Infinite Information in a Finite World
July 24, 2020 6:46 PM Subscribe
"But information is physical. Modern research shows that it requires energy and occupies space. Any volume of space is known to have a finite information capacity (with the densest possible information storage happening inside black holes). The universe’s initial conditions would, Gisin realized, require far too much information crammed into too little space. “A real number with infinite digits can’t be physically relevant,” he said."Physicist Nicolas Gisin is seeking to describe a physics that doesn't presume infinitely precise knowledge of initial conditions. (Atlantic)
Leaving the language of classical mathematics behind for the language of intuitionistic mathematics. (Science Daily press release)
Paywalled Nature Physics paper
Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real? (paper by Nicolas Gisin)
Leaving the language of classical mathematics behind for the language of intuitionistic mathematics. (Science Daily press release)
Paywalled Nature Physics paper
Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real? (paper by Nicolas Gisin)
Like the condition of poor Mr Schrodingers kitty, there just are some things that are unknowable.
Can't know it but we (for a certain aproximation of 'we') can prove it.
posted by sammyo at 8:10 PM on July 24, 2020
Can't know it but we (for a certain aproximation of 'we') can prove it.
posted by sammyo at 8:10 PM on July 24, 2020
....An extra dimensional thumb drive!
posted by clavdivs at 11:12 PM on July 24, 2020 [1 favorite]
posted by clavdivs at 11:12 PM on July 24, 2020 [1 favorite]
here's the manuscript: https://arxiv.org/abs/2002.01653
posted by aribaribovari at 11:29 PM on July 24, 2020 [1 favorite]
posted by aribaribovari at 11:29 PM on July 24, 2020 [1 favorite]
Very interesting piece. Thanks, blue shadows.
posted by Paul Slade at 11:55 PM on July 24, 2020
posted by Paul Slade at 11:55 PM on July 24, 2020
I wonder what the difference between intuitionistic/constructible and computability are, the concepts sound very similar. Also it sounds like both finite digits and quantization have in common are they are attempts to give physical reality an epistemically founded constraint
posted by polymodus at 12:01 AM on July 25, 2020 [1 favorite]
posted by polymodus at 12:01 AM on July 25, 2020 [1 favorite]
Interesting stuff. The relationship between information and physical reality always makes me think about the relationship between physics and thought. If thinking is, to some extent, a physical process, then physical laws, to some extent, must influence the scope and nature of thought. And if physical laws are fully deterministic, then it is hard to see how thinking could be anything but fully deterministic.
Except that is not what thinking feels like, at all. It feels like a directed effort to think, like something that we can abandon, or fail to do properly. It doesn't feel like something that's predetermined. Of course, that feeling may be illusory or deceptive. Perhaps just like loose rocks naturally tumble towards the center of the earth, all thought tumbles naturally towards the One, and the feeling of directed effort is just how we experience that tumbling.
But if that is true, then I find it hard to understand why anyone should be credited for the thinking that leads to that conclusion, or, for that matter, for any thinking at all. After all, the conclusion was inevitable, and the thinking that led to it would have happened regardless. So it strikes me that committed determinists and many-world advocates are almost always just a little bit too smug to convince me they really believe what they're saying. But perhaps they can't help it.
posted by dmh at 3:00 AM on July 25, 2020 [10 favorites]
Except that is not what thinking feels like, at all. It feels like a directed effort to think, like something that we can abandon, or fail to do properly. It doesn't feel like something that's predetermined. Of course, that feeling may be illusory or deceptive. Perhaps just like loose rocks naturally tumble towards the center of the earth, all thought tumbles naturally towards the One, and the feeling of directed effort is just how we experience that tumbling.
But if that is true, then I find it hard to understand why anyone should be credited for the thinking that leads to that conclusion, or, for that matter, for any thinking at all. After all, the conclusion was inevitable, and the thinking that led to it would have happened regardless. So it strikes me that committed determinists and many-world advocates are almost always just a little bit too smug to convince me they really believe what they're saying. But perhaps they can't help it.
posted by dmh at 3:00 AM on July 25, 2020 [10 favorites]
steve hsu, debating finitism, muses: "We experience the physical world directly, so the highest confidence belief we have is in its reality. Mathematics is an invention of our brains, and cannot help but be inspired by the objects we find in the physical world. Our idealizations (such as 'infinity') may or may not be well-founded. In fact, mathematics with infinity included may be very sick, as evidenced by Godel's results, or paradoxes in set theory. There is no reason that infinity is needed (as far as we know) to do physics. It is entirely possible that there are only a (large but) finite number of degrees of freedom in the physical universe."[1,2,3]
more here:
also btw...
more here:
...the primacy of physical reality over mathematics (usually the opposite assumption is made!) -- the parts of mathematics which are simply models or abstractions of "real" physical things are most likely to be free of contradiction or misleading intuition. Aspects of mathematics which have no physical analog (e.g., infinite sets) are prone to problems in formalization or mechanization. Physics (models which can to be compared to experimental observation; actual "effective procedures") does not ever require infinity, although it may be of some conceptual convenience. Hence one suspects, along the lines above, that mathematics without something like the "axiom of infinity" might be well-defined. Is there some sort of finiteness restriction (e.g., upper bound on Godel number) that evades Godel's theorem? If one only asks arithmetical questions about numbers below some upper bound, can't one avoid undecidability?@johncarlosbaez: "you *can* define truth for sentences with at most n quantifiers."
also btw...
- Information, information processing and black holes - "How much information in the universe?"
I discuss fundamental limits placed on information and information processing by gravity. Such limits arise because both information and its processing require energy, while gravitational collapse (formation of a horizon or black hole) restricts the amount of energy allowed in a finite region. Specifically, I use a criterion for gravitational collapse called the hoop conjecture. Once the hoop conjecture is assumed a number of results can be obtained directly: the existence of a fundamental uncertainty in spatial distance of order the Planck length, bounds on information (entropy) in a finite region, and a bound on the rate of information processing in a finite region. In the final section I discuss some cosmological issues related to the total amount of information in the universe, and note that almost all detailed aspects of the late universe are determined by the randomness of quantum outcomes.
- Minimum length and quantum gravity - "An implication of the result is that there may only be a finite number of degrees of freedom per unit volume in our universe - no true continuum of space or time. This means that there is only a finite amount of information or entropy in our universe (or at least in any finite patch of it)."
- Feynman and Everett - "A couple of years ago I gave a talk at the Institute for Quantum Information at Caltech about the origin of probability -- i.e., the Born rule -- in many worlds ("no collapse") quantum mechanics. It is often claimed that the Born rule is a consequence of many worlds -- that it can be derived from, and is a prediction of, the no collapse assumption. However, this is only true in a particular (questionable) limit of infinite numbers of degrees of freedom -- it is problematic when only a finite number of degrees of freedom are considered."
- Horizons of truth - "[Gregory Chaitin:] I think it's reasonable to demand that set theory has to apply to our universe. In my opinion it's a fantasy to talk about infinities or Cantorian cardinals that are larger than what you have in your physical universe. And what's our universe actually like?"
a finite universe?posted by kliuless at 3:16 AM on July 25, 2020 [7 favorites]
discrete but infinite universe (ℵ0)?
universe with continuity and real numbers (ℵ1)?
universe with higher-order cardinals (≥ ℵ2)?
Does it really make sense to postulate higher-order infinities than you have in your physical universe? Does it make sense to believe in real numbers if our world is actually discrete? Does it make sense to believe in the set {0, 1, 2, ...} of all natural numbers if our world is really finite?
CC: Of course, we may never know if our universe is finite or not. And we may never know if at the bottom level the physical universe is discrete or continuous...
GC: Amazingly enough, Cris, there is some evidence that the world may be discrete, and even, in a way, two-dimensional. There's something called the holographic principle, and something else called the Bekenstein bound. These ideas come from trying to understand black holes using thermodynamics. The tentative conclusion is that any physical system only contains a finite number of bits of information, which in fact grows as the surface area of the physical system, not as the volume of the system as you might expect, whence the term "holographic."
Hot take based on the abstract of the Science Daily article: "He suggests making changes to the mathematical language to allow randomness and indeterminism to become part of classical physics."
So, let's slap some Bayesian Statistics onto it?
posted by JoeXIII007 at 8:14 AM on July 25, 2020
So, let's slap some Bayesian Statistics onto it?
posted by JoeXIII007 at 8:14 AM on July 25, 2020
I am not a theoretical physicist, but those physical equations that I know of, which describe reality, seem to depend on various constants that are not finite, or cannot be described with a finite number of digits. Does finitism or what is described in the linked post that uses finite numbers use other values?
posted by They sucked his brains out! at 8:20 AM on July 25, 2020
posted by They sucked his brains out! at 8:20 AM on July 25, 2020
The article says that the major constants are still the same, assuming that's what you're talking about. Even though PI is infinitely long, every digit can be determined by a formula so it is essentially determinable.
posted by Think_Long at 8:37 AM on July 25, 2020 [1 favorite]
posted by Think_Long at 8:37 AM on July 25, 2020 [1 favorite]
Aren't those formulas (for pi, let's say) just approximations, if you stop after some number of iterations? To express pi as a Taylor series, for instance, or a sum of reciprocal of roots, you need an infinite number of terms.
These phrases from the article popped out: “intuitionist mathematics”... rejects the existence of numbers with infinitely many digits; and, “A real number with infinite digits can’t be physically relevant”.
But physics discovers those real numbers everywhere in nature — pi, e, Feigenbaum constants, etc. — so it seems odd to call them irrelevant, even if we can't practically compute or measure them to infinite precision.
posted by They sucked his brains out! at 9:11 AM on July 25, 2020
These phrases from the article popped out: “intuitionist mathematics”... rejects the existence of numbers with infinitely many digits; and, “A real number with infinite digits can’t be physically relevant”.
But physics discovers those real numbers everywhere in nature — pi, e, Feigenbaum constants, etc. — so it seems odd to call them irrelevant, even if we can't practically compute or measure them to infinite precision.
posted by They sucked his brains out! at 9:11 AM on July 25, 2020
> But physics discovers those real numbers everywhere in nature — pi, e, Feigenbaum constants, etc. — so it seems odd to call them irrelevant, even if we can't practically compute or measure them to infinite precision.
i am bad at physics and really at any math that involves numbers, but in reality isn't the limit for (for example) the precision of pi-as-it-exists-in-nature determined by the planck constant?
posted by Reclusive Novelist Thomas Pynchon at 9:37 AM on July 25, 2020 [2 favorites]
i am bad at physics and really at any math that involves numbers, but in reality isn't the limit for (for example) the precision of pi-as-it-exists-in-nature determined by the planck constant?
posted by Reclusive Novelist Thomas Pynchon at 9:37 AM on July 25, 2020 [2 favorites]
Has physics ever discovered a number in nature? Physicists have certainly made good use of numbers in modelling nature, but that's not the same thing.
posted by Panthalassa at 9:40 AM on July 25, 2020 [4 favorites]
posted by Panthalassa at 9:40 AM on July 25, 2020 [4 favorites]
right like isn't the seeming efficacy of math as a tool to model the world rather than just a very fun intellectual game kind of a philosophical scandal?
posted by Reclusive Novelist Thomas Pynchon at 9:48 AM on July 25, 2020
posted by Reclusive Novelist Thomas Pynchon at 9:48 AM on July 25, 2020
Imagine if you lined up a million billiard balls in a straight line and hit the first one into the second (ignoring friction and elasticity... we could easily arrange a boost at each collision). The idea is to have the balls all go perfectly forward. Any error in the first strike is worse on the second and it quickly escalates after that. To predict the results at a further ball requires vastly more information about the initial strike. It’s implausible to presume that it makes any physical sense to do so indefinitely. So even in classical physics we have an information problem in that in some sense the solution is harder than the problem it “solves.”
I find this approach fascinating and while I would never have made the connection to intuitionistic math, it makes sense to me especially as information theory is increasingly being connected to physics. It’s also connected to computable numbers, the actual subject of Turing’s famous proof. It’s been pointed out that while pi is irrational, it is computable. And there are only countably many computable/constructible numbers. There is reason to doubt that the continuum is, um, “real,” however useful it may be as a formalism and for things like, well, calculus.
posted by sjswitzer at 9:50 AM on July 25, 2020 [3 favorites]
I find this approach fascinating and while I would never have made the connection to intuitionistic math, it makes sense to me especially as information theory is increasingly being connected to physics. It’s also connected to computable numbers, the actual subject of Turing’s famous proof. It’s been pointed out that while pi is irrational, it is computable. And there are only countably many computable/constructible numbers. There is reason to doubt that the continuum is, um, “real,” however useful it may be as a formalism and for things like, well, calculus.
posted by sjswitzer at 9:50 AM on July 25, 2020 [3 favorites]
But physics discovers those real numbers everywhere in nature — pi
Where in nature was this perfect circle that you speak of discovered?
Joking aside, I think Gisin's key point is precisely that real numbers don't occur in nature, because their physical representation would require infinite space. Conversely, any real, physical system can only be approximated to some finite limit, beyond which quantum randomness sets in. Consequently Gisin argues that real numbers are better thought of as random numbers:
Where in nature was this perfect circle that you speak of discovered?
Joking aside, I think Gisin's key point is precisely that real numbers don't occur in nature, because their physical representation would require infinite space. Conversely, any real, physical system can only be approximated to some finite limit, beyond which quantum randomness sets in. Consequently Gisin argues that real numbers are better thought of as random numbers:
Accordingly, to name them “real number” is seriously confusing. A better terminology would be to call them “random numbers”. Unfortunately, Descartes named them “real” to contrast them with the complex numbers, those numbers that include the square root of −1, traditionally denote i. Hence:posted by dmh at 9:54 AM on July 25, 2020
Mathematical real numbers are physical random numbers.
There are some (so far) irreducibly fundamental constants, some even dimensionless, that might qualify as numbers discovered in nature. We are not quite sure what to make of them, though.
posted by sjswitzer at 9:54 AM on July 25, 2020 [1 favorite]
posted by sjswitzer at 9:54 AM on July 25, 2020 [1 favorite]
I'm gonna have to read his paper, but I'm not sure based on the post alone I understand what he's getting at?
The conditions at the beginning were a pure unified field at least according to my understanding, pure symmetry, a unity....
To talk about space or time at this point is irrelevant? Is he talking about the period after the "rupture" (or breaking of the symmetry) but before the inflationary epoch?
Off to read I guess!
posted by symbioid at 10:03 AM on July 25, 2020
The conditions at the beginning were a pure unified field at least according to my understanding, pure symmetry, a unity....
To talk about space or time at this point is irrelevant? Is he talking about the period after the "rupture" (or breaking of the symmetry) but before the inflationary epoch?
Off to read I guess!
posted by symbioid at 10:03 AM on July 25, 2020
right like isn't the seeming efficacy of math as a tool to model the world rather than just a very fun intellectual game kind of a philosophical scandal?
Oh totally. There are very live debates about it in the philosophy of maths. But maths isn't just a given; we've carefully shaped it so as to facilitate it being as interesting and fruitful as possible. For example, the replacement of naive set theory with ZFC, widely considered a less intuitive set of axioms.
posted by Panthalassa at 10:03 AM on July 25, 2020 [1 favorite]
Oh totally. There are very live debates about it in the philosophy of maths. But maths isn't just a given; we've carefully shaped it so as to facilitate it being as interesting and fruitful as possible. For example, the replacement of naive set theory with ZFC, widely considered a less intuitive set of axioms.
posted by Panthalassa at 10:03 AM on July 25, 2020 [1 favorite]
Has physics ever discovered a number in nature?
Including those I mentioned? There's the fine-structure constant, I think, albeit still being refined.
posted by They sucked his brains out! at 10:06 AM on July 25, 2020
Including those I mentioned? There's the fine-structure constant, I think, albeit still being refined.
posted by They sucked his brains out! at 10:06 AM on July 25, 2020
Speed of light would be another, probably.
posted by They sucked his brains out! at 10:09 AM on July 25, 2020
posted by They sucked his brains out! at 10:09 AM on July 25, 2020
In correct units, the speed of light is 1. :) Best to go looking for dimensionless constants, e.g. the mentioned fine structure constant.
posted by sjswitzer at 10:13 AM on July 25, 2020 [2 favorites]
posted by sjswitzer at 10:13 AM on July 25, 2020 [2 favorites]
In correct units, the speed of light is 1.
Ha, sure, fair enough.
posted by They sucked his brains out! at 10:27 AM on July 25, 2020
Ha, sure, fair enough.
posted by They sucked his brains out! at 10:27 AM on July 25, 2020
I’ve observed that the most problematic/interesting things in math and science occur when comparing a set and its power set. Russell’s paradox, Goedel and Turing, P vs NP, the number of states in a universe of N particles, natural vs real numbers. These things seem connected to me somehow, but just how is way out of my reach. It’s no wonder Gregor Cantor went mad.
posted by sjswitzer at 11:00 AM on July 25, 2020
posted by sjswitzer at 11:00 AM on July 25, 2020
Is this like that Steven Wright routine? "I bought a map of the US that's actual size. The scale reads 1 mile = 1 mile. I spent all summer re-folding it."
So a thorough description of the current state of the universe, would be a data set the size of the universe, requiring a universe's worth of matter and energy to store? Do I have it right, more or less?
posted by bartleby at 11:00 AM on July 25, 2020 [4 favorites]
So a thorough description of the current state of the universe, would be a data set the size of the universe, requiring a universe's worth of matter and energy to store? Do I have it right, more or less?
posted by bartleby at 11:00 AM on July 25, 2020 [4 favorites]
I believe it’s much worse than that. The data set describing the universe would be unimaginably larger than the universe... without the kind of reconceptualization being explored here.
posted by sjswitzer at 11:07 AM on July 25, 2020 [4 favorites]
posted by sjswitzer at 11:07 AM on July 25, 2020 [4 favorites]
we could easily arrange a boost at each collision
Isn't it easier to just pretend there is no friction than to award ourselves a million little, what, thruster jets on the billiard balls?
posted by thelonius at 11:22 AM on July 25, 2020
Isn't it easier to just pretend there is no friction than to award ourselves a million little, what, thruster jets on the billiard balls?
posted by thelonius at 11:22 AM on July 25, 2020
I’ve observed that the most problematic/interesting things in math and science occur when comparing a set and its power set. Russell’s paradox.....
I don't see a connection to power sets? Russell's paradox arises from the idea that any predicate defines a set. You could probably generate an instance with a predicate that refers to the power set of a set, but that does not look essential to me....
posted by thelonius at 11:24 AM on July 25, 2020
I don't see a connection to power sets? Russell's paradox arises from the idea that any predicate defines a set. You could probably generate an instance with a predicate that refers to the power set of a set, but that does not look essential to me....
posted by thelonius at 11:24 AM on July 25, 2020
Isn't it easier to just pretend there is no friction than to award ourselves a million little, what, thruster jets on the billiard balls?
Oh absolutely. I was just trying to avoid inherently unphysical assumptions (the spherical elephant in the room).
You could probably generate an instance with a predicate that refers to the power set of a set, but that does not look essential to me.
That is the essence if it, though. "The set of all sets that...."
posted by sjswitzer at 11:33 AM on July 25, 2020
Oh absolutely. I was just trying to avoid inherently unphysical assumptions (the spherical elephant in the room).
You could probably generate an instance with a predicate that refers to the power set of a set, but that does not look essential to me.
That is the essence if it, though. "The set of all sets that...."
posted by sjswitzer at 11:33 AM on July 25, 2020
man if i had time for a longform reply that longform reply would absolutely 100% be about leibniz's monadology.
posted by Reclusive Novelist Thomas Pynchon at 11:46 AM on July 25, 2020 [1 favorite]
posted by Reclusive Novelist Thomas Pynchon at 11:46 AM on July 25, 2020 [1 favorite]
That is the essence if it, though. "The set of all sets that...."
Well, no. A power set of a set is the set of all its subsets. Every set is an member of its power set , of course, since every set is a subset of itself. But the concept of self-membership does not seem to me to be needed at all to define a power set; and that concept is what generates trouble in the Russell paradox. Maybe there is something I don't understand, of course.
posted by thelonius at 11:52 AM on July 25, 2020 [2 favorites]
Well, no. A power set of a set is the set of all its subsets. Every set is an member of its power set , of course, since every set is a subset of itself. But the concept of self-membership does not seem to me to be needed at all to define a power set; and that concept is what generates trouble in the Russell paradox. Maybe there is something I don't understand, of course.
posted by thelonius at 11:52 AM on July 25, 2020 [2 favorites]
a thorough description of the current state of the universe, would be a data set the size of the universe, requiring a universe's worth of matter and energy to store? Do I have it right, more or less?
The point of physics, though, is to generate not thorough descriptions but applicable descriptions; descriptions that require manipulating tractably small amounts of information to yield usefully accurate predictions.
I see good physical law as essentially akin to good lossy compression: such behaviours of reality as it fails to account for, when used within its domain of applicability, are largely inconsequential.
We're quite a long way from having good physical law that describes the behaviour of complex systems, quite possibly because complexity is in and of itself the very quality of being strongly resistant to lossy compression.
posted by flabdablet at 10:31 AM on July 26, 2020
The point of physics, though, is to generate not thorough descriptions but applicable descriptions; descriptions that require manipulating tractably small amounts of information to yield usefully accurate predictions.
I see good physical law as essentially akin to good lossy compression: such behaviours of reality as it fails to account for, when used within its domain of applicability, are largely inconsequential.
We're quite a long way from having good physical law that describes the behaviour of complex systems, quite possibly because complexity is in and of itself the very quality of being strongly resistant to lossy compression.
posted by flabdablet at 10:31 AM on July 26, 2020
Aren't those formulas (for pi, let's say) just approximations, if you stop after some number of iterations?
Yes. But the point is that the thing they're approximations to is a real number that's very much finite and absolutely well-defined.
There is no uncertainty in pi, merely limited precision in any digital representation of it. And if you ever need more precision than the best representation you have to hand can give you, there are well-defined algorithms for improving that precision to an arbitrary extent.
The limit on accuracy for predictions involving pi never needs to reflect a limit on the precision of any approximation of pi that's put in its place in the formulae associated with those predictions, because it's always possible to generate an approximation to pi whose precision is good enough that the effects of noise in the physical measurements used as inputs to those formulae will completely swamp it.
posted by flabdablet at 10:58 AM on July 26, 2020
Yes. But the point is that the thing they're approximations to is a real number that's very much finite and absolutely well-defined.
There is no uncertainty in pi, merely limited precision in any digital representation of it. And if you ever need more precision than the best representation you have to hand can give you, there are well-defined algorithms for improving that precision to an arbitrary extent.
The limit on accuracy for predictions involving pi never needs to reflect a limit on the precision of any approximation of pi that's put in its place in the formulae associated with those predictions, because it's always possible to generate an approximation to pi whose precision is good enough that the effects of noise in the physical measurements used as inputs to those formulae will completely swamp it.
posted by flabdablet at 10:58 AM on July 26, 2020
Consider the weather. Because it’s chaotic, or highly sensitive to small differences, we can’t predict exactly what the weather will be a week from now. But because it’s a classical system, textbooks tell us that we could, in principle, predict the weather a week on, if only we could measure every cloud, gust of wind, and butterfly’s wing precisely enough. It’s our own fault we can’t gauge conditions with enough decimal digits of detail to extrapolate forward and make perfectly accurate forecasts, because the actual physics of weather unfolds like clockwork.
The weather is a complex system, and we don't have good ways of making high-quality long-term predictions about those because we can't know in advance, without measuring the initial state of every particle involved, which of those states will not ultimately matter. And once it would take longer to actually do the required measurements than to just let the system run and see what it does, having a model becomes pointless.
I'm not at all convinced that we need to ban all irrationals from physics to reconcile the intuitive consequences of Relativity and QM; I think mainly what we need to do is make sure that the phrase "in principle" never gets asked to do more heavy lifting than it's actually capable of.
Another good thing to avoid is using clickbait equivalences like "clockwork" when thinking about the determinism, or otherwise, of complex systems. Clockwork is in general not complex; at best it's complicated. Its behaviour also has many more outputs we don't generally care about than weather does.
posted by flabdablet at 11:19 AM on July 26, 2020
The weather is a complex system, and we don't have good ways of making high-quality long-term predictions about those because we can't know in advance, without measuring the initial state of every particle involved, which of those states will not ultimately matter. And once it would take longer to actually do the required measurements than to just let the system run and see what it does, having a model becomes pointless.
I'm not at all convinced that we need to ban all irrationals from physics to reconcile the intuitive consequences of Relativity and QM; I think mainly what we need to do is make sure that the phrase "in principle" never gets asked to do more heavy lifting than it's actually capable of.
Another good thing to avoid is using clickbait equivalences like "clockwork" when thinking about the determinism, or otherwise, of complex systems. Clockwork is in general not complex; at best it's complicated. Its behaviour also has many more outputs we don't generally care about than weather does.
posted by flabdablet at 11:19 AM on July 26, 2020
In a predetermined world in which time only seems to unfold, exactly what will happen for all time actually had to be set from the start, with the initial state of every single particle encoded with infinitely many digits of precision. Otherwise there would be a time in the far future when the clockwork universe itself would break down.
Near as I can tell, this is straightforward ego chauvinism. Having identified and labelled some given particle, I can see no in-principle objection to the idea that encoding its initial state would require infinitely many digits of precision. But them, I'm also completely comfortable with the idea that whatever it is that you're proposing to encode in this way is a measurement of some underlying reality that, in and of itself, does not require encoding in digital form in order to be real or to get on with doing whatever it's going to do.
Pi is not any of its approximate encodings. The fact that we need to generate approximations to it in order to make predictions with it doesn't change that. Likewise, it seems to me that the fact that we would need to generate approximations to the initial states of some unknowable number of particles in order to plug those into some kind of mathematical model whose outputs are predictions about the future state of things does not mean that the particles themselves in any sense need to carry such approximations along with them.
Determinism does not, in my opinion, imply predictability. All it says is that there is only one future that's actually possible, even if we don't and never will have any feasible way to predict which of the futures that seem likely to us, if any, it will actually turn out to be.
Now, if we accept determinism as a working assumption, it does become possible to construct a conceptual four-dimensional* map with well-defined places to put all the physical events, past present and future, that we know about. And that map, to my way of thinking, is the Block Universe.
But the Block Universe is a map. It's not the territory, which is always going to be far more detailed and messy than any literally conceivable map. There will always be a need to modify and update the content of any Block Universe map as we discover more about the territory it's a map of.
If we say "this is how the Universe really is", what we're actually doing is making a usefulness claim about a particular map of it. That's all we're doing because that's all we can do, given our own physical limitations. Which are, it seems to me, far more productive to think through the consequences of than any conceptual untidiness associated with irrational numbers.
*or five-dimensional if we insist on making it Euclidean
posted by flabdablet at 11:42 AM on July 26, 2020 [1 favorite]
Near as I can tell, this is straightforward ego chauvinism. Having identified and labelled some given particle, I can see no in-principle objection to the idea that encoding its initial state would require infinitely many digits of precision. But them, I'm also completely comfortable with the idea that whatever it is that you're proposing to encode in this way is a measurement of some underlying reality that, in and of itself, does not require encoding in digital form in order to be real or to get on with doing whatever it's going to do.
Pi is not any of its approximate encodings. The fact that we need to generate approximations to it in order to make predictions with it doesn't change that. Likewise, it seems to me that the fact that we would need to generate approximations to the initial states of some unknowable number of particles in order to plug those into some kind of mathematical model whose outputs are predictions about the future state of things does not mean that the particles themselves in any sense need to carry such approximations along with them.
Determinism does not, in my opinion, imply predictability. All it says is that there is only one future that's actually possible, even if we don't and never will have any feasible way to predict which of the futures that seem likely to us, if any, it will actually turn out to be.
Now, if we accept determinism as a working assumption, it does become possible to construct a conceptual four-dimensional* map with well-defined places to put all the physical events, past present and future, that we know about. And that map, to my way of thinking, is the Block Universe.
But the Block Universe is a map. It's not the territory, which is always going to be far more detailed and messy than any literally conceivable map. There will always be a need to modify and update the content of any Block Universe map as we discover more about the territory it's a map of.
If we say "this is how the Universe really is", what we're actually doing is making a usefulness claim about a particular map of it. That's all we're doing because that's all we can do, given our own physical limitations. Which are, it seems to me, far more productive to think through the consequences of than any conceptual untidiness associated with irrational numbers.
*or five-dimensional if we insist on making it Euclidean
posted by flabdablet at 11:42 AM on July 26, 2020 [1 favorite]
One might suggest, following on from flabdablet, that the fantasy of complete predictability is closely interwoven with the destructive human impulse of complete controllability, which has done so much to get us into our current predicament?
posted by domdib at 1:14 AM on July 27, 2020 [1 favorite]
posted by domdib at 1:14 AM on July 27, 2020 [1 favorite]
Is this like that Steven Wright routine? "I bought a map of the US that's actual size. The scale reads 1 mile = 1 mile. I spent all summer re-folding it."
This reminded me of Borges’s “On Exactitude in Science,” which according to Wikipedia ‘elaborates on a concept in Lewis Carroll's Sylvie and Bruno Concluded: a fictional map that had "the scale of a mile to the mile."’
posted by LeviQayin at 9:20 AM on July 27, 2020
This reminded me of Borges’s “On Exactitude in Science,” which according to Wikipedia ‘elaborates on a concept in Lewis Carroll's Sylvie and Bruno Concluded: a fictional map that had "the scale of a mile to the mile."’
posted by LeviQayin at 9:20 AM on July 27, 2020
Thge Philosophy of Mathematics is a very real thing that tries to explain how mathematics works on an epistemological level. One of my favorite things to come out of it is Fictionalism - the notion that numbers and operations are cognitive tools to understand mathematics, but do not exist in their own right. It's not a settled and widely accepted notion, but Hatry Field managed to restate Newton's law of Gravitation without using a single number.
posted by Slap*Happy at 7:18 PM on July 27, 2020
posted by Slap*Happy at 7:18 PM on July 27, 2020
Academic philosophy can get so fucking irritating. All those pixels spilt on hair-splitting arguments about whether 2 + 2 = 4 is false on the basis of religiously intense nit-picking about whether or not abstract obects are real.
I swear, an academic philosophical review of 50 Shades of Grey would consist entirely of dense, peer-reviewed and self-congratulatory argument on exactly how many of those shades are really black or white, the awfulness of the narrative being treated as completely irrelevant. Fuxache.
posted by flabdablet at 7:56 PM on July 27, 2020
I swear, an academic philosophical review of 50 Shades of Grey would consist entirely of dense, peer-reviewed and self-congratulatory argument on exactly how many of those shades are really black or white, the awfulness of the narrative being treated as completely irrelevant. Fuxache.
posted by flabdablet at 7:56 PM on July 27, 2020
So here's how it works, OK?
Mathematics is an exploration of the consequences of logic. That's what it is, but that's also all it is.
As soon as you're mounting a logical argument about mathematics, you're applying the very same tools as mathematicians use.
So to the extent that your argument encourages me to conclude that mathematical statements are in general not true, it leads me to conclude the same thing about logical statements in general, and therefore about itself.
Which means that my best response is to have a bit of a giggle at your apparent seriousness about it and the eagerness with which all your most sincere followers fall over themselves to express their admiration for the tailoring of your nonexistent clothes.
posted by flabdablet at 8:18 PM on July 27, 2020 [1 favorite]
Mathematics is an exploration of the consequences of logic. That's what it is, but that's also all it is.
As soon as you're mounting a logical argument about mathematics, you're applying the very same tools as mathematicians use.
So to the extent that your argument encourages me to conclude that mathematical statements are in general not true, it leads me to conclude the same thing about logical statements in general, and therefore about itself.
Which means that my best response is to have a bit of a giggle at your apparent seriousness about it and the eagerness with which all your most sincere followers fall over themselves to express their admiration for the tailoring of your nonexistent clothes.
posted by flabdablet at 8:18 PM on July 27, 2020 [1 favorite]
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posted by Saxon Kane at 8:03 PM on July 24, 2020 [3 favorites]