Comments

  • A simple question
    Imagine the nerve of somebody demanding fair treatment for all kinds of people, even the designated victims! Appalling, innit?Vera Mont

    Indeed it is. I quite share your sensibilities, or at the very least I have great sympathy for them.

    But the larger point is that you have heard about people these days who prefer equity to equality, equality of outcome over equality of opportunity. You in fact seem to happen to be one of those folks.

    But earlier, you claimed there were no such people.

    So I take it that you have conceded my point. I'm not arguing the point of view pro or con; only that the point of view exists. That in fact you exemplify and represent it. So what you initially said, that you did not believe there were many of these people, was not quite true. Have I got that right? I don't want to presume, I may have misunderstood you.

    Secondly, and again purely for conversation, on the issue of criminal justice. Do you follow New York City politics and current events? Do you support Alvin Bragg? Can you see how some people might think that compassion to criminals, no matter how well intentioned, can end up becoming a pronounced lack of compassion for their victims? Some of the folks pushed onto subway tracks by individuals previously treated gently by the criminal justice system might see it that way. Can you at least see that?
  • Fall of Man Paradox
    Indulge me in an analogy.

    I see the Matrix (pictures):
    keystone

    This entire idea was completely lost on me.


    Both perspectives accurately correspond to the simulation. So I agree that sets are fundamental, and I could even be convinced that digital rain is more fundamental than the Matrix.keystone

    Digital rain is more fundamental than the Matrix. That's very poetic.


    But Let's not go there. I'm specifically talking about the (continuous version of the) Matrix where I believe continua are more fundamental than points. But I don't even want to debate this further, I'd rather show you what could be done with a Top-down approach and let you decide.keystone

    You know, it might be better if you would write a draft then edit it. This seems like stream of consciousness. It has a groovy vibe to it but it doesn't say anything.

    I bring up the Matrix because, I want you to know that I recognize the unique purity and precision of the digital rain, but there are times, especially in discussions on geometry, when it's more effective to visually interpret the geometry from within the Matrix. Please allow yourself to enter the Matrix, try to understand my visuals, just for a little while. End of Matrix analogy.keystone

    As it happens I hate that stupid movie. It's a kung-fu flick with silly pretensions to pseudo-intellectuality. Also someone did the calculation and it turns out that humans make lousy batteries. Very inefficient.

    Where is the line between your indulging yourself, and your trying to communicate a clear idea to me?


    Okay, I lost you because I made a mistake. Let me try again:

    Set: { (0,0) , (0,0.5) , (0.5,0.5) , (0.5,1) , (1,1) } where x1 and y1 in element (x1,y1) is a rational number

    Metric: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 |
    keystone

    No idea what you are trying to do, what you are doing, why you are doing it, and why you are telling me about it.

    Upon further consideration, I've decided to significantly restrict my focus to a smaller enclosing set. I am now interested only in what I want to call 'continuous sets' which are those sets where, when sorted primarily by the x-coordinate and secondarily by the y-coordinate, the y-coordinate of one element matches the x-coordinate of the subsequent element. For example, we'd have something like:keystone

    Like a triangular section of the plane? Why?


    You're right, |x-y| doesn't qualify as a metric. Let me try again. Forget about Universal Set. Instead, I aim to define a Continuous Exact Set. A set is defined as an exact set if all elements satisfy |x-y|=0. I propose that within my enclosing set, the only Exact Set is the trivial set, containing just one element. Once again, this isn't a groundbreaking revelation; I am simply emphasizing that rational numbers by themselves are insufficient for modeling a continuum.keystone

    I just don't know what you're doing. You seem to be having fun, and I don't mind because this like taking a rest after the mortal combat of the staircase thread.


    Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.


    The story: Achilles travels on a continuous and direct path from 0 to 1.
    The bottom-up view: During Achilles' journey he travels through infinite points, each point corresponding to a real number within the interval [0,1].
    The top-down view: In this case, where there's only markings on the ground at 0, 0.5, and 1, I have to make some compromises. I'll pick the set defined above and describe his journey as follows:

    (0,0) -> (0,0.5) -> (0.5,0.5) -> (0.5,1) -> (1,1)
    keystone

    Wasted on me, hope you got something from it.

    In words what I'm saying is that he starts at 0, then he occupies the space between 0 and 0.5 for some time, then he is at 0.5, then he occupies the space between 0.5 and 1 for some time, and finally he arrives at 1.keystone

    No idea, eyes glazed long ago.

    Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs.keystone

    Quite a tall order.

    I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation.keystone

    Well Euclid considered points fundamental, along with lines and planes. But modern set-theory based math takes sets as fundamental. In fact there is nothing other than sets. You start with the empty set and the rules of set theory and build up everything else.

    The word point is only used casually, to mean an element of some set, or a tuple in Euclidean space, or a function in a function space.

    I'm familiar with these methods. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them.keystone

    I'm just throwing things out that seem related to what you're saying.

    I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual.keystone

    I'm very glad I can help. What is the digital rain? Do you remember the Church of the Cathode Ray from the movie Videodrome?

    Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual.keystone

    No problem, take your time. I hope you and your guest have a lovely visit.
  • Infinite Staircase Paradox
    That's not quite what I'm saying. The process described by the op has no limit.Metaphysician Undercover

    Oh I had no idea we were still talking about the OP. This thread's gone way beyond that.

    I thought you were making a more general point, that the limit lives in a different kind of conceptual space than the sequence itself, or that the limit was imposed on the sequence by observers.

    If I misunderstood then nevermind. I've long forgotten the staircase problem. I don't think I ever actually understood it.

    That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described processMetaphysician Undercover

    Well 1/2 + 1/4 + 1/8 + ... is a well known convergent sequence. It converges to 1. And surely we've all experience one second going by. So that's the paradox, right?

    I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass.Metaphysician Undercover

    It has not been productive in the past for us to discuss mathematics, and your misunderstanding of limits is not my job to fix. I gave at the office. Nothing personal but you know we have been down this road. I sort of get what you're saying but mostly not. "The process carries on, unlimited, even though there's a limit." I haven't the keystrokes to untangle the myriad conceptual difficulties with that statement, and the beliefs and mindset behind it; even if I had the inclination. I hope you'll forgive me, and understand.

    Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician.Metaphysician Undercover

    LOL. And the meaning of Moby Dick is only because of what we all determined the symbols to mean. Man and His Symbols, Jung. Yes we are symbolic beasts.

    But within the sphere of math, the definition of a limit is as objective as can be. We lay down a definition, you know the business with epsilon and L, and we confirm that the sum converges; just as in the sphere of the English language, Moby Dick is a story about a bunch of guys who go whaling and it mostly doesn't end well.

    To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it.Metaphysician Undercover

    I'm sorry, I can't really talk about the staircase problem specifically, I never paid much attention to it at the beginning. I mostly got interested in this thread when other issues were introduced. But mathematicians are very good at determining limits, and the one in question is perfectly well known to everyone who ever took a year of calculus. You might take a look at the Wiki page on limits.

    That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit".Metaphysician Undercover

    You don't need any esoteric "principles other than those stated in the description of the process" to determine the sum of a geometric series as a particular limit.
  • A simple question
    We must subvert our tendency to competeBenj96

    I just noticed this. What means would you use to bring this about?
  • A simple question
    This appears to be the case.Vera Mont

    Just for light conversation ... when I say that a lot of people these days are advocating for equality of outcome rather than equality of opportunity ... you do not know what I am referring to? The DEI movement, social justice, wokitude, and the like? Disciplinary standards relaxed in schools, admission criteria relaxed in universities, the criminal justice system biased in favor of criminals, massive social change for the purpose of balancing out racial categories?

    This news has not yet reached your province?
  • SCOTUS
    No. What Trump says and does and what the Supreme Court says and does are not the same.Fooloso4

    I only mentioned it because this is a bit of Trumpy thread. A lot of people think the court's on Trump's side and not being judicially impartial. And opinions about that correlate with people's opinions on Trump. So this is really a Trump thread. Or at least a Trumpy thread. That was my thought process anyway. But I'm not actually participating in the thread, so I haven't got any strong feelings, I had just noted that there's a zillion-page long Trump thread, and I assumed that was there to soak up the gusher of opinion on the guy.

    Gotta say, the man was on reality tv for ten years, he knows what the American people love, or love to hate. The historians will have a field day, if we all live that long.
  • Infinite Staircase Paradox
    You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact.Relativist

    You can think of it that way. Or you can think of it "reaching" its limit at a symbolic point at infinity. Just as we augment the real numbers with plus and minus infinity in calculus, to get the extended real numbers, we can do something analogous with the natural numbers, and augment them with a symbolic point at infinity , so that the augmented natural numbers look like this:

    1, 2, 3, 4, ...,

    Now a sequence is just a function that for each of 1, 2, 3, ..., we assign the value of the sequence, the n-th term. And we can simply assign the limit as the value of the function at . It's perfectly legitimate. We can define a function with ANY set as its domains. So a sequence is a function on , and a sequence augmented with its limit (or any other value!) is just a function on .

    This is a key point. I've stated it a number of times recently and I'm not sure I'm getting through. The natural numbers augmented with a point at infinity is a perfectly good domain for a function; and we can use such a function to identify each of the points of a sequence, along with the limit.

    I also think you are misinterpreting the meaning of limit.Relativist

    On a forum our words must speak for themselves. But in this instance I can assure you that nothing could possibly be farther from the truth.

    This article describes it this way:
    In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

    The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L.
    Relativist

    Wiki is not necessarily a good source for mathematical accuracy or subtlety of expression, and in this case they have led you astray.

    I hope very much that you will give some thought to what I wrote about defining the limit of a sequence as the value of some function on the naturals augmented with a symbolic point at infinity; or more concisely, as a function on . A sequence is just a function on , which is a synonym for . You can "attach" the limit to the sequence by extending the same function to one on .

    I hope this is clear. I find it an extremely clarifying mental model of what's going on with a sequence and its mysterious limit. "Where does the limit live?" I get that it's kind of confusing. The limit lives at the point at infinity stuck to the right end of the natural numbers.

    1, 2, 3, 4, ...

    That's how people need to learn how to count in order to better understand supertasks and limits.

    You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
    — fishfry
    No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does?
    Relativist

    Sure, after 1 second. It's perfect obvious from daily existence. When I got up to make a snack I did first walk halfway to the kitchen then halfway again. So now I'm arguing for supertasks. But I could just as well argue against supertasks. So whatever you said, I could probably convince myself to agree with it.

    I think the word "reach" is being abused in this conversation. It comes out of badly taught calculus classes, and Wikipedia.
  • Infinite Staircase Paradox
    There is no limiting process in the premises of the op, nor in what is described by ↪Relativist . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited.Metaphysician Undercover

    Wow that's deep. Deep and wrong at the same time. That's interesting.

    If I am understanding you: You say that if we have a sequence; that if that sequence happens to have a limit, then the limit is not inherent to the sequence, but is rather imposed by the observer.

    I suppose the analogy is color, which is in the eye-brain system of the observer, not in the object or even in the light.

    But actually, the limit can be considered part of the sequence. Just as a sequence is a function defined on the natural numbers; a sequence along with its limit can be defined as a function on the natural numbers augmented with a point at infinity, which I've been calling .

    It's really no different than taking the set {1/2, 3/4, 7/8, ...} and augmenting it with the number 1, to yield the new set {1/2, 3/4, 7/8, ..., 1}. Surely you can see that 1 is a perfectly sensible number on the number line. In many ways it's the ONLY sensible number. All other numbers are derived from it. That and 0. Give me 0 and 1 and I'll build all the numbers anyone needs.

    So if that's what you're saying, I find that a very interesting thought. But there is no reason to imbue limits with mysticism. They're very straightforward. They're just the value of a sequence at the augmented point at infinity; which, if you don't like calling it that, is just adding the number 1 to the 1/2, 3/4, ... sequence.
  • Infinite Staircase Paradox
    This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc.Michael

    Yes, I agree with you that math and physics use different definitions.

    I apologize for getting crabby last night. As I went to bed I was thinking, Why am I snarling at someone about supertasks, I don't even care about supertasks.

    You're right, I was not the one you were originally addressing. I jumped in because I was annoyed by your total lack of logic in claiming that supertasks are metaphysically impossible or logical contradictions. I agree with you that supertasks don't exist physically today, but I allow for the possibility of new physics in the future, just as there's always been new physics in the past. I don't think you've supported your metaphysical or logical arguments. That's why I jumped in.

    Also it's perfectly clear that I can walk a mile, and I first walked the first half mile, etc., so if someone (not me, really!) wanted to argue that supertasks exist on that basis, I'd say maybe they have a point.

    P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.

    P2. 30 + 15 + 7.5 + ... = 60

    C1. The sequence of operations1 described in P1 ends at 60 seconds without ending on some final natural number.

    But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

    So C1 is a contradiction. Therefore, as a proof by contradiction:

    C2. P1 or P2 is false.

    C3. P2 is necessarily true.

    C4. Therefore, P1 is necessarily false.

    And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum.
    Michael

    I think "reciting natural numbers" is a red herring, because it's perfectly clear that there are only finitely many atoms in the observable universe, and that we can't physically count all the natural numbers.

    But let me riddle you this. Suppose that eternal inflation is true; so that the world had a beginning but no end, and bubble universes are forever coming into existence.

    And suppose that in the first bubble universe, somebody says "1". And in the second bubble universe, somebody says, "2". Dot dot dot. And bubble universe are eternally created, with no upper bound on their number.

    Therefore: Under these assumptions, there is no number that doesn't get spoken. And therefore, all the numbers are eventually counted.

    You see we don't have to "reach the end," since we can't do that. All we have to do is show that there is no number that never gets counted. Therefore they all do. It's a standard inductive argument. You show something's true for all natural numbers because there can't be a smallest number where it's not true.

    I remind you that while eternal inflation is speculative but is taken seriously by a lot of smart people.

    Therefore I claim that it is metaphysically possible to physically count the natural numbers; and that no logical contradiction is entailed. I'll grant you that I haven't yet shown how to do it in finite time, and so I have not refuted your point. I'm giving more of a plausibility argument that someday, there might actually be a finite-time supertask. We just don't know. You personally can not know. That's my real point, bottom line.

    You cannot know what future physics will allow or conceptualize. That's my whole argument. That's why I say that supertasks violate contemporary physics, Planck scale and all that. But based on the shocking paradigm shifts of the past, there will be shocking paradigm shifts in the future; and physically actualized infinitary processes are as good a candidate as any for what comes next.

    I wrote a response to @NoAxioms above in which I laid out my thoughts, it might be of interest ... https://thephilosophyforum.com/discussion/comment/900398

    Thanks again for your good cheer in not firing back!
  • Infinite Staircase Paradox
    What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something.noAxioms

    Well physics is of course exempt from math and logic. The world does whatever it's doing. We humans came out of caves and invented math and logic. The world is always primary. Remember that Einstein's world was revolutionary -- overthrowing 230 years of Newtonian physics. The world told us what new math to use. The world is not constrained by math, nor logic, nor by any historically contingent work of fallible man.

    Math and even logic have always been drawn from looking at the world around us. So just as an aside to the main discussion, but responding to this one sentence that caught my eye ... physics IS exempt from math and logic. Meaning that historically, and metaphysically, physics is always ahead of math and logic and drives the development of math and logic.

    But to the main question, the physical/mathematical distinction is important. I can never count all the integers in the physical world (as far as we know -- to be clarified momentarily); but in math I can invoke the axiom of infinity, declare the natural numbers to be the smallest inductive set guaranteed by the axiom, and count its contents by placing it into order-bijection with itself. That is: The identity map on the natural numbers is an order-preserving bijection that shows that the natural numbers are countable.

    The former is a physical activity taking place in the world and subject to limitations of space, time, and energy. The latter is a purely abstract mental activity. How meat puppets such as ourselves come to have the ability to have such lofty abstract thoughts is a mystery. And if we are physical beings; and if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation? That's another good question.

    Perhaps our very thoughts of infinity are nature's way of manifesting infinity in the world.

    So bottom line it's clear to me that we can't count the integers physically, but we can easily count them mathematically. And the reason I say that we can't physically do infinitely many things in finite time "as far as we know," is because the history of physics shows that every few centuries or so, we get very radically new notions of how the world works. Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.

    You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so?noAxioms

    Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.

    I utterly reject the notion that supertasks are a logical contradiction or metaphysical impossibility. They're only a historically contingent impossibility. We split the atom, you know. That was regarded as a metaphysical impossibility once too.

    I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time.noAxioms

    I'm not being specific like that. I'm only saying this:

    There have been radical paradigm shifts in physics in the past;

    There will certainly be radical paradigm shifts in the future; and

    The next shift just may well incorporate some notion of physically instantiated infinities or infinitary processes; in which case actual supertasks may be on the table.


    I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist; then when shown to be logically consistent, considered only a mathematician's plaything, of no use to more practical-minded folk; and then shown to be the most suitable framework for Einstein's radical new geometry of spacetime.

    Mathematical curiosities often become physicists' tools a century or more later. I think it's perfectly possible that physically instantiated infinities may become part of mainstream physics at some point in the future.

    I will close with two contemporary examples of where speculative physics is starting to think about infinity.

    One, eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending. In this eternal multiverse are many bubble universes; either infinitely many, or at least a very large finite number. Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.

    And two, the many-world interpretation of quantum physics. Most people have heard of the Copenhagen interpretation, in which observing a thing causes the thing to be in one state or another; whereas before the measurement, it was neither in one state nor the other, but rather a superposition of the two states.

    In Everett's many-world's interpretation, an observation causes the thing to be in both states in different universes. The universe splits in two, one in which the thing is in one state, and another universe it's in the other state. In some other universe I didn't write this. I know it sounds like bullshit, but Sean Carroll, a very smart guy and a prominent Youtube physicist (he's a real physicist too) is a big believer. He's recently moved away from mainstream physics, and more into developing a new philosophy of physics that incorporates many-worlds. How many worlds are there? Again this is a little vague, infinitely many or a large finite number.

    These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists. Would anyone really bet that they personally can predict the next 200 years of physics?


    By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask.noAxioms

    Well I can walk a mile, and I first walked the first half mile, and so forth, so it's a matter of everyday observation that supertasks exist. That would be an argument for supertasks. Zeno really is a puzzler. I don't think the riddle's really been solved.

    Well that's for reading, there's been a lot of back and forth lately and I hope I was able to at least express what I think about all this.
  • Information and Randomness
    Appeal to consequenceswonderer1

    Thanks.
  • A simple question
    I agreeBenj96

    I'll quit when I'm ahead here then :-)

    A healthy society can have universal healthcareBenj96

    Many issues with long wait times at NIH in Great Britain. And in Canada, they offer assisted suicide for depression. I'd like to see some datapoints where universal health care has worked. Not an expert on health care policy, just repeating anecdotal evidence re Britain and Canada. Not necessarily defending the expensive US system, but it's a complicated issue. Just giving people free stuff is not a panacea. Who pays for the free stuff? As Margaret Thatcher once noted, "The problem with socialism is that you eventually run out of other people's money."
  • Infinite Staircase Paradox
    Which has no bearing on what I'm arguing.Michael

    You are not arguing, you're repeating your lack of argument. I'll let you have the last word, you are incapable of rational discussion.
  • Infinite Staircase Paradox
    I'm not talking about infinite sets and transfinite ordinals. I'm talking about an infinite succession of acts. If you can't understand what supertasks actually are then this discussion can't continue.Michael

    A discussion can't continue when you keep making unsubstantiated, evidence-free claims.

    I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. Yes it's speculative, but nobody is saying it's "metaphysically impossible" or "logically incoherent."

    https://en.wikipedia.org/wiki/Eternal_inflation

    Here's a definition for you: "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time".

    The key parts are "sequence of operations" and "occur sequentially".
    Michael

    Please stop embarrassing yourself.
  • Infinite Staircase Paradox
    If I write the natural numbers in ascending order, one after the other, then it is metaphysically impossible for this to complete (let alone complete in finite time). This has nothing to do with what's physically possible and everything to do with logical coherency.Michael

    It's physically impossible. I have no idea why you keep claiming it's "metaphysically" impossible or logically incoherent. What's logically incoherent about infinite sets and transfinite ordinals? You just keep repeating the same unsupportable claims. You can count the natural numbers by placing them into bijective correspondence with themselves. This is the standard meaning of counting in mathematics.
  • Infinite Staircase Paradox
    And it doesn't address the issue.Michael

    I asked you to consider a hypothetical world and you pretended I was talking about mathematical sets.


    If I write the natural numbers in ascending order, one after the other, then this can never complete.Michael

    Yes, the observable universe is finite. We're agreed on that. How many times are you going to try to convince me of something I've already agreed with many times?

    To claim that it can complete if we just write them fast enough, but also that when it does complete it did not complete with me writing some final natural number, is just nonsense,Michael

    I have not claimed otherwise.

    and so supertasks are nonsense.Michael

    According to current physics. That's as far as we can go.

    That we can sum an infinite series just does not prove supertasks.Michael

    Nor does it disprove their metaphysical possibility. We just don't know at present.
  • Infinite Staircase Paradox
    No, I'm responding to you to explain that your reference to mathematical sets and mathematical limits does not address the issue with supertasks.Michael

    I gave you a mathematical model that puts your unsupported claims into context.


    I've provided arguments, and examples such as Thomson's lamp that shows why.Michael

    Thompson's lamp shows nothing of the sort. I've explained that to you repeatedly as well.
  • Infinite Staircase Paradox
    Would you prefer the term "act"? It is metaphysically impossible for an infinite succession of acts to complete.Michael

    Metaphysically impossible? Repeating a claim ad infinitum is neither evidence nor proof.

    Have you even looked up supertasks? I don't know how you can confuse them with mathematical sets.Michael

    I'm not the one advocating for supertasks, yet you keep arguing with me that they are impossible.
  • A quest chin
    If there are an infinite number of whole numbers, and an infinite number of decimals in between any two whole numbers, and an infinite number of decimals in between any two decimals, does that mean that there are infinite infinities?an-salad

    There are indeed infinitely many infinities, but not by the argument you gave.

    And an infinite number of those infinities? And an infinite number of those infinities? And…(infinitely times. And that infinitely times. And that infinitely times. And…) …an-salad

    Yes. There are many online resources.


    https://en.wikipedia.org/wiki/Georg_Cantor

    https://en.wikipedia.org/wiki/Aleph_number

    https://en.wikipedia.org/wiki/Cardinal_number

    https://en.wikipedia.org/wiki/Ordinal_number
  • Infinite Staircase Paradox
    The task consists of a sequence of actions occurring at intervals of time that decrease by half at each step: 1/2 minute, 1/4, 1/8,.... It is logically impossible for this sequence of actions to reach the 1 minute mark (the point in time at which the descent is considered completed), it just gets increasingly close to it.Relativist

    Zeno again?

    Say (in some hypothetical world, say current math or future physics) that we have a "sequence of actions" as you say, occurring at times 1/2, 3/4, 7/8, ... seconds.

    It's perfectly clear that 1 second can elapse. What on earth is the problem?

    You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing.

    I can't imagine what you are thinking here, to claim that one second of time can't pass.

    I have repeatedly noted in this thread that we can symbolically adjoin a "point at infinity" to any countably infinite sequence, and that's where the limit lives. We can note that 1/2, 3/4, 7/8, ... has the limit 1, which lives in the ordered set {1/2, 3/4, 7/8, ..., 1}.

    We can also do the same thing in the integers as 1, 2, 3, 4, ..., , where can be thought of as a formal symbol that's greater than every natural number. It also has technical importance as the first transfinite ordinal.

    Either way, sequences do "reach" their limit via the limiting process, though the sequence itself does not necessarily attain the limit. It's just semantics.

    You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
  • Fall of Man Paradox
    Based on this picture, what I want to say is that Achilles can occupy any position on the continuous line, but, for this specific example where the ruler only has a few tick marks on it, I'm limited to describing his location using one of five specific intervals:
    (0,0)
    (0,0.5)
    (0.5,0.5)
    (0.5,1)
    (1,1)
    keystone

    Sorry what? We're doing Zeno now? I must pass on that.

    I believe what I want to do is define a 2D metric space on set S={(0,0),(0,0.5),(0.5,0.5),(0.5,1),(1,1)} where each element is an ordered pair (x1,x2).

    While I will eventually explore higher dimensional spaces, for now, let's say that my sandbox is limited to sets of ordered pairs of rational numbers.
    keystone

    I do not know what you are talking about now.

    You're right. Scratch the Universal Metric. If my metric is |x2-x1| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. This is yet another trivial conclusion since we know that rational numbers alone cannot model a continuum.keystone

    Lost me again. In a metric space the distance between two points is 0 if and only if they are the same point.

    Elements of sets are sometimes called points, but it's possible to do set theory without elements!
    — fishfry
    Is it sets all the way down or do you eventually get to points? Anyway, you don't have to answer that question. I'm willing to agree that it doesn't matter which is more fundamental. What matters is what approach yields the most powerful math. Let's move on.
    keystone

    It's sets all the way down. In set theory everything is a set.

    Points are just elements of a set. Sometimes a "point" in a function space can be a function. Sometimes a point is just a tuple of coordinates in Euclidean space. Points aren't fundamental. Perhaps you're thinking of Euclid's original formulation of geometry.

    You are trying to invent something more powerful than contemporary math?

    I don't get the top-down idea. 'Splain me please.
    — fishfry
    I was hoping to get closure on the open topics first, but if you don't have any problems with this post then I think we're there. [/quoote]

    I don't understand what you are doing. Seems like random flailing.

    keystone
    By the way, if you ever feel like my time is running out then please let me know and I'll plow through. But at the current pace I'm extracting a lot of value from our conversation.keystone

    I'm fine.

    By the way I wanted to mention that there are often two ways of describing a mathematical object, internal and external. For example we can define the real numbers internally, by building them up from the empty set to get the naturals, integers, rationals, and finally reals.

    Or, we can define the reals as the unique Dedekind-complete totally ordered set. That characterizes the reals without bothering to construct them. Perhaps you're getting at this.

    You also talked about cuts, and perhaps you're interested in Dedekind cuts, which are used to construct the reals out of sets of rationals.

    https://en.wikipedia.org/wiki/Dedekind_cut

    You seem to want to make points out of cuts in a line, but I don't see where you're going with that.
  • Fall of Man Paradox
    No, I'm only talking about topological metric spaces.keystone

    A metric space is typically just called a metric space. There aren't "nontopological" metric spaces. Any metric space can be made into a topological space by defining the open sets in terms of the metric.

    I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric),keystone

    This is kind of muddled. Typically we start with a set and put some structure on it -- a metric, a topology, whatever. It makes no sense to talk about "outside" the space till we say what set that is. For example, what's outside the real numbers. Well the complex numbers are, but so are all the animals on Old McDonald's farm. The complement of any set is the entire rest of the universe; and if you don't say what universe you're working in, you run in to the "set of all sets" paradox. The unrestricted complement of a set is not a set. So it would be good if you could clarify this point. What's outside your metric space of interest?


    and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects).keystone

    Metric spaces are indistinguishable from metric spaces, yes. But isn't that a trivial remark?

    And as I said, you will have trouble rigorously defining what you mean by outside of your metric space, unless you first say what the enclosing set is. So please do. By analogy, if you wish to discuss what's outside the real numbers, you have to say if you're talking about the complex numbers, the quaternions, or everything in the entire mathematical universe, which turns out to not be a set. Because the set of all sets that don't contain themselves is a member of the "outside" of the real numbers. Hope I'm making this clear.

    Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm referring to a non-trivial universal metric.keystone

    As it happens, the trivial topology is already defined as the opposite idea. The discrete metric has the most possible open sets. The trivial metric has the fewest open sets. Only the empty set and the entire space are open.

    https://en.wikipedia.org/wiki/Trivial_topology

    But you can't just eliminate the one metric that falsifies your idea, there could be other weird ones. You have to say exactly what you mean.

    Also I have no idea what the "universal metric" is. You have not communicated that to me.

    There's a whole SEP article on holes. Deep stuff.
    — fishfry
    Wow, it's a deeper topic than I imagined.
    keystone

    Holes are deep!

    It turns out the photos were more helpful to me than to you. You've helped me realize that what I'm actually discussing are metrics.keystone

    Ok.

    So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this.
    — fishfry
    There are two primary methods for creating core mathematical artifacts:
    keystone

    You just ignored my comment and steamrollered over it. Why do I care which is more fundamental? I don't even know what that means. Sets are fundamental, then you add properties. That's how it works.

    Bottom-up Approach:
    Starts with tiny building blocks to assemble (or at least define) more complex mathematical objects.
    Points are considered fundamental in this approach.
    keystone

    Sets are fundamental, not points. Elements of sets are sometimes called points, but it's possible to do set theory without elements! All you actually need is to describe the relationships among sets, without regard for the internal contents of the sets.


    This method is akin to assemblage art, where separate elements are combined to form a whole.

    Top-down Approach:
    Begins with a larger, unified block and divides it to produce mathematical objects.
    Continua are fundamental in this approach.
    Similar to sculpting, where material is removed from a larger mass to reveal the desired form.
    keystone

    I can't imagine how you would get anything done that way. And you are not getting me to believe you have a coherent idea about it.

    I've observed that orthodox mathematics predominantly favors the bottom-up approach.keystone

    Starting from sets, yes. Lot of mindshare the past century and a quarter. There's also type theory, which I imagine you'd see as another bottom up approach. I don't know what a top down approach to mathematical ontology would look like.

    However, my informal exploration of the top-down method has revealedkeystone

    Not to me. Maybe to you. You have not yet communicated to me what is a top-down development of math. How would you top-down construct or define the real numbers? Unless you mean axiomatically. Is that what you mean?

    a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view.keystone

    Where's the beef? That's handwavy, tells me nothing.

    I'd like to share this perspective with you,keystone

    I'd like to hear it. What is a top-down construction of the real numbers? Of the integers? Of the number 6?

    so you can either help identify any potential flaws (I don't want to waste my time on a dead end) or guide me further (for example, I've already learned from this discussion that I should be describing them as topological metric spaces rather than elastic rulers).keystone

    A metric space is a metric space. If you are interested in metric spaces there's a large literature on the subject.

    I don't get the top-down idea. 'Splain me please.
  • Infinite Staircase Paradox
    ↪jgill That's true, but that just makes it physically impossible. I think it's stronger: logically impossible.Relativist

    @Michael keeps making the same claim, and I do not understand the argument.

    I agree that it's impossible to do infinitely many physical things in finite time according to present physics.

    I do not see what the logical impossibility is.
  • SCOTUS
    Is there good reason why the Supreme Court should not have already quickly and unequivocally ruled that Trump is not above the law?Fooloso4

    Shouldn't this thread go into the Trump thread, which was specifically created so that people could vent their Trump spleen in one place?
  • Infinite Staircase Paradox
    No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them.Michael

    Ok. Clearly this is a matter of semantics.

    Mathematically, if I have a set of events , there's no problem whatsoever.

    You seem to assign some meaning to the word "event" that I don't understand. Must an event be physical? In probability theory we have events that need not be physical, such as the probability of choosing a random real number between 0 and 1/3 from the unit interval. That's an event with no physical meaning at all.

    An infinite sequence of events from the set I defined above would be . No muss no fuss. That's an infinite sequence of events.

    Perhaps you can tell me what an event is, bearing in mind that event is a technical term in probability theory that does not imply physicality.

    https://en.wikipedia.org/wiki/Event_(probability_theory)


    That's not relevant to the claim I'm making.Michael

    The claim you're making is not one I'm disputing.

    I'm saying that if I have finished counting the members of some set then some member must be the final member I counted.Michael

    I disagree. Counting means to place the elements of some set in order-bijective correspondence with the natural numbers, or in a more general context, with some ordinal.

    By that definition, we can easily count the natural numbers. The identity map will do.

    You seem to think counting is a physical process. That's fine for most contexts, but it's not the only meaning of counting.

    For example we have the famous countable/uncountable distinction between infinite sets. A set is countable if it can be placed into bijection with the natural numbers. The natural numbers, the integers, the rational numbers, and the algebraic numbers are all famous examples of countable sets that are infinite.

    If you mean to say that we can't physically count the natural numbers, of course I agree. I personally could not get past 13 or 14 or so without losing interest. We could use a supercomputer, but even that has finite capacity. We could use the entire observable universe, but that contains only atoms. So sure, physical counting is constrained by resources.

    But who's saying otherwise? Perhaps you can explain that to them, since I have never said anything remotely like that.
  • Fall of Man Paradox
    I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill, but as an engineer, my strengths lie more in visual thinking. This is particularly true with mathematics, where I sometimes struggle to express my thoughts precisely in words. Consequently, I tend to rely on illustrations to communicate my ideas. I ask for your patience and flexibility in trying to understand the essense of my message.keystone

    I just don't see where you're going with all this. You're pointing out that some topological spaces aren't metrizable. Right?

    Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"?keystone

    Oh but there is one. For any universe or set, define a metric as follows: d(x,y) = 1 if x and y are different, and 0 if x and y are the same. This is known as the discrete metric.

    You can put the discrete metric on any space of points whatsoever.

    https://en.wikipedia.org/wiki/Discrete_space

    Think of it like this: a hole is an emergent property. To have a hole, you first need an object that can contain a hole. In this sense, the object is more fundamental. We begin with the object, which holds the potential for a hole. Then, once we make a cut, what we have is the same object, but now with an actual hole in it.keystone

    There's a whole SEP article on holes. Deep stuff.

    https://plato.stanford.edu/entries/holes/

    I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points.keystone

    I don't know what's common. Does it matter?

    If you return to my photographs,keystone

    I did not understand the photos.

    you will see that I start with a continous object and put cuts in it. I call those cuts points. Just as an object is more fundamental than the hole, with my view a continua is more fundamental than the cuts (i.e. points). I used k-continua and k-points instead of continua and points because I wanted to avoid a debate over what's more fundamental. In my sandbox the continua are more fundamental. If you want to grant me that, then perhaps we can set aside all this 'k-' terminology.keystone

    Why is "what's more fundamental" important? Do you think I hold one view versus the other? What difference does it make?

    Okay, this feels like progress. Let's iron out the points discussed above and then I'll give you more details on where this is going.

    If it's not obvious, I want you to know that I really appreciate you sticking with me on this.
    keystone

    Ok I'll keep going as long as I can, but I feel like I'm going down in warm maple syrup.

    So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this. Did I argue the contrary at some point?
  • Infinite Staircase Paradox
    Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks.Michael

    Ok.

    I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility.Michael

    Now that's something I disagree with. But I don't care about supertasks much so it's better if I don't engage.

    No physical law can allow for an infinite sequence of events to be completed.Michael

    This is an open question. Of course no physical law currently known allows for supertasks, but you can't say what we will regard as physical law in another couple of centuries.

    The very concept of an infinite sequence of events being completed leads to a contradiction.Michael

    You keep repeating that, but you have no evidence or argument.

    To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense.Michael

    Do you deny infinite mathematical sets?

    If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity.Michael

    Mathematically that's not true. The set {1, 2, 3, 4, ...} contains all the natural numbers, but there's no last number.

    I already agree with you that there are no infinite collections of physical objects according to currently accepted theories of physics. But you can't claim that there will never be any such theory.

    And besides, eternal inflation posits a temporally endless universe. It's speculative, but it's part of cosmology. Serious scientists work on the idea. So at least some scientists are willing to entertain the possibility of a physically instantiated infinity.
  • Infinite Staircase Paradox
    This is an example of a supertask:

    I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.

    According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.

    Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.

    It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote.
    Michael

    You're continuing to argue against a position I don't hold. Why are you doing this? There's no interesting conversation to be had. Supertasks are not consistent with known physics. We're agreed on that.

    I would, however, disagree with you that being inconsistent with known physics is the same as logical impossibility. Known physics changes all the time, sometimes radically.
  • A simple question
    I hope you exaggerate.Ludwig V

    Not by much.
  • A simple question
    I have not seen it demonstrated that anyone demands similarity of outcomes.Vera Mont

    I can't read you the news. I don't think you and I live in the same reality if you believe what you wrote.
  • Fall of Man Paradox
    At this stage, I'm making such minor points that perhaps you are confused why it took me so many words (and pictures) to express it. If that is the case, my apologies.keystone

    I'm still concerned about that screwdriver ...

    I think what I'm trying to say is the following:

    1) Topological spaces have no sensible notion of distance.
    keystone

    Perfectly standard.

    2) Topological metric spaces have a sensible notion of distance.keystone

    By the definition of a metric space, right? Also perfectly standard.

    3) If you lived outside a topological metric space, you wouldn't be able to use it as a measuring tool on external objects (i.e. the metric qualities of the space are not applicable to objects outside of the topological metric space).keystone

    Yes ok.

    4) If you lived inside a topological metric space, you'd perceive it as a metric space,[/quoute]

    Little unclear. Who is the perceiver? How do they perceive they're in a metric space? I suppose by applying the basic definition that there exists a distance function satisfying the usual requirements. In which case an internal perceiver and an external perceiver would use exactly the same method of determining that a space is a metric space.
    keystone
    where the topological qualities aren't obvious in everyday experiences. For instance, if our world were a topological metric space and everything, including the space, ourselves and our measuring tools, suddenly grew twice as big, we wouldn’t detect the change because all our measurements would scale up too.keystone

    Yes that's true.

    5) If it is always possible for an object to exist outside of a topological metric space, it's notion of distance cannot be universally applied to all objects. I phrased this as, 'there cannot exist a Universal Elastic Ruler'.keystone

    Ok, but "universal elastic ruler?" That part I don't get.

    6) I'm constructing a topological metric space from the ground up, rather than examining one that already exists in completion. So, in my example, it's a very crude ruler and there is no mention of real numbers. Does this qualify as a topological metric space?keystone

    Is it a topological space? Is there a metric? Then yes.

    Aside from the topological discussion, I also made the following point:

    7) I'm treating continua as fundamental objects and points as emergent objects which become actualized when I make cuts.
    keystone

    Emergent objects become actualized? Bit vague for me.


    I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points.keystone

    Losing me.

    Perhaps you wouldn't characterize your viewpoint in these exact terms; you might regard points and continua as simply coexisting without one preceding the other. However, it's undeniable that the conventional approach primarily describes continua in terms of points rather than the reverse.keystone

    Ok.

    Is there disagreement or confusion on any of these points?keystone

    Not much disagreement, only confusion about where this is all going. It's perfectly clear that some topological spaces are metrizable and others aren't.
  • The hole paradox I came up with
    Read the whole thing, but it did not counter my post any. Unless of course you didn't read my whole post and just assumed I said things I didn't, misunderstanding the context, and using the strawperson argument.Echogem222

    Just posting the link since SEP has a lot to say about holes. Guilty as charged on he rest of it. Guess I'll go crawl back into my, uh, hole.
  • The hole paradox I came up with
    How can we define a hole as a type of nothing when empty space itself is considered a positive value?Echogem222

    https://plato.stanford.edu/entries/holes/
  • Fall of Man Paradox
    Are you with me? I know this seems extremely basic (and perhaps inconsequential), but I'm laying the groundwork for a more consequential idea so I hope you stick with me.keystone

    Wow. Man. I read your post. I have no idea what you're talking about. A metric space has a metric. Some topological spaces are metric spaces. But a topological space without a metric can not have a sensible notion of distance. Topological spaces by definition are stretchable. But you can't measure distance consistently in them.

    You have totally lost me. I don't know what point you are making.

    I would really appreciate that. I don't plan to have many photographs in my subsequent posts. This was just my way of laying the groundwork.keystone

    You lost me totally. I have no idea what your point is other than that you're stretching a topological space and noting that there's no sensible notion of distance.
  • A simple question
    What's that to do with equality or equity? Outcomes owe a whole lot to beginnings. It doesn't mean that everything (??) should be the same or that everyone should be the same, it means that everyone should have the same chance of a positive outcome.Vera Mont

    Some people in the public square these days would burn you at the stake for arguing for equality of opportunity versus equality of outcome. And for exactly the reason you mention, that outcomes are highly influenced by the random social circumstances of one's beginnings.

    I agree with you in principle that equality is good, but these days that's not enough for a lot of people, and you seem to be denying that's the case.

    I'm confused by your post. @Ludwig V said, "The idea that equality means that everyone is the same, or should be treated in the same way in all contexts is little more than political propaganda. No-one believes that." And I responded by noting that these days A LOT of people believe that. Then you kind of jumped in and defined equality, ignoring my point that many these days reject equality in favor of equity, which is equality of outcomes combined with grievances against racial groups they think are holding them down.
  • Infinite Staircase Paradox
    Right! It's not the sequence described in the scenario! There is a background temporal sequence, as the clock ticks, that reaches 1, but we aren't mapping the step counting to the ticks of the clock. The step-counting sequence occurs only at points of time <1. In real analysis, this is called a "right open interval" (i.e.it's open on the right= the endpoint is not included in the interval). 1 is the endpoint, but not included within this interval.Relativist

    I agree it's about a right-open interval. We have 1/2, 3/4, .. in (0,1). We can adjoin 1 to work in (0,1].

    The limit of the series is "reached" only in the sense that we can reach a mathematical answer.Relativist

    But I'm not talking about anything else! This is purely a mathematical problem! There is no lamp that switches in arbitrarily small intervals of time. Adding time to this problem confuses the issue. It makes people think there's a physical component to the problem when there isn't. It's purely mathematical.

    The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers.Relativist

    There isn't any physical process to speak of. The lamp is fictional. Purely mathematical, a function on {1/2, 3/4, ...} with its completion defined in {1/2, 3/4, ..., 1}

    Deriving the limit just means we've identified where the sequential process leads.Relativist

    It may "lead" somewhere but there's no law that constrains the final state. It may be discontinuous, like Cinderella's coach that's a coach at 1/2, 1/4, 1/8, ... seconds before midnight, then becomes a coach at midnight. That's why it's perfectly possible that the lamp becomes a pumpkin after 1 second.

    In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario?Relativist

    Lost me there, limit of what is infinity? If you put a symbolic "point at infinity" after the natural numbers and you define a function on the augmented set 1, 2, 3, 4, ..., , you can define a function on the augmented set whose value at infinity is anything you like.

    The meaning is entailed by the fact there are infinitely many natural numbers, so it means the process continues without end. It can mean nothing else.Relativist

    Kind of lost me here. The process 1, 2, 3, ... never ends, but we can still stick a symbolic point at infinity. Just like we can add the point 1 to the set {1/2, 3/4, 7/8, ...} to make {1/2, 3/4, 7/8, ..., 1}
  • Infinite Staircase Paradox
    There is a difference between saying that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 and saying that one can write out every 1/2n in order. The latter is not just a physical impossibility but a metaphysical impossibility.Michael

    Of course it's not a physical possibility.

    If by metaphysical you really mean physical, then it's not a metaphysical possibliity.

    But clearly we humans have the ability to conceptualize infinite sets and infinite processes, and we can even formalize the idea and get freshman calculus students to get a passing notion of the idea. If metaphysics includes abstract concepts created by humans, then infinite sets and mathematically infinitary processes are definitely part of metaphysics.

    But it depends on what you mean by metaphysics. There is no doubt in my mind whatsoever that infinite sets, infinite sequences, and the theory of convergent infinite series have mathematical existence. Whether you include that in your metaphysics is up to you, but the mathematical existence of convergent infinite series is beyond dispute.

    Some say that the latter is not a metaphysical impossibility because it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity, and so that this infinite sequence of events (writing out every 1/2n) can complete (and in a finite amount of time).Michael

    You are now talking about a physical process. Of course we can not write out infinitely many terms of the series. That has nothing to do with the mathematical truth expressed as 1/2 + 1/4 + ... = 1.

    Examples such as Thomson's lamp show that such supertasks entail a contradiction and so that we must reject the premise that it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity.Michael

    Nobody is writing anything down and this is not a physical process and you are entirely wasting your time trying to convince me that we can't physically write down an infinite series because I already know that.

    If you want to say that supertasks are possibleMichael

    In reality? In the physical world? No, I deny them entirely. It's tiresome to argue against your representation of positions I don't hold.

    but then have to make up some nonsense final state like "pumpkin" then I think this proves that your claim that supertasks are possible is nonsense and I have every reason to reject it.Michael

    I never claimed any such thing. I have no idea why you think I claimed any such thing. Supertasks can be defined abstractly, as in limiting processes. They are not physically instantiable as far as we currently know.

    As far as the "final state," think of it as a function on the ordered set

    {1/2, 3/4, 7/8, ..., 1}.

    We can define a function any way we like. We can assign 1 to 1/2, 0 to 3/4, 1 to 7/8, and so forth.

    We are then entirely free to define the value of the function at 1. We can call it pumpkin if we simply declare pumpkin to be an element of our output set.

    There is no requirement that the value of a function at any point is required to be any particular thing. Functions are pretty much arbitrary. Just like Cinderella's coach. A coach at 1/2 second before midnight. A coach at 1/4 second before midnight. Dot dot dot. And then a pumpkin at the stroke of midnight.

    It's a perfectly legal function. It just doesn't happen to be continuous. But it's perfectly legal to define a function that's a coach at each of infinitely many elements of a sequence, and then a pumpkin at the final limit point.

    Mathematically it's just a function

    where is just the order type of the set {1/2, 3/4, 7/8, ..., 1}.
  • Fall of Man Paradox
    Should I abbreviate my explanation, you might resort to conventional thinking to bridge the gap, which could lead to misunderstanding.keystone

    Should you abbreviate it, I might have a chance at reading it.

    My first impression -- again, forgive me, but I'm finding some virtue in just telling you the truth of my own experience of seeing this -- my first impression is the overwhelming passion that you have toward this subject; passion that is admirable in you, but makes me reluctant to even try to engage. Pictures, and rulers, and pencils ... it's a little ... off putting. That I would be engaging with someone too obsessed for their own good. I would feel that I need to tread cautiously.

    Now I do want to try to give this a fair reading. I have a few other mentions to get to tonight and I'll put this aside for later. But I must say one thing. It is impossible to prove anything mathematically using physical constructions. There's no way you and I are even having the same conversation, if you think your opening salvo should involve pencil and paper and scissors and ... is that a screwdriver? I don't have to come down to the basement to see this, do I? Why are you closing that door behind me ...? Aiiiiiyyyyy.

    Pass the popcorn, please. I am sitting in the bleachers watching with interest. :chin:jgill

    I wish I was up there with you.
  • Information and Randomness
    Ontological randomness may be logically possible but it's philosophically repugnant.Metaphysician Undercover

    Is there a name for the logical fallacy that "P is repugnant, therefore not-P." That happened with non-Euclidean geometry. A priest worked out the implications of rejecting the parallel postulate, and derived results that he regarded as geometrically repugnant. So he rejected them.

    Later mathematicians realized that although his conclusions were seemingly repugnant, they were nevertheless logically consistent; and perfectly true, in certain axiom systems.

    Looked it up. Giovanni Girolamo Saccheri. He discovered non-Euclidean geometry in 1733 but rejected it because, "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines".

    Striking that he used the same word you used, repugnant. But repugnance is not a logically argument, and what's repugnant in 1733 may turn out to be exactly what's needed to model general relativity in 1915. You never know.

    The problem being that if something is deemed as random, it is in that sense unintelligible. So if something is deemed as ontologically random, and it is considered to be unintelligible, then there is no will to attempt at figuring it out.Metaphysician Undercover

    This is the argument from despair. If the universe is random my life is meaningless so I might as well kill myself. Isn't this what the Existentialists try to figure out? The problem of living in a meaningless world.

    So one can despair of the meaninglessness; or one can go to the beach. It's a personal choice. We're alive, we might as well make the most of it. We know that life can be extinguished in an instant, so we make the most of it. Either way you look at it, it's not an argument against the randomness of the universe. It's another argument from "feelings, nothing more than feelings ..."

    Now the problem is that if something appears to be random there is no way of knowing whether it is epistemologically random, or ontologically random, because of the unintelligibility of it.Metaphysician Undercover

    I agree with this.

    So we won't know which until we figure it out, therefore we must assume it to be epistemologically random.Metaphysician Undercover

    Fair enough.

    And even if it is ontologically random, we will still never know that this is the case, so we will always have to assume that it is epistemologically random, and try to figure it out. The category of "ontological randomness" is absolutely useless.Metaphysician Undercover

    I agree. That's the existentialist solution, isn't it? I actually don't know the details of the philosophy. But if you are saying that, "We don't know what life is, but we might as well make the most of it," I perfectly well agree.

    In this sense you are putting the idea of a random universe in the same category of solipsism. You can't prove it's false, but it's pointless to believe it because it leads nowhere. Therefore we should reject it on that basis. They're both essentially nihilistic ideas.

    I think I might disagree with that. Solipsism really is nihilistic. But if the underlying physical reality is random, there's still the question of what it means to create and perceive order.

    In any event, I conclude that it's still logically possible that the true nature of the universe, if there even is such a thing, is random. And then we can still wonder ... how does all this apparent order arise from underlying randomness? So the philosophers would still have something useful to do, even in a fully random world.
  • A simple question
    The idea that equality means that everyone is the same, or should be treated in the same way in all contexts is little more than political propaganda. No-one believes that.Ludwig V

    If only. The new word is "equity" and it DOES mean that everything should be the same. Equality of outcome and not just opportunity; and if outcomes are unequal, call people racists. Tear down statues And a lot of people think that way these days. So "no one believes that" is false. Marxism is coming back into vogue, whichever side of of the matter one may happen to be on.