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
Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already.
— fishfry
— Michael
There isn't one. I've answered this several times already. That's what it means for me to accept P1.
But you need to prove P2. You haven't done so. — Michael
So we're back to my post here:
a. I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum — Michael
Believe it or not, that's an incredibly helpful remark. — Ludwig V
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. — Ludwig V
I had to look Tegmark up. — Ludwig V
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. — Ludwig V
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. — Ludwig V
My apologies. I should have restricted my remark to those who dream up paradoxes. — Ludwig V
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. — Ludwig V
unless you mean the original line of Euclid, "A line is breadthless length."
— fishfry
Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental entities. Thanks for pointing this out. — keystone
What is a line? What does the notation [0, 0.5] mean?
— fishfry
Euclid also said that "The ends of a line are points." When I describe a path as 0 U (0,1) U 1:
(0,1) corresponds to the object of breadthless length and
0 and 1 correspond to the points at the end. — keystone
It seems that some people intepret Euclid as saying that a line without endpoints extends to infinity. I do not think this is necessarily the case. While (-inf,+inf) is a valid line, I believe (0,1) is also a valid line in and of itself. — keystone
Please give the following figure a chance as it captures a lot of what I'm trying to say: — keystone
I believe that someone even as intelligent and knowledgeable as yourself is not qualified to discuss the bottom-up philosophy of a continuum because it is flawed. — keystone
I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy. — keystone
You're right, I did say that the endpoints were necessarily rational numbers. (-inf, +inf) has no endpoints. While there are scenarios where it is useful to include points at infinity, for this discussion, let's agree that the points at -inf and +inf are not real points. I'm only using infinity as a shorthand. I should have been clearer. — keystone
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
But if my consciousness itself is simulated, then the simulation argument requires that consciousness is computational, a point I strenuously disagree with, with Penrose and Searle on my side.
— fishfry
Why do you think it's not computational? — RogueAI
Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?
— fishfry
I start with a few.
1) It's not all a lie. I mean, I can't know that, but if it's all crap, then I can know nothing regardless of how I interpret the lies, so I have no choice but to give weight to the empirical.
2) It's not about me. If I am the center of the universe, the rest is probably a lie. So I pretty much find that any view that puts me, humanity, Earth, the universe itself, as the center of something larger, to be unproductive. — noAxioms
Descartes apparently worried about it all being a lie. I reject that road only because it is untravelable, not because it is wrong. But it seems that modern science has thrown a cold paid of doubt on the validity of "I think therefore I am". — noAxioms
Not sure what you mean by empirical testing here.
As I said, one can empirically examine the causal chain that makes the body walk for instance. In a VR, it does not originate in the brain of the avatar, but external, from the mind controlling the body. Say you're playing tomb raider. Open up Lara Croft's head. No brain in there, or if there is, it's just a prop. None of the stuff she does has its cause originating from there.
Why does nobody pursue such investigations? Is technogoly still so backwards that it can't be done? They already have machines that can detect a decision having been made before you are aware of having done so yourself. — noAxioms
Trendy, yes. Kind of dumbs down the validity of any scientific discovery. Why would a simulation choose to display CMB anisotropy if that isn't what a real universe would look like? — noAxioms
I think that example was being used as an illustration of Moore's law, and not as support for a VR hypothesis. — noAxioms
But you are the one saying that you only have rationals.
— fishfry
No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5). — keystone
Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set. — keystone
I don't know anything about set theory with urlements.
— fishfry
Urelements are indivisible 'atoms'. The lines that I'm working with are divisible. — keystone
You only believe in rationals. Where are you getting these things?
— fishfry
That is not what I believe. I can define a line using no rationals: (-inf,+inf). — keystone
I see this line as a single object (a line). — keystone
It is not populated by rational points. It is not populated by any points for that matter. — keystone
I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2. — keystone
You would call this green line the 'real number line'. You see this as 2^aleph_0 points. You believe that to walk any length on this green line you need limits. I understand what you're saying. We're just starting from different starting points. You're starting from the bottom and I'm starting from the top. — keystone
Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration. — keystone
Currently, I don't believe I can persuade you that a continuum can exist without points. — keystone
However, I've come to realize that convincing you of this isn't necessary. Here’s my new approach:
1) Start at the bottom
2) Build up to the top
3) 'Start' at the top
4) Approach the 'bottom' from the top — keystone
I see this equivalent to starting at the bottom of the S-B tree -> working our way to the top of the tree -> then proceeding back down to approach the bottom. I know you won't see it that way, which is fine. But to be clear, I still believe things are very ugly at the bottom filled with pumpkins. Nevertheless I do understand how mathematicians think things work at the bottom and if starting at the bottom is the only way you'll allow me to get to the top then I'll go with it. I understand your criticisms of starting at the top, I just don't accept them. Once you allow me to get to (3) I endeavor to show you that (3) and (4) alone fully satisfy our needs and if I'm careful (e.g. by only allowing for rational endpoints) that (1) and (2) are not only superfluous but undesirable. Is that a fair approach? — keystone
But now only the endpoints are rational, leaving me baffled as to what those objects are.
— fishfry
Yes, the endpoints are rational, — keystone
and the object between any pair of endpoints is simply a line. — keystone
It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points. — keystone
However, I view the line as a primitive object. — keystone
Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification. — keystone
A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.
— fishfry
I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on those—the one from the comment where you thought I was just trolling. — keystone
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
This is what I mean by reciting backwards:
If I recite the natural numbers <= 10 backwards then I recite 10, then 9, then 8, etc.
If I recite the natural numbers <= 100 backwards then I recite 100, then 99, then 98, etc.
If I recite all the natural numbers backwards then I recite ... ?
It's self-evidently impossible. There's no first (largest) natural number for me to start with. — Michael
It is Achilles' run but with time reversed: https://plato.stanford.edu/entries/spacetime-supertasks/#MissFinaInitStepZenoWalk — Lionino
I accept this:
P1. If we can recite forward 1, 2, 3, ... at successively halved intervals of time then we can recite all natural numbers in finite time
But I reject these:
P2. We can recite forward 1, 2, 3, ... at successively halved intervals of time
C1. We can recite all natural numbers in finite time — Michael
If you want to claim that C1 is true then you must prove that P2 is true. You haven't done so. — Echarmion
I would put things differently. We have clearly made tremendous progress in simulating all manner of physical processes, including those happening inside brains. Where we have made no progress is in developing a conceptual framework for connecting such physical processes with the subjective experience of consciousness. — Echarmion
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). — Echarmion
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
Perhaps this will bring us closer to understanding our own consciousness, but perhaps not. — Echarmion
Does Bostrom actually address this distinction?
— fishfry
Bostrom seems to presume that consciousness is computational, and leaves it undefended.
In such a simulation, nobody is being fooled. — noAxioms
In a VR, is it a lie to have the subject experience a world that is not the same world as the reality in which the mind exists? If so, most forms of dualism are arguably deceptions. — noAxioms
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.
— fishfry
They can't both be right? — noAxioms
You're agreeing with my point.
I think I am, yes. — noAxioms
Anything analog can be approximated with digital. But anything digital can be perfectly implemented with analog. Searle is perhaps referencing property dualism? I don't know if I got that right. Can't seem to articulate the differences between the variants. — noAxioms
I guess I'm even more skeptical than Descartes. I win! I didn't pick my handle for no reason. I try not to leave anything unquestioned. — noAxioms
VR says that all you know is potentially lies. You are not of this universe, but rather you are experiencing it. All very dualistic. The 'brain' in the body (if there is one at all, have you ever checked?) is not what's making any of the decisions.
If you think about it, the view can be empirically tested. Not so much with the simulation hypothesis. — noAxioms
It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.
Definitely the former. But Elon musk is arguing for VR, and references Bostrom's paper to support it, so he has no idea what he's talking about. — noAxioms
The comment above (and my reply) belongs in the other topic. I see you posted more or less the same question there. — noAxioms
There is never a final tick in an infinite sequence, even if the sequence has a limit.
or not a first tick. Zeno's dichotomy very much as a final tick. I can make a scenario that has a first and last, and gets singular in the middle somewhere. Just illustrating the classical snippet: Never say never. — noAxioms
You’re suggesting that my line, which already extends in space, requires additional points, which themselves individually have no length, to actually have length. I wish you could appreciate the irony in your position. — keystone
If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length. — keystone
Our problem is that you are only allowing points in your sets. — keystone
Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set? — keystone
Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?
— fishfry
I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer. — keystone
Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm. — keystone
My preference is that you accept non-points into sets, — keystone
however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 points — keystone
(which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points). — keystone
In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you? — keystone
All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly? — keystone
...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)... — keystone
So for example, can you allow me to say that there are 5 objects depicted below? You can go on thinking that 2 of the objects are composite objects and I'll go on thinking that all 5 objects are fundamental (they're either 0D or 1D continua). — keystone
He doesn't seem to know the difference between the simulation argument (Bostrom is a good example of this) and a virtual reality argument (the Matrix is the typical example). — noAxioms
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. — Ludwig V
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. — Ludwig V
I fully understand your criticism. The problem is that you are missing my point (or perhaps I should say you are missing my 'continua'). — keystone
Let's continue to work with the path defined as [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] as depicted below. — keystone
I say that (0,0.5) and (0.5,1) contain no points so you think I'm only working with three objects - the points as depicted below. The length of all points within my system is 0 so you think the objects I'm working with have zero length. — keystone
I say that (0,0.5) and (0.5,1) describe continua so I say I'm working with 5 objects as depicted below. The length of all points within my system is indeed 0 but the length of the continua within my system add up to 1. — keystone
I prefer working with such simple paths as described above but let's do the impossible and say that somehow I could cut my unit line aleph-0 times such that there is a point for each rational number between 0 and 1. — keystone
You say that the length of all these rational points adds up to 0. I agree. — keystone
You say that there are gaps between these points. I disagree. In between each pair of neighbouring points would lie an infinitesimally small continua. — keystone
If I add up the lengths of all of these tiny continua it would add up to 1. These infinitesimally small continua are indivisible. — keystone
I'm not fond of discussing impossible scenarios as they tend to lead to incorrect conclusions. Indeed, rational points do not have neighbors, and continua are inherently divisible (unless we're treating points as 0D continua, in which case they are indivisible). Therefore, we shouldn't lend too much credence to this example, but I thought it was necessary to address your points more directly. — keystone
The problem is that you're not allowing continua to be valid objects in themselves. — keystone
It is as if you are only allowing points to be valid objects. — keystone
So I figured out a better way to talk about this instead of using metric spaces. Instead, it is better to use Graph Theory.
... [stuff omitted]
To travel from vertex 0 to vertex 4 we simply walk the connected path. One nice thing about this view is that it's clear that no limits are required to walk these graphs. — keystone
I doubt that consciousness is computable
— fishfry
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. — noAxioms
After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation?
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. — noAxioms
We are water. The vast majority of mass would be lost (as would consciousness) if the water was taken away. Lots of pipes going here and there. It's a pretty good description for the Roman days. — noAxioms
Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network?
In the process. — noAxioms
Yes, I saw a domino logic gate on Youtube a while back.
Gawd, I spelled it 'Turning' machine. More typos.
Anyway, yes, the discussion was inspired by that. Any moron can create a domino or gate, but creating a nor gate gets tricky. Any gate can only be used once, so it's impossible to create say a flip flop, normally a trivial thing created with a pair of nor gates.
I've not seen the video, but mention of it inspired me to design a Turing machine with the technology. Can dominos be used to run a physical simulation? I think it's possible since I found not obvious roadblocks. I'm tempted to start a topic on it, but not here since it isn't a philosophy topic at all.[/quoet]
I don't know about dominos. The pencil and paper argument is stronger.
— noAxioms
Perhaps it's some kind of analog computation, but that's not the same thing.
I've also programmed analog computers in school, never on the job. It's a different sort of thing, I tell ya. — noAxioms
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.
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). — noAxioms
I was imagining a clock that speeds up in its ticking to ape a convergent geometric series.
— fdrake
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. — noAxioms
Example please?
— fishfry
Capital, the fetishistic worship thereof. — Vera Mont
Maybe I'm not being clear, so I'll try one more time. — Michael
If you want to argue that the first supertask can end ... — Michael
So I ask again: can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? — Michael
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
I don't wish for poor kids to be deprived of an education.
— fishfry
Only because you seem to be so vehemently against letting them off some of the accumulated compound interest on their student loans. — Vera Mont
And maybe because you seem hell-bent on putting an unfair burden of putative working class taxpayers. — Vera Mont
And thirdly, because you pretend that government is responsible for everything it cannot possibly control. — Vera Mont
And lastly, because you appear to have a peculiarly skewed view of the working class, even as you advocate for its supposed interest. — Vera Mont
I would prefer if Congress would pass a law to have high income earners fund college costs.
— fishfry
Well, who wouldn't? — Vera Mont
But Congress and Senate are protecting high earners - perhaps because they themselves are high earners?
Both the Senate and the House have now passed a bill to block President Joe Biden’s student loan forgiveness program, which promises to cancel up to $20,000 of debt for millions of borrowers but has been held up by courts. CNN
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
But you haven't got a continuum if your intervals contain only rational numbers.
— fishfry
Ok, this was an excellent post! — keystone
I better understand your criticism. It lies in the fact that I'm using the term 'interval' in an unorthodox manner. I use the term interval to describe the objects (whatever they may be) lying between the upper and lower bounds. — keystone
Let's consider the interval (0,0.5). — keystone
From a bottom-up perspective, the objects within the interval are aleph-1 actual points. — keystone
From a top-down perspective, the object within the interval is a single continua. — keystone
It doesn't contain the rational points between 0 and 0.5, it contains no points. — keystone
However it holds the potential for rational number points between 0 and 0.5. — keystone
It's only deep from a bottom-up perspective. From the top-down perspective it is elementary. — keystone
Do you believe in the number 1/3 then?
— fishfry
I believe that I could use the Stern-Brocot algorithm to generate a 3 layer tree whose third layer will contain a node described by LL and having all the properties that we generally attribute to 1/3. — keystone
Consider one of your rational intervals [0,1]. What is its length?
— fishfry
The length of continuum (a,b) is b-a. So consider the continuum defined by interval (0,0.3). It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone
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
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
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
Student borrowers are taxpayers. The question is, which taxpayers are having to pay more? — Vera Mont
You say the working class; I say the high earners. — Vera Mont
Would it be so very terrible if people making over $400,000 a year (many of whom are in the money-lending business) had to pay a little more so that the children of orderlies and fish-packers could get an education? — Vera Mont
You accept some rational numbers. Not much of a continuum you have there. You understand that, right?
— fishfry
I concur that rational numbers alone, represented as points, are insufficient for constructing a continuum. That's not the argument I'm making. You keep thinking I'm trying to build a continuum. No, I'm starting with a continuum, defined by the interval notation we have discussed, and working my way down to create points. — keystone
There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3
— fishfry
Oh no, the classic debate about whether 0.9=1. — keystone
I know you dislike the S-B tree but it makes the top-down and bottom-up views very clear. Maybe use some eyedrops? :P — keystone
Bottom-up view: Using a supertask, — keystone
I'm pretty sure that you won't like my depiction of the bottom-up view as I frame it in a way that make's it clearly problematic. I'm fine with not investing further on this specific topic at this time as it really will just be a distraction from the main topic. — keystone
I'm not questioning the mathematics itself, but rather the philosophical underpinnings of the mathematics. For instance, I recognize Cantor's remarkable contributions to math, even though I personally do not subscribe to the concept of infinite sets. His contributions have a valuable top-down interpretation. — keystone
I think you are an intuitionist.
— fishfry
You make a good point. However, I'm not sure about the details of the constructivist approach - my impression is that a typical intuitionist would say that the number 42 permanently exists once we've intuited it. So while I'm hesitant to label myself hastily, I do think that broadly speaking I fit into this camp. — keystone
You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?
— fishfry
I totally accept and am in awe with the algorithm. I just don't think the algorithm can be run to completion to return a number. I also don't think it has to be run to completion to be valuable. — keystone
If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum.
— fishfry
I agree, but isn't that what I've been doing all along? Doesn't [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] define a continuum? — keystone
Maybe it would be valuable if you detail what you think a continuum must be. For example, will you only accept the definition if it is composed solely of points (and no intervals)? — keystone
I'd like to move forward since we haven't yet reached the most interesting topics — keystone
, but if you believe that I'm not defining a continuum, then there's no point in proceeding further. — keystone
On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:
Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics. — jgill
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
You can't play it in reverse
— fishfry
So you're saying that it's possible to have recited the natural numbers in ascending order and possible to have recorded this on audio but impossible to then replay this audio in reverse? That seems like special pleading. Am I metaphysically incapable of pressing the rewind button? — Michael
I am presenting two versions of your argument; one in which I have recited the natural numbers in ascending order and one in which I have recited the natural numbers in descending order. I am using the second version to illustrate the flaw in the first version. — Michael
No, once again you recited the natural numbers in ascending order.
— fishfry
No, I'm reciting them in descending order. I'll repeat it again and highlight to make it clear:
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 – e.g. my recitation ends with me saying "3" at 12:00:07.5 then "2" at 12:00:15 then "1" at 12:00:30 and then "0" at 12:01:00. — Michael
Notice that even if the conclusion follows from the premise that the argument fails because the premise is necessarily false. It is impossible, even in principle, for me to have recited the natural numbers in the manner described. — Michael
Even if the conclusion follows from the premise I do not accept that the premise can possibly be true. Like with the previous argument, I think that it's impossible, even in principle, for me to have recited the natural numbers in the manner described. — Michael
I have attempted at least to explain why this is impossible (e.g. with reference to recording us doing so and then replaying this recording in reverse), but as it stands you haven't yet explained why this is possible. If you're not trying to argue that it's possible – only that I haven't proved that it's impossible – then that's fine, but if you are trying to argue that it's possible then you have yet to actually do so. — Michael
Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? — Michael
It's not something that we can just assume unless proven otherwise. — Michael
Even Benacerraf in his criticism of Thomson accepted this. — Michael
√ω has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about.
— fishfry
OK. I'll accept that. I do believe somebody has shown no limit to the potential cardinality of some sets. — noAxioms
I worked a great deal of my career writing code for multiple processors operating under the same address space. It gets interesting keeping them from collisions, with say two of them trying to write different data to the same location. — noAxioms
Anyway, not sure what you mean by your statement. It seems on the surface to say two processors is no more powerful than one, which isn't true, but two also isn't twice as powerful. — noAxioms
You didn't read my comment then. Ability to move is a given (an axiom, not something that can be proven). — noAxioms
Given that, doing so is a supertask only if Zeno's premise holds, that for any starting point, one must first move halfway to the goal. I can't prove that it holds, but I can't prove that it doesn't hold either. — noAxioms
I defined the terminal lamp state as a plate of spaghetti.
Yes, the PoS solution. — noAxioms
Does 'bottom of the stairs' imply a bottom step? If every other step was black and white, what color is the bottom step? PoS, I know. Same problem from where I stand. — noAxioms
I'll look at that. I have all the respect for the PSE guys, who blow everybody else away. Quora stands somewhat at the opposite end of that spectrum. — noAxioms
You convinced me. Let's transfer the legally contracted debt of people who signed for it, to those who never took out that debt, never saw any of the money, and are busy working while the kids are partying it up in school.
— fishfry
That's not happening and nobody's planning it. — Vera Mont
Which is quite reasonable. Plumbers make about $60,000; a welder's average is $47,000. Still not vast, and they don't start out $50,000 in the hole.
If their graduate kids make a little more, they can buy their old parents a cruise of something. — Vera Mont
Student loaninterest forgiveness for low earners. — Vera Mont
So long as the workers are being oppressed. — Vera Mont
Once social justice and balance are established, — Vera Mont
there are no sides and classes. — Vera Mont
Everybody shares the resources and contributes to the community. — Vera Mont
That means, every child has the opportunity to learn as much as he or she is able to and wants to, without penalties. A just society would have no such thing as student debts, or any other kind of debt-load that keeps growing, even while you're paying. A just society would outlaw compound interest and 90% of the other financial legerdemain on Wall street. — Vera Mont
You're make a big show of defending the workers - represented by a skilled occupation, the holder of which probably considers himself middle class, anyway - while assuming that the working class is a static, unchangeable entity: nobody in, nobody out, beleaguered forever by white collar workers.
That's as gross a misrepresentation as that of NY crime and that of Biden's policies. — Vera Mont
That is the inevitable outcome, every cycle. Boom, growth, consolidation, wealth concentration, political corruption, bust, depression, protest, repression or revolution. — Vera Mont
Transfinite ordinal numbers are numbers.
Are they? Does √ω have meaning? — noAxioms
It does for numbers. It's a serious question. I am no expert on how transfinite ordinal numbers are treated. It seems like a different species, like having a set {1, 2, 3, ... , green} which is also a valid set, and countable. — noAxioms
Ordering irrelevant. The set supposedly needs to be countable, and it is. Michael's definition of supertask came from wiki, and that definition says it is countable, else it's a hypertask. The SEP definition of supertask omits the 'countable' part and seemingly groups the two categories under one word. — noAxioms
The definition also includes 'sequential', meaning parallel execution of multiple steps is not allowed. — noAxioms
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?
Clearly it isn't a supertask if it is impossible to go only half the remaining distance for some intervals. If that is possible, then it must be a supertask. — noAxioms
It violates thebijunction
— noAxioms
I take that back. It doesn't violate the bijection. And I spelled it wrong too. So many errors. — noAxioms
Note that I no longer have an order-preserving bijection.
That's fine. The rational numbers are both ordered and countable, but they cannot be counted in order. — noAxioms
Sounds like the lamp problem is unsolved. It is still 'undefined'. — noAxioms
Another note: The paradox of the gods that I occasionally bring up is fun to ponder, but it isn't a supertask since it cannot be completed (or even started). Progress is impossible. Ditto with the grim reaper 'paradox' where I die immediately and cannot complete the task. — noAxioms
Your ω might help with the stairs. The guy is at 'the bottom' and there is but the one step there, labeled ω. No steps attached to it, but step on that one step and up you go, at some small finite numbered step after any arbitrarily small time. — noAxioms
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.
His assertion isn't justified, I agree. — noAxioms
Some speculative physicists (at least one, I believe) think the world is a large finite grid
So much for the postulates of relativity then. I kind of thought we demolished that idea with some simple examples. It seems to be a 'finite automata' model, and the first postulate of SR is really hard (impossbile) to implement with such a model, so a whole new theory is needed to explain pretty much everything if you're going to posit something like that. I haven't read it of course, so any criticism I voice is a strawman at best. — noAxioms
The chessboard universe sounds very classical, and it's been proven that physics is not classical, so I wonder how this model you speak of gets around that. — noAxioms
If supertasks are impossible and motion is possible then motion isn't a supertask.
— Michael
This evaded the question ask. Sure, we all agree that if supertasks are impossible, then supertasks are impossible. He asked how you justify the impossibility of a supertask. All your arguments seem to hinge on a variant that there isn't a largest natural number. — noAxioms