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) by hendrikboom on Wednesday April 22 2020, @09:18PM (3 children)
Classically, you can prove P OR Q without ever getting, or even being able to get, a clue which of the two holds.
To be specific, classically, P OR NOT P (it's the law of the excluded middle) holds for any proposition P whatsoever, and you don't even have to be able to find out whether P holds or NOT P holds to be able to assert P OR NOT P.
Long after Brouwer discovered intutionistic mathematics (about the beginning of the 1900's) Godel proved there were sentences that could neither be proved not disproved. Such a Godel sentence G is still considered classically to satisfy G OR NOT G. Hence the statement that there are true but unprovable statements. (Classically, either G is one, or NOT G is one.)
Now proofs can be complicated (just like computer programs) so intuitionistically, once you have proven P OR Q, it may not be obvious which of the two holds, but it is possible to unravel the proof and find out which. It may take a long time, but will not take infinite time.
Now even intuitionistically there are propositions P for which P OR NOT P holds. In particular, it holds for arithmetic P that can be decided by simple, termination, deterministic calculation.
But it doesn't hold for *all* propositions.
(Score: 2) by martyb on Thursday April 23 2020, @11:59PM (2 children)
Thank you for that! I was exposed to the Godel numbering theorem in college and later tool it upon myself to read Godel, Esher, Bach. Perfect example and explanation.
My brain hurts a little, but I could follow right along. I very much appreciate the time and effort you put into your reply!
Wit is intellect, dancing.
(Score: 2) by hendrikboom on Saturday April 25 2020, @12:20PM (1 child)
I think I should use this question-answer session to inform the content of the two pages on my website that relate to constructivism. Thanks for the discussion.
(I believe those two pages are linked from my original post, in case you haven't seen them yet)
Or feel free to write me offline at hendrik at topoi.pooq.com
(Score: 2) by martyb on Tuesday April 28 2020, @01:56PM
I enjoyed the discussion; thanks for the feedback! There was something about what you posted that poked at my understanding of Boolean Algebra. Your clarifications were sufficient, but I'm glad our chat could help your constructivism entry. That's good enough for me.
Wit is intellect, dancing.