• Infinite Staircase Paradox
    The series of 1/2 + 1/4 + 1/8 + ... equals 1. The sequence an=1−0.5n
    ...
    converges to 1, yet 1 is not part of the sequence. As you agreed, there is no ∞-th item. Cool.
    Lionino

    I said no such thing!! If you like, you can think of the limit as being the -th item.

    That is, if 1/4, 3/4, 7/8, ... are the first, second, third, etc. terms of an infinite sequence with limit 1, then 1 may be sensibly taken as the -th item, or as I've been calling it, the item at , which is traditional in this context.

    So I believe I've been trying to get across the opposite of what you thought I said. There is an -th item, namely the limit of the sequence.

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

    The issue that I see is:
    1 – if we admit that time is infinitely divisible;
    2 – and we admit that an=1−0.5n
    [bad markup omitted]
    gives us the lenght covered by Achilles in the Zeno Walk at each step;
    the walk only finishes if it accomplishes an infinite amount of steps. Right?
    Lionino

    I think trouble ensues when you try to apply abstract math to the physical world. I certainly can walk across the room, clearly accomplishing infinitely many Zeno-steps in finite time. I have no explanation nor does anyone else. The common explanation that calculus lets us sum an infinite series, I reject. Because that's only a mathematical exercise and has no evidentiary support in known physics.

    If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1?Lionino

    No.

    If not, how is the walk ever completed?Lionino

    In math? Via the standard limiting process. In physics? I don't know, I'm not a physicist. But the physicists don't know either. They don't regard it as a meaningful question.


    ; if so, is there not a corresponding state for the mechanism when the full time elapses?Lionino

    Nobody knows the answer to any of these questions.
    In other words, by admitting that the result of an infinite series is necessarily true¹, how do you justify at the same time that the state is really undefined at 1 while also defending that Achilles can finish the run?Lionino

    The justification is purely mathematical. Physics doesn't support these notions since we can't reason below the Planck length.

    I want to emphasise that I am not arguing about the mathematics, but about the (meta)physical meaning of some mathematical concepts.Lionino

    The metaphysical meaning is perfectly clear within the math. I don't know how it works in the physical worldl

    Does that make sense?Lionino

    The math is clear. The physics is unknown. But motion is commonplace.

    1 – Is that also the case for non-standard analysis and arithmetic?Lionino

    .999... = 1 is a theorem of nonstandard analysis. I don't see how it could help. I don't know if anyone's thought about applying NSA to these puzzles.
  • Infinite Staircase Paradox
    OK, that other meaning of 'count'.

    I think we're talking past each other. When asked for the difference between a mathematical and physical supertask, you seem to focus on two different definitions of countable: The assignment of a bijection, and calling or writing down each of the numbers.
    noAxioms

    I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute?

    I'm talking about a physical supertask as described by Zeno, which arguably has countably (first definition) steps performed in finite time. Nobody is posited to vocalize the number of each step as it is performed.noAxioms

    Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.

    Bit off on the lore. It turns into a pumpkin, and at the 12th stroke, where presumably midnight is the first stroke, but I googled that and could not find an official ruling on the topic.noAxioms

    Point is you can define the state at the limit of a sequence to be anything you want. The lamp could turn into a pumpkin too. The premises of the problem don't forbid it.
    I like Bernadete's Paradox of the Gods because it doesn't make those mistakes, and thus seems very much paradoxical since motion seems prevented by a nonexistent barrier.noAxioms

    I looked it up, didn't seem to find a definitive version.

    For educational purposes concerning how infinity works, I like Littlewood-Ross Paradox because it is even more unintuitive, but actually not paradoxical at all since it doesn't break any of the above rules. It shows a linear series (effectively 9+9+9+...) being zero after the completion of every step.noAxioms

    Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not. If you number them 1, 2, 3, ... then the vase is empty at the end, since every ball eventually gets taken out. But if they're not numbered, the vase will have infinitely many balls because you're always adding another 9. Is that about right?
  • A simple question
    And because, even if he doesn't have children, he's helping to train up the well taxed caregivers and inventors of helpful products for his old age.Vera Mont

    The working class should fund the education of the cognitive elite who will vastly out-earn them in their respective lifetimes? Did I understand you correctly?

    Pretty good deal for the Eloi, but once in a while they get eaten by the Morlocks.
  • Infinite Staircase Paradox
    I think you are both mistaken to rely on physics to define what one wants to get at in this context. Physics is not only limited by the current state of knowledge, but also by its exclusion of much that one would normally take to be both physical and real. Somewhere near the heart of this is that there is no clear concept that will catch what we might mean by "whatever exists that is not mathematics" or by "whatever applied mathematics is applied to".Ludwig V

    Which includes magic lightbulbs and staircases? I'm open minded, I don't think I can predict the future. Even a few hundred years ago nobody could imagine the science and technology of today.

    I'm sorry. I didn't mean to gross you out. Perhaps if you think of death as a least upper bound, you'll be able to think of it differently. It is, after all, an everyday and commonplace event - even if, in polite society, we don't like to mention it.Ludwig V

    Yes ok
    Yes. I was just drawing out the implications. You might disagree.Ludwig V

    Not too strenuously. As I mentioned I don't place as much metaphysical import on these puzzles.
    Yes. In the context of the Achilles problem that's fine and I understand that you are treating that and the natural numbers as parallel.Ludwig V

    Have mostly been talking about the lightbulb. Haven't talked about Achilles or Zeno.

    But it's not the natural numbers that are parallel. It's the natural numbers augmented by the point at infinity. That's my conceptual setting for these problems.


    It's not clear to me that it really works. It makes sense to say that "1" limits "1/2, 1/4, ..." But I'm not at all sure that it makes sense to say that <omega> limits the sequence of natural numbers. "+1" adds to the previous value. "<divide by 2>" reduces from the previous value. The parallel is not complete. There are differences as well as similarities.Ludwig V

    There's an order-isomorphism between the ordered sets {1/2, 3/4, 7/8, ..., 1} and {1, 2, 3, ..., . One of the virtues of abstraction is that it lets us see that two seemingly different things are really the same, when we only focus on certain attributes. Both these sets are an infinite sequence followed by an extra element. Their order properties are the same. It's no different than playing chess on a board with purple and green squares versus red and black squares. It's the same game with respect to the rules of the game, even though they're different in other respects.

    How can it be out of reach? I went to the supermarket today. I walked from one end of the aisle to the other. I reached the end. I did indeed evidently sum a convergent infinite series.
    — fishfry
    Did you "get to the limit by successors" or "get there by a limiting process"? I don't think so. You are just not applying that frame to your trip.
    Ludwig V

    But if I did apply that frame, then Zeno would have a good point. I did somehow either 1) accomplish infinitely many tasks in finite time; or b) The world's not continuous like the real numbers.

    I think Zeno had a very good point, and I don't accept the common wisdom that summing an infinite series solves the problem.
    I've met other mathematicians who agree that Achilles is not interesting. But I'm fascinated that you think the arrow is interesting. I don't. Starting is a boundary condition and so not part of the temporal sequence, any more than the boundary of my garden is a patch of land. End of problem.[or/quote]

    If time is made of instants, then from instant to instant, how do things know what to do next? Where is the momentum and velocity information stores? It's like a computer program where an object has associated with it a data structure containing information about the object. If I shoot an arrow, where is the arrow's data structure stored? I think it's a good question. But I've never really given a lot of thought to the matter. It all seems to work out.
    Ludwig V
    But this may be interesting in the context of what we are talking about. A geometrical point does not occupy any space. It is dimensionless. One could say it is infinitely small. But it is obvious that there is no problem about passing an infinite number of them. It is a question of how you think about them. This is not quite the same as Zeno's problem, but it is close.Ludwig V

    That's a good question too. How do dimensionless points form lines and planes and solids?

    That is a perfectly sensible answer to the question, "What is the state at the limit?" It's perfectly sensible because the conditions of the problem don't specify the value at the limit. And since the lamp is not physical, it can turn into anything we like at the limit. It's no different than Cinderella's coach, which is a coach at 1/2 second before midnight, 1/4 second before midnight, and so on, and turns into a coach at midnight.
    — fishfry
    I agree with that.
    Ludwig V

    If you agree, I'm happy, because that's the only point I'm making. I've never written more just say less. My only point is that in the lamp and these other problems, we're not defining the state at the limit. Therefore the choice of state is pretty much arbitrary. If we thought about it that way it might be more clear.


    Perhaps then, these problems are not mathematical and not physical, but imaginary - a thought experiment. (The Cinderella example shows that we can easily imagine physically impossible events) That suggests what you seem to be saying - that there are no rules. (Which is why I posited another infinite staircase going up). But if there are no rules, what is the experiment meant to show?Ludwig V

    Aha. You'd have to ask those who care so much. I think they only show that underspecified problems can have arbitrary answers. But others see deeper meanings.

    The only restriction I can think of is that it needs to be logically self-consistent - and the infinite staircase is certainly that. I guess the weak spot in the supertask is the application of a time limit.Ludwig V

    If I can walk from one end of the grocery aisle to the other, I don't see why you can't get down the staircase, infinitely many steps or not.

    However, I also want to say that I cannot imagine an endless staircase, only one that has not ended yet - once I've imagined that, I can wave my hand and say, that is actually an infinite staircase.Ludwig V

    It's a thought experiment. There are no infinite staircases.

    But I did walk through infinitely many inverse powers of two lengths at the grocery store. I did sum an infinite series in finite time. So there's something interesting going on.
  • A simple question
    I'm sorry about this rant, but I don't know how else to respond.

    It depends on your philosophy of education. The thinking behind all education is a mess; the thinking behind higher education is even more of a mess; and the thinking about adult education is practically non-existent. You can think about in terms of vocational (career) benefits and non-vocational ("for fun") programmes and a combination of private benefits (for the student) and public benefits (for society in general). There's also an issue about benefits to employers, but these are rarely thought about in their own right.
    Ludwig V

    I appreciate your heartfelt thoughts about education, but the subject is the morality and economics of student loan "forgiveness," which is a gaslighting euphemism for passing the costs on to the taxpayers.

    If I, as a competent adult, borrow $100k and sign a legally binding contract to pay it back; and then the government declares that YOU should pay it back instead; I take it you would object. One, it's not your debt; and two, it creates a moral hazard. Why shouldn't I go out and borrow another $100k in the expectation that the government will favor me again?

    Clearly the morality and economics are no different if the cost of covering my $100k debt "forgiveness" is spread out among a few tens of millions of taxpayers, instead of you personally.

    In the case of Biden's recent loan forgiveness, it also happens to be illegal. Only Congress can profligately waste taxpayer money. The House of Representatives has the "power of the purse." The Supreme Court ruled that Biden's earlier loan "forgiveness" program was illegal. He's just flouting the law because nobody will call him on it.
    Underwater basket weaving looks like a bad career choice, but possibly a good choice for fun. Either way, the student should pay.Ludwig V

    We have no disagreement then. People should pay off their own loans that they knowingly agreed to pay.

    Some programmes, like IT skills (and mathematical ones) lead to extremely profitable careers in the finance industry; again, the student should pay. But if there's a serious shortage of welders, such that various industries cannot find the workers they need, there's good reason why employers, and/or the state, might want to pay.Ludwig V

    That's exactly the point. Welders don't have student loans. If you want to have a program where everyone who graduates from high school gets a couple hundred thousand in cash, to go to college or start a welding business, then let Congress pass such a law.

    Then there are programmes like social work and nursing, which require specialized professional training, but don't pay well. Isn't there a good case for state support? What abaout high-level professional careers which could be financed by students, but where that is impractical because of their high costs whether in infrastructure or time required; again, public subsidy makes sense. Another category is risky careers, like acting or archaeology or philosophy; again, there's a case for public subsidy, not only to ensure a supply for the labour market, but because the existence of those careers is a public good.Ludwig V

    If state support is a social good, let Congress pass a law. There is no law authorizing Biden to "forgive," always in quotes, student loans, and pass the costs onto the taxpayers without Congress having authorized such an expenditure.

    If you think the government should fund college and trade school for everyone, that would be a reasonable question. But that's not the loan bailouts we're talking about here. This is a transfer of wealth to the students camping out on university lawns this week, coming from the workers who had to pick up the trash after the police raids. All the more reason the students at "elite" colleges should not have their debts paid by the working class.

    If you thought that was a mess, consider the non-vocational subjects, or those subjects which can be studied for vocational reasons and can also be studied for fun. The catch here is that all the specific vocational careers presuppose some level of basic, general skills and knowledge, which enables people to function in society in general, both within and without their vocations; these skills are also the basis of good citizenship. These include reading, writing, and arithmetic, but also extend (In the UK) to Science, Technology, Engineering and Mathematics (practice needs theory, after all) and to various skills under the heading of good citizenship - philosophy, literature and history and the arts. Those last four are often regarded as purely for fun, so I don't claim that the idea that they are not just for fun is uncontentious. Perhaps the most effective argument for them is that democracy cannot function properly without them. J.S. Mill recognized this, but it seems now to be ignored, which is a pity. Mind you, the idea that an understanding of the humanities was essential for a decent society took a very serious knock in WW2. But it is far from dead.Ludwig V

    I don't disagree with anything you say. The question isn't the virtue of the liberal arts. The question is Biden's bailouts of college students at the expense of everyone else.

    Underwater basket weaving? Probably not. Philosophy? Fine Art? There's at least a case to think about, isn't there?Ludwig V

    Think about having the taxpayers assume the legal obligations of privileged university students? No, I don't think so. Forgive me, am I mistaken that we were talking about Biden's student loan bailouts as opposed to the general virtues of education? We're not having the same conversation any more.

    PS. I forgot to explain how students should pay when they need to. Through the tax system. If their career choice pays off, they will pay increased taxes, so the public purse will benefit and their debt "repaid" - or, if you prefer, the public investment in their career pays off. Where their career does not pay off in that way, the public (and employers) will benefit from an increased supply of highly qualified labour. Where their career is not directly developed by their qualification, it will have been helped by the "transferable skills" developed in their programme and by the improved contribution they can make by their contribution to social and political life.
    In other words, payment through the tax system is perfectly well justified by the multiple benefits provided by higher education. Nobody has a problem with that way of paying for schools. Why would higher education be any different?
    Ludwig V

    I haven't given these matters enough thought to converse sensibly about how post-secondary education should be paid for. I apologize if I've given that impression somehow. I have only been talking about Biden's transfer of billions in student debt to the taxpayers, most of whom didn't go to college. And this debt transfer, which is illegal by US law since only Congress can spend money, is happening not for any lofty goal of a more educated citizenry; but as a direct bribe to students to vote for Biden in an election year.

    That process - what was liberal and new, becomes old hat, and conservative. That what's happened to feminism, etc. The agenda has moved on. It's very disappointing to those of us who thought the problems were solved. But there are unsolved and unconsidered issues and big gaps in even the basic rights that one thought had been established.Ludwig V

    Yes, a lot of positions have changed in the past couple of decades.

    There's no rehabilitation going on. There's a revolving door of people committing violent crimes, being put back on the street, and re-offending.
    — fishfry
    If that's so, there is a problem.
    Ludwig V

    I follow New York City news. It's a revolving door. It's actually a bad situation.
  • Infinite Staircase Paradox
    I dealt with this already. If you restrict the meaning of "physical" to that which abides by the law of physics, then every aspect of what we would call "the physical world" which violates the laws of physics, dark energy, dark matter, for example, and freely willed acts of human beings, would not be a part of the "physical" world.Metaphysician Undercover

    Don't be silly. The rotational rate of galaxies is physical, even if our current theory of gravity doesn't explain it. We don't say it's not physical just because we don't have a theory of dark matter or modified gravity yet.

    That's not true at all. It does not correctly represent how we use the word "physical". "Physical" has the wider application than "physics". We use "physical" to refer to all bodily things, and "physics" is the term used to refer to the field of study which takes these bodily things as its subject. Therefore the extent to which physical things "obey the known laws of physics" is dependent on the extent of human knowledge. If the knowledge of physics is incomplete, imperfect, or fallible in anyway, then there will be things which do not obey the laws of physics. Your claim "a physical thing must obey the known laws of physics" implies that the known laws of physics represents all possible movements of things. Even if you are determinist and do not agree with free will causation, quantum mechanics clearly demonstrates that your statement is false.Metaphysician Undercover

    I think I can't play these word games. If you want to pretend not to know what a physical thing is, I can't argue with you. Bowling balls falling down was a physical phenomenon two thousand years ago even if Aristotle's physics didn't explain it sufficiently.

    I gave you an example. A human body moving by freely willed acts violates Newton's first law.Metaphysician Undercover

    How can a human body move by free will? You're the determinist here. You reject randomness. How does this "will" influence the body? Good questions. I don't know. You don't either.

    "Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia."Metaphysician Undercover

    Sorry what? I didn't say that, is that someone else's quote?

    There is no such "external force" which causes the freely willed movements of the human body. We might create the illusion that the violation can be avoided by saying that the immaterial soul acts as the "force" which moves that body, but then we have an even bigger problem to account for the reality of that assumed force, which is an "internal force". Therefore Newton's first law has no provision for internal forces, and anytime such forces act on bodies, there is a violation of Newton's laws.Metaphysician Undercover

    Can you remind me what is the point of all this? I haven't the heart to discuss Newtonian metaphysics.

    That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature.
    — fishfry

    If you understand this, then you ought to understand that being physical in no way means that the thing which is physical must obey the laws of physics.
    Metaphysician Undercover

    Fine. The speed of the rotating galaxies is physical but we don't have a law of physics that explains it yet.

    Can you remind me why we're having this discussion? I think if you wrote less I could respond more deeply. This flood of deepity is a bit much for me.


    It is not the case that we only call a thing "physical" if it obeys the laws of physics, the inverse is the case. We label things as "physical" then we apply physics, and attempt to produce the laws which describe the motions of those things. Physical things only obey the laws of physics to the extent that the laws of physics have been perfected.Metaphysician Undercover

    I think if you wrote a shorter post I'd engage. Do you think bowling balls falling down was not physical before Newton? Before Einstein? Before the next genius not yet born?

    By your reasoning, nothing at all is physical, since all physical theories are only approximate.

    Ok, now we're getting somewhere. The point, in relation to the "paradox" of the thread is as follows. There are two incompatible scenarios referenced in the op. Icarus descending the stairs must pass an infinite number of steps at an ever increasing velocity because each step represents an increment of time which we allow the continuum to be divided into. In the described scenario, 60 seconds of time will not pass, because Icarus will always have more steps to cover first, due to the fact that our basic axioms of time allow for this infinite divisibility. The contrary, and incompatible scenario is that 60 seconds passes. This claim is supported by our empirical evidence, experience, observation, and our general knowledge of the way that time passes in the world.Metaphysician Undercover

    Ok. I haven't engaged with the staircase at all. Can't argue it. But if 60 seconds of time can't pass, how did I walk from the living room to the kitchen for a snack?

    What I believe, is that the first step to understanding this sort of paradox is to see that these two are truly incompatible, instead of attempting to establish some sort of bridge between them. The bridging of the incompatibility only obscures the problem and doesn't allow us to analyze it properly. Michael takes this first step with a similar example of the counter ↪Michael, but I think he also jumps too far ahead with his conclusion that there must be restrictions to the divisibility of time. I say he "jumps to a conclusion", because he automatically assumes that the empirical representation, the conventional way of measuring time with clocks and imposed units is correct, and so he dismisses, based on what I call a prejudice, the infinite divisibility of time in Icarus' steps, and the counter example.Metaphysician Undercover

    I can't argue with you about your analysis of what someone else said. Nor can I argue about the staircase. I haven't really said much in this thread about the staircase. The lamp is much more clear to me.

    I insist that we cannot make that "jump to a conclusion".Metaphysician Undercover

    I made it to the kitchen and back. How do you account for that?

    We need to analyze both of the two incompatible representations separately and determine the faults which would allow us to prove one, or both, to be incorrect. So, as I've argued above, we cannot simply assume that the way of empirical science is the correct way because empirical science is known to be fallible. And, if we look at the conventional way of measuring time, we see that all the units are fundamentally arbitrary. They are based in repetitive motions without distinct points of separation, and the points of division are arbitrarily assigned. That we can proceed to any level, long or short, with these arbitrary divisions actually supports the idea of infinite divisibility. Nevertheless, we also observe that time keeps rolling along, despite our arbitrary divisions of it into arbitrary units. This aspect, "that time keeps rolling along", is what forces us to reject the infinite divisibility signified by Icarus' stairway to hell, and conclude as Michael did, that there must be limitations to the divisibility of time.Metaphysician Undercover

    I actually agree with you that the mathematical real numbers may well not be an accurate representation of the ultimate nature of reality. If that's what you mean about infinite divisibility.

    Now the issue is difficult because we do not find naturally existing points of divisibility within the passage of time, and all empirical evidence points to a continuum, and the continuum is understood to be infinitely divisible. So the other option, that of empirical science is also incorrect. Both of the incompatible ways of representing time are incorrect. What is evident therefore, is that time is not a true continuum, in the sense of infinitely divisible, and it must have true, or real limitations to its divisibility. This implies real points within the passage of time, which restrict the way that it ought to be divided. The conventional way of representing time does not provide any real points of divisibility.Metaphysician Undercover

    If you're arguing for a discrete universe, maybe so.

    "Real divisibility" is not well treated by mathematicians.Metaphysician Undercover

    It's profoundly and beautifully and logically rigorously treated by mathematicians. I can't imagine what you mean here. You're flat out wrong.

    We may not know the ultimate nature of the world, but the mathematical real numbers are treated very well indeed.


    The general overarching principle in math, is that any number may be divided in any way, infinite divisibility.Metaphysician Undercover

    Arbitrary, not infinity. I can divide 1 into 1/2, into 1/4, into 1/8. I can divide any finite number of times, and there are infinitely many numbers. but I can't "divide infinitely." That's imprecise and essentially wrong.

    However, in the reality of the physical universe we see that any time we attempt to divide something there is real limitations which restrict the way that the thing may be divided. Furthermore, different types of things are limited in different ways. This implies that different rules of division must be applied to different types of things, which further implies that mathematics requires a multitude of different rules of division to properly correspond with the divisibility of the physical world.Metaphysician Undercover

    Not at all. That's the physicists' job. Mathematicians need not concern themselves with the physical world at all.

    But mathematicians do have "different rules of division." The rules of division in the integers are very different than the rules of division for the real numbers.

    Without the appropriate rules of divisibility, perfection in the laws of physics is impossible, and things such as "internal forces" will always be violating the laws of physics.Metaphysician Undercover

    There can never be perfection in any physical law. You know that. You lost me with internal forces.

    The Planck limitations are just as arbitrary as the rest, being based in other arbitrary divisions and limitations such as the speed of light. The Planck units are not derived from any real points of divisibility in time.Metaphysician Undercover

    The math is pretty solid, it's based on Fourier series as I understand it. I think you're a little out over your skis here. But your complaints are about physics. I'm not qualified to help.

    No, the point of the puzzle is to demonstrate that the sum is always less than one, and that the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum.Metaphysician Undercover

    You're just typing stuff in. What you wrote isn't true. "the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum." Not even wrong.

    The assumption that the sum is equivalent to one is what creates the paradox.Metaphysician Undercover

    The math is beyond dispute.

    the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.
    — fishfry

    I hope you're joking,
    Metaphysician Undercover

    I've never meant anything more seriously. It was more than 200 years of intellectual struggle from Newton and Leibniz to Weierstrass, Cauchy, Cantor, and Zermelo.

    but based on our previous discussions, I think you truly believe this. What a strangely sheltered world you must live in, under your idealistic umbrella.Metaphysician Undercover

    Based on our previous discussions, you're still an ignorant troll. I'm done here. What is wrong with you?
  • A simple question
    That was attempted back in 2016 with the whole "DESTROYS sjw with facts and logic", it only got worse. Some of the people on that "side" are victims, I imagine they don't even have an inner monologue so they can't even filter what information is fed to them.
    Regardless of whether they even know what they are saying, the time to be subtle with people who want you gone and your culture burned was long ago, nobody cares about being called racist/sexist/theosophist any longer. There is no god anymore, everything goes.
    Lionino

    Yeah I care. I spend most of my time on this forum in mathematically-oriented threads. I'm dipping my toes in the political waters over here and treading lightly. So far, anyway.
  • Infinite Staircase Paradox
    However, how do you arrive at that conclusion?Lionino

    By the conditions of the problem. Is this about the lamp? The problem says it's on at 1/2, on at 3/4, off at 7/8, etc.. The problem itself doesn't define the state at 1. So I'm free to define the state at 1 any way I like. Because the problem itself leaves that information unspecfified.

    The two options that I can think of is by admitting that the sum of an infinite series is an approximation instead of the exact value,Lionino

    No. The mathematics is pristine. 1/2 + 1/4 + 1/8 + ... = 1 in the same sense that 1 + 1 = 2. Two names for the same thing. May be used interchangeably. Exactly equal. Denote exactly the same real number.

    or by casting some doubt on the idea of an ∞-th item of a series.Lionino

    There is no ∞-th item of a series. There is the limit of a series (or sequence, I think you may mean here). That's not the same thing. In the sequence 1/2, 3/4, 7/8, 15/16, ... there is no last element. The sequence has a limit of 1. But 1 is the limit, it's not a member of the sequence.

    The latter seems to cause more problems than solve them for me. Did you use a different reasoning?Lionino

    Not sure what you mean. Reasoning in terms of what? The final lamp state is not defined. I can arbitrarily define any function at a point where it's not defined, especially when there's no natural reason (such as continuity) to prefer one limit state over another.

    Am I understanding your question correctly? I didn't quite understand what you mean by asking if I used different reasoning.

    It's like one of those "fill in the missing number" puzzles, like 1, 2, 8, 16. They want you to say 32. But mathematically, you can put in anything you want. If you don't tell me the lamp state at the limit, I can define it any way I want, especially since there's no way to define it in such a way that the sequence attains its limiting value in a continuous manner.
  • Infinite Staircase Paradox
    OK. I remembered WIttgenstein's oracular remark that death is not a part of life. My concern that the limit is not generated by the defining formula isn't the problem I thought it might be.Ludwig V

    Jeez that's kind of creepy ... true, I suppose. Death is the least upper bound of the open set of life.


    I don't really believe in "possible" without qualification. There's logically possible, (is mathematically possible the same or something different? Does is apply here?), physically possible, and a range of others, such as legally possible. So what kind of possibility is a supertask?
    Ludwig V

    In the future, if physics ever figures out how to work with physically instantiated infinities, supertasks might be possible. Way too soon to know.

    So your reply is that it is neither. It suggests a combination of physical and mathematical rules which is incoherent but generates an illusion.Ludwig V

    I just mentioned that I could argue it either way.

    But then you say
    On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot
    Obviously, as each stage gets smaller, I will complete it more quickly. But still, it will take some period of time, and the final step looks out of reach. That looks like a combination of physical and mathematical rules.
    Ludwig V

    How can it be out of reach? I went to the supermarket today. I walked from one end of the aisle to the other. I reached the end. I did indeed evidently sum a convergent infinite series. Except for the fact that nobody knows if spacetime below the Planck length is accurately modeled by the mathematical real numbers. Maybe it's not. We just don't know. In any event, I don't know.

    It isn't a real problem because I can analyze the task in a different way. I can complete the first yard, the second yard.... When I have completed 1760 yards, I have completed the task. But the supertasks seem not to permit that kind of analysis. Is that the issue?Ludwig V

    I don't think it's much of a problem. It doesn't keep me up at night. I just saw a Youtube video of an interview of Graham Priest, a famous philosopher. He thinks Zeno and other paradoxes are important. a lot of people think they're important.

    I think some of Zeno's other paradoxes are more interesting. When you shoot an arrow, it's motionless in an instant. How does it know where to go next, and at what speed? I think that's a more interesting puzzle. Where are velocity and momentum "recorded?" How does the arrow know what to do next?
  • Infinite Staircase Paradox
    To count a set means to place it into bijection with:
    — fishfry
    OK, that meaning of 'count'. In that case, I don't see how mathematical counting differs from physical counting. That bijection can be done in either case. In the case with the tortoise, for any physical moment in time, the step number of that moment can be known.
    noAxioms

    If I stand in a parking lot and call out "one, two, three, ..." and keep going, I can never count all the natural numbers.

    In math, I can say, Let {1, 2, 3, ...} be the set of natural numbers whose existence is guaranteed by the axiom of infinity. Now I've counted the natural numbers.

    I see a difference in those scenarios. I suppose someone could say that my conceptualization of {1, 2, 3, ...} is a physical process in my brain. Is that what yu mean that there's no difference between mathematical and physical counting?

    I am saying that Zeno describes a physical supertask, that Achilles must first go to where the tortoise was before beginning to travel to where the tortoise is at the end of that prior step.
    Zeno goes on to beg the impossibility of the task he's just described, so yes, he ends up with a contradiction, but not a paradox.
    noAxioms

    I'm probably not in a position to differ. Clearly we can walk from one place to another. Maybe that's a supertask. I don't know.

    I also would hate to have to talk about the poor kilometerage that Bob's truck gets.noAxioms

    Do British people talk about kilometerage? I've actually never heard that usage but I suppose it makse sense.

    It [the even-oddness of ω]is neither, and who's asking such a thing?
    — fishfry
    The lamp scenario asks it, which is why the comment was relevant.
    noAxioms

    It's not a physical lamp, since no physical circuit could switch that fast. Therefore it's an imaginary lamp. Its state is defined at each of the times 1/2, 3/4, 7/8, ... But its state is not defined at 1. Therefore we may define its state as 1 as becoming a plate of spaghetti.

    It's just like Cinderella's coach. It's a coach at midnight minus 1/2, midnight minus 1/4, etc. At at exactly midnight, it turns into a coach.

    The Planck-scale defying lamp circuit is every bit as fictional as Cinderella's coach. Since the state at 1 is not defined, I'm free to define it as a plate of spaghetti. That's the solution to the lamp problem.

    The reason it's as sensible as any other solution is that there is no final state that makes the lamp function continuous.

    If, for example, the lamp is on at 1, on at 3/4, on at 7/8, and so forth, then we could still define the state at 1 as a plate of spaghetti; but if we defined the lamp to be on at 1, that would have the virtue of making the lamp function continuous. Continuous functions are to be preferred.

    But no possible value for the final state of the lamp makes the problem continuous. Therefore any old arbitrarily solution will do just as well as any other.

    As far as I'm concerned, that solves the problem. Until, of course, some future genius not yet born figures out how to implement a switching circuit that makes the lamp physical.
  • Infinite Staircase Paradox
    See here:

    As Salmon (1998) has pointed out, much of the mystery of Zeno’s walk is dissolved given the modern definition of a limit. This provides a precise sense in which the following sum converges:

    Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers.
    Michael

    I am not sure why you think it's necessary to point that out to me.

    A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series.Michael

    I took the class. I read a couple of different books, not that particular one. But I'm not sure why you are taking the time to explain to me the basics of convergent infinite series in real analysis.

    From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity.Michael

    Well no, not really, because a convergent infinite series does not have a temporal component. There's no notion of "add the next thing then add the next thing ..." Rather, the sum of the series is 1 in a single moment if you will. Rather, there are no moments at all. 1/2 + 1/4 + ... = 1 in the same sense that 1 + 1 = 2. They are two expressions that mean the same thing.

    We can contrast convergent infinite series with loops in a programming language. Loops are executed in time, consume energy, and produce heat in the computational substrate. Mathematical series don't do any of that. Mathematical operations happen atemporally. They are, they don't do.

    When you try to put it into a physical context, that's where the confusion comes in.

    ]Suppose we switch off a lamp. After 1 minute we switch it on. After ½ a minute more we switch it off again, ¼ on, ⅛ off, and so on. Summing each of these times gives rise to an infinite geometric series that converges to 2 minutes, after which time the entire supertask has been completed.Michael

    No physical lamp can switch that fast, so there's nothing physical about this thought experiment.

    Consider a function f defined on the ordered set {1/2, 3/4, 7/8, ..., 1}. This is a perfectly well defined set of rational numbers. Suppose f(1/2) = 0, f(3/4) = 1, f(7/8) = 0, and so forth; and f(1) = a plate of spaghetti.

    That is a perfectly sensible answer to the question, "What is the state at the limit?" It's perfectly sensible because the conditions of the problem don't specify the value at the limit. And since the lamp is not physical, it can turn into anything we like at the limit. It's no different than Cinderella's coach, which is a coach at 1/2 second before midnight, 1/4 second before midnight, and so on, and turns into a coach at midnight.

    That is literally the answer to the Thompson's lamp puzzle.

    I have been arguing that it is a non sequitur to argue that because the sum of an infinite series can be finite then supertasks are metaphysically possible.Michael

    You have not so argued. You have so claimed. You have not provided any proof or even evidence that a supertask is "metaphysically impossible." Maybe it is, maybe it isn't. The Planck scale and the physics of switching circuits preclude the existence of the lamp, according to current physics. A century or two from now, who can say? In particular, how can you personally know that future physics won't let us peer below the Planck scale?

    The lack of a final or a first task entails that supertasks are metaphysically impossible.Michael

    I just showed you the final state. A supertask is a function on the set {1/2, 3/4, 7/8, ..., 1}. The final state of Cinderella's coach is pumpkin. The final state of Thompson's lamp is plate of spaghetti.

    I do not see the problem. Neither the lamp nor the coach are physical entities. This is purely an abstract thought experiment, and my solution is mathematically sound.


    I think this is obviousMichael

    That's not a proof. If you had a proof you'd give it, instead of simply claiming how obvious it all is to you that something is "metaphysically impossible." How do you know what's metaphysically possible? None of us are given to know the ultimate nature of the world. Not currently, anyway, and maybe never.

    if we consider the supertask of having counted down from infinity, and this is true of having counted up to infinity as well.Michael

    Bit of a subject change, not sure where you're going with this.

    We can also consider a regressive version of Thomson's lamp; the lamp was off after 2 minutes, on after 1 minute, off after 30 seconds, on after 15 seconds, etc. We can sum such an infinite series, but such a supertask is metaphysically impossible to even start.Michael

    I think you must have your own private meaning for "metaphysically impossible," because after all our conversation, you have not convinced me of your point. I think literal, physically instantiated supertasks may or may not turn out to be possible. I base my opinion on the long history of revolutions in scientific understanding. The earth turned out to revolve around the sun. We split the atom. We evolved from more primitive animals. Many things formerly thought to be "metaphysically impossible" turned out to be not only possible, but true.

    What makes you think you can see the indefinitely far away scientific future?
  • A simple question
    Not so. I don't know what data is available to you, but perhaps you should look around. All I'm saying is that you cannot assume that every vocational programme provides marketable qualifications nor that every non-vocational programme does not. It's up to the market to decide what it wants.
    Equally, it is up to students to decide what they want, even if they make choices that you think are unwise. It's not as if we can predict and provide what the economy wants.
    Ludwig V

    Perfectly correct. Then why should the taxpayers shoulder the burden of those who make bad choices? Doesn't that create what they call a moral hazard?

    "In economics, a moral hazard is a situation where an economic actor has an incentive to increase its exposure to risk because it does not bear the full costs of that risk."

    Why shouldn't I take out $100,000 US in loans to study underwater basket weaving, if I'm reasonably sure some future administration is going to transfer my loans to the taxpayers? "Cancelling" student loans, a deliberately misleading euphemism for transferring the debt to the taxpayers, incentivizes more bad choices.

    Point being that pipefitters shouldn't be shouldering the cost of the loans forgiven for social justice majors.
    — fishfry
    Well, if the cost is funded by general taxation, the contribution will depend on their income. That doesn't seem unreasonable - unless you think that people should not study social justice. But I think it is a good thing that as many people as possible should understand what social justice is.
    Ludwig V

    I ask again. Why should a pipefitter pay off someone else's student loans? And if someone borrows large amounts of money to major in a financially unrewarding pursuit, why shouldn't they bear the burden of their own choice?

    It's complicated. In the UK, liberals in the 19th century were, by and large, members of the elite. They were never particularly enthusiastic about supporting the working classes. They were much more interested in free trade, political issues like voting rights and moral/social issues like divorce, gay rights &c. (Conservatives supported protection and social conservatism). The working classes, by and large, had to fight their own battles, which they did through the Trade Unions.
    But I'm sure the alignments were different in the USA.
    Ludwig V

    I think it's the same. "Classical" liberalism, which is more like conservatism today. Although conservatism is pretty muddled, what are they really for? Hard to tell these days.

    Violent criminals are being put back on the street to re-offend. That's not fair to the victims. Violent criminals belong behind bars.
    — fishfry
    People can't re-offend if they're locked up.
    — fishfry
    You can't imprison violent criminals forever, unless you can prove them criminally insane. Sooner or later, they have to hit the streets again. That's why rehabilitation is so important.
    Ludwig V

    You can put them away for a little bitty while, can't you? In major US cities, soft-on-crime DA's won't even do that. And their victims are starting to notice.

    Perhaps I just spend to much time following NYC politics. They're having a problem with soft-on-crime politicians leading to a great decrease in public safety.
    — fishfry
    Nothing wrong with that, so long as you are open to new ideas occasionally.
    Ludwig V

    I'm very open to new ideas. I used to be way more of a liberal. I still am. It's the liberals who have gone way too far recently.

    I don't know the details, but my instinct is to suggest that if the rehabilitation programmes in NYC aren't working, find a better programme, don't give up on the attempt. Money spent on effective programmes to keep people out of prison is a good investment. Back that up by improving detection and arrest, which is by far the most effective deterrent. Tossing people out of prison into the general population will not work and putting them back in prison later on is very expensive, not only in running the prisons, but also in the damage inflicted on families and children.Ludwig V

    There's no rehabilitation going on. There's a revolving door of people committing violent crimes, being put back on the street, and re-offending.
  • Fall of Man Paradox
    I appreciate you asking a specific question about my explanation instead of dismissing it outright. I believe this has helped us move forward.keystone

    I always engage directly with anything I understand. I don't dismiss the rest, I clearly say I don't understand. This has been true all along.

    What does (.5, .5) represent?
    — fishfry
    Yes, it represents the point we would conventionally label 0.5.
    keystone

    And why do you do that, he asked ...

    Step one involves defining the journey through the use of intervals.keystone

    Journey through the use of intervals, I don't understand. Not a dismissal. Direct statement that you said something I can't understand. Can't relate it to anything in my experience or knowledge.

    Step two entails describing these intervals within the framework of a topological metric space.keystone

    It's just a metric space. It's like saying that I petted my cat mammal.

    Secondly, "describing these intervals within" etc? First, they're not intervals. The notation (0,0) denotes an empty interval. So you are not communicating.

    To successfully carry out step two, it's crucial that all elements involved are of the same type. For instance, I assume that defining a metric on a set that includes both points and intervals is not straightforward.keystone

    You haven't defined any intervals, you have (0,0) and (.5, .5) and say these are intervals. But as intervals, they are both empty. They denote the empty set. You haven't got any intervals.

    Secondly, to define a metric, you need a distance function that satisfies some properties. You haven't done that here. And there's already a perfectly good metric on the real numbers, the usual one.

    The real numbers include points and intervals, and the usual metric is a perfectly good metric, and it's very easy to define. distance(x, y) = |x - y|.

    Have you read this?

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

    As mentioned earlier, rather than defining continua in terms of points, I am defining points in terms of continua, utilizing intervals (at least in the 1D case).keystone

    All your intervals so far are degenerate, denoting the empty set. Do you understand that?

    Don't the standard real numbers already "describe continua with arbitrarily fine precision?
    — fishfry
    Before I answer your question, I want to ensure we are on the same page. Do you understand how each of the five steps along the journey from 0 to 1 is represented by intervals, and that the union of these five intervals describes a continuous journey from 0 to 1?
    keystone

    No, because all of your interval notations denote the empty set and I can't figure out what you are doing. The usual metric on the real numbers seems perfectly satisfactory and you are somehow obfuscating it. You are being unclear. That's not a dismissal. I'm telling you that you are not saying anything clearly that I can figure out.

    I can define a continuous "journey," whatever that means, using the identity function on the real numbers f(x) = x.
  • Well that doesn't sound like a good idea.
    From birth, children are taken to this daycare+school+apartment+prison complex to learn life skills such as trigonometry.Scarecow

    If only. It's the gender theory taught to five year olds that worries people about the government schools these days.
  • Fall of Man Paradox
    I'm taking the Google Maps directions/map and making them more 'mathematical'. Let me try iteration 0 and tell me if this is clear:

    Iteration 0
    1) Start at 6445-6451 Peel Regional Rd 1
    2) Travel the road Erin Mills Pkwy/Peel Regional Rd 1 N towards McDonalds
    3) Arrive at intermediate destination: McDonalds
    4) Travel the road Millcreek Dr towards 6335-6361 Millcreek Dr
    5) Arrive at destination: 6335-6361 Millcreek Dr
    keystone

    I find this incredibly annoying. Can't you get to the point?

    Do you honestly not see how this relates to the Google Maps screenshot I sent a few posts back?keystone

    I don't see the point.

    I'm developing a framework that applies topological metric spaces to describe continua with arbitrarily fine precision. [ /quote[

    Oh. Ok. I understand that. I appreciate this clear, simple statement of what you are doing.

    Question: Don't the standard real numbers already do a fine job of exactly that?
    keystone
    This might seem esoteric,keystone

    It's not esoteric, it's basic high school math. The real number line.

    but achieving this involves turning everything upside down—without dismissing any past mathematical progress. This approach offers a powerful new perspective on mathematics.keystone

    You aren't making a case for that.

    It begins with this map example because I want to (1) describe the continuous journey using intervals and (2) show how those intervals can be described by a topological metric space. However, you're not even letting me do step (1).keystone

    I"ll stipulate to the real number line.

    I'm still confused by your describing points on the real line with two coordinates.



    Please tell me which iteration you are tripping up on: 0, 1, 2, 3, or 4?
    keystone


    I get that you start at 0, land at .5, and end up at 1. Is that sufficient for your purposes?

    I still don't see why you use two coordinates to describe a point on the real line.

    I'm using interval notation. It's an interval.keystone

    It's an interval?? What? You are labeling locations on the real line as intervals? That makes little sense. Google maps doesn't do that.

    No, that's not right. I referred back to your picture. You described the origin on the line as (0,0). What is the meaning of that?

    You described the point commonly notated as .5 as (.5, .5). What am I supposed to take from that?

    In fact the notation (.5, .5) is a degenerate open interval. It denotes the empty set. If I take (.5, .5) as interval notation, there are no points at all in it. Do you see that?

    So we have two specific questions on the table.

    1) What does (.5, .5) represent? In standard mathematical notation, it's the empty set. At best, [.5, .5] would simply be the point .5. But why do that?

    2) Don't the standard real numbers already "describe continua with arbitrarily fine precision?" [
  • Fall of Man Paradox
    Although I didn't plan to start with directions and maps, I'm glad we ended up here. It's an excellent starting point.keystone

    I don't even get a mention now? That's the only way I know when someone's talking to me.

    I don't know what you are talking about. I swear to God, I do not understand what you are doing, what you're talking about, why you're doing this. I am totally lost. I went back over the thread, i simply don't understand what you are talking about. And you flat out refuse to tell me. At one point you were talking about Achilles, so is this something to do with one of Zeno's paradoxes?

    Why can't you just give me the top-line summary of what you are doing? A while back we were talking about metric spaces and topological spaces, that at least made some sense. The rest of this, the map, the grid, I just don't know what you are doing. I don't know what is the overall point being made, what I'm supposed to be getting from this. It's very frustrating.

    I'll stipulate that you can traverse a grid. Or a line. Your coordinates have two components yet appear on a straight line. That's a little odd. What is your point?

    I do have one specific question. Why do your points on a straight line have two coordinates? What does that denote?
  • Information and Randomness
    The word is "logic", and I think it's pretty important to a discussion like this, to have good agreement as to what this word means.Metaphysician Undercover

    Well, it's not logical to reject a possibility on the grounds that you don't like it.

    If I simply assert, as if a true proposition, "chocolate is better than vanilla", there is not logic here. But if I state my premises, I am allergic to vanilla, and to have an allergic reaction is bad, then my stated preference "i prefer chocolate to vanilla" is supported by logic and is logical. Do you agree? .Metaphysician Undercover

    You're allergic to randomness? Is that the argument you're making?

    If you said that believing in randomness makes you break out in a rash, and therefore you prefer to not believe in randomness, that would be logical. But it would be a logical argument for why you hold that belief. It would not be a logical argument against randomness.
  • Fall of Man Paradox
    I've been trying to build towards a more important point but I feel like I have to keep going simpler and simpler to find a common ground with you. I'm hoping interpreting a map is the common ground where we can start from. If you acknowledge that you understand how directions and maps work then I will advance with my point.keystone

    Please start any time. I simply have no idea what your overall point is, nor have I understood any of your examples. Start from the top. "I wish to reform the entire corpus of modern mathematics." Then tell me what are numbers, sets, functions, relations, etc.

    I just can't figure out what you are doing.

    Explain to me as you would if I were standing in front of you, what point you are making with the map.
  • Fall of Man Paradox
    It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised.keystone

    I'm unable to understand their point.

    I apologize if this appears to diverge from your interests, but focusing on the image below, can you see how the instructions on the left relate to the image on the right? (This is not a trick question)keystone

    McDonalds, Sushi, Wok and Roll. Now I'm hungry.

    Once again you leave me utterly baffled as to why you posted this.
  • Information and Randomness
    Slow down, you are not taking the time to understand what I said. In the application of logic, there is two aspects to soundness, the truth or falsity of the premises, and the validity of the logical process.Metaphysician Undercover

    We're just arguing about a word. If you want to claim that "I prefer chocolate to vanilla" is an example of logical reasoning, what is the point of my arguing with you about a thing like that?

    Therefore, we must respect the fact that moral arguments can proceed with valid logic,Metaphysician Undercover

    In the sense that we can argue a conclusion from moral premises, I agree.

    But "I find such and so repugnant, therefore such and so is logically false," is simply bad logic. Or not even logic. As you prefer. But it's not good logic. I'm certain of that.

    I appreciate that you wrote a lot. I haven't the heart to continue this line of discussion, my apologies. "Randomness is repugnant to me therefore it's false" is not good logic. As to whether it's bad logic, or not even logic, I'll leave open.
  • Infinite Staircase Paradox
    I really don't see how there could be a staircase which is not physical.Metaphysician Undercover

    If I understood the OP, the walker spends arbitrarily small amounts of time on each step, 1/2 second, 1/4 second, etc. That violates the known laws of physics. So it's not a physical situation. It's a cognitive error to think we're contrasting math to physics. There is no physics in this problem.


    That really makes not sense. However, just like in the case of the word "determine", we need to allow for two senses of "physical". You seem to be saying that to be physical requires that the thing referred to must obey the laws of physics.Metaphysician Undercover

    Well yeah. To be a fish a thing has to obey the known laws of fishes. Note that I include the word "known." Biologists could discover a new fish that extends our concept of what's a fish, just as physicists refine their laws from time to time. But a physical thing must obey the known laws of physics. This seems a very trivial point, i can't imagine what you mean by questioning it.


    But the classic definition of "physical" is "of the body".Metaphysician Undercover

    Wasn't that a classic Star Trek episode? "Are you of the body?" And if you weren't, they zapped you with an electric stick.


    And when a body moves itself, as in the case of a freely willed action, that body violates Newton's first law.Metaphysician Undercover

    Sorry, what? Given me an example of something that violates Newton's laws, unless it's an object large enough, small enough, or going fast enough to be subject to quantum or relativistic effects.

    A freely willed action? Can you give me an example? You mean like throwing a ball? You kind of lost me here.

    Therefore we have to allow for a sense of "physical" which refers to things which are known to violate the laws of physics, like human beings with freely willed actions.Metaphysician Undercover

    I'll be happy to consider any specific examples you have of human beings whose actions violate the laws of physics. If you mean actions caused by mentation, that's a bit of a puzzler, but I'm not sure how to violate the laws of physics. If I set out today to violate Newton's laws, I don't know how I could do that.

    What is implied here is that the laws of physics are in some way deficient in their capacity for understanding what is "physical" in the sense of "of the body".Metaphysician Undercover

    Do you mean something like, "I think about raising my right hand and my right hand goes up, how does that happen?" If so, I agree that nobody understands the mechanism.


    That's why people commonly accept that there is a distinction between the laws of physics and the laws of nature.Metaphysician Undercover

    That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature.

    The laws of physics are a human creation, intended to represent the laws of nature, that is the goal, as what is attempted.Metaphysician Undercover

    Agreed.

    And, so far as the representation is true and accurate, physical things will be observed to obey the laws of physics, but wherever the laws are false or inaccurate, things will be observed as violating the laws of physics.Metaphysician Undercover

    Still waiting for specific examples. I believe the muons were misbehaving a while back and it made the news. Of course there are things we don't understand, like dark matter, dark energy, a quantum theory of gravity.

    Evidently there are a lot of violations occurring, with anomalies such as dark energy, dark matter, etc., so that we must conclude that the attempt, or goal at representation has not been successful.Metaphysician Undercover

    Ok. Scary that you and I are thinking along the same lines. What is your point here with respect to the subject of the thread?

    Sure, it's a conceptual thought experiment, but the interpretation must follow the description. A staircase is a staircase, which is a described physical thing,Metaphysician Undercover

    The walker spends ever smaller amounts of time on each step, and that eventually violates the Planck scale.

    just like in Michaels example of the counter, such a counter is a physical object,Metaphysician Undercover

    Which counter? The lamp? The lamp is not physical. No physical circuit can switch in arbitrarily small intervals of time.

    and in the case of quantum experiments, a photon detector is a physical object. And of course we apply math to such things, but there are limits to what we can do with math when we apply it, depending on the axioms used. The staircase, as a conceptual thought experiment is designed to expose these limits.Metaphysician Undercover

    It's designed to confuse people who mis-learned a little calculus and don't know what's allowed or disallowed by the laws of physics.

    OK sure, but that's a limit created by the axioms of the mathematics. So it serves as a limit to the applicability of the mathematics. The least upper bound is just what I described as "the lowest total amount of time which the process can never surpass". Notice that the supposed sequence which would constitute the set with the bound, has already summed the total. This is not part of the described staircase, which only divides time into smaller increments. It is this further process, turning around, and summing it, which is used to produce the limit. The limit is in the summation, not the division.Metaphysician Undercover

    Ok I guess. No walker can traverse a staircase as described by the premises of the problem. So if I said the staircase was not physical, I should have said the walker is not physical. Better?

    It is very clear therefore, that the bound is part of the measurement system, a feature of the mathematical axioms employed, the completeness axiom, not a feature of the process described by the staircase descent. The described staircase has no such bound, because the total time passed during the process of descending the stairs is not a feature of that description. This allows that the process continues infinitely, consuming a larger and larger quantity of tiny bits of time, without any limit, regardless of how one may sum up the total amount of time. Therefore completeness axioms are not truly consistent with the described staircase.Metaphysician Undercover

    I don't see why not. The whole point of the puzzle is to sum 1/2 + 1/4 + ... = 1, and then to ask what is the final state. Which, as I have pointed out repeatedly, is not defined, but could be defined to be anything you like.

    However, since our empirical observations never produce a scenario like the staircase, that inconsistency appears to be irrelevant to the application of the mathematics, with those limitations inherent within the axioms. The limitations are there though, and they are inconsistent with what the staircase example demonstrates as logically possible, continuation without limitation. Therefore we can conclude that this type of axiom, completeness axioms, are illogical, incoherent.Metaphysician Undercover

    I'm sorry that you fine the completeness axiom of the real numbers incoherent. On the contrary, the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.


    The real problem is that as much as we can say that the staircase scenario will never occur in our empirical observations, we cannot conclude from this that the incoherency is completely irrelevant.Metaphysician Undercover

    The premises violate the known laws of physics, specifically the claim that we can know the walker's duration on each step even though that duration is below the Plancktime.

    We have not at this point addressed other scenarios where the completeness axioms might mislead us. Therefore the incoherency may be causing problems already, in other places of application.Metaphysician Undercover

    Modern math is incoherent. Is it possible that you simply haven't learned to appreciate its coherence?
  • Infinite Staircase Paradox
    Yes. I got enough from it to realize a) that ω is one of a class of numbers and b) that it comes after the natural numbers (so doesn't pretend to be generated by "+1")Ludwig V

    Yes exactly. comes into existence via a limiting process. The idea is that the natural numbers are generated by successors, and the higher ordinals are generated by successors and limits. So we're adding a new rule of number formation, if you like. We go 1, 2, 3, ... by successors, and then to by taking a limit, then , , etc., then eventually we get to by taking a limit, then we keep on going. I don't want to go too far afield, but the idea is that we can take successors and limits to get to all the higher ordinals.


    This business about actions is what confuses people.
    — fishfry
    Certainly. That's what needs to be clarified, at least in my book. There's a temptation to think that actions must, so to speak, occur in the real world, or at least in time. But that's not true of mathematical and logical operations. Even more complicated, I realized that we continually use spatial and temporal terms as metaphors or at least in extended senses:-
    Ludwig V

    Right. A lamp that cycles in arbitrarily small amounts of time is not physical. A staircase that we occupy for arbitrarily small intervals of time is not physical. So trying to use physical reasoning is counterproductive and confusing. That's my objection to all these kinds of puzzles. People say there's a conflict between the math and the physics ... but as i see it, there's no physics either.


    By the way, ω is the "point at infinity" after the natural numbers
    — fishfry
    What does "after" mean here?
    Ludwig V

    Follows in order. Given 1, 2, 3, 4, ..., we can adjoin "at the end." What do I mean by that? I mean that we extend the "<" symbol so that

    1 < , 2 < , 3 < , and so forth. So that conceptually, every natural number is strictly smaller than . Does that make sense?

    If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work
    — fishfry
    Yes, but it seems to me that this is not literally true, because numbers aren't objects and a set isn't a basket. (I'm not looking for some sort of reductionist verificationism or empiricism here.)
    Ludwig V

    I can always form a set out of a collection of objects. Not following your objection.

    {1/2, 3/4, ...} is a set, and {1} is a set, and I can surely take the union of the two sets, right?

    {1/2, 3/4, ..., 1} is just a particular subset of the closed unit interval [0,1].

    If you are not sure about what I'm saying we should stay on this point. I can definitely form a set out of any arbitrary collection of other sets. And each of 1, 2, 3, ... and can be defined as particular sets.

    Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.
    — fishfry
    In that respect, yes. But I can't help thinking about the ways in which they are different.
    Ludwig V

    Of course {1, 2, 3, ..., } is a different set that {1/2, 3/4, ..., 1}. But strictly in terms of their order, they are exactly the same. And with ordinals, all we care about is order.

    That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.
    — fishfry
    Yes. But it doesn't end in the sense that we can't count from any given natural number up to the end of the sequence.
    Ludwig V

    The sequence is endless, and there's an extra point that's defined to be strictly greater than all the others. We can't get to the limit by successors, but we can get there by a limiting process.

    I try not to mention this in public, but the fact is that I never took a calculus class, nor was I ever taught to think about limits or infinity in the ways that mathematicians sometimes do. I did a little formal loic in my first year undergraduate programme. Perhaps that's an advantage.Ludwig V

    You're far better off. People who take calculus and then engineering math end up confused about limits and the nature of the real numbers. Taking logic and not calculus is actually helpful, in that you haven't mis-learned bad ideas about limits.

    Calculus is focussed on the computational and not the philosophical aspects of limits, and calculus students often end up a little confused about some of the technical details. I was actually referring to the other poster who you noted was talking past me and vice versa.


    I have the impression that you don't think that they are mathematically possible either. (I admit I may be confused.) So does that mean you don't think that supertasks are possible?Ludwig V

    I've convinced myself both ways. On the one hand we can't physically count all the natural numbers, because there aren't enough atoms in the observable universe. We're finite creatures.

    On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot.

    I have no strong belief or opinion about supertasks. I have strong opinions about some of the bad logic and argumentation around supertasks.
  • A simple question
    H'm. In principle, that is a valid complaint. But, back when I was involved, something like 60% of vacancies for graduates (i.e. those requiring a BA degree or higher) did not specify the subject. That may have changed. But you might be surprised at where Eng. Lit. and Fine Arts graduates end up.Ludwig V

    In the DEI departments of university administrations I imagine.

    I'm not sure how education for professions and trades differs now; there's a lot of emphasis on training all the way up to BA level and higher. Many Universities are re-casting their non-vocational qualifications as vocational and there's effort going in to tracking what level of job graduates actually get. I've heard anecdotes that some vocational programmes don't do very well. It's complicated. I suspect that the identity of the awarding institution is more important than the subject. Whether it is question of reputation, prestige or snobbery depends on how polite I'm feeling.Ludwig V

    Point being that pipefitters shouldn't be shouldering the cost of the loans forgiven for social justice majors.

    Oh, I wondered why that business about the student loans was happening now. Not pretty, but then, one has to please one's voters.Ludwig V

    It's a scandal. The executive branch (Biden) actually has no authority to forgive those loans and foist them on the taxpayers. The Supreme Court already ruled on that. Biden's actions are illegal. Just election year pandering. And of course we're seeing this week who those students are.

    It has happened gradually over two or three decades. I hesitate to get too detailed. It's mainly about social liberalism/conservativism - abortion, gay rights &c. Curiously, the Conservative party now seems to be at least as socially liberal as the Labour party, if not more so. There is certainly an issue in the Labour party that the liberal metropolitan elite now vote for Labour and this often clashes with the conservative social values of many "working class" people (not a politically correct classification any more.)Ludwig V

    Right. The "liberals" used to be for the working classes. Now the liberals support the elite against the working classes. Bit of a puzzler.

    Originally the Labour party was explicitly a party for the working class - it was founded by the Trade Union movement. The Conservative Party tended also to have foundations in the "higher" parts of the class system; but now it's more about economics - free market vs state intervention (not Socialism as such). It does seem that many people in what used to be the working class who might well have voted Labour in the past now vote Conservative. This is all not very reliable. I'm not an expert.[/qgge.uote]

    Me either, I was making a much more limited point earlier, and the poster I was making it to has chosen not to engage.
    Ludwig V
    Compassion for criminals is anti-compassion for their victims.
    — fishfry
    I don't see why it has to be. Except, of course, that a victim may be more vengeful than the system is. But I don't see that as a question of compassion or not. Support for victims (in the UK at least) has been pathetic, but is now improving (but not nearly perfect).
    Ludwig V

    Violent criminals are being put back on the street to re-offend. That's not fair to the victims. Violent criminals belong behind bars.

    I think the first duty of civic authorities is to provide for civic order.
    — fishfry
    Of course that's true. Part of the argument is that sympathetic ("humane") treatment of criminals and addicts gets better results in preventing recidivism - and a huge proportion of crime is recidivism. There's empirical evidence for that.
    Ludwig V

    People can't re-offend if they're locked up.

    Another part is that more severe sentences are not effective in preventing crime. Effective detection and police work is much more effective. It makes sense. 20 years in jail is not much of a deterrent if you aren't going to get caught. But if you know you won't get away with, you know also that you won't benefit much, whatever the penalty. (Some crimes are not deterred even by the high likelihood of getting caught, but those are unlikely to be deterred by severe penalties.) I know, I know, justice demands.... That, in my book, is not about justice; it is about revenge. Prevention is more important than revenge.Ludwig V

    Perhaps I just spend to much time following NYC politics. They're having a problem with soft-on-crime politicians leading to a great decrease in public safety.
  • Infinite Staircase Paradox
    And as I keep explaining, the issue with supertasks has nothing to do with mathematics. Using mathematics to try to prove that supertasks are possible is a fallacy.Michael

    But I'm not doing that. I haven't been doing that. Are you deliberately misunderstanding me or am I being unclear?

    "Using mathematics to try to prove that supertasks are possible is a fallacy"

    Who did that? Are they in the room with us right now?
  • SCOTUS
    I do not think it is some secret plan. They are anti-regulation, anti-LGBT rights, pro-discrimination on the basis of religious freedom, and pro-gun.Fooloso4

    But then why did they saddle the GOPs with a five to ten point deficit in every election at every level everywhere in the country for years, by overturning Roe? Now abortion's always on the ballot. That hurts every cause you listed. That's my point. Dobbs was profoundly counterproductive for the right.
  • Infinite Staircase Paradox
    I have more or less dropped out due to the repetitive assertions not making progress, but thank you for this post.noAxioms

    Thanks.

    the set {1/2, 3/4, 7/8, ..., 1}
    — fishfry
    Interesting. Is it a countable set? I suppose it is, but only if you count the 1 first. The set without the 1 can be counted in order. The set with the 1 is still ordered, but cannot be counted in order unless you assign ω as its count, but that isn't a number, one to which one can apply operations that one might do to a number, such as factor it. That 'final step' does have a defined start and finish after all, both of which can be computed from knowing where it appears on the list.
    noAxioms

    Of course it's a countable set. It's a subset of the rationals, after all. You are right that it's not order-isomorphic to 1, 2, 3, ...

    This is not radical. The rational numbers are countable, but not if counted in order, so it's not a new thing.noAxioms

    Right. Exactly right. Point being that contemplating a set that includes an infinite sequence along with an extra point is nothing strange at all. And it serves as a nice conceptual model for supertask puzzles.

    If Zeno includes 'ω' as a zero-duration final step, then there is a final step, but it doesn't resolve the lamp thing because ω being odd or even is not a defined thing.noAxioms

    There is no final step. There is a point at infinity. Not quite the same. Unless you allow the limiting process itself as a step. It's just semantics.

    and we inquire about the final state at ω
    Which works until you ask if ω is even or odd.
    noAxioms

    It's neither, and who's asking such a thing? Even and odd apply to the integers.

    Anyway if this is repetitive feel free to not reply. I just go through my mentions everyday trying to reply best I can. And I do have a thesis, which is that the ordinal is the proper setting for the mathematical analysis of supertask puzzles. So I'll repeat that every chance I get.
  • A simple question
    Constructive or healthy modes of competition. We cannot eliminate our desire to win or outcompete one another. We like reward, acknowledgement and status. All we can do is steer the compulsion away from competition that worsens the the wellbeing or basic rights of the losing group.Benj96

    What ruling body decides on that? Steer the compulsion away if it "worsens the wellbeing?" Are their commissars for that? Your idea sounds like top-down authoritarianism in the guise of being caring. "We're crushing your competitiveness for your own good, Comrade. Enjoy your stay at the Gulag."
  • Infinite Staircase Paradox
    The relativity thing was more of a refinement and had little practical value for some time. Newtonian physics put men on the moon well over a half century later.
    QM on the other hand was quite a hit, especially to logic. Still, logic survived without changes and only a whole mess of intuitive premises had to be questioned. Can you think of any physical example that actually is exempt from mathematics or logic?[/quot]

    Relativity more of a refinement? Not a conceptual revolution? I don't think I even need to debate that. In any even it's a side issue. It's clear that the universe doesn't care what mathematics people use. In that sense, the laws of nature are exempt from mathematics. Historically contingent human ideas about the world are always playing catch up to the world itself. But if you disagree that's ok, it's a minor sidepoint of the discussion.
    noAxioms
    QM is also the road to travel if you want to find a way to demonstrate that supertasks are incoherent.
    Zeno's primary premise is probably not valid under QM, but the points I'm trying to make presume it is.
    noAxioms

    I don't really care much about supertasks and haven't argued that they're coherent or incoherent. I'm mostly trying to clarify some of the bad reasoning around them.


    If you mean mentally ponder each number in turn, that takes a finite time per number, and no person will get very far. That's one meaning of 'count'. Another is to assign this bijection, the creation of a method to assign a counting number to any given integer, and that is a task that can be done physically. It is this latter definition that is being referenced when a set is declared to be countably infinite. It means you can work out the count of any given term, not that there is a meaningful total count of them.noAxioms

    Ok. I think some of the quoting got mangled since things I said ended up as part of your post.

    But if anyone thinks I can't count all the natural numbers 1, 2, 3, ... by mathematical means, please identify the first one I can't count.

    Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics?noAxioms

    To count a set means to place it into bijection with:

    a) A natural number; or

    b) the set of natural numbers, to establish countability; or

    c) some ordinal number, if one is a set theorist or logician or proof theorist.

    That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things.noAxioms

    Lost me. As far as we know takes into account the great conceptual revolutions of the past, as evidence that there will be more such in the future.

    They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one.noAxioms

    Ok. I think I'm a little lost in the quoting and not actually sure what we are talking about here. I'm not strenuously defending whatever ideas you're concerned with.

    Octonians shows signs of this sort of revolution.noAxioms

    Well ... ok.

    Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.noAxioms

    But that's exactly my point. If speculative physics is starting to take physically instantiated infinity seriously, then it's perfectly reasonable that in the future, physically instantiated infinity may become a core aspect of physics; in which case supertasks may be on the table.


    It is a mistake to talk about 'time creating these other universe'.noAxioms

    Was this for me? I never said any such thing nor quoted anyone else saying it.

    Time, as we know it, is a feature/dimension of our one 'universe' and there isn't that sort of time 'on the outside'. There is no simultaneity convention, so it isn't meaningful to talk about if new bubbles are still being started or that this one came before that one.noAxioms

    I'll have to plead ignorance on the question of whether there's a meta-universal time that transcends the bubble universes. Good question though.

    All that said, the model has no reason to be bounded, and infinite bubbles is likely. This is the type-II multiverse, as categorized by Tegmark. Types I and III are also infinite, as is IV if you accept his take on it. All different categories of multiverses.noAxioms

    You are completely agreeing with my point. That if speculative physic already includes infinity, then mainstream physics may include infinity in the future.

    And two, the many-world interpretation of quantum physics.
    That's the type III.
    noAxioms

    You are agreeing with my point.

    Observation for one is a horrible word, implying that human experience of something is necessary for something fundamental to occur. This is only true in Wigner interpretation, and Wigner himself abandoned it due to it leading so solipsism.noAxioms

    Nothing to do with my point, which is that speculative physics already includes infinity, therefore mainstream physics may include infinity after the next scientific revolution.

    I don't buy into MWI, but bullshit is is not. It is easily the most clean and elegant of the interpretations with only one simple premise: "All isolated systems evolve according to the Schrodinger equation". That's it.noAxioms

    You're agreeing with me again. Why are you typing this stuff in? You've kind of lost me.

    Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable.noAxioms

    Ok.

    I would have loved to see Einstein's take on MWI since it so embraces the deterministic no-dice-rolling principle to which he held so dear.noAxioms

    Ok.

    Ah, local boy. I am more used to interacting with those who walk a km. There's more of em.noAxioms

    Depends on the exchange rate.


    And suppose that in the first bubble universe, somebody says "1".
    The universes in eternal inflation theory are not countable.
    noAxioms

    Wow. You have evidence for that? My understanding is that it is an open question in eternal inflation as to the cardinality of the bubbles: finite, countably infinite, or uncountable. But either way my point about reciting the integers stands. I don't actually get the sense that you're engaging with anything I wrote.

    Yes, each step in a supertask can and does have a serial number. That's what countably infinite means.noAxioms

    That's not the definition of supertask others are using. But I used the example of bubble universes to illustrate the possibility of counting the natural numbers physically.

    Anyway sorry if I got lost in the quoting and didn't really understand some of your responses.
  • A simple question
    The issue behind the student loan question is the question how far state-funded free education should go. If you want a level playing field in careers, everyone who can benefit should get higher education - and that means that almost everybody should be entitled to have a go. At the same time, if people benefit financially, there is a good case for saying that some of that benefit should go back to whoever funded it. Ironically, in the UK, the financial benefit from higher education is rapidly shrinking and, some say, has disappeared, mainly because it has been extended so widely. The proportion of student loans that is actually repaid is astonishingly low. (I can't remember the actual figures.)Ludwig V

    That's right. So the students majoring in unmarketable majors are subsidized by people who skipped school and went into the trades. That doesn't seem fair. It's just that the college grads vote for Democrats and the tradesmen vote for Republicans, so the Democratic administration forgives billions in student loans -- illegally, as the Supreme Court has already ruled -- in an election year.

    And not just that. The Democratic party use to be the party of the tradesmen and no longer is. When did the left abandon the workers, and why? I gather the Labour party in the UK has undergone a similar transition, is that right?

    So is it possible that a different version of the social justice approach might be more effective? Is it possible that other places may be implementing it in a better way?Ludwig V

    Compassion for criminals is anti-compassion for their victims. New York City is a great lesson in restorative justice gone too far. I think the first duty of civic authorities is to provide for civic order. What's weird is that the voters themselves vote for the faux-compassion that ends up hurting them.

    So the real solution to our problems is better voters!
  • Infinite Staircase Paradox
    I've watched this debate for a long time - though I don't claim to have understood all of it. But I think those two quotes show that you are talking past each other.Ludwig V

    He'll come around :-)

    I didn't like ω at all, when it was first mentioned. I'm still nowhere near understanding it. But the question whether a mathematical symbol like ω is real and a number is simply whether it can be used in calculations. That's why we now accept that 1 and 0 are numbers and calculus and non-Euclidean geometries. ω can be used in calculations. So that's that. See the Wikipedia article on this for more details.Ludwig V

    This paragraph gratified me. If you are struggling to understand my posts then I'm getting through to at least one person. My talk about is something most people haven't seen, but the ideas aren't that hard. For what it's worth there's a Wiki page on ordinal numbers. The page itself isn't all that enlightening, but it does at least show that the ordinal numbers really are a thing in math, I'm not just making it all up.

    You have a great insight that what makes a mathematical concept real is, in the end, its utility. Sometimes not even to anything practical, but just to math itself. We want to solve the equation x + 5 = 0 so we invent negative numbers. That kind of thing. In that sense, the ordinals exist.

    But another way to think about it is that it's just an interesting new move in a game. As if you were learning chess and they told you how the knight moves. You don't say, "Wait, knights slay dragons and rescue damsels, they don't move like that." Rather, you just accept the rules of the game. You can think of ordinals like that. Just accept them, work with them, and at some point they become real to you. Just like the moves of the chess pieces any other formal game.

    But I have tried to give a very concrete, down to earth example of how this works.

    Suppose that we have the sequence 1/2, 3/4, 7/8, ... It converges to 1.

    Now we can certainly form the set {1/2, 3/4, 7/8, ..., 1}. It's just some points in the closed unit interval.

    But it gives us a model, or an example, of a set that contains an entire infinite sequence that "never ends" blah blah blah, and also contains its limit.

    If you believe in the set {1/2, 3/4, 7/8, ..., 1}, then you should have no trouble at all believing in the set

    {1, 2, 3, ..., }. That's also just a set that contains an entire infinite sequence, along with its limit. We typically don't encounter this concept in the math curriculum that most people see, but it's perfectly standard once you go a little further. Also a lot of people have seen the extended real numbers with and nobody complains about that, or do they?

    It's true that the distances are different inside the two sets. But in terms of order, the two sets are exactly the same: an infinite series, along with its limit.

    Anyway, this framework is very handy for understanding supertask type problems. That's why I'm mentioning it.

    So if you don't like , that' s no problem. Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.

    Does that help?

    But it is also perfectly true that a recitation of the natural numbers cannot end.Ludwig V

    That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.

    And if we can't imagine that, we can certainly imagine {1/2, 3/4, 7/8, ..., 1}. There's nothing mysterious about that. An entire infinite sequence is in there, along with an extra point. It's a legitimate set.

    If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work. They are containers for infinite collections of things.

    By the way, is the "point at infinity" after the natural numbers. And is the name for the set {1, 2, 3, 4, ..., }.

    is the natural setting for all supertask puzzles. We have the state at each natural number, and we inquire about the final state at .

    That's why I like as a mental model for these kinds of problems.

    As I said earlier, it is remarkable that we can prove it. Yet we cannot distinguish between a sequence of actions that has not yet ended from one that is endless by following the steps of the sequence. So we are already in strange territory.Ludwig V

    This business about actions is what confuses people. They set up scenarios that violate the laws of physics, like the lamp that switches in arbitrarily small intervals of time, and then they try to use physical reasoning about them. Then they get confused.

    In the way I'm describing this, you may think that the difference is between the abstract world (domain) of mathematics and another world, which might be called physical, though I don't think that is right.Ludwig V

    Well yes, you are correct to feel that it's not quite right. Because there is nothing physical about the lamp or the staircase. So it's a category error to try to use everyday reasoning about the physical world. That's why people get confused.

    I'm very puzzled about what is going on here, but I'm pretty sure that it is more about how one thinks about the world than any multiverse.Ludwig V

    I think it all comes down the fact that calculus classes care about computation and not theory. That, and the fact that we don't know the ultimate nature of the world, and there's are good reasons to think it's not anything like the mathematical real numbers.

    So on the one hand, the continuity of the world is an open question. And two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion.
  • Information and Randomness
    There is no such thing as "going by pure logic", toward understanding the nature of reality. [/quore]

    Agreed. But that does not justify using some means OTHER than logic to understand reality, and calling it logic! That's @Michael's fallacy. Saying something's a logical contradiction when it merely makes no sense to him. You agreed with me earlier that this is a fallacy. But you defend it when YOU do it.

    To be clear: I have no objection to using extra-logical means of understanding reality. But then don't turn around and all it logic.

    Metaphysician Undercover
    "Pure logic" would be form with no content, symbols which do not represent anything. All logic must proceed from premises, and the premises provide the content. And premises are often judged for truth or falsity. But as explained in the passage which ↪wonderer1 referenced, in the case of an "appeal to consequences", there is no fallacy if the premises are judged as good or bad, instead of true or false. That's why I said that this type of logic is very commonly employed in moral philosophy, religion, and metaphysical judgements of means, methods, and pragmatics in general. So for example, one can make a logically valid argument, with an appeal to consequences, which concludes that the scientific method is good. No fallacy there, just valid logic and good premises.Metaphysician Undercover

    Call it anything you like, but not logic! Logic means something else. That term is already taken. You are using extra-logic. Morality, right or wrong, productive/nonproductive. All well and good, but not logic. If logic is to mean anything, it has to mean something.

    Therefore it is not the case that the reasoning is "extra-logical", it employs logic just like any other reasoning. What is the case is that the premises are a different sort of premises, instead of looking for truth and falsity in the premises we look for good and bad. So this type of judgement, the judgement of good or bad, produces the content which the logic gets applied to.Metaphysician Undercover

    Let's agree to disagree on that point.

    No, that is not the case, because there are two very distinct senses of "determined". One is the sense employed by determinism, to say that all the future is determined by the past. The other is the the sense in which a person determines something, through a free will choice. In this second sense, a choice may determine the future in a way which is not determined by the past. And, since it is a choice it cannot be said to be random. Therefore it is not true that if the world is not random then it's determined (in the sense of determinism), because we still have to account for freely willed acts which are neither determined in the sense of determinism, nor random.[/qouote]

    You can't have determinism and free will. Frankly if the world is random and we have some kind of influence on it through our will, or spirit, I find that much more hopeful than a universe in which I'm just a pinball clanging around a well-oiled machine.

    Determinism is the nihilistic outlook, not randomness. In randomness there is hope for freedom. Say that's a pretty catchy saying. The church of Kolmogorov. In randomness lies the hope of freedom.
    Metaphysician Undercover
    As I said above, it is not a matter of transcending logic, the conclusions are logical, but the premises are judged as to good or bad rather than true or false. So from premises of what is judged as good (rejecting repugnant principles), God may follow as a logical conclusion.Metaphysician Undercover

    God's going to hurl thunderbolts at you for so blithely enlisting him on your side to make such a specious argument. If I'm choosing good versus bad I'm not using logic, I'm using feelings. Logic says kill the one rather than the million. But if the one's you or yours, you kill the million. It's been done. Feelings trump logic. But your feelings are not logic!!


    No I was not arguing that. In that case I was arguing that the idea ought not be accepted (ought to be rejected) unless it is justified. In the case of being repugnant, that in itself is, as I explained, justification for rejection. You appear unwilling to recognize what wonderer1's article said about the fallacy called "appeal to consequences". It is only a fallacy if we are looking for truth and falsity. If we are talking principles of "ought", it is valid logic. Therefore the argument that the assumption of randomness ought to be rejected because it is philosophically repugnant, cannot be said to be invalid by this fallacy, and so it may be considered as valid justification.Metaphysician Undercover

    Yes but the contrary proposition of determinism is even more repugnant, as I've noted. Shouldn't we (logically!) choose the lesser of two repugnancies?

    But Michael did not show that supertasks are philosophically repugnant.Metaphysician Undercover

    And you have not shown randomness philosophically repugnant. By the time I thought about it a little, I realized that randomness is our only hope for salvation. It's the only way we're not automatons. Clockwork oranges. So you haven't made your point here. I am a proud randomite.


    He showed that they are inconsistent with empirical science,Metaphysician Undercover

    There's no empirical science in these silly omega sequence paradoxes like the effing lamp and the effing staircase. That's the massive category error everyone makes. They posit these physics-defying scenarios then claim they're talking about the physical world.

    and his prejudice for what is known as "physical reality" (reality as understood by the empirical study of physics) influenced him to assert that supertasks are impossible.Metaphysician Undercover

    I believe I made the same claim, but qualified it to "presently known physics."


    As I explained in the other thread, in philosophy we learn that the senses are apt to mislead us, so all empirical science must be subjected to the skeptic's doubt. So it is actually repugnant to accept the representation of physical reality given to us by the empirical sciences, over the reasoned reality which demonstrates the supertask. And this is why that type of paradox is philosophically significant. It inspires us to seek the true reasons for the incompatibility between what reason shows us, and what empirical evidence shows us. We ought not simply take for granted that empirical science delivers truth.Metaphysician Undercover

    This is way past the lamp. The lamp is not a physical thing. These puzzles have no bearing on physical reality. That's a cognitive error everyone makes about them.

    Also, there's more bad reasoning than "reason" in the discussions about these problems.

    As explained above, I am not taking a standpoint of determinism. There are two very distinct senses of "determine", one consistent with determinism, one opposed to determinism (as the person who has a very strong will is said to be determined). I allow for the reality of both.Metaphysician Undercover

    You say randomness and determinism are compatible, and your justification is to use an alternate and unrelated meaning of the word determined?

    But as of now, in this very post, I've convinced myself that I'm a randomist. But then again I've always suspected I'm a Boltzmann brain, and that's how randomists come into existence.
  • Infinite Staircase Paradox
    So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number.Relativist

    No. My thinking about limits is extremely precise and perhaps a bit more general than what you're accustomed to. I have never said that a series (or sequence if that's what you mean here) reaches infinity. I would not say that, and I did not say that.

    What I said was that there is a mathematical view that sheds light on the subject, and makes it clear in where the limit of a sequence lives. The sequence 1/2, 3/4, 7/8, ... has the limit 1. Of course it never "reaches" 1. But you would have no objection to my putting {1/2, 3/4, ..., 1} into a set together. After all, I am allowed to take unions of sets: and {1/2, 3/4, ...} U {1} = {1/2, 3/4, ..., 1}. So it's a legit set.

    Now 1 is in no way a "point at infinity," after all it's just the plain old number 1. And no member of the sequence ever "reaches" it. But it does live there as the limit; as the result of a well-defined limiting process.

    I have suggested this mathematical model as a thought aid to these kinds of paradoxes. If you find it helpful all to the good, but if not, that's ok too. I find it helpful.

    For what it's worth, in math, the natural numbers have an upward limit, called , that plays the same role for the sequence 1, 2, 3, ... that the number 1 is for the sequence 1/2, 3/4, ...

    It's the limit. It's more general notion of limit, one that allows us to reason about a "point at infinity." Which is exactly what these puzzles are about. That's why it's a handy framework for thinking about these kinds of puzzles.

    You have a sequence that's defined (on/off, on a step, whatever) at each member of a convergent sequence; and you want to speculate on the definition at the limit. is exactly what you need; or rather, a set called , which is like the set {1/2, 3/4, ,,,, 1}. It's a set that contains an entire infinite sequence and its limit. It's exactly what we need to analyze these problems.

    If it helps, here's the Wiki page on ordinals, at least so that you know they're a real thing. You can "keep counting past the natural numbers," and you get some very cool mathematical structures. Ordinals find application in proof theory and mathematical logic.

    Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark.Relativist

    I'm going to defer talking about supertasks today, had enough for a while.

    There is a mathematical (and logical) difference between the line segments defined by these two formulae:
    A. All x, such that 0<=x < 1
    B. All x, such that 0<=x <= 1
    Relativist

    Please reread what I wrote. This is not on topic if you understand what I'm saying.

    Your blurred analysisRelativist

    I'm doing my best to fit you with a sharper pair of mathematical eyeglasses to unblur your vision ... but you keep making a spectacle of yourself!!

    conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers.Relativist

    You might consider using words like "reach" and "approach" with precision. They are not part of the mathematical definition of a limit. They're casual everyday synonyms that you are allowing to confuse you.
  • Infinite Staircase Paradox
    Then rather than recite the natural numbers I recite the digits 0 - 9, or the colours of the rainbow, on repeat ad infinitum.

    It makes no sense to claim that my endless recitation can end, or that when it does end it doesn't end on one of the items being recited – let alone that it can end in finite time.
    Michael

    The natural numbers do not end, yet they have a successor in the ordinal numbers, namely . This is an established mathematical fact.

    I regard this as a helpful point of view when analyzing these kind of puzzles. I've explained it as best I can.

    "It makes no sense" is not a logical argument. It's only a description of your subjective mental state. Once, violating the parallel postulate or the earth going around the sun or splitting the atom made no sense. You are not making an argument.

    So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible.Michael

    If you only demonstrated the reductio. All you have is "it makes no sense," and that is not an argument.
  • Infinite Staircase Paradox
    Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them.Ludwig V

    I think a lot of people feel that way.
  • A simple question
    The problem with Margaret Thatcher is that she thought that a dumb quip is a substitute for serious thinking. But then, she was a politician. She also believed that there is no such thing as society.Ludwig V

    I thought it was on point. People in the US like "forgiving student debt." But every nickel is just passed on to the taxpayers. Government doesn't have any money that it doesn't take from someone else. Or borrow and print, that's a nice game that has to end at some point too.

    I agree that equality of outcome is not a reliable index of equality of opportunity and that people often talk, lazily, as if they were. But if equality of opportunity does not result in changes to outcomes, then it is meaningless. The only question is, how much change is it reasonable to expect? If 50% of the population is female and only eight of UK's top 100 companies are headed by women (Guardian Oct. 2021), don't you think it is reasonable to ask why? I agree that it doesn't follow that unfair discrimination is at work, but it must be at least a possibility. No?Ludwig V

    I agree. We need a balance between trying to homogenize society, and old-fashioned notions of merit.

    Perhaps it's a matter of pendulum swinging and patience.


    There are always issues with the NHS in the UK. But that's not about universal health care or not. It's about what can be afforded, what priority it has. Difficult decisions, indeed, but anyone with sense knows they must be made. That's why we have the national institute of clinical excellence. It is not perfect, but it is an attempt to make rational decisions; other systems do not even attempt to do that.
    Of course, when my life, or my child's life, is at stake, I will put the system under as much pressure as I can to try everything. And to repeat, it's not about charity or robbing the rich. It's about insurance.
    Ludwig V

    Health care policy's hard, I agree. I've only heard anecdotal evidence about NHS.
  • A simple question
    I have no reason to give a flying fig about New York politics.Vera Mont

    It's a beautiful living experiment in what's known as restorative justice.

    Crime is rampant and the DA is busy prosecuting the victims. People don't feel safe. It's going to sink Mayor Adams's once-promising political career.

    I would think that many people interested in politics do follow New York City politics. But if you don't, that's cool. Not sure you are qualified to comment on the social justice approach to crime, though. It's failing in New York City in a very obvious way.
  • Infinite Staircase Paradox
    I can explain it very easily. There is two different senses of "limit" being used here. One is a logical "limit" as employed in mathematics, to describe the point where the sequence "converges". And "unlimited" is being used to refer to a real physical boundary which would be place on the process, preventing it from proceeding any further. There is no such "limit" to a process such as that described by the op. The appearance of paradox is the result of equivocation.Metaphysician Undercover

    Mathematicians would just refer to it as an "upper bound."

    But you talk about a "real physical boundary." Here you imagine that the staircase is physical. It's not. The conditions of the problem violate known laws of physics.

    It's only a conceptual thought experiment. And why shouldn't math apply to that?

    But anyway, it's an upper bound. If it's a least upper bound, it's a limit.
  • SCOTUS
    I do think that there are members of the court who have an agenda. It is not that they are on Trump's side but that they see Trump as useful to their side. An expedient for attaining their conservative goals.Fooloso4

    Maybe. Dobbs certainly. But that's been a disaster for the GOP. It cost them the 2022 red wave and many local and special elections. It put abortion back on the table as a political issue. Centrist voters that could trend GOP on the economy, crime, and immigration, now have to vote Dem for abortion.

    The percentage of the public for whom abortion is their top issue, represents a certain percent of the vote lost to the GOP and won by the Dems in every election at every level of politics forever, till Congress hashes out a law everyone can live with. And good luck with that.

    In the long run, the conservative justices have done more harm to the conservative cause than if they'd left Roe in place. The Dems are going to win a lot of races they'd otherwise have lost, as long as abortion is on the ballot. And overturning Roe put abortion back on every ballot.

    If this is part of some secret plan by the conservative Supes, I wonder what it is.
  • A simple question
    They do, they are just playing dumb.Lionino

    LOL. I'm just trying to take the subtle approach.
  • Information and Randomness
    "Repugnant", is a commonly used word in philosophy. The argument I gave is logical, but what is concluded is that the assumption, "there is ontological randomness" is philosophically repugnant, because it would be counter-productive to the desire to know. Therefore it's more like a moral argument. The desire to know is good. The assumption of ontological randomness hinders the desire to know. Therefore that assumption is bad and one ought not accept it.Metaphysician Undercover

    I can agree with your reasoning that one "ought" not to accept it, but the reason is extra-logical. That is, if we are going by pure logic, you have not argued against it. It's like solipsism. Can't refute but pointless to believe it.

    But consider: If the world is not random, then it's determined. And is that not equally repugnant? Nothing matters because we have no choice.

    What do you say to that? It's repugnant either way. Either there's no meaning or ... there's no meaning. Is there a way out?


    Since the argument concerns an attitude, the philosophical attitude, or desire to know, you're right to say that it is an argument concerning "feelings". But that's what morality consists of, and having the right attitude toward knowledge of the universe is a very important aspect of morality. This is where "God" enters the context, "God" is assumed to account for the intelligibility of things which appear to us to be unintelligible, thereby encouraging us to maintain faith in the universe's ability to be understood. Notice how faith is not certainty, and the assumption that the universe is intelligible is believed as probable, through faithMetaphysician Undercover

    God transcends logic, fair enough. But again, that's not a logical argument.

    Not only is it pointless to believe it, but I would say it is actually negative. Choosing the direction that leads nowhere is actually bad when there are good places to be going to.Metaphysician Undercover

    Determinism is worse.


    I agree that it is very important to leave as undecided, anything which is logically possible, until it is demonstrated as impossible. Notice what I argue against is the assumption of real randomness, that is completely different from the possibility of real randomness.Metaphysician Undercover

    In that case we are entirely in agreement. I never pretend to know the ultimate nature of the world. It may be random, it may be determined, it may be a combination of both, or it may be something entirely else such that the random/determined dichotomy is rendered meaningless.

    That we ought to leave logical possibilities undecided was the point I argued Michael on the infinite staircase thread. Michael argued that sort of supertask is impossible, but I told him the impossibility needed to be demonstrated, and his assumption of impossibility was based in prejudice.Metaphysician Undercover

    But yes!! Here you are arguing that just because an idea is repugnant is no logical reason to reject it! So you should apply the same reasoning to randomness.

    I believe that paradoxes such as Zeno's demonstrate an incompatibility between empirical knowledge, and what is logically possible.Metaphysician Undercover

    I think it's highly unlikely that the world will turn out to be a mathematical continuum like the real numbers. The real numbers are far too strange.

    Most people will accept the conventions of empirical knowledge, and argue that the logically possible which is inconsistent with empirical knowledge is really impossible, based on that prejudice. But I've learned through philosophy to be skeptical of what the senses show us, therefore empirical knowledge in general, and to put more faith and trust in reason. So, to deal with the logical possibility presented in that thread, we must develop a greater intellectual understanding of the fundamental principles, space and time, rather than appeal to empirical knowledge. Likewise, here, to show that the logical possibility of ontological randomness is really impossible, requires a greater understanding of the universe in general.Metaphysician Undercover

    I agree with you there. I agree with most of what you wrote. Still I do want to understand why you see that @Michael is wrong to say that supertasks are logically impossible, when they are merely repugnant; yet you seem to reject that same reasoning when applied to randomness.

    Also, don't you think determinism is at least as repugnant as randomness?