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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: 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"?

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  • (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.

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    • (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.

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      • (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.

              --
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              • (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.