Does physics require a continuum?
September 12, 2021 6:54 AM   Subscribe

Finitism and Physics - "A brief precis: Gravitational collapse limits the amount of energy present in any space-time region. This in turn limits the precision of any measurement or experimental process that takes place in the region. This implies that the class of models of physics which are discrete and finite (finitistic) cannot be excluded experimentally by any realistic process. Note any digital computer simulation of physical phenomena is a finitistic model. We conclude that physics (Nature) requires neither infinity nor the continuum. For instance, neither space-time nor the Hilbert space structure of quantum mechanics need be absolutely continuous. This has consequences for the finitist perspective in mathematics..." (previously)

re: finitism: "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:
...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."
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?

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."
also btw...
How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. - "For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise."
posted by kliuless (88 comments total) 37 users marked this as a favorite
 
Note any digital computer simulation of physical phenomena is a finitistic model.

I think I’m missing something because simulating a continuous model requires a bit more abstraction but is perfectly doable.
posted by Tell Me No Lies at 7:07 AM on September 12 [1 favorite]


I don’t understand like 95% of this but it’s very exciting, as a former pedantic child who had a real issue with the concept of infinity.
posted by showbiz_liz at 7:47 AM on September 12 [2 favorites]


Gravitational collapse limits the amount of energy present in any space-time region. This in turn limits the precision of any measurement or experimental process that takes place in the region. This implies that the class of models of physics which are discrete and finite (finitistic) cannot be excluded experimentally by any realistic process. Note any digital computer simulation of physical phenomena is a finitistic model. We conclude that physics (Nature) requires neither infinity nor the continuum. For instance, neither space-time nor the Hilbert space structure of quantum mechanics need be absolutely continuous. This has consequences for the finitist perspective in mathematics...


This is what I've been saying all along. Finally, mainstream scientists are listening to me.
posted by Balna Watya at 8:00 AM on September 12 [10 favorites]


Have you ever thought about just how stupid the universe really is? It's just really fucking goofy. A wack-ass system of precisely tuned utter nonsense with a bunch of emergent properties that in a very complicated, roundabout way, essentially do nothing but convert information into heat, that bleeds away. What the hell even is that?
posted by seanmpuckett at 8:13 AM on September 12 [40 favorites]


If this is true, how far down do the turtles go?
posted by mule98J at 8:23 AM on September 12 [2 favorites]


A wack-ass system of precisely tuned utter nonsense with a bunch of emergent properties that in a very complicated, roundabout way, essentially do nothing but convert information into heat, that bleeds away. What the hell even is that?

Bitcoin
posted by OMGTehAwsome at 8:27 AM on September 12 [63 favorites]


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.

This is the kind of nonsense you get when physicists do psychology/neuroscience. There is no 'direct' experience of reality; 'reality' is an invention of our brains.
posted by logicpunk at 8:29 AM on September 12 [13 favorites]


Something in one of the links kind of blew my mind: “decoherence is merely the mechanism by which the different Everett worlds lose contact with each other!”
I also didn’t know that Feynman asserted he developed Many Worlds independently, but of course he did, considering Feynman diagrams.
posted by waytoomuchcoffee at 8:33 AM on September 12 [2 favorites]


ever notice how a crack in a rockface looks just like the path taken by a lightning bolt?

lightning we see in the sky isn't actually a giant electrical spark, it's the human-perceivable part of a 4-dimensional crack formed when different electrospace-magnetime worlds shift relative to each other and settle again.

we just treat it like a big spark so our kids will stop asking us questions about it
posted by glonous keming at 8:41 AM on September 12 [4 favorites]


There is no 'direct' experience of reality; 'reality' is an invention of our brains.

That sounds criminally anthropocentric to me. Did reality not exist until human brains did? Or even the first brains? I feel like rocks might have something to say about that.
posted by showbiz_liz at 8:51 AM on September 12


Reality is the invention of all brains, big or small.
posted by njohnson23 at 8:54 AM on September 12 [2 favorites]


Whatever existed before consciousness isn’t the same as the reality that is the phenomenal experiences had by consciousness. In other words, the reality that we perceive is not identical to the causes of those perceptions. You see or feel a rock. That perception is entirely in your consciousness. The thing (or things or events) that prompted your brain to have a rock-experience may be very different from the rock-experience.
posted by oddman at 9:10 AM on September 12 [8 favorites]


It is entirely possible that there are only a (large but) finite number of degrees of freedom in the physical universe.

Also entirely possible that a properly complete model of the universe would have no degrees of freedom at all, being completely constrained to model only what has happened and is happening and will happen, as distinct from modelling what we have reason to expect could happen in circumstances defined and/or controlled to some arbitrary degree.

Degrees of freedom are what we get when we model parts of reality with language and/or art and/or mathematics designed to cover a bunch of cases that we have interpreted as recurring and/or related elements of some pattern. But any perceptible pattern (even one as fundamental to our ways of thinking as causality itself) is nothing more than a compressed description of what's really there.

No two parts of reality are exactly alike, because if they were, they wouldn't be distinguishable at all, and would therefore be the same part.

And when there's a difference between in-theory and in-practice, we're far better served by believing in practice than in theory.
posted by flabdablet at 9:17 AM on September 12 [1 favorite]


we just treat it like a big spark so our kids will stop asking us questions about it

Why do we do that?
posted by flabdablet at 9:23 AM on September 12


"No two parts of reality are exactly alike, because if they were, they wouldn't be distinguishable at all, and would therefore be the same part."

That you Wilhelm?*


*This is some deep, insider, philosophy humor. If you get this joke, you are too deep in the weeds. There is no help for you.
posted by oddman at 9:30 AM on September 12 [4 favorites]


"No two parts of reality are exactly alike, because if they were, they wouldn't be distinguishable at all, and would therefore be the same part."

except for Nickelback songs
posted by philip-random at 9:35 AM on September 12 [7 favorites]


A wack-ass system of precisely tuned utter nonsense with a bunch of emergent properties that in a very complicated, roundabout way, essentially do nothing but convert information into heat, that bleeds away. What the hell even is that?

Dwarf Fortress
posted by Zargon X at 9:41 AM on September 12 [19 favorites]


There are only two Nickelback songs.
posted by flabdablet at 9:44 AM on September 12 [2 favorites]


Whatever existed before consciousness isn’t the same as the reality that is the phenomenal experiences had by consciousness. In other words, the reality that we perceive is not identical to the causes of those perceptions. You see or feel a rock. That perception is entirely in your consciousness. The thing (or things or events) that prompted your brain to have a rock-experience may be very different from the rock-experience.

Totally with you on this, but isn’t what you’re calling “the causes of those perceptions” reality?
posted by showbiz_liz at 10:35 AM on September 12 [3 favorites]


There is no 'direct' experience of reality; 'reality' is an invention of our brains.
posted by logicpunk at 8:29 on September 12


There may be no “direct experience of reality”, but said experience is not an “invention” of reality but an “interpretation”.

There is a concrete reality out there that we are all experiencing and living in. Our brains filter that incoming experience and information and form an interpretation of said reality.

But we aren’t “inventing” anything, unless we’re hearing voices or having some other form of psychiatric crisis.
posted by Pirate-Bartender-Zombie-Monkey at 10:37 AM on September 12 [6 favorites]


There is always a SMBC. In this case there are two.
posted by lalochezia at 10:47 AM on September 12 [4 favorites]


Physical equations and models drip with irrational and transcendental numbers, which we can only specify with some finite precision when used to make predictions or validate theories.

We can come up with vague grammatical equations to describe such numbers ("a unit circle's ratio of circumference to radius") but that isn't a real quantitative entity, it is just words that further bake in imprecision as soon as any measurement is done.

The fact that we seem to require these numbers to describe reality would imply either that physics is wrong to use them, which seems unlikely given how useful they are, or that such numbers and their inherent property of being able to be evade calculation to full precision reflects some deeper properties of reality.

Perhaps we may not be able to understand those properties, and at best only be able to make approximate calculations within the bounds of what processing capacity our piece of the universe affords us. I'm unsure this is a proof that physics requires finitism, only an observation of our apparent limitations, based on what we know, so far.
posted by They sucked his brains out! at 11:44 AM on September 12 [6 favorites]


"We experience the physical world directly, so the highest confidence belief we have is in its reality." It is common knowledge this is untrue, the physical world experienced by us is unique in each case, and a lot of what each of us experiences is a fictional account, offered up by our sensory neurons and hastily rebuilt moment to moment by memory. The universe we awaken into each moment is subjective, yeah, yeah, yeah, there are trains, planes, and automobiles, a shambled clockwork, but there are hills slowly melting to the seas, guided only by what is, vast canyons carved through everything which happened before. The truly ugly secrets surrounding our willing destruction of this miracle, will rise to light too late, no one is coming to check the math, to show how we missed the mark by an infinitessimal degree, and therefore all is forgiven. There will be no pats on the head, no luncheon, served by patient girls and boys.
posted by Oyéah at 11:53 AM on September 12


There may be no “direct experience of reality”, but said experience is not an “invention” of reality but an “interpretation”.

I propose we all agree that, while it is not strictly necessary to assume that our experiences are somehow related to a shared reality, you're not going to get very far if you don't. As the famous philosopher Descartes once said: "Like, who the hell even knows man."

We're all just hallucinating clouds in space anyways.
posted by Spiegel at 11:54 AM on September 12 [1 favorite]


isn’t what you’re calling “the causes of those perceptions” reality?

The issue is that the word "reality" isn't used to refer to whatever that noumenal thing might be, but rather to a sense-impression derived model which may or may not bear some relationship to what you're positing as "reality". It seems to me that the only time we ever associate the word "reality" with the noumenal is in making the philosophical assertion that they are associated. For all other purposes, the thing referred to by "reality" is the model we derive from phenomena.
posted by howfar at 11:58 AM on September 12 [2 favorites]


But we aren’t “inventing” anything, unless we’re hearing voices or having some other form of psychiatric crisis.

Oh, we're all inventing something about our perception of reality, whatever that may be.

Statements like this and the usual rebuttals dive deep into metaphysics, psychology and the nature of consciousness and awareness so fast and deep it makes marine geologists turn various shades of green with jealously and blue with hypoxia.

Even once we get past the nature of our brains and bodies inventing things through pattern recognition to deal with information overload and cutting perceptual corners, if we start diving into the nature and physics of time as related to the question of "what really is reality?" things start to get deeply weird and uncomfortable in a hurry.

As far as I know we currently have no reliable way to understand the nature or perceptions of someone having a "psychiatric crisis" and what is actually happening to them, especially with regards to what is termed as schizophrenia or psychosis.

Sure, we can identify that appears to be biological and chemical in nature, that it may have genetic causes - but what if some people experiencing these things were somehow able to perceive (or not perceive) aspects of time that we just don't understand because everyone else who is - for lack of better terms - sane or rational because our brains simply can't handle the concept of not perceiving time as a linear arrow?

For all we know those hallucinations are real, and not just to the person perceiving them.

I'm not bringing this up to dismiss nor praise the very real cultural and social issues of mental illness but to ask a very strange question:

What exactly do we do if we find out that people experiencing these things are perceiving real things that are out of place in either time or space? Would we view our contemporary medical treatment of them as barbaric? What if at some point in the far future that we found some kind of temporal therapies that re-integrated people experiencing this theoretical "time sickness"?

Our universe is an incredibly strange place. Richard Feynman was aware of this, and it may explain some of his dabbling and dealing with modern magic and the occult.

And, shoot, I should probably read Gödel, Escher, Bach: An Eternal Golden Braid again.
posted by loquacious at 12:19 PM on September 12 [5 favorites]


Now what if the Riemann hypothesis is proven false?
posted by sammyo at 12:32 PM on September 12


It occurs to me reading this thread, that our use of finite rationals which can only approximate the transcendental and algebraic numbers and infinities of our mathematics is very like our use of our brains and senses to approximately perceive a presumed shared reality.

Mock turtles all the way down.
posted by jamjam at 12:46 PM on September 12 [5 favorites]


Nearly all scientific observation is mediated by electronic machines, not your fuzzy senses and weird brains made of meat. Human senses are terrible for precision and repeatability, so we've mostly eliminated them from our study of our universe. This has led to great technological advances.
posted by ryanrs at 12:49 PM on September 12


This is the kind of nonsense you get when physicists do psychology/neuroscience. There is no 'direct' experience of reality; 'reality' is an invention of our brains.

Well, the 'real' is a metaphysical concept, and just how metaphysical does not seem to be apparent to people who just use 'reality' as a synonym for 'experience' (that's most people, it seems, too.) I think that many scientifically educated people take it as just obvious that the real and the physical are the same thing, but that is really something that you need to do some work to support, if you wanted to defend it as your position. Its not just common sense, even if it temporarily seems to be.

I'm more sympathetic to physicalism than are some of the usual posters in philosophy-ish threads. But I've got a big side eye for the idea that it should constrain mathematics so strongly. There are many mathematical objects (other than infinite sets) that do not have a physical analogue. Are they all unreal, or just the ones that turn out to generate weird results? I suppose this kind of puzzle is a main reason that most (I think) mathematicians are formalists and don't concern themselves with problems about what "real" objects their concepts correspond to.
posted by thelonius at 12:54 PM on September 12 [4 favorites]


First there is the infinite, unbounded.
Something, we know not what, tore what amounts to a unity in nothingness, an infinity of all (which includes the infinity of none).
This fundamental rupture of unity creates a division, a bound.
This bound creates the surface of each "thing".
Unitarity is no longer the universal thing, but each unto its appointed thing.
The finity which now exists comes about from this rupture, the division between things, the perfect symmetry torn, and the fabric of space time folding in on itself.
The Planck resolution describes the limit of our ability to measure, there is your finity. And that finity lays precisely in the overall composition of the universe "stuff" itself, how much energy there is, the process of rate of decay/entropification of the universe, etc... (That is to say "time and space and matter and energy").

Complex interacting and fields of degrees of "existence" (probabilities) all merging and waving and sloshing about until they all reach the energies which are bound and attracted and affect each other (both in the positive and negative).

I guess my point is, yes, the universe is infinite, but we cannot know this infinity, likewise, there may be a god, but we cannot know this god. (sorry, Theists, you do you).

We see the finite, and we can expand and refract the knowledge and possible knowledge and explore as we can using the models we have, all we can do or just become 100% agnostic, but ...

pass me the bong man.
posted by symbioid at 12:58 PM on September 12 [1 favorite]


Can't we just say that our sense impressions of the world are more immediate than our mathematical abstractions without asserting that they are "direct" (as if there is only an either/or, direct or not-direct)?

It seems like they are saying mathematics ought to line up with the most direct experiences of the world that we have.
posted by straight at 1:02 PM on September 12 [1 favorite]


I'm just looking at precis, but whatever they're saying is just wrong. He says it implies that finitistic models are not excluded. It does not follow, which he claims in the next breath, that nature excludes, or does not require, nonfinitistic computation. The observation that we cannot rule out physical finitism says nothing about whether nature requires nonfinitism or not.

But maybe it's just a badly written paragraph, I assume this is not what the authors mean and they know what they're talking about.
posted by polymodus at 1:05 PM on September 12 [1 favorite]


"Totally with you on this, but isn’t what you’re calling “the causes of those perceptions” reality?"

That depends. If you're seeing a red ball about 10 ft. in front of you, what is it that you want to label with the term "reality?" The experience of there being a red ball 10 ft. in front of you or the unknown cause that prompts that experience? In common, every day use, we mean the experience represented by the statement "I see a red ball over there." We don't mean the ontological characteristic(s) that evoke the experience but which are unknowable in themselves. We usually only mean that when we're talking strictly about metaphysics. There's nothing wrong with agreeing to use "reality" in that stricter sense, we just usually don't. We are more often in the pretheoretical position taken by the physicist who offered that original quote (this is sometimes referred to as the naive position, but that can come across as patronizing).

"There is a concrete reality out there that we are all experiencing and living in. "

See, that's the thing. We don't know this. Indeed there may not be anyway to know what's underneath our perceptions. There could be a concrete world that very closely matches the features of our perception or we could be in a world of information (there are a few different types of this) and that information could be embedded in something analogous to our perceived reality (a computer program with a computing substrate) or it might be abstract (hey there Plato) or some kind of ideal reality (Berkeley has some ideas (sorry that was shameless of me)).

We can't say with certainty that the things(s) causing our experiences are even coherent, consistent, constant or non-arbitrary. I mean, if you want to get into some really odd metaphysical scenarios we can't even insist that our senses are good at keeping us alive. Mind you, I'm in favor of the view that we get enough right to suggest that the universe is at least consistent in the ways that we interact with it, but there's just no proof of that.


"Our brains filter that incoming experience and information and form an interpretation of said reality."

This is an interesting point. I think that if we could establish whether our experiences are inventions or interpretations, it might reduce the number of reasonable ontological options. However, as I understand it (I don't do contemporary epistemology) I'm not sure there's a particularly meaningful distinction between interpretation and invention in the given situation where we do not know that which excites our senses. I mean, it could even be the case that we interpret things by inventing phenomenal placeholders that stand for X where X may not have any properties in common with the variable. (Or to tackle the idea from a different angle (even though it amounts to the same position): we have to know about the world beyond our senses to know whether sensations are interpretive or inventive.)
posted by oddman at 1:30 PM on September 12 [10 favorites]


Amazingly enough, Cris, there is some evidence that the world may be discrete, and even, in a way, two-dimensional.

Chaitin ought to know better than to say this, because of course discrete/continuous and dimensionality are orthogonal concepts. The holographic principle says nothing about whether the universe may be discrete.

Note any digital computer simulation of physical phenomena is a finitistic model.

It's not doable in this universe, but there's no reason why the meta-universe couldn't be capable of computing with real numbers. (That said, I still think "simulationism" is basically just an intellectual fad.)
posted by nosewings at 1:32 PM on September 12 [1 favorite]


Also, I hope I'm not coming across as patronizing or a know it all. This happens to be my field of expertise (literally I have a PhD in theories of substance). The various claims I've responded to are good ideas that have been popular at one time or another in the history of philosophy (and some sciences). Some very good philosophers have heartily endorsed versions of the naive position and have good arguments for it (even if I don't find them convincing).

The current view, heavily indebted to Locke, Kant and some others, is more or less the view I've been defending, but it wouldn't shock me if we swerved back toward idealism or something else at some point in the future.
posted by oddman at 1:36 PM on September 12 [9 favorites]


This happens to be my field of expertise

Thank you for coming and engaging, then! All us Boltzmann Brains are just avid for your input, even if it's impossible to tell the difference between it and random fluctuations in the chaos.
posted by notoriety public at 1:41 PM on September 12 [3 favorites]


nth-ing this by loquacious...

And, shoot, I should probably read Gödel, Escher, Bach: An Eternal Golden Braid again.

posted by hwestiii at 3:21 PM on September 12


I say this with love, of course, but you realize you're all a bunch of wankers, right?
posted by BigBrooklyn at 4:10 PM on September 12 [2 favorites]


I've always interpreted the finitist perspective in mathematics as people that don't like infinity and therefore refuse to incorporate it into mathematics, and spend the rest of their lives trying to demonstrate that everything which can be proved with infinity can be proved without it. Which is fine, but it does not prove that infinity is an nonsense concept.
posted by plonkee at 4:13 PM on September 12 [4 favorites]


And also, surely physics just uses whichever mathematics seems to work best for that particular bit. The choice of tool is interesting to some people I guess, but hardly seems the point.
posted by plonkee at 4:15 PM on September 12


Metafilter: you realize you're all a bunch of wankers, right?
posted by automatronic at 4:15 PM on September 12 [7 favorites]


1) The claim that Goedel's incompleteness theorems are evidence that conventional mathematics is "very sick", requires interpreting them in a very strong way. Most mathematicians view those theorems as putting bounds on what can be accomplished with and expected of the usual mathematics, not as evidence that the whole thing needs to be discarded.

2) As others have mentioned above, even if the universe is finite (in whatever sense), we seem to need infinity to describe it precisely--see, e.g., pi and e, which, numerically, are limits of certain infinite sequences.

3) Mathematics has never been an exact representation of the physical world. Consider, for example, the circle, as a geometric shape. A perfect circle doesn't actually exist anywhere in the universe, as far as we know, but rather the geometric object is an abstraction of shapes we do observe. Given that mathematics simply models the world, it is reasonable to endow it with objects and properties that are convenient for modeling, even if they may not be physically realistic. The circle is one example, there being infinitely many natural numbers is another.

4) Mathematics doesn't just exist to model physical phenomena. It is used extensively in computer science, finance, etc. So the suggestion that it should be completely bound by whatever physicists discover is just silly. For example, the Turing machine is our theoretical basis for doing computation on machines. When discussing limits of computation, it is very convenient to consider Turing machines with infinite tapes, even if everyone understands that they cannot be physically built.

5) Beyond modeling, mathematics is its own field. There are many objects in pure mathematics that were never intended to model anything in the physical world or other fields of study. Consider, for example, rings, categories, and topological spaces. These did come about as abstractions of more down-to-earth constructions like the integers and real numbers, but they were explicitly meant to be abstract. It so happens that some of them did eventually get used by physicists, but even without that they'd still have a purpose to mathematicians. A physicist or philosopher coming along to declare these things (or, rather, infinite examples thereof) off limits would be rather presumptuous.

6) Folks seem to get a bit too excited about what should and shouldn't be allowed in math. But to most mathematicians, the game is about precisely stating your assumptions and rules of inference, and then seeing what can be concluded from them. In that context it's fair game to assume that there are infinite sets, that there are no infinite sets, that there are infinite sets of different sizes, or whatever else. It's all about what you can prove with that, and how it fits with other existing mathematics. Saying that some of these possibilities are somehow too offensive to be considered places undue significance on mathematical objects, in my opinion.
posted by epimorph at 8:02 PM on September 12 [20 favorites]


... Bekenstein bound ...

When I read about it, it was the Bekenstain bound.
posted by bryon at 8:19 PM on September 12 [8 favorites]


There is a concrete reality out there that we are all experiencing and living in.

Easy to say, hard to prove. People could be spread across three or four realities but as long as our interpretations of our realities meshed the differences might never come up.
posted by Tell Me No Lies at 8:22 PM on September 12 [1 favorite]


Metafilter: A wack-ass system of precisely tuned utter nonsense with a bunch of emergent properties that in a very complicated, roundabout way, essentially do nothing but convert information into heat, that bleeds away. What the hell even is that?
posted by otherchaz at 10:15 PM on September 12 [3 favorites]


There is a concrete reality out there that we are all experiencing and living in.

I enjoy reading "realist" theoretical physicists like Lee Smolin, Sean Carroll, and Einstien.

Even those realists don't regard the rock you perceive as what's "really real".

For instance, Carroll says, "the only reality is the Wave Function of the Universe." Anything else is a simplification we experience due to our temporal and physical limitations.
posted by Mei's lost sandal at 10:30 PM on September 12 [2 favorites]


"the only reality is the Wave Function of the Universe."

That reminds me. I should really finish trying to capture the single electron shared by the entire universe. If my calculations are correct and I can manage to capture and destroy it or at least stop it from moving around so much we might see some serious shit.
posted by loquacious at 2:25 AM on September 13 [4 favorites]


I assume this is not what the authors mean and they know what they're talking about

That's exactly what they want you to think.
posted by flabdablet at 2:47 AM on September 13 [1 favorite]


I'm just getting around to digging into the articles (as a dilettante) , but, isn't the first article just a restatement of Heisenberg?
posted by OHenryPacey at 7:53 AM on September 13


> mule98J:
"If this is true, how far down do the turtles go?"

All the way, obviously.
posted by ArgentCorvid at 8:27 AM on September 13 [1 favorite]


And they get smaller as they go so that they can all fit.
posted by flabdablet at 8:34 AM on September 13


Further to which: continua might not be required but the way they let you make models with places to fit literally everything without imposing arbitrary restrictions of their own makes them super handy.

Which is pretty much the entire point and purpose of them. The real numbers are an abstracted ideal for measurements in much the same way that geometric figures are abstracted ideals for shapes. There are no perfect circles, for example, in nature; everything gets lumpy and wobbly and a bit indistinct if you examine it carefully enough, but that doesn't make the idea of circles useless because there are loads of things that are near enough to circular for all practical purposes.

there's no reason why the meta-universe couldn't be capable of computing with real numbers.

I don't care how meta your universe is, it won't be doing digital computation with real numbers, almost all of which can be specified digitally only by using representations of infinite length.
posted by flabdablet at 8:49 AM on September 13


Tell Me No Lies: I think I’m missing something because simulating a continuous model requires a bit more abstraction but is perfectly doable.

This was my initial reaction as well. Now I have second thoughts.

You can certainly set up the functions that build a model from a continuous point of view, but if you're evaluating those functions with a digital computer (as we know them), there will be a finite number of possible inputs and therefore a finite number of possible outputs.

Simplistic example: If you're modelling the height h of a falling object as a function of time t, you might write a function that you'd call h(t). Because this is a function, it can't have more output values than input values. If t has 32 bits, there can be no more than 2^32 results for h, even though we generally conceive of a falling object as passing through all of an infinite number of h values. Even though h(t) was conceived in a continuous way, evaluating it digitally will limit it to a finite number of discrete outputs.

In fact it's hard to imagine any kind of mechanical calculation device that wouldn't have the same issue. Because electric charge is quantized, even an analog computer would have a finite number of discrete inputs to and outputs from the model it's evaluating. The quantization of the reality we're modeling creeps into the modeling devices themselves.

In truth I feel thoroughly unqualified to comment here, maybe I'm missing something.
posted by Western Infidels at 8:56 AM on September 13 [7 favorites]


I feel thoroughly unqualified to comment here

Who is?

Dive on in, the water's fine.
posted by flabdablet at 9:04 AM on September 13 [2 favorites]


I’m starting to wonder (like, just in the last ten minutes looking through these comments) if our entire concept of discrete numbers isn’t a flawed artifact of being extra-smart monkeys that’s getting in the way of our understanding even as it’s our only real tool for doing so. Natural numbers are really useful for dealing with the objects of our immediate experience, and extend trivially to wholes and then integers once the monkeys invent debt. But at every turn where we’ve pushed beyond our immediate experience, the universe has thrown us curves where extrapolations of our intuitions based on immediate experience entail problematic edge cases (e.g., infinity) that also pretty reliably turn out to be entirely wrong when we invent ways to measure objects outside of our immediate experience.

The number line implies Euclidean space, which just isn’t how space on the large scale turned out to work. Likewise quantum effects appear absolutely insane from the perspective of a naive observer expecting something that conforms to natural experience. We build ever more complicated math (still rooted in the same discrete foundations, because it’s all we’ve got) that only the smartest monkeys understand to model the behaviors we observe, but keep running into the problematic idealizations. Meanwhile, the universe itself seems to do a weirdly good job of insulating itself from those problems, e.g., time dilation means a black hole evaporates before it ever needs to answer the question what “infinitesimal” means.

Hell, even economics, entirely a product of our own genius (or masochism, or both), really gets away from us at scale. A dollar is a dollar is supposed to be a dollar until you start trying to assume the same dollar means the same thing to a panhandler as it does to Jeff Bezos. There’s once again something lurking at the edges of our experience that isn’t well represented by applying a simple number line to it.

What’s the alternative? Hell if I know. I’m just another monkey, and one that has only dabbled in relevant education at that. But it does give me a somewhat disturbing glimpse of what it might mean to postulate an intelligence that is to humans what humans are to dogs or ants or amoebae, that there are possibly hard limits to our ability as a species to ever understand what’s going on around us.
posted by gelfin at 9:11 AM on September 13 [2 favorites]


Discrete numbers are for counting things. The reals are non-discrete, an abstraction for measurement rather than quantity.
posted by flabdablet at 9:13 AM on September 13 [1 favorite]


Isn't this where pragmatism enters the picture? it's fine, in principle, to refine and re-define abstractions , but in the end what we want are usable tools to help us navigate what we define as our existence and our future. Math, and by extension physics, computing, economics are tools we've come up with so far, and they continue to be useful. Many of these "arguments" seem to be retreading familiar ground.
posted by OHenryPacey at 9:18 AM on September 13 [1 favorite]


The reals are an abstraction. If you’re talking about the natural universe, every real is an integer with a decimal place and a limit to measurement precision. Irrationals are one of the edge cases that arise from our intuition that, say, there is any such thing as a perfect circle.
posted by gelfin at 9:20 AM on September 13


the question what “infinitesimal” means

I've always taken "infinitesimal" to mean exactly "smaller than anything that matters".

there are possibly hard limits to our ability as a species to ever understand what’s going on around us

I distinctly remember realizing, at about the age of eight, that I didn't actually need to keep going to school because I already knew everything, and I could prove it because everything I could think of, I knew that!

Good times.
posted by flabdablet at 9:22 AM on September 13 [4 favorites]


Irrationals are one of the edge cases that arise from our intuition that, say, there is any such thing as a perfect circle

or even perfect squares, which turn out to have sides and diagonals whose measurements can't both be exactly enumerated in the same units.
posted by flabdablet at 9:26 AM on September 13 [1 favorite]


I've always taken "infinitesimal" to mean exactly "smaller than anything that matters".

That’s fair, but you could likewise get out of a lot of mathematical confusion by defining “infinite” as “bigger than anything that matters.”
posted by gelfin at 9:31 AM on September 13 [1 favorite]


> > there's no reason why the meta-universe couldn't be capable of computing with real numbers.

> I don't care how meta your universe is, it won't be doing digital computation with real numbers, almost all of which can be specified digitally only by using representations of infinite length
This isn't really true. We have lots of mathematical software that computes with real numbers symbolically. There are lots of limitations to computing with real numbers that are transcendental (detecting 0 can be undecidable e.g.) but in practice a lot of things work. You don't keep track of "infinite decimal expansions", you keep track of the defining formulae of the numbers you are working with.

You might argue that this does not give a uniform representation for every possible real number but instead limits you to only real numbers that you can actually specify. Which is fair, but it is not something that limits our ability to compute with real numbers in any way.
posted by 3j0hn at 9:40 AM on September 13 [1 favorite]


I would class a limit on our ability to compute with real numbers that restricts us to working only with those we have well specified symbolic representations for to be a pretty damn important limit.

It gives us no way, for example, even to tackle a simple three-body problem in Newtonian dynamics.
posted by flabdablet at 9:46 AM on September 13 [1 favorite]


The fundamental issue is that computation with actual genuine real numbers, as opposed to the rational approximations thereof that we actually use in practice, runs aground as soon as we require the value of a real number that can only be specified as the limit of some infinite series or other.

If we're computing with rational approximations, we just plug away calculating terms of the infinite series until our end result stops seeming to change, and call it Good Enough. Using genuine real numbers, we don't get to do that; actually summing an infinite series, rather than simply specifying it, requires infinite time and generates infinite information.

And sure, we can attempt to sidestep this kind of issue simply by plugging in a symbolic specification for the infinite series instead of actually calculating it, but except in the infinitesimal minority of cases where most of this shit cancels out, what we're going to end up with as end results for our "computation" is a bunch of incomprehensible symbolic formulae that are far too complicated to actually use for anything, which kind of defeats the purpose of computation.
posted by flabdablet at 10:03 AM on September 13 [1 favorite]


In other words: the computable numbers, like the rationals, are a mere countable subset of the reals; what's available for computation in any physically realizable computer is a finite subset of one of those countable subsets; and I see no evidence that a finite subset of the computable numbers makes a more useful basis for general computation than the customary finite subset of the rationals. And in neither case are we genuinely talking about computation using arbitrary reals.

Physics might or might not be finitistic, but digital computers certainly are.
posted by flabdablet at 10:12 AM on September 13 [1 favorite]


As a few people have pointed out, most of the discussion of mathematics on computers is limited to digital computation. You actually have a whole different set of constraints with Quantum and Analog Computers. These don't point one way or the other to the argument about the validity of infinity, but they show it's possible to more directly use physical phenomena as the means of computation. They only take the discrete value at the end of a series of computations using potentially continuous phenomena.
posted by axlan at 10:30 AM on September 13 [1 favorite]


"Actually" summing a infinite series to a real number is really no different than defining "s" to the value of the sum. Any question you have about "s" can be answered by referring back to it's definition as a sum. An "incomprehensible symbolic formulae" is the computational path back to any information one wants about that value. If nothing cancelled then mostly likely it's just because it's complicated and having "actual" real numbers there would be no more informative.

It is the case, that if you had a computer that could do constant time arithmetic with real numbers then you could do a lot of really cool (impossible) things. But we can't even do constant time arithmetic on rational numbers or even integers. Computation time always scales with the "size" of the numbers involved and certain problems in arithmetic are always going to be undecidable.

I guess I am agreeing that:
> there's no reason why the meta-universe couldn't be capable of computing with real numbers.
isn't true, but it's not because digital computers can't represent real numbers. It's because of other fundamental limits on computing (all tracing back to the halting problem).
posted by 3j0hn at 10:38 AM on September 13 [1 favorite]


I fail to recall why (universal) Turing machines are justified to have an unbounded tape, but I do remember that Alan Turing's idea of computers originally consisted of just human secretaries doing simple tasks given arbitrary amounts of scratch paper.

Also I vaguely recall examples of putting a real decimal on a TM so as pointed out the problem with computing reals is undecidability, not inaccuracy.
posted by polymodus at 11:52 AM on September 13 [1 favorite]


Can't we just say that our sense impressions of the world are more immediate than our mathematical abstractions ...

I'm not sure we can, actually. Abstract ideas, like mathematical concepts, can arguably be apprehended by the mind more directly than the physical world can.
posted by Artifice_Eternity at 2:18 PM on September 13 [2 favorites]


You don't keep track of "infinite decimal expansions", you keep track of the defining formulae of the numbers you are working with.

ISTR that determining whether two formulae are equivalent is a Hard Problem. I don't know how the difficulty scales in practice, but I imagine a general-purpose symbolic computation engine would spend a lot of time reducing formulae and a lot of time figuring out whether it needs to reduce formulae.
posted by Joe in Australia at 6:59 PM on September 13


Metafilter: It's tools all the way down.

MetaMetafilter: Waaaaay too much time on our hands.
posted by Pouteria at 3:33 AM on September 14


Old but relevant talk on advances in real computation by Turing award recipient. I skimmed it and one important basic point is that the vast majority of real numbers are uncomputable.

That's philosophically interesting because as someone who assumes the extended Church-Turing thesis (because it is folk consensus amongst computer scientists), then it doesn't make sense for the laws of the universe to use some more powerful form of computation, because that would mean the universe could solve undecidable problems. So coming from that perspective/assumption that the physical universe cannot mechanistically be more powerful than a Turing machine, it's an obvious consequence that the universe/physics doesn't require Reals.
posted by polymodus at 4:12 AM on September 14


> ISTR that determining whether two formulae are equivalent is a Hard Problem
> one important basic point is that the vast majority of real numbers are uncomputable
Checking whether two formulae are equivalent and computing with general are both undecidable problems. But, my Ph.D supervisor (a complexity theory expert) liked to say that undecidability results usually don't tell you anything about practical computing -- one often has to bend over backwards to formulate the halting problem in ways that are unlikely to arise in practice.

So, it's okay that most reals are uncomputable -- there are A LOT of real numbers -- it could still be the case that all real numbers arising in practical physics (say) are computable.
posted by 3j0hn at 11:13 AM on September 14


I've been doing some reading on this, and I have discovered a remarkably marginal proof that 1 is not a computable number.

Enumerate all the natural numbers in their binary form:
  .
  1
 10
 11
100
  …

There are countably infinite rows here, each one corresponding to some computable number. In fact, these rows contain every possible computer program, and therefore every computable number can be generated by a computer running some row as input.
Now use Cantor's diagonalisation technique to find a number not in this table:
    1
   11
  110
 1 11
1 100
    …

If you place all these numbers adjacent to each other you have one that differs from every number in that table, and consequently is not among the computable numbers. If we place a radix point (what would be a "decimal point" for decimal numbers) in front of this number we have the infinite fraction .111…

Call this fraction 'x'.
2x-x = 1.111… - .111… = 1
Consequently, x=1, which means that 1 is not a computable number. QED.
posted by Joe in Australia at 4:41 PM on September 14 [1 favorite]


I don't see how you justify putting a radix point "in front of" the new number you generated by diagonalization, when you haven't put one similarly "in front of" any number on the original list.

In fact, given that your diagonalized digit string is of infinite length by construction, not only is it not a "new" natural number, it isn't any natural number; all natural numbers share the property of specifiability using a finite number of digits.

And all that aside, at some point you seem to have conflated a number's occurrence in a putatively exhaustive list of computer programs with its occurrence in a list of outputs from such programs.

Did the construction of this proof involve actual drugs, or was it just sleep deprivation? :-)
posted by flabdablet at 6:06 PM on September 14 [1 favorite]


it could still be the case that all real numbers arising in practical physics (say) are computable.

Wouldn't be so sure about that. Pi turns up all the time, and it isn't.
posted by flabdablet at 6:10 PM on September 14


Oops. Yes it is. My bad.
posted by flabdablet at 6:37 PM on September 14


I guess the point is supposed to be, you need only a finite number of decimal places of pi to model or compute or even to actually instantiate any possible physical process?

I don't think there are any perfectly circular physical objects, are there? None at the observable scale. But what counts as a physical object becomes harder to say as you go deeper into physics, of course, and I don't actually know that there are no perfectly circular subparts to a quark or whatever, or even if that's a coherent thing to speculate on.
posted by thelonius at 6:28 AM on September 15


Now use Cantor's diagonalisation technique to find a number not in this table

I thought the Cantor thing involved altering (by incrementing, decrementing, or inverting) the digits that appear on the diagonal in the infinite list, so that the constructed result would always differ from the Nth number in the list in the Nth place and therefore would differ from every number in the infinite list.

An infinitely long string of 1s should be in the infinite list already, shouldn't it?
posted by Western Infidels at 7:24 AM on September 15


Not if it's an infinite list of natural numbers. Every natural number is representable using a finite number of digits, so an infinitely long string of nonzero digits has no place in that list.

Were this not the case, diagonalization could be used to demonstrate that the set of natural numbers can't be put into 1:1 correspondence with itself, which is absurd.
posted by flabdablet at 8:29 AM on September 15 [2 favorites]


Cantor's diagonal argument was based on an infinite list of infinitely long sequences of digits. If you eliminate the "infinitely long sequences of digits" part, then a complete listing of all possible sequences up to N digits long is also finite. Diagonalization will only cover N rows of the list and won't produce a new sequence.

It seems clear that the list Joe suggests will contain "1" and "11" and "111" and so on without bound; I would guess it fair to say it includes an infinite sequence of 1s but I don't know what a mathematician would say. Infinity doesn't follow the rules we're all used to.
posted by Western Infidels at 10:25 AM on September 15


the list Joe suggests will contain "1" and "11" and "111" and so on without bound; I would guess it fair to say it includes an infinite sequence of 1s

No. It will contain an infinite collection of strings of 1s, each individual string being of finite length and representing a specific whole number of the form 2length-1. This is not at all the same thing as an infinite string of 1s, which does not represent a whole number unless you prefix it with a radix point, in which case it represents (by definition) the number 1.
posted by flabdablet at 11:49 AM on September 15 [1 favorite]


If you have an infinitely large set that consists of every possible sequence of 1's with no repeats you, of necessity, will end up with an infinite sequence of 1's as an element of that set.
You seem to want to have your infinite set but artificially remove any infinite elements. And, as pointed out above, the diagonal argument doesn't work for finite sets.
posted by thatwhichfalls at 1:29 PM on September 15


Sagan's Contact had a fun bit about :pi: ... a bit of trolling from the higher-ups, "we made dis"
posted by seanmpuckett at 2:05 PM on September 15


"If you have an infinitely large set that consists of every possible sequence of 1's with no repeats you, of necessity, will end up with an infinite sequence of 1's as an element of that set."

No, flabdablet is quite correct. It's easy to make a (countably) infinite set, no element of which is itself infinite. The set of all natural numbers is not finite but has no member that is not finite.

Incidentally, a binary string of (countably) infinite 1's can be given a perfectly rigourous definition as a number, it turns out to equal "-1", add 1 to it to see why. The general theory here is the p-adic numbers which turn out to have all sorts of interesting properties.
posted by Proofs and Refutations at 3:35 PM on September 15


I would like to observe that there are several classless metawishes in this thread, but that's normal for this kind of discussion.
posted by loquacious at 9:25 PM on September 15


This is old ground for me, fwiw.
posted by flabdablet at 9:44 PM on September 15


« Older Bruce the Tool Using Parrot   |   Sondheim's Six Ladies In Red Newer »


You are not currently logged in. Log in or create a new account to post comments.