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posted by martyb on Tuesday April 14 2020, @09:38PM   Printer-friendly
from the how-many-digits-in-a-real-number? dept.

In a number system where the real numbers could not have an infinite number of digits, how would our physics models change?

Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.:

Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.

In Albert Einstein's theory of relativity, for example, time is woven together with the three dimensions of space, forming a bendy, four-dimensional space-time continuum — a "block universe" encompassing the entire past, present and future. Einstein's equations portray everything in the block universe as decided from the beginning; the initial conditions of the cosmos determine what comes later, and surprises do not occur — they only seem to. "For us believing physicists," Einstein wrote in 1955, weeks before his death, "the distinction between past, present and future is only a stubbornly persistent illusion."

The timeless, pre-determined view of reality held by Einstein remains popular today. "The majority of physicists believe in the block-universe view, because it is predicted by general relativity," said Marina Cortês, a cosmologist at the University of Lisbon.

However, she said, "if somebody is called on to reflect a bit more deeply about what the block universe means, they start to question and waver on the implications."

Physicists who think carefully about time point to troubles posed by quantum mechanics, the laws describing the probabilistic behavior of particles. At the quantum scale, irreversible changes occur that distinguish the past from the future: A particle maintains simultaneous quantum states until you measure it, at which point the particle adopts one of the states. Mysteriously, individual measurement outcomes are random and unpredictable, even as particle behavior collectively follows statistical patterns. This apparent inconsistency between the nature of time in quantum mechanics and the way it functions in relativity has created uncertainty and confusion.

Over the past year, the Swiss physicist Nicolas Gisin has published four papers that attempt to dispel the fog surrounding time in physics. As Gisin sees it, the problem all along has been mathematical. Gisin argues that time in general and the time we call the present are easily expressed in a century-old mathematical language called intuitionist mathematics, which rejects the existence of numbers with infinitely many digits. When intuitionist math is used to describe the evolution of physical systems, it makes clear, according to Gisin, that "time really passes and new information is created." Moreover, with this formalism, the strict determinism implied by Einstein's equations gives way to a quantum-like unpredictability. If numbers are finite and limited in their precision, then nature itself is inherently imprecise, and thus unpredictable.

Physicists are still digesting Gisin's work — it's not often that someone tries to reformulate the laws of physics in a new mathematical language — but many of those who have engaged with his arguments think they could potentially bridge the conceptual divide between the determinism of general relativity and the inherent randomness at the quantum scale.

[...] The modern acceptance that there exists a continuum of real numbers, most with infinitely many digits after the decimal point, carries little trace of the vitriolic debate over the question in the first decades of the 20th century. David Hilbert, the great German mathematician, espoused the now-standard view that real numbers exist and can be manipulated as completed entities. Opposed to this notion were mathematical "intuitionists" led by the acclaimed Dutch topologist L.E.J. Brouwer, who saw mathematics as a construct. Brouwer insisted that numbers must be constructible, their digits calculated or chosen or randomly determined one at a time. Numbers are finite, said Brouwer, and they're also processes: They can become ever more exact as more digits reveal themselves in what he called a choice sequence, a function for producing values with greater and greater precision.

By grounding mathematics in what can be constructed, intuitionism has far-reaching consequences for the practice of math, and for determining which statements can be deemed true. The most radical departure from standard math is that the law of excluded middle, a vaunted principle since the time of Aristotle, doesn't hold. The law of excluded middle says that either a proposition is true, or its negation is true — a clear set of alternatives that offers a powerful mode of inference. But in Brouwer's framework, statements about numbers might be neither true nor false at a given time, since the number's exact value hasn't yet revealed itself.

In work published last December in Physical Review A, Gisin and his collaborator Flavio Del Santo used intuitionist math to formulate an alternative version of classical mechanics, one that makes the same predictions as the standard equations but casts events as indeterministic — creating a picture of a universe where the unexpected happens and time unfolds.


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  • (Score: 5, Interesting) by Immerman on Tuesday April 14 2020, @10:11PM (24 children)

    by Immerman (3985) on Tuesday April 14 2020, @10:11PM (#982804)

    E=mc^2 is not relativity - it's the formula for one tiny aspect of relativity, the latent energy of a mass at rest. There's also no numbers in it. None. You only get numbers when you plug in specific values for the various symbols. Values that will be finite. Not just finite in size, but also finite in precision, which is a big part of what they just said intuitionism is about. Pi would be a good example - it's commonly understood to have a definite value of finite size but infinite precision, which ituitionism would reject.

    Also, there's an *immense* amount of evidence that General Relativity is wrong - it predicts the completely wrong rotational curves for galaxies for starters. We use "Dark Matter" and "Dark Energy" as explanations for those discrepancies - but in the absence of any direct evidence for either, they're really just a well defined "here there be dragons" description of the ways in which current theories of gravity fail.

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  • (Score: 4, Interesting) by DrkShadow on Tuesday April 14 2020, @10:31PM (23 children)

    by DrkShadow (1404) on Tuesday April 14 2020, @10:31PM (#982807)

    I'm not supporting or denying relativity. I have my own problems with that.

    I'm saying that mathematics is mathematics. We're using laws, as they relate between things. At the end, we can plug numbers in and get numbers out, but whatever their precision is doesn't impact the laws from which those numbers came. There's the concept of significant figures which applies here -- whatever physical constants you put in can only go so far. (Yes, they're taken to be "infinitely-precise", but that would be an error. There are error bars on our physical constants, and that is taken into account with calculations.)

    Pi, as an example, is not a "number with infinite precision". It is the relation between the circumference of an object and its radius. There are no numbers here -- it has no precision. It's just a relation. If you want to use it in a numerical form, then it is absolutely bounded by the number of digits that you can verify. Probably, that number is greater than any of the other numbers you're using in your calculations, so it's effectively "infinitely precise". It doesn't matter, as long as it's "at least as precise as the other numbers I'm using." "Infinitely precise" conversion factors are a high-school concept for simplification.

    Similarly, for any experiment that you're running, it is incorrect to calculate results based on a number lacking error bars. "Five-sigma" certainty applied to physical discoveries. There are error bars attached, and it's accepted that it may be wrong, however it's accepted that such an error would be very surprising and probably isn't worth considering.

    Perhaps this paper is all about stating what everyone already knows, and takes into account. Computing the rotation of galaxies is built on averages, using very tight conversion factors, sure; it's a model of the world, after all -- not the actual world.

    • (Score: 3, Interesting) by Anonymous Coward on Tuesday April 14 2020, @11:08PM (5 children)

      by Anonymous Coward on Tuesday April 14 2020, @11:08PM (#982814)

      Pi, as an example, is not a "number with infinite precision"

      Look, if you have a rational-valued diameter in a system where pi has finite precision (eg. in "base pi" instead of decimal "base 10" or binary etc), then the relation will have an irrational circumference. The "just a relation" embeds that irrationality in the relationship, no matter where you stash it.

      Look up proofs of irrationality of sqrt(2) for examples. Note that "square root of 2" is "just a relation" with an input, but I assure you, putting n=2 into sqrt(n) gives a different class of number than putting n=4 in.

      • (Score: 4, Interesting) by c0lo on Tuesday April 14 2020, @11:29PM

        by c0lo (156) Subscriber Badge on Tuesday April 14 2020, @11:29PM (#982825) Journal

        The author of TFA has a problem with both, saying [arxiv.org] that no construct of this Universe is going to allow you to measure π or sqrt(2) with infinite precision.

        1. “Truncated real numbers”.
        A first possibility is to consider physical variables as takingvalues in a set of “truncated real numbers”. This, as already noted by Born, would ensure the empirical indistinguishability from the standard classical physics: “a statement like x=πcm would have a physical meaning only if one could distinguish between it and x=πncm for every n, where πn is the approximation of πn by the first n decimals. This, however,is impossible; and even if we suppose that the accuracy of measurement will be increased in the future,n can always be chosen so large that no experimental distinction is possible”
        ...

        2. Rational numbers.
        Another possibility is to consider that physical quantities take value in the rational numbers, Q. Even if this sounds somewhat strange, one can argue that, in practice, physical measurements are in fact only described by rational numbers. Moreover, the use of rational numbers leads to those that can be named “Pitagora’s no-go theorems”. Indeed, positing a physics based on rational numbers, would rule out the possibility of constructing a physical object with the shape of a perfect square with unit edge or a perfect circle with unit diameter. In fact, by means of elementary mathematical theorems, their diagonal and circumference, respectively, would measure √2 and π, hence resulting to be physically unacceptable

        So, yes, DrkShadow is right in this case pointing the non-story character of TFA - "that's not mathematics, that's arithmetic. No physicist is going to pretend we'll be able to measure everything with infinite precision"

        --
        https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
      • (Score: 3, Interesting) by HiThere on Tuesday April 14 2020, @11:31PM (3 children)

        by HiThere (866) Subscriber Badge on Tuesday April 14 2020, @11:31PM (#982827) Journal

        The proofs of irrational existence all rely on the continuity of the real number line. If you deny that, then the proofs fall apart. And the proofs of that are basically "we can't see any other way it could work" or "well, you just keep repeating this operation an infinite number of times". But invoking infinity that way means it's not an algorithm. I.e., it's not something that could actually be done.

        P.S.: If you're allowed an infinite number of operations, then I have a method for trisecting an angle with only compass and straight edge. But it's invalid, because you aren't allowed an infinite number of operations.

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        • (Score: 2) by legont on Wednesday April 15 2020, @03:17AM (2 children)

          by legont (4179) on Wednesday April 15 2020, @03:17AM (#982901)

          Can we build a math where this continuity is violated?
          Let's call the smallest difference between two numbers a Plank's number p.
          Question. How many dots a circle with radius equal to one has? One can't use our regular formula with this precision because the dots on the circle would be on a curve that would have to comply with the same p.
          Let me be more precise. Say we have dots 1 and 2 next to each other on the circle. We also have dot 2a on tangent line going through dot 1 and next to 1. Distance between 2 and 2a can't be smaller than p as well.

          --
          "Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
          • (Score: 2) by HiThere on Wednesday April 15 2020, @03:10PM (1 child)

            by HiThere (866) Subscriber Badge on Wednesday April 15 2020, @03:10PM (#983082) Journal

            Yes. Various consistent approaches can be devised. Many of them use modular arithmetic, so that the nearly as large as possible flows smoothly into the nearly as small a possible. A friend of mine is convinced that this is correct, and has published formal theorems and proofs.

            Well, he's proved that the math is consistent. I'm not convinced that he's proved that we live in a space-time that matches his formulae. But it *is* consistent. I'm sure that other approaches exist, but I don't know anyone who's published proofs on them.

            --
            Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
            • (Score: 2) by legont on Wednesday April 15 2020, @04:38PM

              by legont (4179) on Wednesday April 15 2020, @04:38PM (#983114)

              This is actually great, I think. I am a mathematician by training but did not do any math for 30 year. I did always felt though that continuity approach was a shortcut. Greeks knew - or at least suspected - that the world is granular, but they ignored it in math for, I believe, simplicity. We may have a chance to build a more interesting math.

              --
              "Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
    • (Score: 2) by c0lo on Tuesday April 14 2020, @11:16PM (13 children)

      by c0lo (156) Subscriber Badge on Tuesday April 14 2020, @11:16PM (#982819) Journal

      Perhaps this paper is all about stating what everyone already knows, and takes into account.

      Not just perhaps, it becomes certain if you go read the arxiv paper linked [soylentnews.org] by maxwell deamon (surprisingly, TFA is not very obfuscated).

      His conclusion seems to be: "No matter what you do, you aren't going to experimentally obtain infinitely precise numbers; furthermore, I posit that any non-quantum physics can live with that, so stop pretending we can theoretically measure all things with infinite precision, the Universe doesn't work this way".

      Doh, I hope that he can sleep better now that he understood the idea of error bars, because I think this will be the only consequence of this article on the development of physics.

      --
      https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
      • (Score: 2) by HiThere on Tuesday April 14 2020, @11:41PM (12 children)

        by HiThere (866) Subscriber Badge on Tuesday April 14 2020, @11:41PM (#982832) Journal

        Actually, the claim that physics works quite well without the continuity of the real numbers is a very strong claim, with strong implications about what the theories mean. But (judging by the summary) he's allowing his philosophical disposition to color his understanding of what those implications are. It doesn't mean the EWG multi-verse isn't a correct interpretation, but it may imply that the possible universes are finite in number. Of course, finite doesn't mean small. AFAIKT it also doesn't remove any of the other standard interpretations of quantum physics except, possibly, super-pre-determinism. And I'm not sure about that one.

        N.B.: If the powerset of all energy-states of all locations in the universe is a finite number, then you've limited what can happen. But you probably haven't limited what you have even a faint possibility of noticing. What you've done is limit the passage of time to some trajectory between energy states in that powerset. (In practice it would be a lot more limited than that, of course.)

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        • (Score: 2) by c0lo on Wednesday April 15 2020, @12:33AM

          by c0lo (156) Subscriber Badge on Wednesday April 15 2020, @12:33AM (#982855) Journal

          Actually, the claim that physics works quite well without the continuity of the real numbers is a very strong claim, with strong implications about what the theories mean.

          I don't thinks so, not in any practical sense or even epistemological sense.

          N.B.: If the powerset of all energy-states of all locations in the universe is a finite number, then you've limited what can happen.

          I still have no problem with that. Some examples:

          • finite as it may be, the limits will be so humongous that the collapse of giant stars in a neutron ones (when the electron degeneracy can no longer support the gravitational pressure) is still allowed (see? an example of how a limited number of state postulated by the Fermi statistics actually explains/models what we observe in reality)
          • the ergodic hypothesis [wikipedia.org] posits that, given enough time, you will witness the event of the pieces of a broken teacup jump from the floor and reconstitute themselves in the original teacup. If the entire Universe would be made just from the floor, teacup and table, after the finite time of > 1020 the time of our Universe, such an event may happen. Do you think this would make any difference in how the human science is going to look like?
            (Recall that a Laplace demon cannot exist embedded in the Universe it is meant to predict [wikipedia.org])

          In other words, even finite limits is not going to modify the models that physics propose for the reality at human scales, if those limits are larger that the scientific humanity can handle. We're still going to use error bars and make predictions based on those models, even when they are accurate only at 20 bits (= 6·σ).

          ---

          Now, extension, do you think it would matter much on how we model the Universe if we don't consider "limited number of states" but the only restriction we set is that "the phase space of this Universe is totally quantified and all the values are countable (instead of being of the power of the continuum) but still infinite"?

          --
          https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
        • (Score: 2) by nishi.b on Wednesday April 15 2020, @01:36AM (10 children)

          by nishi.b (4243) on Wednesday April 15 2020, @01:36AM (#982877)

          I read an interview of this guy a few month ago, and it seems his motivation was his faith-based refusal to accept determinism because of free will.
          It was more or less like "I know I have free will. But according to the standard model of physics with cause and consequences, all my actions are determined by the past, therefore I have no free will. Therefore something must be wrong in physics".
          I did not really try after this to read more, it might still be interesting but the math aspects seem above my level.

          • (Score: 2) by nishi.b on Wednesday April 15 2020, @01:38AM

            by nishi.b (4243) on Wednesday April 15 2020, @01:38AM (#982879)
          • (Score: 3, Interesting) by HiThere on Wednesday April 15 2020, @03:14AM

            by HiThere (866) Subscriber Badge on Wednesday April 15 2020, @03:14AM (#982900) Journal

            Well, I tend to be a finitist, myself, to the point of only considering continuity to be a useful kludge in calculation. But I don't really see that this has any consequence WRT free will.

            FWIW, anyone who takes finiteness seriously runs into just as many weird things along the edge as those who accept continuity. And as both are consistent with anything we can observe, which you choose is a matter of taste.

            --
            Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
          • (Score: 3, Interesting) by maxwell demon on Wednesday April 15 2020, @07:45AM (6 children)

            by maxwell demon (1608) on Wednesday April 15 2020, @07:45AM (#982970) Journal

            I think this is based on a (very common) misunderstanding of free will. Randomness isn't free will, quite the opposite. If my actions are random, I don't do them because I want to, but I do them for no reason at all. An unintentional action is not an act of free will, it is an accident.

            Free will means that my actions are determined by my will. And as such, it is not contradicted by determinism. Free will is not about whether my actions are determined, it is about what determines my action. Namely whether the cause of my action is me. If I am the cause of my action, it is an act of free will. If something else (or nothing at all) is the cause of my action, it is not an act of free will.

            --
            The Tao of math: The numbers you can count are not the real numbers.
            • (Score: 2) by nishi.b on Wednesday April 15 2020, @06:25PM

              by nishi.b (4243) on Wednesday April 15 2020, @06:25PM (#983152)

              I agree with that, but from your own explanation it just moves the question to what constitutes "me". For people who believe in an immaterial soul that drives the body, "I" may be independent of the "physical" past, therefore seem to be random even when knowing the past. In my opinion, the past (my genes, nutrition, all life events) is what constitute "me" so having this determine my decisions is free will.

            • (Score: 3, Insightful) by Immerman on Thursday April 16 2020, @03:55AM (4 children)

              by Immerman (3985) on Thursday April 16 2020, @03:55AM (#983447)

              I would say the fundamental concept of free will, is that you have the ability to choose your actions.

              If your actions are pre-determined, so that it was known at the beginning of time exactly how much syrup you put on your pancakes this morning, then in what sense do you have any choice in your actions? You have the illusion of choice, granted by your limited perspective being unable to accurately see the future, but there was never any possibility of you choosing anything other than what you did. Or alternately the Many Worlds version, where all possible options *are* taken, and you have the illusion of choice because of your inability to see that you occupy all world-lines, and no choice was actually made.

              Randomness is certainly not free will either, not on its own. But in the blending of chaos and order there is at least the possibility of choice. In particular, if the probability of randomness can be intentionally manipulated (which theory and experiment say is the case at the quantum level), then there are at least a couple of possibilities:
                - an immaterial, metaphysical "soul" that is manipulates the randomness to steer our material self - that's a popular one I think
                - ordered systems operating in a chaotic environment can create a semi-predictable feedback system, such as perhaps a mind. A bad analogy would be a boatman sailing across a bay on a stormy day. They have an intended destination, but the fact that they're navigating an unpredictable environment makes their path unpredictable, even if their actions might be completely predictable within a predictable environment. Of course reality is more complicated than that - chaos is interwoven into our existence on a subcellular level - essentially the boatman and the sea are inseparably interwoven.

              Another interesting possibility is that the randomness isn't actually random, but only appears that way as it's the the result of choices made with truly free will. In essence, some primitive flecks of consciousness are actively exerting free will at a subatomic level. And just as our body is an emergent symbiotic cooperative of trillions of individual cells, each living its own life ignorant of the super-organism that it enables, so our consciousness may be an an emergent cooperative of the countless tiny flecks of consciousness of our particles. If our particles have free will then, insofar as we *are* our particles, so do we. That also tidily sidesteps some really thorny questions of how animate consciousness emerges from inanimate material: it doesn't - it's simply a question of how well different organizations of particles enables large-scale emergent behavior from networks of smaller consciousnesses to create a cooperative super-consciousness.

              • (Score: 2) by maxwell demon on Thursday April 16 2020, @06:03AM (3 children)

                by maxwell demon (1608) on Thursday April 16 2020, @06:03AM (#983473) Journal

                In particular, if the probability of randomness can be intentionally manipulated (which theory and experiment say is the case at the quantum level)

                I know quantum theory pretty well (I've worked professionally in that field for years), and I don't see any place in the theory where intentional manipulation of probabilities is possible (apart from the obvius one, if you manipulate the state through physical means, that of course also affects the probabilities). And I'm also not aware of any peer-reviewed experiments that probabilities can be intentionally manipulated.

                --
                The Tao of math: The numbers you can count are not the real numbers.
                • (Score: 2) by Immerman on Thursday April 16 2020, @01:19PM (2 children)

                  by Immerman (3985) on Thursday April 16 2020, @01:19PM (#983562)

                  Two examples that spring to mind
                  - the probability of radioactive decay changes extremely non-linearly over very short time periods, so that if you repeatedly measure whether it has decayed fast enough, you can make it arbitrarily unlikely that it will decay over a longer time period, effectively extending its half-life indefinitely. (as I recall the nonlinearities appear at *very* short timescales - fractions of a us. Longer timescales between measurements don't change the overall probability curve)
                  - If you measure the spin of a particle on one axis you "erase" all information about its spin on a perpendicular axis so that it will be 50/50 what you'll measure. However, if you measure its spin at a very small angle it's a near-certainty that it will be spinning the same direction as on the original axis - by making many such small incremental measurements you can choose the perpendicular spin with a high probability of success.

                  • (Score: 2) by maxwell demon on Thursday April 16 2020, @01:40PM (1 child)

                    by maxwell demon (1608) on Thursday April 16 2020, @01:40PM (#983573) Journal

                    Ah, that's what you mean. Yes, that's of course real, but that's changing the state through physical means (the measurement interaction). In particular, there is not necessarily intention involved; the very same can be done by a mindless computer-controlled measurement apparatus.

                    --
                    The Tao of math: The numbers you can count are not the real numbers.
                    • (Score: 2) by Immerman on Thursday April 16 2020, @02:08PM

                      by Immerman (3985) on Thursday April 16 2020, @02:08PM (#983587)

                      Certainly - but it's an channel that *might* be used by a non-deterministic immaterial observer (soul) capable of manipulating particle states in order to "drive" a material body.

                      And of course, all sorts of interesting things can emerge from feedback loops where random and deterministic systems mutually influence each other.

                      Not *will* emerge of course - but any feedback system that incorporates manipulable randomness at least has the possibility that it *might* offer genuine choice to an emergent consciousness - a choice which can't exist in either a purely deterministic or purely random system.

          • (Score: 1, Interesting) by Anonymous Coward on Wednesday April 15 2020, @12:58PM

            by Anonymous Coward on Wednesday April 15 2020, @12:58PM (#983032)

            Actually, I think that the standard model (of QED) allows for free will, while GR denies it.

            This is the philosophical rift of our two primary physical theories.

            It is funny because they both seem to work well to describe everything we see, but they fundamentally cannot both be correct.

            This has been the big problem with physics for over a century.

    • (Score: 4, Insightful) by Immerman on Tuesday April 14 2020, @11:28PM (2 children)

      by Immerman (3985) on Tuesday April 14 2020, @11:28PM (#982823)

      Yes, pi ids defined as the ratio of circumference to diameter - but try to compute the exact value of it, through any of a multitude of techniques, and you'll find that you'd need infinitely many digits to represent it.

      And, while it seems at first glance that the existence or non-existence of infinitely precise numbers would be irrelevant to computing formulas - that overlooks the fact that those formulas are themselves constructed based on a multitude of mathematical laws that themselves assume that infinitely precise numbers exist. The Law of the Excluded Middle which they mention for example, is a very common, and occasionally essential, tool in proving the validity of other laws. If the Law of the Excluded Middle is false, then so is every mathematical law which relies on it as an essential element of the proof. Which means you can no longer use those laws (or any laws that rely on them) when manipulating formulas. Whole families of algebraic manipulations that were believed to be possible without changing the truth of a statement, actually couldn't be, as the manipulated statement would implicitly incorporate the falsehood of the Law of the Excluded Middle.

      Basically, if infinitely precise numbers can't, in theory, exist, then you've fundamentally altered vast swaths of the underlying structure of mathematics. Eliminating vast swaths of algebraic manipulations that were falsely believed to preserve the truthhood of a statement, while introducing vast swaths of undiscovered new manipulations that we would currently dismiss as trivially false.

      You could still plug numbers into the old formula - but the formula would no longer be accepted as accurately representing the underlying theory.

      • (Score: 2) by c0lo on Wednesday April 15 2020, @01:38AM (1 child)

        by c0lo (156) Subscriber Badge on Wednesday April 15 2020, @01:38AM (#982880) Journal

        And, while it seems at first glance that the existence or non-existence of infinitely precise numbers would be irrelevant to computing formulas - that overlooks the fact that those formulas are themselves constructed based on a multitude of mathematical laws that themselves assume that infinitely precise numbers exist.

        The author of TFA doesn't have a problem with maths, it has a problem with applying maths to a chaotic universe in which we can't measure things with absolute precision.

        He seems to imply that's a big deal, but me thinks is a non-problem, at least not in physics (philosophers may philosophate, the Universe doesn't care)

        --
        https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
        • (Score: 2) by MostCynical on Wednesday April 15 2020, @02:46AM

          by MostCynical (2589) on Wednesday April 15 2020, @02:46AM (#982893) Journal

          number of possible 'next states' is either infinite or finite.

          If infinite, author of TFA thinks he has free will

          if finite, author of TFA thinks he can't have free will

          if finite, but so uncountably large, that the heat death of the universe will occur before you count them, is that close enough to infinite not to matter to humans? Author of TFA thinks the distinction matters. Anyone not trying to justify having free will or a soul or whatever doesn't care.

          "huge" is close enough to "infinite" for most people. Drawing any circle is possible, so then announcing that it is a unit size of whatever units equal one diameter of that circle proves a unit circle can be drawn.

          Author needs to let go of his hang ups.
          Here, have some soul [youtube.com]

          --
          "I guess once you start doubting, there's no end to it." -Batou, Ghost in the Shell: Stand Alone Complex