what, because consciousness is not a physical process, or that physical processes cannot be simulated? You seem to be in the former camp. If that's the case, then no, it probably isn't computable.I doubt that consciousness is computable — fishfry
Pretty much 1-1 odds. That's when the terminology became part of our language. You describe yourself in terms of the things you know.After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation?
In the process.Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network?
Gawd, I spelled it 'Turning' machine. More typos.Yes, I saw a domino logic gate on Youtube a while back.
I've also programmed analog computers in school, never on the job. It's a different sort of thing, I tell ya.Perhaps it's some kind of analog computation, but that's not the same thing.
Your view of consciousness is modelled by a VR. One big distinction is that a VR cannot be implemented with paper and pencil (or dominos).ps -- I checked out the Simulation thread and from there, saw your initial post in the "What is the Simulation Hypothesis" thread, and I agree with everything you said. I especially appreciated the distinction between simulation and VR, which is something a lot of the simulation discussions miss.
OK, that would be pretty much what has been the topic of discussion this whole thread. If it completes in finite time, it's a supertask. Don't forget the inverse case where the clock starts fast and slows down to its final tick.I was imagining a clock that speeds up in its ticking to ape a convergent geometric series. — fdrake
Correct, but a second unstated premise must be assumed in order to draw this conclusion, since without it, one can only say that the tortoise cannot be overtaken at any particular step. That second premise might well be that supertasks cannot be completed. That premise is indeed in contradiction with the first premise and empirical observation. At least one of the three is wrong.I think you misunderstand Zeno's paradoxes. Zeno concluded that Achilles cannot overtake the tortoise. That is explicit. — Metaphysician Undercover
Fundamental axioms? None of the premises are that. They're both easily doubted.even though the logic proceeding from fundamental axioms proves
Or the premise of supertasks being uncompletable is wrong, or that empirical evidence isn't as strong as is asserted.Due to the strength of the empirical evidence, we are led toward the conclusion that the fundamental axioms concerning the continuity of space and time, and the infinite divisibility of those continuums, must be faulty.
OK, that would be pretty much what has been the topic of discussion this whole thread. — noAxioms
Which would be odd, seeing as such an object has a model in set theory — fdrake
How can a sequence of operations in which each operation is performed only after the previous operation is performed complete without there being a final operation? — Michael
I am so sorry. I started a hare by mistake. The horse first appeared in this commentThe Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too. — fishfry
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.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
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.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 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.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
Well my quote above is not given from authority. Planck units are just a standard of natural units. A Plank length is a small distance, but the fact that they know that distance down to at least 7 significant digits means that far smaller space units are meaningful. Still, wiki says "Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length", which is similar to what I was trying to convey.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. — Ludwig V
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.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?
Somebody still suggests that matter is continuous? I mean, that sort of went out the window a couple centuries ago.If what happened to the question whether matter was continuous or not is anything to go by, I think that a third alternative is most likely.
Actually, I've been asking about the distinction between those two. Nobody has really answered. A nice example (not a supertask example if possible) of something that is one but not the other would be nice.I imagined you lot were talking about metaphysical rather than logical possibility. — fdrake
It may grind against your intuitions, but no logical argument against it has been presented. That you personally find it 'evidently absurd' carries no weight.So you’re claiming that it’s logically possible to have recited the natural numbers in descending order. That’s evidently absurd. — Michael
Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility! — fdrake
It is logically impossible to have recited every natural number in descending order because it is logically impossible to even start such a task. — Michael
Correct, but a second unstated premise must be assumed in order to draw this conclusion, since without it, one can only say that the tortoise cannot be overtaken at any particular step. — noAxioms
That second premise might well be that supertasks cannot be completed. — noAxioms
That premise is indeed in contradiction with the first premise and empirical observation. At least one of the three is wrong. — noAxioms
How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down. — noAxioms
Actually, I've been asking about the distinction between those two. Nobody has really answered. A nice example (not a supertask example if possible) of something that is one but not the other would be nice. — noAxioms
So when I hear Michael talking about the impossibility of a geometric series "completing" (so to speak) due to being unable to recite the terms in finite time... — fdrake
How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down. — noAxioms
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