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, Insightful) by Anonymous Coward on Tuesday April 14 2020, @09:44PM (1 child)
But stop hogging the doobie and pass it around.
(Score: 3, Funny) by Runaway1956 on Wednesday April 15 2020, @02:11AM
Is that you, Uncle Larry? No one else talks like that anymore. I'll just get off your raggedy looking lawn . . .
(Score: 5, Informative) by maxwell demon on Tuesday April 14 2020, @09:53PM
The paper without paywall: https://arxiv.org/abs/1909.03697 [arxiv.org]
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by DrkShadow on Tuesday April 14 2020, @09:56PM (36 children)
E=mc^2.
Yes, there's a number in it. Two. Not infinitely-large -- it is, indeed, impossible to compute with infinitely-large numbers.
m=E/c^2
Mathematics is the relation and manipulation of things. Sometimes simple numbers come in as a simplification:
y=2x
We could write them without the numbers (y = x * x), but... why. Y is equal to two of X. Mass is equal to energy proportional to the square of the speed of light. Mass is equal to energy proportional to the speed of light in two dimensions.
Ooph. In English, it's hard to say. So we derived mathematics for a rigorous way of saying things. Sometimes that leads to incomprehensible English: sqrt(-1). (More ugliness: there are actually two answers to sqrt(2): -sqrt(2) and +sqrt(2). -1.41420.. and 1.41420.. . 2 = (+/- sqrt(2))^2.)
We're not using "infinitely-large numbers". We're using relations between things. The physical world is _things_, and they relate, and we've stated "mathematical laws" for how they relate. All of these are proportions between things, and sometimes a few simple numbers come out of them.
Thus far, this has worked relatively well. Even now, there is no research that I'm aware of saying that these laws fail at any given scale -- just observations that the laws don't tend to emerge at one scale or another, but in a way that doesn't preclude these mathematical relations.
So what was this article saying about numbers and arithmetic, again?
(Score: 5, Interesting) by Immerman on Tuesday April 14 2020, @10:11PM (24 children)
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.
(Score: 4, Interesting) by DrkShadow on Tuesday April 14 2020, @10:31PM (23 children)
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)
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
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.
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)
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.
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, @03:17AM (2 children)
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)
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
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)
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)
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.)
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 2) by c0lo on Wednesday April 15 2020, @12:33AM
I don't thinks so, not in any practical sense or even epistemological sense.
I still have no problem with that. Some examples:
(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)
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
I found a similar link to the one I read :
https://www.unige.ch/communication/communiques/en/2020/une-physique-indeterministe-pour-un-monde-plus-ouvert/ [unige.ch]
(Score: 3, Interesting) by HiThere on Wednesday April 15 2020, @03:14AM
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)
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
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)
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)
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)
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)
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
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
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)
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)
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
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
(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) ?
(Score: 1, Insightful) by Anonymous Coward on Wednesday April 15 2020, @12:44AM (2 children)
"E=mc^2.
Yes, there's a number in it."
No, there's not a number in it. It's used to denote a square operation.
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @04:50AM (1 child)
I sure as hell see the number 2 there, it is literally a number used to write down the last character of the equation. A number "2" at that!
(Score: 3, Insightful) by Immerman on Thursday April 16 2020, @04:09AM
Integer exponents are just shorthand for multiplication
E=m*c*c
If you look at the original derivation of the formula, you'll almost always find that they first appear as multiplication before being re-written in shorthand
As a matter of fact you can say the same about most fractional exponents, becoming fractional when an equation is manipulated solved for a symbol that had been multiplied by itself.
(Score: 0) by Anonymous Coward on Thursday April 16 2020, @07:19PM
(y=2x) != (y = x * x)
you either meant:
(y=x^2) == (y = x * x)
or you meant:
(y=2x) == (y = x + x)
Math is confusing enough before not proofing your logical equivalence :)
(Score: 3, Funny) by looorg on Tuesday April 14 2020, @09:56PM (22 children)
What Einstein May Have Gotten Wrong? His choice of a barber? Beyond that I guess it was mostly fine.
(Score: 0) by Anonymous Coward on Tuesday April 14 2020, @11:07PM (2 children)
He was wrong about Λ. Not the first time, the time he second-guessed himself about it.
(Score: 2) by maxwell demon on Wednesday April 15 2020, @07:51AM (1 child)
Actually he was wrong the first time. because he introduced it for the wrong reason: Because he believed the universe to be static. Which was then disproved by observation, which is why he then removed that constant.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @08:21PM
Thing is we use Λ in theory, not just by setting it to match observation, but by setting it to match Einstein's desired universe and others too. Getting the guessing game wrong is nothing compared to leaving it out. Someone would have to come along and *fix* his work in the latter case.
(Score: 2) by PartTimeZombie on Tuesday April 14 2020, @11:13PM (16 children)
Ethanol-Fueled will be along shortly to point out how Einstein was a Jew, and therefore relativity is just a plot by Zionists to help Mexicans destroy the purity and essence of our natural fluids.
Also, Einstein should have just stayed in Berlin, nothing bad would have happened to him, oh no.
(Score: 0) by Anonymous Coward on Tuesday April 14 2020, @11:22PM (4 children)
He got fired, he's probably close to death by alcohol poisoning by now.
(Score: 1, Troll) by Ethanol-fueled on Tuesday April 14 2020, @11:33PM (3 children)
Like every other time I got fired in history. And the only thing I can actually point out here besides anti-Jew shitposting is that the Jews who wrote Star Trek: The Next Generation mentioned the cosmological constant in the episode where Broccoli becomes smart. Wasn't that the cosmological constant discredited even by Einstein himself well before then? Or did that re-become relevant because Stephen "Getting his limp dick sucked on Epstein Island" Hawking -re-proved it along with the egg that unbreaks and magically jumps back to the countertop in the future?
That's why I leave this kind of shit to the real boss niggers of math. They know what they're doing.
(Score: 2, Touché) by Anonymous Coward on Wednesday April 15 2020, @12:04AM (1 child)
Not quite dead yet, but the signs of irreversible brain damage are manifesting. He's arguing with TV shows now.
(Score: 3, Funny) by MostCynical on Wednesday April 15 2020, @02:49AM
at least it's Star Trek, and not Judge Judy
"I guess once you start doubting, there's no end to it." -Batou, Ghost in the Shell: Stand Alone Complex
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @05:13AM
Kyle Larson, is that you?
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @12:01AM (10 children)
EF is an irrational number.
(Score: 3, Touché) by c0lo on Wednesday April 15 2020, @01:43AM (9 children)
FTFY
@TMB, BTW, why the heck <strike> doesn't seem to work anymore. At least in "Preview" doesn't
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 3, Informative) by martyb on Wednesday April 15 2020, @04:07AM (8 children)
Our code to "balance tags" has a bug. If I open a <i> in a comment and fail to provide a closing </i>, then all text on the page that follows it would be in italics. The balance_tags routine detects the omission and automatically inserts one, in the appropriate place, before the comment is saved to the DB. It should handle the strike tag in the same way. It fails to do so. An unclosed strike tag affects all subsequent text on the page.
So, strike has been removed from the list of "allowed tags".
Use the DEL tag, which *does* get auto-balanced, instead.
Wit is intellect, dancing.
(Score: 2) by c0lo on Wednesday April 15 2020, @04:39AM (5 children)
Thanks.
Could you adjust the
below the "Preview/Submit" buttons?
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 2) by martyb on Wednesday April 15 2020, @05:11AM (4 children)
Wit is intellect, dancing.
(Score: 3, Informative) by coolgopher on Wednesday April 15 2020, @08:02AM (3 children)
Oh now you take note? Back when I posted about it [soylentnews.org] nobody seemed to care... ;P
(On the up side, I learned about the "del" tag)
(Score: 2) by martyb on Wednesday April 15 2020, @03:21PM (2 children)
Wit is intellect, dancing.
(Score: 2) by coolgopher on Thursday April 16 2020, @02:21AM (1 child)
Haha, all good. If I'd remembered I should've sent an email about it too when I didn't see a follow-up.
(Score: 2) by martyb on Thursday April 16 2020, @07:45PM
No problem! If nothing else, it provided a forum in which some of how the site works could be explained.
Wit is intellect, dancing.
(Score: 2) by maxwell demon on Wednesday April 15 2020, @07:55AM (1 child)
Is there a reason why the balance_tags routine wasn't fixed instead?
The Tao of math: The numbers you can count are not the real numbers.
(Score: 3, Funny) by c0lo on Wednesday April 15 2020, @10:28AM
It's stochastic, chaotic and has free will. One step remote from conscience; would it not be Perl, it would have evolved into a self-driving AI and became an Uber gig driver. Don't fix it, it may make it worse and turn into a stable genius, evil by necessity, and will strike everything in this world.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 3, Touché) by HiThere on Tuesday April 14 2020, @11:46PM
His barber? How many people do you think have as recognizable a photo as Einstein? Even if all you can see is his hair you can recognize him.
Sometimes I wonder if he carried along a portable Van de Graaff generator.
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, @12:43AM
My guess — drawing on the philosophical slant to the present thread — is that he went to Russell’s barber [wikipedia.org] (the one who shaved all people who don’t shave themselves).
But of course that barber promptly disappeared in a puff of logic, leaving Einstein with his unkempt hair.
(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
(Score: 1, Insightful) by Anonymous Coward on Wednesday April 15 2020, @12:06AM (1 child)
What, specifically, is the reason for any "Alternative Interpretations of Classical Physics" when we *already know* that "classical physics" is but a simplification, i.e. in any precise formulation it is just plain *wrong*?
(Score: 1, Interesting) by Anonymous Coward on Wednesday April 15 2020, @03:59AM
This is a good point, about classical vs. quantum physics. It's hard to see the motivation to re-do a classical theory with motivation from a quantum one (and I don't think this is even a good way of doing it.) Some theoreticians *are* trying to describe QM in terms of GR, but most are going the other way.
As far as being wrong, we don't have any physical theories that give us precise answers. An extreme case is how QM doesn't have anything coherent to say about gravity, for example. It basically takes spacetime for granted (even more so than GR) and we're seeing problems because of it. There is some really cool work being done in this area by people who are trying to build spacetime from quantum entanglement. It has the advantage of dispensing with the whole "local realism" problem with entanglement.
(Score: 2) by Snotnose on Wednesday April 15 2020, @12:52AM (4 children)
It's been a while, but it's more like e = lambda mc^2, where lambda is based on how fast you're moving relative to light speed to an observer. For most of us our velocity is close enough to zero that lambda is 1. But you get to, I don't remember, 80% of light speed or more and lambda grows at an exponential rate.
How I can use this knowledge in my normal life I have no idea. I've tried to use it to pick up chicks at the bar with no luck, and I have yet to convert the mass of my cat's turds into energy.
It's probably why social isolation resembles my normal life so much.
When the dust settled America realized it was saved by a porn star.
(Score: 0) by Anonymous Coward on Wednesday April 15 2020, @02:54AM
> I've tried to use it to pick up chicks at the bar with no luck,
That's because you lack an ego like Zaphod Beeblbrox. Nothing wrong with your pitch, it's the delivery.
(Score: 2) by MostCynical on Wednesday April 15 2020, @03:03AM
Einstein (1907) M = μ + E0/c^2
Lorentz (1913) ε = Mc^2 and ε0 = mc^2
Einstein (1946) E = m c^2 (but only at rest, as it removes kinetic energy)
E = m0c^2 for an object at rest, or E = mrelc^2 when the object is moving.
it was only an elegant simplification when he wrote it - Einstein never claimed it was "right" or complete. But it is very good for certain approximations.
"I guess once you start doubting, there's no end to it." -Batou, Ghost in the Shell: Stand Alone Complex
(Score: 2) by maxwell demon on Wednesday April 15 2020, @08:04AM
That factor is usually denoted with γ (gamma), not λ.
And no, it does not grow at an exponential rate; rather it has a pole at v=c.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by hendrikboom on Tuesday April 21 2020, @11:49AM
Actually, lambda = 1 / sqrt( 1 - v^2 / c^2 ), which goes to infinity as v goes to c, but not exponentially.
And the e = mc^2 formula shows up as the first term of a power series approximation in v of some formula involving lambda. I'm no longer sure of the details. Play with it. I think the Newtonian kinetic energy shows up in that power series as the second term.
My vaguely remembered reference? Einstein's book The Meaning of Relativity, which I read more than 50 years ago. I think it's available online as a pdf. (The epub version I've seen is useless -- it displays the math as gibberish).
-- hendrik
(Score: 3, Interesting) by hendrikboom on Wednesday April 15 2020, @04:15AM (7 children)
Intuitionism doesn't say that precise real numbers don't exist.
pi certainly does exist, and it has a precise value.
The fact that sin(pi) = 0 does not depend on pi being computed.
Intuitionism considers a real number to be given when you have a procedure that will produce arbitarily close approximations to it.(there is some philosophical wiggle room about the word of 'procedure')
Limited precision comes into it when you try to compare two real numbers.
If you can prove that their procedures will always generate compatible approximations, you know they are equal.
If you can prove that their procedures will generate incompatible approximations, you know they differ.
But you can't assert, in general, that one of these situations will always occur.
In theory, this gives you an infinitesimal fuzziness,
In practice, you get a finite fuzziness, because our computational resources are finite.
Also, intuitionists reject nonconstructive existence proofs.
The basic method for demonstrating nonconstructive existence is proof by contradiction:
-- suppose the thing doesn't exists.
-- lots of reasoning,leading to:
-- contradiction
-- Therefore the thing must exist.
This form of "proof" doesn't actually find the thing. So this form of proof is ruled out.
SO proof by contradiction goes out the window.
And with it, the law of the excluded middle.
The intutitionist's P or Q:
To prove P or Q, you must either
* prove P, or
* prove Q.
Thus you actually have to be able to figure out which of the two holds.
This is a subtly different meaning for 'or' than the classical one.
Just as there's a subtly different meaning to existence (must be constructive)
The term 'constructive' is often used nowdays for the mathematics of intuitionism.
L.E.J.Brouwer's original term intuitionism referred to a philosophy of mind he used to present his mathematics.
Constructivists (like me) use the same mathematics without his philosopyy.
-- hendrik
See also my web [pooq.com] page [pooq.com]
(Score: 2) by martyb on Tuesday April 21 2020, @08:36AM (6 children)
Wit is intellect, dancing.
(Score: 2) by hendrikboom on Tuesday April 21 2020, @10:41AM (5 children)
I did not mean that. If P is true and Q is true, then P OR Q is also true.
And if P OR Q is true, investigating which one is true will indeed report either that P is true or that Q is true (likely depending on how P OR Q was proved in the first place), but that doesn't imply that the other is false. The other might well turn out to be true as well.
(Score: 2) by martyb on Tuesday April 21 2020, @07:25PM (4 children)
Thanks for the reply!
So, I'll refer back to:
What is the 'subtly different meaning':
I had thought the subtle difference was that it was an 'exclusive or' instead of an 'inclusive or'. Your clarification (thank you!) throws that out the window, so now I'm at a loss as to your intention. Would you mind elaborating, perhaps with an example? Thanks!
Wit is intellect, dancing.
(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.
-- hendrik
(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
-- hendrik
(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.
--martyb
Wit is intellect, dancing.
(Score: 2, Interesting) by BlackGriffin on Wednesday April 15 2020, @08:43AM (1 child)
Numbers are tools we use, not the mystical things that have an existence of their own. So, arguments about what you're allowed to call a number are unable to reveal anything about reality. Bottom line behind the masked man fallacy, is the argument that space-time doesn't have infinite resolution. Say hello to loop quantum gravity [wikipedia.org].
(Score: 2) by hendrikboom on Tuesday April 21 2020, @11:20AM
Right. The mathematical existence of real numbers has nothing to do intrinsically with the nature of space. It's just that to a high degree of approximation, encompassing everyday phenomena, it works. This, and everyday experience, leads us in the intuitive belief that real numbers are reality.
What the referenced article does is present an analogy -- and it's only an analogy -- between the arbitrarilyy improvable approximations that are the staple of intuitionism with the unfolding reality of quantum time evolution.
And it bolsters it with talk of something he calls propensities that are a philosophers way of talking about probabilities without any of the baggage associated with that word -- the baggage in question being the frequentist interpretation.
And that's an entirely different argument. The intuitionists accept that a real numbers are approximate, but they also accept that further computation will produce arbitrarily close approximations.
The grains of space in loop quantum gravity (LQG) are a specific bound that say you cannot divide space further -- flat-out contradiction of the arbitrarily close approximability of the intuitioninsts.
The only way to get more precise numbers that are capable of meaning anything in measurement of LQG's space is by statistical averaging. Repeated measurement of the "same" thing will yield a cluster of values (uncertainty) because the same thing is never really the same thing, but they can average out to more and more precise values (the central limit theorem in statistics). Ultimately, our capability to measure reaches a limit of repeatability (the universe may reach an end, or not have enough physical space to store all those observational data), and physics once again gets in the way.
Theoretical physics is indeed looking at other models of reality the real numbers. There's a massive hunt for other kinds of mathematical objects to use for expressing theoretical physics. Extra dimensions, odd logics, strange geometries, abstract topologies, cohomology, topos theory (Whose internal logic is intuitionistic, by the way), ... All these are candidates as they seek something better. See John Baez's blog This Week's Finds [ucr.edu] for a smattering, just to whet your appetite.
Someone said last century that string theory was far too difficult and that it was next century's mathematics. We're living the attempts to do it now.
-- hendrik