9 ms·

Struggles with the Continuum

63 points by hackandthink 3y ago

dcow 3y ago
I find the idea of a universe with a discrete substrate/structure both natural and compelling. I wish we’d explore it more. If you really think about it, it’s hard to imagine nature would just happen into a system with irrational numbers as we naïvely (undergraduate level college and below) understand them. Things clearly look infinite on a macro level, but “what type of discrete systems give rise to the observed macro behavior” is a field that needs more exploration. The maths analog is alternative number systems, which at least seem to be gaining interest and popularity. The metaphor for programmers: is the universe just a beautifully seeded Conway’s game of life?
- greatquux 3y ago
  Couldn't agree more, though when I see this brought up, I've always seen "people who seem to know more than me" saying that it just leads to other problems, and they kind of hand-wave it away... but I still think there should be some more serious exploration of this. How could "real" numbers _truly_ exist, any more than "imaginary" ones?
  - housecarpenter 3y ago
    I think anybody who thinks real numbers exist will also say that imaginary numbers exist. "real" and "imaginary" in the context of numbers are just arbitrary labels assigned for historical reasons, they're not meant to convey any philosophical judgements about existence.
    - AnimalMuppet 3y ago
      I don't think so. You can have imaginary rationals, or even imaginary natural numbers (that is, a+bi, where both a and b are rational numbers or integers). Although I will admit that a big part of the point of the reals was solutions to polynomials, and it takes complex numbers to be able to solve them all.
- imglorp 3y ago
  Maybe someone has a survey paper? Two for discussion would be Loop Quantum Gravity and honorable mention for Wolfram's hypergraph.
  - eigenspace 3y ago
    > Maybe someone has a survey paper? Two for discussion would be Loop Quantum Gravity What would you like to know about it? > honorable mention for Wolfram's hypergraph. I'd call it more of a dishonourable mention. If you're looking for niche theories in physics with discrete spacetime, I guess you'd want to look at https://en.wikipedia.org/wiki/Causal_sets https://en.wikipedia.org/wiki/Causal_sets
    - imglorp 3y ago
      Thanks. Curious why the hypergraph diss? Regardless of their process, it sounds like they came up with a model which has some alignments with conventional theory.
    - aeonik 3y ago
      Why dishonorable? His work is pretty interesting to me: graph taxonomy of all possible rules and rules spaces (what Wolfram calls the Ruliad) seems like a very natural extension of Computer science and Analysis.
- cwmma 3y ago
  Sounds like your describing complex systems thoery https://en.wikipedia.org/wiki/Complex_system https://en.wikipedia.org/wiki/Complex_system
  - dcow 3y ago
    Not quite. I’m describing a system with discrete structure that gives rise to macro level complex system in the same way the game of life can implement arbitrary computation.
- bmacho 3y ago
  > I find the idea of a universe with a discrete substrate both natural and compelling. I really like the idea of a finite, ever growing, and totally knowable universe. What Wolfram is searching for. The universe was a small finite graph at the beginning (which we can possibly know by looking at the sky), and it grows since that, according to some deterministic rules. This means that the universe is perfectly replayable, we can replay history, observe the thoughts of famous people, and such.
  - dcow 3y ago
    I believe there may be ways to introduce entropy into a system like that so it need not completely deterministic, but can’t quite remember the name for the topic…
- nyrikki 3y ago
  The real numbers are irrational almost everywhere, they are uncomputable 'almost everywhere' also. “what type of discrete systems give rise to the observed macro behavior” is what we have been doing forever through Laplacian determinism. There are no alternative numbers systems that get around this. That is the whole point of the problem of the cardinality of the continuum. We know that Laplacian determinism fails, with quantum mechanics being the easiest counterexample but Cantor diagonalization is another. The sofar unsuccessful efforts to produce string theory is an example of this effort, changing world lines into sheets. With the discovery of strange non chaotic attractors and the discovery that our numerical systems are subject to the WadA property in time delayed and Hamilton systems it is possible that a unified model may not exist. None of this is undergrad though, due to how we have chosen to teach it. But it is very possible that existential quantifiers are what math limits is to building models of.
rnjailamba 3y ago
Previous discussion (2016) - https://news.ycombinator.com/item?id=12441990 https://news.ycombinator.com/item?id=12441990
xanderlewis 3y ago
I remember telling my tutor in the first year of a (pure) mathematics degree that I didn’t really enjoy analysis because I “couldn’t see why anyone should believe in the real numbers”. He looked at me as if I was slightly mad. At the time I’m not sure I realised I wasn’t totally alone in having such worries. It seems that most professional mathematicians have never really given it much thought (or at least no more than other ontological, ‘non-mathematical’ questions). Of course, when it comes to fundamental physics it all starts to become more relevant. edit: My comment shouldn’t be taken as a recommendation. I’m not suggesting mathematicians should worry about such things (it may well be a waste of their time!). I’m simply surprised that more don’t find it as unnerving as I did at the time.
- agnosticmantis 3y ago
  Bring us another complete ordered field and I’ll pay handsomely for it.
- whatisyour 3y ago
  Professional mathematicians have given it a lot of thought. Your tutor probably just didn't wonder enough about it. I did my graduate studies in numerical methods of mathematical physics, and everyone I talked to agreed, that the continuum assumption creates a lot of problems. But even so, the nice things they give in the space of functions and numerical convergence has been more important than the difficulties they cause. And hence, continuum analysis it is. In one of my papers, I actually assumed that two features in material have to be minimum epsilon distance away to show that the method convergences in a reasonable amount of computational effort. (where epsilon is arbitrarily small) So, I did break the continuum barrier to do some useful physics ones.
  - xanderlewis 3y ago
    Some have given it a lot of thought, but not nearly as many as I would have expected. Despite such questions being philosophical in nature it seems to mostly be mathematical physicists and applied mathematicians who spend time on it. Pure mathematicians mostly don’t care; they’re not interested in existence or ‘reality’ — if there’s complexity in an idea it’s worth studying whether it has anything to say about models of the universe or not. I guess what I’m trying to say is just that I’ve always been surprised by the lack of interest most pure mathematicians show towards questions of ‘what exists’ or ‘what the purpose of mathematics is’ and so on. I would have expected such (clearly) intellectually curious people to be less myopic. Because the world is a big place, one can find countless essays on these sorts of topics. But it’s still a minority sport. John Baez is a brilliant outlier in many ways.
Aardwolf 3y ago
As far as I know, just like matter and energy, information density can also contribute to a black hole. A real number has infinite precision, so even a single particle with a real number position or weight, would have infinite information and create a black hole. Would that be a valid reasoning to say why it can't be real numbers?
- eigenspace 3y ago
  No.
  - Aardwolf 3y ago
    > No. Please elaborate
    - eigenspace 3y ago
      Because what you're describing is coordinate or unit system dependent. I could invent a unit system where the mass of an electron is pi units, and another unit system where the mass of the electron is 1 unit. Me using a different unit system doesn't affect the physics of the system.
      - Aardwolf 3y ago
        But the particles can move around, they won't always be exactly on values of pi, 1, etc... in any coordinate system. And pi and 1 happen to be examples that can be described with a finite amount of symbols. But a continuous particle has to move through any value in between.
        aeonik 3y ago
        Your idea interests me. I think it's related to the Plank Scale, but what you're talking about would possibly be a "Plank Precision"? I'm having a hard time seeing if this is fundamentally the same thing as the Plank Length, or if it's something different entirely. https://en.wikipedia.org/wiki/Planck_units#Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_length Very cool idea though.
red_trumpet 3y ago
> For example: how many real numbers are there? The continuum hypothesis proposes a conservative answer, but since this is independent of the usual axioms of set theory, the question remains open: there could be vastly more real numbers than most people think. If don't agree with that. There are exactly as many real numbers as there are inifnite sequences consisting of zeroes and on (0,0,1,0,1,...). That is the same as the set of subsets of the natural numbers. That is well-defined. The continuum hypothesis asks if there are infinities smaller than this one, but bigger than the infinity of the natural numbers.
- nyrikki 3y ago
  The constructable numbers, which is what you are describing are aleph naught, or countable. The Cantor set is a way to think about this.
  - red_trumpet 3y ago
    Not true. If you think about binary representation of numbers it is clear that there are as many infinite binary sequences as there are real numbers between 0 and 1. So that is uncountably infinite. Alternatively, apply the usual diagonal argument directly to the set of sequences to see that there are uncountably many of them. This has not much to do with constructible numbers, which are those (complex) numbers which can be constructed using compass and straightedge constructions starting from 0 and 1.
    - nyrikki 3y ago
      The definable numbers, no matter if you use the constructible, algebraic, or computable numbers are countable infinities, or aleph naught. Any segment of the real line, no matter how small you make it is aleph_1 or an uncountable infinity. I realize that those concepts are difficult to build intuitions around, but what I said was correct. definable,algebraic, and computable numbers have such a tiny percentage of the real line that added together their infinity of representations are still countable and equal in size to the natural numbers. You could map them all 1 to 1 to natural numbers and they would be the same size as the natural numbers, which is tiny compared to the reals. The reals being uncomputable 'almost everywhere' is well proven at this point. Cantor diagonalization being the typical way of showing it. 'Almost everywhere' as used above has a formal definition: A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero. The reals are uncomputable 'almost everywhere', the reals are not constructable 'almost everywhere' etc...
at_a_remove 3y ago
So, I only escaped with a bachelor's in physics and a bit more math than was usually required for the degree, but I have often wondered if "we" went astray by spending time on concepts like infinity. Summing 1 + 2 + 3 + 4 + ... to get -1/12 using various infinities and "regularization" just seems, well, like some kind of sleight of hand, and certainly divorced from physical reality.
- jjgreen 3y ago
  Ramanujan summation (1 + 2 + ... = -1/12) is used in the derivation of Casimir effect I'm afraid.
  - at_a_remove 3y ago
    I wonder if there's an alternate way to get there without that monstrosity, which manages to be triply-offensive.
audunw 3y ago
The way renormalisation is mostly covered in popular media, made me think you can’t do quantum physics without dirty tricks to get rid of infinites. Then I read The Road to Reality by Roger Penrose and in the chapter on renormalisation it mentions just “by the way“ that the problem goes away if you assume the universe has a minimum scale, or something like that. Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation? Why is the default assumption that the universe is continuous? Doesn’t seem like we have a good reason for it, other than the fact that we use integration as a tool and that assumes continuity by default. Does it make mathematical analysis easier all-in-all? Why not just pick the Planck length as a reasonable assumption for the quantisation of space? Might not be correct but probably a more reasonable guess than assuming that it’s continuous?
- eigenspace 3y ago
  > Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation? It's because of the historical development where it really did start off as a nasty bag of mathematical tricks. It wasn't until much later that Wilson and co put renormalization procedures on firm footing, and by that time it had a quite bad reputation.
- ndsipa_pomu 3y ago
  I thought it was due to relativity which assumes a continuous space. There's a lot of evidence to back up relativity, though it does have issues with not working in a black hole, so it would be questionable to throw it out without something better to replace it.
- BoiledCabbage 3y ago
  > other than the fact that we use integration as a tool and that assumes continuity by default. Is there really no equivalent of calculus that works on a "quantized"/discreet numberline? The seems really hard to believe.
- ko27 3y ago
  > Why is the default assumption that the universe is continuous? Why is your default assumption that physicists haven't tried to make it discrete? You need to break Lorentz invariance to have the universe be discrete, which is an effort that's worth a Nobel prize. > Doesn’t seem like we have a good reason for it We have many good reasons for it. https://www.forbes.com/sites/startswithabang/2020/04/17/this-is-why-space-needs-to-be-continuous-not-discrete/ https://www.forbes.com/sites/startswithabang/2020/04/17/this... From the article: "If space is discrete, then the principle of relativity is wrong"
jerf 3y ago
If you look at the history of physics, including not just the final breakthroughs but the previous misunderstandings, there's a long history of "Well, we thought it had to be either A or B, but it turns out it's a novel and hard-to-imagine combination of both of those things." For example, consider the universal speed limit that we now call the speed of light. Prior to relativity, when space and time were kept rigidly separated, there was a bit of a problem. The math of Newtonian physics and their corresponding frame transforms admit no particular speed limit; no matter how fast a thing is going, a force could always be applied to it to make it go yet faster. So a speed limit seemed to be impossible. But observationally, it became more and more clear there was one, and what's more, it seemed to be unrelated to your frame of reference. So apparently there was a speed limit, and not only that, but one that was essentially uncharacterizable in Newtonian physics. But if something was going the speed limit, what happened if you applied further forces on that thing? The universe seemed contradictory; it seemed to have a speed limit, yet it couldn't have one, and if it did have one it seemed it would lead to contradictions as well. So, does the universe have a speed limit or not? And the answer is basically that that question, in the context of a Newtonian view of the universe, is simply ill formed and has no meaning, because the universe isn't Newtonian. With the better (if not necessarily entirely correct) view that Einstein introduced, what we get is actually something that even now a lot of people don't really get, which is that the universe both does and does not have a speed limit. The former is generally understood; c is the limit. You can not witness anything in the universe going faster than c from your perspective. But there is also a sense in which the universe doesn't have a speed limit; no matter how fast you are going, you can always accelerate in any direction and go faster in that direction. There isn't a speed limit in the sense that there is no speed that you can be going where the universe will say "Nope, stop, you can't go any faster that way because physics won't allow it." (Other things, like blueshifting the cosmic microwave background into arbitrarily high-energy gamma rays from your point of view may stop you, but "physics" itself won't.) Subjectively, time dilation means that there is a sense in which you can head to distant location as fast as you like; you can cover what was originally a million light years in arbitrarily short amounts of subjective time, and there is a real sense in which that becomes your "speed" and it can greatly exceed lightspeed in your original reference frame. And just try explaining to a 17th century physicist the contortions the universe will go through in order to make it that even that arbitrarily-fast object from a certain point of view will still not witness anything going faster than light, with length contraction, time dilation, all that jazz. If you hadn't been taught it, you wouldn't ever come with it on your own. That is, it's a complicated mixture of both having and not having a speed limit that, when fully understood, makes sense. But if you try to put the answer in the context of the original Newtonian understanding it makes no sense. Personally I expect the base of our universe to have something similar when it comes to "is it continuous or discrete?" I expect it's going to end up being something that has elements of both, in some manner that is essentially impossible to understand if you insist on fitting it into that strict dichotomy but all works together and makes sense on its own terms. Loop quantum gravity is the best effort in this direction I've seen, but since it isn't "done" yet doesn't seem to be the right solution. (There's been a lot of things in science history where there was some sort of "Is this thing rigidly This or That?" and the answer turns out to be "well, yes, and also no, it's more complicated and fluid than you thought" in some manner that would have been difficult to guess in advance.)

63 points by hackandthink 3y ago

dcow 3y ago

I find the idea of a universe with a discrete substrate/structure both natural and compelling. I wish we’d explore it more. If you really think about it, it’s hard to imagine nature would just happen into a system with irrational numbers as we naïvely (undergraduate level college and below) understand them. Things clearly look infinite on a macro level, but “what type of discrete systems give rise to the observed macro behavior” is a field that needs more exploration. The maths analog is alternative number systems, which at least seem to be gaining interest and popularity. The metaphor for programmers: is the universe just a beautifully seeded Conway’s game of life?

greatquux 3y ago
Couldn't agree more, though when I see this brought up, I've always seen "people who seem to know more than me" saying that it just leads to other problems, and they kind of hand-wave it away... but I still think there should be some more serious exploration of this. How could "real" numbers _truly_ exist, any more than "imaginary" ones?
- housecarpenter 3y ago
  I think anybody who thinks real numbers exist will also say that imaginary numbers exist. "real" and "imaginary" in the context of numbers are just arbitrary labels assigned for historical reasons, they're not meant to convey any philosophical judgements about existence.
  - AnimalMuppet 3y ago
    I don't think so. You can have imaginary rationals, or even imaginary natural numbers (that is, a+bi, where both a and b are rational numbers or integers). Although I will admit that a big part of the point of the reals was solutions to polynomials, and it takes complex numbers to be able to solve them all.
imglorp 3y ago
Maybe someone has a survey paper? Two for discussion would be Loop Quantum Gravity and honorable mention for Wolfram's hypergraph.
- eigenspace 3y ago
  > Maybe someone has a survey paper? Two for discussion would be Loop Quantum Gravity What would you like to know about it? > honorable mention for Wolfram's hypergraph. I'd call it more of a dishonourable mention. If you're looking for niche theories in physics with discrete spacetime, I guess you'd want to look at https://en.wikipedia.org/wiki/Causal_sets https://en.wikipedia.org/wiki/Causal_sets
  - imglorp 3y ago
    Thanks. Curious why the hypergraph diss? Regardless of their process, it sounds like they came up with a model which has some alignments with conventional theory.
  - aeonik 3y ago
    Why dishonorable? His work is pretty interesting to me: graph taxonomy of all possible rules and rules spaces (what Wolfram calls the Ruliad) seems like a very natural extension of Computer science and Analysis.
cwmma 3y ago
Sounds like your describing complex systems thoery https://en.wikipedia.org/wiki/Complex_system https://en.wikipedia.org/wiki/Complex_system
- dcow 3y ago
  Not quite. I’m describing a system with discrete structure that gives rise to macro level complex system in the same way the game of life can implement arbitrary computation.
bmacho 3y ago
> I find the idea of a universe with a discrete substrate both natural and compelling. I really like the idea of a finite, ever growing, and totally knowable universe. What Wolfram is searching for. The universe was a small finite graph at the beginning (which we can possibly know by looking at the sky), and it grows since that, according to some deterministic rules. This means that the universe is perfectly replayable, we can replay history, observe the thoughts of famous people, and such.
- dcow 3y ago
  I believe there may be ways to introduce entropy into a system like that so it need not completely deterministic, but can’t quite remember the name for the topic…
nyrikki 3y ago
The real numbers are irrational almost everywhere, they are uncomputable 'almost everywhere' also. “what type of discrete systems give rise to the observed macro behavior” is what we have been doing forever through Laplacian determinism. There are no alternative numbers systems that get around this. That is the whole point of the problem of the cardinality of the continuum. We know that Laplacian determinism fails, with quantum mechanics being the easiest counterexample but Cantor diagonalization is another. The sofar unsuccessful efforts to produce string theory is an example of this effort, changing world lines into sheets. With the discovery of strange non chaotic attractors and the discovery that our numerical systems are subject to the WadA property in time delayed and Hamilton systems it is possible that a unified model may not exist. None of this is undergrad though, due to how we have chosen to teach it. But it is very possible that existential quantifiers are what math limits is to building models of.

rnjailamba 3y ago

Previous discussion (2016) - https://news.ycombinator.com/item?id=12441990 https://news.ycombinator.com/item?id=12441990

xanderlewis 3y ago

I remember telling my tutor in the first year of a (pure) mathematics degree that I didn’t really enjoy analysis because I “couldn’t see why anyone should believe in the real numbers”. He looked at me as if I was slightly mad. At the time I’m not sure I realised I wasn’t totally alone in having such worries. It seems that most professional mathematicians have never really given it much thought (or at least no more than other ontological, ‘non-mathematical’ questions). Of course, when it comes to fundamental physics it all starts to become more relevant. edit: My comment shouldn’t be taken as a recommendation. I’m not suggesting mathematicians should worry about such things (it may well be a waste of their time!). I’m simply surprised that more don’t find it as unnerving as I did at the time.

agnosticmantis 3y ago
Bring us another complete ordered field and I’ll pay handsomely for it.
whatisyour 3y ago
Professional mathematicians have given it a lot of thought. Your tutor probably just didn't wonder enough about it. I did my graduate studies in numerical methods of mathematical physics, and everyone I talked to agreed, that the continuum assumption creates a lot of problems. But even so, the nice things they give in the space of functions and numerical convergence has been more important than the difficulties they cause. And hence, continuum analysis it is. In one of my papers, I actually assumed that two features in material have to be minimum epsilon distance away to show that the method convergences in a reasonable amount of computational effort. (where epsilon is arbitrarily small) So, I did break the continuum barrier to do some useful physics ones.
- xanderlewis 3y ago
  Some have given it a lot of thought, but not nearly as many as I would have expected. Despite such questions being philosophical in nature it seems to mostly be mathematical physicists and applied mathematicians who spend time on it. Pure mathematicians mostly don’t care; they’re not interested in existence or ‘reality’ — if there’s complexity in an idea it’s worth studying whether it has anything to say about models of the universe or not. I guess what I’m trying to say is just that I’ve always been surprised by the lack of interest most pure mathematicians show towards questions of ‘what exists’ or ‘what the purpose of mathematics is’ and so on. I would have expected such (clearly) intellectually curious people to be less myopic. Because the world is a big place, one can find countless essays on these sorts of topics. But it’s still a minority sport. John Baez is a brilliant outlier in many ways.

Aardwolf 3y ago

As far as I know, just like matter and energy, information density can also contribute to a black hole. A real number has infinite precision, so even a single particle with a real number position or weight, would have infinite information and create a black hole. Would that be a valid reasoning to say why it can't be real numbers?

eigenspace 3y ago
No.
- Aardwolf 3y ago
  > No. Please elaborate
  - eigenspace 3y ago
    Because what you're describing is coordinate or unit system dependent. I could invent a unit system where the mass of an electron is pi units, and another unit system where the mass of the electron is 1 unit. Me using a different unit system doesn't affect the physics of the system.
    - Aardwolf 3y ago
      But the particles can move around, they won't always be exactly on values of pi, 1, etc... in any coordinate system. And pi and 1 happen to be examples that can be described with a finite amount of symbols. But a continuous particle has to move through any value in between.
      - aeonik 3y ago
        Your idea interests me. I think it's related to the Plank Scale, but what you're talking about would possibly be a "Plank Precision"? I'm having a hard time seeing if this is fundamentally the same thing as the Plank Length, or if it's something different entirely. https://en.wikipedia.org/wiki/Planck_units#Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_length Very cool idea though.

red_trumpet 3y ago

> For example: how many real numbers are there? The continuum hypothesis proposes a conservative answer, but since this is independent of the usual axioms of set theory, the question remains open: there could be vastly more real numbers than most people think. If don't agree with that. There are exactly as many real numbers as there are inifnite sequences consisting of zeroes and on (0,0,1,0,1,...). That is the same as the set of subsets of the natural numbers. That is well-defined. The continuum hypothesis asks if there are infinities smaller than this one, but bigger than the infinity of the natural numbers.

nyrikki 3y ago
The constructable numbers, which is what you are describing are aleph naught, or countable. The Cantor set is a way to think about this.
- red_trumpet 3y ago
  Not true. If you think about binary representation of numbers it is clear that there are as many infinite binary sequences as there are real numbers between 0 and 1. So that is uncountably infinite. Alternatively, apply the usual diagonal argument directly to the set of sequences to see that there are uncountably many of them. This has not much to do with constructible numbers, which are those (complex) numbers which can be constructed using compass and straightedge constructions starting from 0 and 1.
  - nyrikki 3y ago
    The definable numbers, no matter if you use the constructible, algebraic, or computable numbers are countable infinities, or aleph naught. Any segment of the real line, no matter how small you make it is aleph_1 or an uncountable infinity. I realize that those concepts are difficult to build intuitions around, but what I said was correct. definable,algebraic, and computable numbers have such a tiny percentage of the real line that added together their infinity of representations are still countable and equal in size to the natural numbers. You could map them all 1 to 1 to natural numbers and they would be the same size as the natural numbers, which is tiny compared to the reals. The reals being uncomputable 'almost everywhere' is well proven at this point. Cantor diagonalization being the typical way of showing it. 'Almost everywhere' as used above has a formal definition: A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero. The reals are uncomputable 'almost everywhere', the reals are not constructable 'almost everywhere' etc...

at_a_remove 3y ago

So, I only escaped with a bachelor's in physics and a bit more math than was usually required for the degree, but I have often wondered if "we" went astray by spending time on concepts like infinity. Summing 1 + 2 + 3 + 4 + ... to get -1/12 using various infinities and "regularization" just seems, well, like some kind of sleight of hand, and certainly divorced from physical reality.

jjgreen 3y ago
Ramanujan summation (1 + 2 + ... = -1/12) is used in the derivation of Casimir effect I'm afraid.
- at_a_remove 3y ago
  I wonder if there's an alternate way to get there without that monstrosity, which manages to be triply-offensive.

audunw 3y ago

The way renormalisation is mostly covered in popular media, made me think you can’t do quantum physics without dirty tricks to get rid of infinites. Then I read The Road to Reality by Roger Penrose and in the chapter on renormalisation it mentions just “by the way“ that the problem goes away if you assume the universe has a minimum scale, or something like that. Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation? Why is the default assumption that the universe is continuous? Doesn’t seem like we have a good reason for it, other than the fact that we use integration as a tool and that assumes continuity by default. Does it make mathematical analysis easier all-in-all? Why not just pick the Planck length as a reasonable assumption for the quantisation of space? Might not be correct but probably a more reasonable guess than assuming that it’s continuous?

eigenspace 3y ago
> Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation? It's because of the historical development where it really did start off as a nasty bag of mathematical tricks. It wasn't until much later that Wilson and co put renormalization procedures on firm footing, and by that time it had a quite bad reputation.
ndsipa_pomu 3y ago
I thought it was due to relativity which assumes a continuous space. There's a lot of evidence to back up relativity, though it does have issues with not working in a black hole, so it would be questionable to throw it out without something better to replace it.
BoiledCabbage 3y ago
> other than the fact that we use integration as a tool and that assumes continuity by default. Is there really no equivalent of calculus that works on a "quantized"/discreet numberline? The seems really hard to believe.
ko27 3y ago
> Why is the default assumption that the universe is continuous? Why is your default assumption that physicists haven't tried to make it discrete? You need to break Lorentz invariance to have the universe be discrete, which is an effort that's worth a Nobel prize. > Doesn’t seem like we have a good reason for it We have many good reasons for it. https://www.forbes.com/sites/startswithabang/2020/04/17/this-is-why-space-needs-to-be-continuous-not-discrete/ https://www.forbes.com/sites/startswithabang/2020/04/17/this... From the article: "If space is discrete, then the principle of relativity is wrong"

jerf 3y ago

If you look at the history of physics, including not just the final breakthroughs but the previous misunderstandings, there's a long history of "Well, we thought it had to be either A or B, but it turns out it's a novel and hard-to-imagine combination of both of those things." For example, consider the universal speed limit that we now call the speed of light. Prior to relativity, when space and time were kept rigidly separated, there was a bit of a problem. The math of Newtonian physics and their corresponding frame transforms admit no particular speed limit; no matter how fast a thing is going, a force could always be applied to it to make it go yet faster. So a speed limit seemed to be impossible. But observationally, it became more and more clear there was one, and what's more, it seemed to be unrelated to your frame of reference. So apparently there was a speed limit, and not only that, but one that was essentially uncharacterizable in Newtonian physics. But if something was going the speed limit, what happened if you applied further forces on that thing? The universe seemed contradictory; it seemed to have a speed limit, yet it couldn't have one, and if it did have one it seemed it would lead to contradictions as well. So, does the universe have a speed limit or not? And the answer is basically that that question, in the context of a Newtonian view of the universe, is simply ill formed and has no meaning, because the universe isn't Newtonian. With the better (if not necessarily entirely correct) view that Einstein introduced, what we get is actually something that even now a lot of people don't really get, which is that the universe both does and does not have a speed limit. The former is generally understood; c is the limit. You can not witness anything in the universe going faster than c from your perspective. But there is also a sense in which the universe doesn't have a speed limit; no matter how fast you are going, you can always accelerate in any direction and go faster in that direction. There isn't a speed limit in the sense that there is no speed that you can be going where the universe will say "Nope, stop, you can't go any faster that way because physics won't allow it." (Other things, like blueshifting the cosmic microwave background into arbitrarily high-energy gamma rays from your point of view may stop you, but "physics" itself won't.) Subjectively, time dilation means that there is a sense in which you can head to distant location as fast as you like; you can cover what was originally a million light years in arbitrarily short amounts of subjective time, and there is a real sense in which that becomes your "speed" and it can greatly exceed lightspeed in your original reference frame. And just try explaining to a 17th century physicist the contortions the universe will go through in order to make it that even that arbitrarily-fast object from a certain point of view will still not witness anything going faster than light, with length contraction, time dilation, all that jazz. If you hadn't been taught it, you wouldn't ever come with it on your own. That is, it's a complicated mixture of both having and not having a speed limit that, when fully understood, makes sense. But if you try to put the answer in the context of the original Newtonian understanding it makes no sense. Personally I expect the base of our universe to have something similar when it comes to "is it continuous or discrete?" I expect it's going to end up being something that has elements of both, in some manner that is essentially impossible to understand if you insist on fitting it into that strict dichotomy but all works together and makes sense on its own terms. Loop quantum gravity is the best effort in this direction I've seen, but since it isn't "done" yet doesn't seem to be the right solution. (There's been a lot of things in science history where there was some sort of "Is this thing rigidly This or That?" and the answer turns out to be "well, yes, and also no, it's more complicated and fluid than you thought" in some manner that would have been difficult to guess in advance.)