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.
(Score: 2, Interesting) by Anonymous Coward on Tuesday April 14 2020, @11:36PM (7 children)
This cannot possibly affect a single physical calculation. If you can't construct the numbers you can't use them and you can't get them as results. But if you take this train of logic to its extreme, you run into the impossibility of enumerating the *rest* of the numbers unless you pick an arbitrary basis that rules out some computable numbers (this is the halting problem in action — by enumerating the computables we could approximate Chaitin's constant.) That sounds like a bigger ontological problem to me, and most people will probably agree that its better to remain agnostic about the ontology of the continuum than it is to introduce more cognitive overhead that still leaves us questioning which set of numbers to include in our ontology.
There are interesting ideas in the paper, but the only connection it has to physics is an attempt to explain something nobody is worried about anymore (spontaneous collapse — nowhere in the math is that a real thing) and/or as an introduction of new physics with no empirical motivation.
(Score: 2) by captain normal on Tuesday April 14 2020, @11:46PM
All the above comment sounds a lot like 2 Dimensional beings trying to figure out a 3 Dimensional universe.
https://en.wikipedia.org/wiki/Flatland [wikipedia.org]
When life isn't going right, go left.
(Score: 2) by HiThere on Tuesday April 14 2020, @11:50PM (3 children)
IIUC, there is a theoretical possibility that the universe is only meta-stable, and that it could collapse into a more stable form. But I don't think that conjectures about the continuity of the real number line would affect that, and last I heard nobody could think of any way to test whether it was true or not. In a way it's similar to "eternal inflation". It's an idea that's consistent with all that we know, but we know of no way to test either.
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 1, Interesting) by Anonymous Coward on Wednesday April 15 2020, @02:51AM
Vacuum collapse (or decay) is another beast, and even more unconcerned with incomputables. I mean the main way you get from one vacuum to another is by quantum tunnelling, and that doesn't really care much about intervening values. Where you tunnel to may be different than where you end up, but the difference is just gradient descent.
I was referring to wave function collapse, which the paper references. It was added, by edict, to allow us to pretend that the small portion of the total wave function that we seem to experience is all there is. It's not a genuine product of the physics, the physics more or less just says that our experience is also part of the wave function and our experience of the total wave function is spread over the wave function.... classical information isn't shared, so the net effect is each experiential continuity is isolated. (Sorry that sounds so awkward, I'm trying to describe it without talking about "Many Worlds." The point is there is a natural description of what is happening that has nothing to do with wave function collapse yet still explains the way the world appears to us.)
(Score: 2) by Muad'Dave on Wednesday April 15 2020, @12:30PM (1 child)
> there is a theoretical possibility that the universe is only meta-stable, and that it could collapse into a more stable form.
That sounds like it would be moderately painful.
(Score: 2) by HiThere on Wednesday April 15 2020, @03:04PM
Well, IIUC, it would spread somewhat faster than light, so you wouldn't even notice it. But you wouldn't be around afterwards to notice that it had happened.
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @10:33AM (1 child)
There'd be quite a lot to explore. For instance if we accept nondeterminism, is all nondeterminism created equally? In other words are things always nicely distributed, or might it be the case that factors might somehow influence the nondeterminism which could suggest a another argument against physical isotropy throughout the universe. And if we begin to reject isotropy it kind of upends basically everything we've ever known about anything about the cosmos.
(Score: 2) by c0lo on Wednesday April 15 2020, @12:45PM
We don't even know if ε0 and/or μ0 have the same value everywhere, we never measured it anywhere except in an exceedingly small part of this universe.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford