Comments

  • What did you mean by "believe"?
    Not this stupid shit again
  • 3 orbiting black holes can break temporal symmetry
    ↪Pfhorrest
    Classically, the three-body problem is time-reversible, and this result doesn't prove otherwise. Indeed, qualitatively this result doesn't prove anything new: the three body problem is already known to be chaotic in the technical sense, i.e. for a certain class of initial initial conditions, any disturbance, no matter how small, results in an unbounded divergence over time.

    The direct result of this work pertains to numerical simulations of a class of three-body problems. The claimed physical relevance comes from making numerical errors (which act as perturbations) smaller than the Planck length.
  • 3 orbiting black holes can break temporal symmetry
    ↪jorndoe
    Thanks, this is interesting!

    Here is the full paper: Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length

    I'm not clear about this. I've always assumed (and I could be very mistaken) that "time reversibility" is just a quirk arising when describing a physical process using mathematics. The two are not the same.

    "And they have shown that the problem is not with the simulations after all."

    Well, they're doing computer simulations in an environment of exceptional chaotic behavior. So I don't know what to think about reversing the actions.     — jgill

    Physics enters the picture when they show that in some fraction of initial configurations the sensitivity to initial positions is so high that a displacement of a magnitude less than the Planck length can result in divergent solutions. They interpret this result as the system being "fundamentally unpredictable" when it starts from one of those configurations.

    Mathematically, if we don't take into account the Planck length limitation, the system is still only chaotic at most, and therefore fully time-reversible.

    By the way, the choice of supermassive black holes is only for astronomical verisimilitude, because in their solution they still use the Newtonian approximation, as in the classic n-body problem.

    As a concrete application of our result, we consider three black holes, each of a million solar masses, and initially separated from each other by roughly one parsec. Such a configuration is not uncommon among supermassive black holes in the concordance model of cosmology and hierarchical galaxy formation... [W]e estimate that the closest approach between any two black holes is on average between 10-2.5 and 10-2 parsec, during which the Newtonian approximation still holds. A parsec equals 1051 Planck lengths. Hence... we estimate that up to 5 percent of triples with zero angular momentum are irreversible up to the Planck length, thus rendering them fundamentally unpredictable. — Boekholt et al.
  • If max speed of light (C) is constant does that mean distance measurements are consistent?
    Treated separately by who? Stephen Hawkings nor my Physics Professor ever said that there were not absolute points in space. — christian2017

    I can readily believe that they never said that, because they wouldn't even know what that means. You can't even explain what you mean, so I suspect that you don't know what you mean either.

    I'm currently reading Einstein's book called "Relativity". It will probably take me 2 years to read that book. — christian2017

    It's a popular book aimed at non-physicists, so you shouldn't have so much trouble with it. But I think you (and Jeff from Youtube) should start from the basics: non-relativistic classical physics. For instance, the question of what it would be like for someone to move at a constant speed - whether they would feel any different than if they were staying put - was considered by Galileo back in the 17th century. Einstein only refined that treatment, but to understand what Einstein did and why, you first need to understand Galilean relativity.
  • Notes From The Underground- Dostoyevsky
    Also in the paragraphs where he accuses the Jews for their demonic power of hatred towards the Russians in particular and Humanity in general? Do you enjoy these paragraphs? Also in the poems in which he manifests a doglike submission to the divine presence of the Tsar? — David Mo

    I don't recall reading either of these, although I am aware of such sentiments by reputation. It is odd though that you should expend so much energy digging up the worst. What is your interest in Dostoevsky ?

    Can aesthetic pleasure silence moral outrage? — David Mo

    I don't know if I would call the experience of reading Dostoevsky an esthetic pleasure. He was not a fine stylist in the usual sense (for that try someone like Turgenev). There is a wicked pleasure to be had in his caustic humor, but when Dostoevsky is in his more serious mood, reading him is about as pleasurable as a hallucinatory fever.
  • Currently Reading
    I found Blindness by José Saramago to be the most terrifying thing I have ever read.
    Its perfect logic sticks to everything I wonder about.     — Valentinus

    I tried reading it a while ago, but... ugh.
  • Notes From The Underground- Dostoyevsky
    The problem is different for me: How can a rational man enjoy the writings of a fanatical believer in God and the Czar, such as Dostoevsky? Can aesthetic pleasure be separated from ideological fanaticism? — David Mo

    I haven't read Notes From The Underground, but I have read some of his other works (C&P, Karamazovs, Idiot, and a few others), so I can comment on those. Dostoyevsky the writer transcends Dostoyevsky the thinker. You (and I) may not much care for his politics, his religion, his philosophy, but that does not detract from the power of his best works. Reading a Dostoyevsky novel is a life experience; you don't walk away from it unaffected.

    The same can be said about Tolstoy - and about any great artist. That is what makes them great: we value them not (or not only) for their ideas, but for their art.
  • The Diagonal or Staircase Paradox
    Because the OP does not specify an axiomatic system but describes the problem essentially in Euclidean geometry. — boethius

    Well, no, it doesn't, because there isn't any problem so long as we stay with Euclidean geometry (as
    ↪tim wood
    rightly noted). The apparent problem only arises when we introduce the notion of a limit, and perhaps other implicit assumptions.

    Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points? How can this be? — jgill

    maybe we're interested in investigating the corners and want to deal with what happens when, trying to take the limit of shrinkifying the stair lengths, essentially every point becomes non-differentiable (that the object is "only corners", or at least all the rational points are defined as corners or some kind of scheme like this; may or may not be of interest to people here). — boethius

    "Almost none" of the limit points on the diagonal (let's just call it that for brevity) is a corner point, for the simple reason that there is only a countable number of them. Also, keep in mind that the diagonal (which we interpret as the limit point of the sequence of curves) is not itself part of the sequence and does not have the same properties. Every member of the sequence is piecewise-differentiable, while the diagonal is, of course, everywhere differential.
  • Coronavirus
    This is exactly what I explain in the sentence you reference. If in some time frame of interest (such as "until now"), the data fits an exponential growth curve, scientists will say "it is growing exponentially". — boethius

    There is always some time frame in which data fits an exponential growth curve! Or logarithmic. Or linear. Or better yet polynomial - it can be made to fit any curve over any time scale. But no scientist in their right mind would propose an exponential growth model just because you can fit an exponential curve to two consecutive points. This is not how scientific modeling works.
  • Coronavirus
    Does this satisfy your doubts that the scientific community describes things as growing exponentially if, in some time frame their interested in, the phenomena does grow exponentially? — boethius

    Yeah, the scientific community describes things as growing exponentially for as long as they grow exponentially. You may insist on exponential growth if you think you have a good handle on the causal mechanism, and can account both for the function and for the changing exponent, without having to make retrospective adjustments after each new measurement. What you don't do is say: "Oh it's still exponential, because we can still express it as a percentage increase." Because that is just cargo cult science.
  • Coronavirus
    ↪fdrake
    It's great, isn't it? If you keep readjusting your model as you go along, you can do pretty much anything. You can make your model linear, or logarithmic, or... why even settle on one function? You can make up a new one for each iteration.
  • Coronavirus
    Apparently, it was originally China's idea. — Baden

    Any bets on whether Trump will start calling chloroquine The Chinese Cure?
  • Coronavirus
    is fatal only among the already very compromised — Hanover

    Yeah, those sick old people had it coming - good riddance!

    (I hear this "point" surprisingly often from people who argue that the threat has been blown out of proportion.)

    Italy had 475 deaths yesterday. With population of 60.5 million and annual mortality rate of 10.2 per 1000, that's 25% increase of mortality rate (even if we allow that, say, 10% of those who died from coronavirus would have died from other causes within the same period).
  • The Diagonal or Staircase Paradox
    ↪boethius
    OK, I didn't notice that you said continuously differentiable in a later post, so sorry about that. But the staircase function is worse than not continuously differentiable - it is not differentiable at all in some points. In any case, no one has actually tried to show that not being continuously differentiable is even a sufficient criterion for the non-convergence of the length difference, and in fact it is easy to construct counterexamples. So all this arguing is kind of a waste of time.

    As for solving any of them. You'll need to do so relative an axiomatic system. If it's Euclidean geometry — boethius

    I am not sure why you keep talking about Euclidean geometry, which, as you admit, doesn't even have the notion of a limit. You may as well be talking about group theory. Yes, I think it's pretty obvious that we are talking in the context where limits and such are defined; real analysis will do for the purpose.

    For any continuous function like whose arclength for a <= x <= b is greater than b-a, its scaled down versions will still have the same ratio of arclength to b-a. So just about any continuous function at all that's not a constant. — Daz

    No, if the arc length decreases any faster than in the examples that have been considered so far, it will converge to the length of the diagonal, as we intuitively expect. This is easy to do with any function whose distance from the diagonal can be scaled. For example, take the half-circle function and scale the peaks down by a factor of n - it will converge like a champ.

    But yes, it is evident that whether the length error is constant, converging to zero or growing without bound is pretty precarious. You have to work to make sure that you get the "intuitive" result, because a lot of the times you will get something else entirely.

    You can obtain the result of the other "paradox" by drawing a symmetrical sawtooth graph on [0,1] that collapses as n increases, and whose length increases without bound. I leave this as an exercise for those interested. — jgill

    Yep, if your sawtooth graph doesn't have this property out of the box, you can easily make it so by multiplying it by some uniformly increasing function of n.


    The reason this looks very counterintuitive to me is because if we put aside analysis and just look at what we get in the limit, every point on the staircase converges towards a matching point on the straight line - which of course has the length of the straight line. So what gives? Well, the formal answer is that the limit towards which the sequence is converging is not an element of the sequence: the limit points do not themselves lie on any staircase curve. This is not so unusual; for example, most converging rational number sequences do not converge to rational numbers.

    Still, it just looks... wrong :)
  • The Diagonal or Staircase Paradox
    My main purpose, as mentioned, was just to explain the definition of "discontinuous" and that normal calculus concepts may not apply. — boethius

    Er, your terminology is all over the place. A continuous function has left and right limits converging to each of its points. The staircase function is, strictly speaking, discontinuous as it is pictured in the OP, but that is just an artefact of the coordiate system. If you tilt the X-Y axes, it will become continuous.

    A differentiable (or smooth) function has the first derivative at each point; the half-circle function is differentiable (again, modulo axis orientation).

    An infinitely differentiable function has all derivatives; the sine function is infinitely differentiable.

    There are also piecewise- versions of all these (piecewise-continuous, etc.).

    You are right, it's a sufficient condition for the failure of the arc-length functional to respect the limiting procedure, not a necessary one. I believe the staircase could be approximated by some differentiable curve (replace the discontinuities with regions of sufficiently high growth, I believe polynomials would work) and cause the same issues. — fdrake

    Yes, that's just what I did with the half-circle curve, and I think the sine curve (with proper scaling) would work as well.

    Do you know a sufficient and necessary condition that characterises this sort of pathology? Other than stating "the arc-length map of the limit of the approximating series of functions is not necessarily the limit of the arc-length map of the approximating series of functions". — fdrake

    Interesting question, but beyond my modest pay grade, I am afraid :)
  • The Diagonal or Staircase Paradox
    ↪boethius
    The half-circle wave is smooth though, i.e. the tangent (first derivative) exists everywhere. I used it for simplicity, but if we want an honest to goodness infinitely differentiable curve, we can use the sine function as in John's other example - just scale it by an additional factor of to make it behave. (I think this will work...)
  • The Diagonal or Staircase Paradox
    ↪fishfry
    It's in the second half of
    ↪jgill
    .

    There must be some neat identity for elliptical functions at work here, because otherwise I wouldn't know how to calculate such a limit.


    By the way,
    ↪fdrake
    and
    ↪boethius
    seemed to suggest that the key to the staircase "paradox" is in some pathology of the shape, namely its corners, where the curve is not differentiable. But this is not so. Consider a similar example, where in place of straight lines there are smooth curves. I'll use half-circle arcs for simplicity:

    FyE4o.png

    No corners here, the curve is everywhere differentiable (although the second derivative does jump around at the intersections). As with the staircase, the amplitude of the wave tends to zero as the number of crests increases without bound. But as with the staircase, the length of the curve does not approach the length of the diagonal. We don't even have to do the calculation to see that the length of the curve does not depend on the number of crests (this is because the length of each half-circle is proportional to its diameter, and the total length of all diameters is the length of the diagonal). And so the length of the wavy curve is always .

    What's more, with a simple modification we can make the length of the wavy curve increase without bound, just as in @jgill's example. Just replace half-circles with ellipses whose major axis is perpendicular to the diagonal and scaled by a factor of .
  • Is Bong Joon Ho's Parasite Subversively Conservative?
    I thought back to Parasite after seeing another film with a somewhat similar theme. To be honest, for all its sleek execution and obvious talent, Parasite turned me off by its heavy-handed messaging (in retrospect, the Oscar win wasn't very surprising). The dream sequence at the close of the film is a good illustration of that. At first I thought that the filmmakers were finally going for a little subtlety by ending the film right there at the happy reunion. Let the viewers work it out for themselves and then congratulate themselves on their perspicacity. But no, they had to drive the message home, lest someone might be confused by the ambiguity, and so they cut back to the half-crazed boy sitting in his squalid banjiha.

    The socially conscious film that I would like to recommend as an antidote is Shoplifters (2018) by the Japanese director Hirokazu Kore-eda. And if you like that, see Nobody Knows (2004) and Still Walking (2008), which I think are even better. I hear that Kore-eda's output is uneven, but these three are excellent.

    Also, anything by the Dardenne brothers.
  • Q. on Fallacy of False Dichotomy
    ↪Tkpa007
    Well, if someone says something to the effect of (A or B), and it is not the case that (A or B), then a logical fallacy has been committed. How damaging that is to their overall argument depends on the context. Sometimes, as you say, not very damaging, and the argument can be repaired without much difficulty.
  • Regulating procreation
    the popular myth merely unilateral or blanket statements based on some silly and highly questionable pop cultural myth or axiom accepted or taken for granted on the basis of faith, nonsensical circular reasoning and rote regurgitation outdated 19th century myths and archaisms archaic and highly debatable or questionable or easily disprovable and contradictroy — IvoryBlackBishop
  • Thought as a barrier to understanding
    ↪Antidote
    I'll try not to think about what you just said.
  • Thought as a barrier to understanding
    If thought were the natural outcome or effect, brought on by confusion, then the more you think, the more confused you will get. — Antidote

    That's a nice example of a self-undercutting argument. If the premise is assumed, then everything that follows from it can be dismissed as confused ramblings. No need to go any further.
  • Intuitions About Time
    Think Heraclitus and Parmenides. — Pneumenon

    I was looking for something in the way of critical reflection, but I find only free-floating metaphors here. I've also been thinking about the metaphysics of time lately, but I prefer a more grounded approach.
  • Intuitions About Time
    Take these two:

    1. Reality is fundamentally flux, and permanency is constructed
    2. Reality fundamentally is, and change is an illusion     — Pneumenon

    I don't even know what either of these statements are saying. It seems like they make some substantive claims about reality, but when I try to nail these claims, they just slip out of my hands.
  • The Diagonal or Staircase Paradox
    I find no error in this. — tim wood

    There is no error, hence the apparent paradox.

    Here is another take I just thought of. We can define two functionals, one that gives some measure of the distance between the points of the stair functions and the diagonal (e.g. the average distance), and the other that gives the length of those functions. The shape functional steadily converges to zero, but the length functional does not.
  • The Diagonal or Staircase Paradox
    ↪jgill
    Nice.

    he number of "stairs" tells something similar how polygons start resembling a circle: — ssu

    The difference is that in the case of polygons approximating a circle, with each successive step the error decreases (don't ask me for a proof - it's pretty messy, from what I remember), whereas in the case of the staircase the error stays the same throughout.
  • Can I deal with 'free will' issue like this?
    People define free-will in different ways. And so they argue about different things. But it really goes back to the concept of "you". You like others, will say you have a body, you have a brain, you have... maybe a spirit or soul... two arms and two legs. Who is "you"? The idea of there being a "you" and the continuation of self is intertwined with all definitions of free-will. — Malice

    Nature as in, our exact state. — Malice

    Our nature is not identical with our physical state. I am not making some dualistic statement here - I am just agreeing with what you said earlier that I quoted here. What constitutes "you" or "your nature" isn't simply reducible to your physical body (or else your personal identity would be extremely volatile). It is a somewhat vague notion rooted more in psychology and culture than in physics - you know, like "free will."
  • Can I deal with 'free will' issue like this?
    The solution to deal with its demoralizing power. ‘Solution’ sounds confusing, I’ll change that — Rystiya

    What does "its" refer to here? You never say.
  • Secular morality
    What do you even mean by "being moral"? — Pfhorrest

    Making right moral decisions. What is right is, of course, the very subject of morality.

    The criteria for the success of what? A moral science, or generally any system of morality? The criteria for success of those things is to provide a means of answering questions about morality. When someone wonders what is moral, how do they figure it out? When two people disagree about what is moral, how do they resolve those difference? Answering how to do that, how to figure out those answers to questions about morality, is the criteria for the success of a system of morality. — Pfhorrest

    So any procedure for answering moral questions will do, as long as it is comprehensive? No other criteria of success are required?

    That you think I'm even trying to do that shows you haven't understood a word that I've said so far. — Pfhorrest

    If by that you mean that I haven't read your articles on morality, then no, I haven't. That wasn't the subject of this thread.

    I predict you'd respond here "aha! So you're starting with a system of morality already, your 'ought' premises, just like I said!" But no, no more than the physical sciences start with some set of unquestionable "is" premises. — Pfhorrest

    That sort of Cartesian scheme that you outline doesn't remotely resemble the way science is done. But even if your approach was better at aping science, that still wouldn't make it any better than a cargo cult, because you still aren't thinking about why you do what you do. Why should morality resemble science?
  • Do professional philosophers take Tegmark's MUH seriously?
    It's certainly incompatible with materialism. A mathematical ontology isn't compatible with there being stuff, so I don't see how it's physical. But I guess if we're allowed to redefine the meaning of "physical" to be whatever is consistent with physical models. — Marchesk

    There is nothing to redefine here, because there aren't any commonly established definitions for physicalism or materialism. All these arguments over this or that being compatible with physicalism/materialism is pure wankery, in my opinion. Argue substance, not isms. (Same goes for realism, of course. Crispin Wright famously wrote that "a philosopher who asserts that she is a realist about theoretical science, for example, or ethics, has probably, for most philosophical audiences, accomplished little more than to clear her throat.")
  • Can I deal with 'free will' issue like this?
    It seems that it's hard to say whether we have free will or not. — Rystiya

    It's hard to say what free will is. It's hard to even get people to think seriously about the question: they would rather argue endlessly about "free will" than think about the question that ought to be addressed before anything else.

    The solution is simple — Rystiya

    The solution to what?
  • Secular morality
    ↪Pfhorrest
    Because while you busy yourself with procedural details of how to reduce morality to a utilitarian optimization, you don't ask what any of that has to do with being moral. What are the criteria of success (other than aping the superficial trappings of science)? How do you jump the is-ought gap?
  • Mathematics is 75% Invented, 25% Discovered
    ↪Nagase
    Thanks, I got it when I thought about it again. I got into a habit of thinking of numbers as abstract objects that just satisfy certain requirements, like Peano axioms, whose representation and internal structure need not concern us. I lost track of the obvious idea that a number can also be thought of as a collection with that many distinct objects - a cardinal*. When you mentioned von Neumann construction, I thought about Zermelo's construction with its Russian doll structure - that wouldn't work so well. I guess that's one of the advantages of the von Neumann model, right?

    * Heh, now that I think about it, that was exactly how numbers were introduced to us in preschool, using little sacks with different amounts of marbles :)
  • Mathematics is 75% Invented, 25% Discovered
    ↪GrandMinnow
    Being correct and being clear is not the same thing. I took ten or so different courses of mathematics in college. It's been a long time and I have forgotten much of it, but I still understand basic combinatorics, thank you very much.
  • Secular morality
    ↪Pfhorrest
    "Methodically" is the key word here. Yes, you have a method, but that's not enough to qualify as "science." If you don't understand why you do what you do, then your method is no more scientific than astrology or divination.
  • Mathematics is 75% Invented, 25% Discovered
    ↪GrandMinnow
    This wasn't very helpful.

    ↪Nagase
    It's still not clear to me how the use of this set theoretic representation explains "the number of functions from 2 into 2 is 4". Whatever representation you use, you still have one number (in this case, one set) in the domain and one number (one set) in the codomain. I mean, I can see (after your explanation) what he is doing, but I am not even sure how to formulate that correctly without specifically referring to the internal structure of a von Neumann ordinal.
  • Mathematics is 75% Invented, 25% Discovered
    The number of functions from 2 into 2 is 4. — GrandMinnow

    I don't understand.



    How is this four functions?

    (FWIW in calculus texts the exponential function is usually defined generically as a power series.)
  • Secular morality
    All we need for an "ethical science" is that kind of broad agreement. — Pfhorrest

    Why do we need an "ethical science"? You never stop to ask yourself this question. This is cargo cult.
  • Mathematics is 75% Invented, 25% Discovered
    2^0 = 1

    The number of functions from 0 into 2 is 1.     — GrandMinnow

    x^y = the cardinality of {f | f is a function & domain(f) = y & range(f) is a subset of x}. — GrandMinnow

    OK, but by the same token 2^2 = 1, because the number of functions from 2 into 2 is 1. What am I missing?
  • Mathematics is 75% Invented, 25% Discovered
    x^y may be defined as the number of functions from y into x. — GrandMinnow

    What do you mean by that?
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