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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.
(Score: 2, Interesting) by Anonymous Coward on Tuesday April 14 2020, @11:03PM (6 children)
Ok I hear you, but. It's not to do with infinite size; it's that there's infinite precision in the irrationals.
Q=rationals (expressible as fractions), Q=reals (which include irrationals and rationals). Q and R are not the same. If I have two elements, A and B, in Q, A*B and A-B and A^B and A/B so on are in Q.
Some mathematical constructs take an input in Q and return an irrational (in R, not in Q). However, those constructs ('functions' or other terms) either:
1) embed an irrational (eg. f(x) = x * 2 * pi)
2) use a limit (but 'two sided' proofs, where the limits in either direction get arbitrarily close to a result, define equality for limits as a corrollary of excluded middle iirc, BUT in this proposed system, the excluded middle proofs don't apply, which is easy to see since "arbitrarily close" means arbitrary precision and this system claims bounded precision, and 'choice sequences', like finding the Nth digit of Pi, can only ratchet by finite amounts, not from Q into R)
There are probably other ways but afaik they all fall into corrollaries or parallels of these.
(Score: 2) by martyb on Wednesday April 15 2020, @04:42AM (4 children)
Exactly! Two words: Planck Length [wikipedia.org].
Assuming we had a perfect measuring device, how many digits of precision could we measure the diameter of the visible universe? And, assuming we wanted to compute its volume, how many digits of pi would we need?
See, too: Orders of magnitude [wikipedia.org].
60 decimal digits should be enough for anyone!
Wit is intellect, dancing.
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @12:32PM
256 bits/value should be enough for everybody.
(Score: 1, Interesting) by Anonymous Coward on Thursday April 16 2020, @12:32AM (2 children)
"And this is basically how I think space in the universe works. Underneath, it’s a bunch of discrete, abstract relations between abstract points. But at the scale we’re experiencing it, the pattern of relations it has makes it seem like continuous space of the kind we’re used to. It’s a bit like what happens with, say, water. Underneath, it’s a bunch of discrete molecules bouncing around. But to us it seems like a continuous fluid."
https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/ [stephenwolfram.com]
search for 'What is space?' under that link and I think you'll be quite happy to take that section in. This is the basic concept behind Wolfram's big drop of papers this month (well summarized at https://science.slashdot.org/story/20/04/14/2112228/stephen-wolfram-presents-a-path-to-the-fundamental-theory-of-physics [slashdot.org] - maybe we lentils should plunder that?)
(Score: 2) by martyb on Thursday April 16 2020, @07:56PM (1 child)
Thanks for the links, I hope I can make time to check them out!
As for "plundering" Slashdot? Nope. If, by chance the same story should happen to be submitted to both sites, we have no way to see their pending story queue, nor can they see ours. So stuff may happen. That said, we have no interest in linking to a story on their site, or in having someone scrape a story appearing there and then submitting it here as their own. On the other hand, Slashdot is not entirely off-limits. If a story should break that they, for example, declared bankruptcy or had their site compromised, then that would be a different matter!
In short, we both target a tech-oriented audience, so there's bound to be some incidental overlap, but by no means do we encourage any kind of plagiarism.
That said, I'm guessing we are already running a story on the announcement? See: Stephen Wolfram: The Universe Runs on Automata Theory [soylentnews.org].
Wit is intellect, dancing.
(Score: 0) by Anonymous Coward on Saturday April 18 2020, @12:52AM
Hey I missed that we were running it. Thanks!
And by "plundering" I meant not automatically, but rather for this specific article, since their writeup was relatively good and well-linked. But no need, now, and I'll use the story submit function itself next time.
Thanks again for adminning this site and for being intellectually honest and pleasant. You're one of the IDs whose comments I look forward to reading, when I see them.
(Score: 2) by Muad'Dave on Wednesday April 15 2020, @12:26PM
> Q=rationals (expressible as fractions), Q=reals (which include irrationals and rationals).
Did you mean Q=rationals (expressible as fractions), R=reals (which include irrationals and rationals) ?