6 ms·

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this artic

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but representing the state even of a very simple system involves continuous quantities. But the Cantor vision of the real numbers is just wrong and completely unphysical. The idea of arbitrary precision is intrinsically broken in physical reality. Instead I am off the opinion that computation is the relevant process in the physical universe, so approximations to continuous quantities are where the "Eternal Nature" line lies, and the abstraction of the continuum is just that -- an abstraction of the idea of having perfect knowledge of the state of anything in the universe.

empath75 1y ago
> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".
- andrewla 1y ago
  There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant). The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
  - EthanHeilman 1y ago
    > On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers. > The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
  - nh23423fefe 1y ago
    > the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
    - andrewla 1y ago
      That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
      - oskaralund 1y ago
        Saying that two sets have the same cardinality is equivalent to saying there is a bijection between them. I don't understand how the latter can be useful but not the former?
      - sfpotter 1y ago
        It isn't very interesting or useful... to you.
  - SabrinaJewson 1y ago
    > For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful. I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
- shkkmo 1y ago
  > They're unphysical, and yet the very physical human mind can work with them just fine. Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
  - SabrinaJewson 1y ago
    We can work with unnameable things very easily. Take, for instance, every known theorem that quantifies over all real numbers. If you try to argue that proving theorems about these real numbers does not constitute “working with” them, it seems you have chosen a rather deficient definition of “working with” that does not match with how that phrase is used in the real world.
    - shkkmo 1y ago
      I would argue that all of those theorems work with nameable sets of real numbers but not with any unnamable real numbers themselves.
- Eddy_Viscosity2 1y ago
  The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.
  - SabrinaJewson 1y ago
    I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there? In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
    - Eddy_Viscosity2 1y ago
      You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.
      - drdeca 1y ago
        The decimal places are just a way of representing it.
        Eddy_Viscosity2 1y ago
        The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
- tialaramex 1y ago
  > They're unphysical, and yet the very physical human mind can work with them just fine Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
  - ysofunny 1y ago
    the rationals are 3 naturals with in a "2,1" structure. the first 2 naturals form an integer. that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble) what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to be really real Reality. in the mathematical realm, finite things are "impossible", they become "zero", negible in the presence of infinities. it's impossible for the primes to be finite (by contradiction). it's impossible for things (numbers or functions of mathematical objects) to be finite. but in the physical reality, it's the "infinite things" which become impossible. the "decimal point" (i.e. scientific notation i.e. positional numeral systems) is truly THE wonder of the world. for some reason I want something better than such a system... so I'm still learning about categories
    - tialaramex 1y ago
      Huh?
NooneAtAll3 1y ago
if we are arguing that natural numbers are made from abstraction, then we must apply that to real numbers as well - quantum values are complex numbers, that only become real once we start asking "what is position of the thing" or "what's its velocity"
blueplanet200 1y ago
>I'm an enthusiastic Cantor skeptic A skeptic in what way? He said a lot.
- andrewla 1y ago
  Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.
  - blueplanet200 1y ago
    Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
    - andrewla 1y ago
      I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
      - CyLith 1y ago
        I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
        1y ago
chasd00 1y ago
> The idea of arbitrary precision is intrinsically broken in physical reality. you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.
- LegionMammal978 1y ago
  > wherever there is a circle, Is there a circle in physical reality? Or only approximate circles, or things we model as circles? In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
- Dylan16807 1y ago
  Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?
  - IAmBroom 1y ago
    A black hole.
    - andrewla 1y ago
      A non-rotating black hole. Or a rotating black hole with zero charge. Or a rotating black hole with non-zero charge no external magnetic fields. Or a rotating black hole with non-zero charge with non-time-varying external magnetic fields. Or a wart on a frog on a bump on the log on a hole on the bottom of the sea.
    - tromp 1y ago
      A black hole is no more a perfect sphere than a sun is. Would gravity from the nearest other black hole not have a deforming effect of at least 2^-64 ?
    - Kranar 1y ago
alphazard 1y ago
> Nature and the universe is all about continuous quantities One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.
- adrian_b 1y ago
  The quantity of matter and the quantity of electricity are discrete, but work, time and space are continuous, like also any quantities derived from them. There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts. Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, or more precisely the models that we can use to predict what happens in nature are still based mostly on continuous quantities, despite the fact that about a century and a half have passed since the discreteness of matter and electricity has been confirmed.
  - gpm 1y ago
    > but work, time and space are continuous I'm under the impression that all our theories of time and space (and thus work) break down at the scale of 1 plank unit and smaller. Which isn't proof that they aren't continuous, but I don't see how you could assert that they are either.
    - dhosek 1y ago
      Matter and energy are discrete. The continuity or discreteness of time and space are unknown. There are arguments for both cases, but nobody really knows for sure. It’s fairly easy to go from integers to many subsets of the reals (rationals are straightforward, constructible numbers not too hard, algebraic numbers more of a challenge), but the idea that the reals are, well real, depends on a continuity of spacetime that we can’t prove exists.
      - adrian_b 1y ago
NoahZuniga 1y ago
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
- IAmBroom 1y ago
  > You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.
  - NoahZuniga 1y ago
    The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
    - oskaralund 1y ago
      This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
      - BalinKing 1y ago
        AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.
kazinator 1y ago
> representing the state even of a very simple system involves continuous quantities. But that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.
wizardforhire 1y ago
1/137
daxfohl 1y ago
But what if the expansion of the universe is due to some banach-tarski process?
tomasson 1y ago
“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ” Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!
- griffzhowl 1y ago
  Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies
  - svnt 1y ago
    Energies, yes, but the concept of energy quanta is inverted to e.g. time and length in that we have a maximum, not a minimum, where our understanding/models are limited, right?

empath75 1y ago

> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".

andrewla 1y ago
There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant). The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
- EthanHeilman 1y ago
  > On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers. > The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
- nh23423fefe 1y ago
  > the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
  - andrewla 1y ago
    That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
    - oskaralund 1y ago
      Saying that two sets have the same cardinality is equivalent to saying there is a bijection between them. I don't understand how the latter can be useful but not the former?
    - sfpotter 1y ago
      It isn't very interesting or useful... to you.
- SabrinaJewson 1y ago
  > For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful. I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
shkkmo 1y ago
> They're unphysical, and yet the very physical human mind can work with them just fine. Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
- SabrinaJewson 1y ago
  We can work with unnameable things very easily. Take, for instance, every known theorem that quantifies over all real numbers. If you try to argue that proving theorems about these real numbers does not constitute “working with” them, it seems you have chosen a rather deficient definition of “working with” that does not match with how that phrase is used in the real world.
  - shkkmo 1y ago
    I would argue that all of those theorems work with nameable sets of real numbers but not with any unnamable real numbers themselves.
Eddy_Viscosity2 1y ago
The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.
- SabrinaJewson 1y ago
  I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there? In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
  - Eddy_Viscosity2 1y ago
    You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.
    - drdeca 1y ago
      The decimal places are just a way of representing it.
      - Eddy_Viscosity2 1y ago
        The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
tialaramex 1y ago
> They're unphysical, and yet the very physical human mind can work with them just fine Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
- ysofunny 1y ago
  the rationals are 3 naturals with in a "2,1" structure. the first 2 naturals form an integer. that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble) what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to be really real Reality. in the mathematical realm, finite things are "impossible", they become "zero", negible in the presence of infinities. it's impossible for the primes to be finite (by contradiction). it's impossible for things (numbers or functions of mathematical objects) to be finite. but in the physical reality, it's the "infinite things" which become impossible. the "decimal point" (i.e. scientific notation i.e. positional numeral systems) is truly THE wonder of the world. for some reason I want something better than such a system... so I'm still learning about categories
  - tialaramex 1y ago
    Huh?

NooneAtAll3 1y ago

if we are arguing that natural numbers are made from abstraction, then we must apply that to real numbers as well - quantum values are complex numbers, that only become real once we start asking "what is position of the thing" or "what's its velocity"

blueplanet200 1y ago

>I'm an enthusiastic Cantor skeptic A skeptic in what way? He said a lot.

andrewla 1y ago
Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.
- blueplanet200 1y ago
  Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
  - andrewla 1y ago
    I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
    - CyLith 1y ago
      I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
      - 1y ago

chasd00 1y ago

> The idea of arbitrary precision is intrinsically broken in physical reality. you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.

LegionMammal978 1y ago
> wherever there is a circle, Is there a circle in physical reality? Or only approximate circles, or things we model as circles? In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
Dylan16807 1y ago
Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?
- IAmBroom 1y ago
  A black hole.
  - andrewla 1y ago
    A non-rotating black hole. Or a rotating black hole with zero charge. Or a rotating black hole with non-zero charge no external magnetic fields. Or a rotating black hole with non-zero charge with non-time-varying external magnetic fields. Or a wart on a frog on a bump on the log on a hole on the bottom of the sea.
  - tromp 1y ago
    A black hole is no more a perfect sphere than a sun is. Would gravity from the nearest other black hole not have a deforming effect of at least 2^-64 ?
  - Kranar 1y ago

alphazard 1y ago

> Nature and the universe is all about continuous quantities One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.

adrian_b 1y ago
The quantity of matter and the quantity of electricity are discrete, but work, time and space are continuous, like also any quantities derived from them. There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts. Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, or more precisely the models that we can use to predict what happens in nature are still based mostly on continuous quantities, despite the fact that about a century and a half have passed since the discreteness of matter and electricity has been confirmed.
- gpm 1y ago
  > but work, time and space are continuous I'm under the impression that all our theories of time and space (and thus work) break down at the scale of 1 plank unit and smaller. Which isn't proof that they aren't continuous, but I don't see how you could assert that they are either.
  - dhosek 1y ago
    Matter and energy are discrete. The continuity or discreteness of time and space are unknown. There are arguments for both cases, but nobody really knows for sure. It’s fairly easy to go from integers to many subsets of the reals (rationals are straightforward, constructible numbers not too hard, algebraic numbers more of a challenge), but the idea that the reals are, well real, depends on a continuity of spacetime that we can’t prove exists.
    - adrian_b 1y ago

NoahZuniga 1y ago

You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?

IAmBroom 1y ago
> You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.
- NoahZuniga 1y ago
  The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
  - oskaralund 1y ago
    This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
    - BalinKing 1y ago
      AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.

kazinator 1y ago

> representing the state even of a very simple system involves continuous quantities. But that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.

wizardforhire 1y ago

1/137

daxfohl 1y ago

But what if the expansion of the universe is due to some banach-tarski process?

tomasson 1y ago

“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ” Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!

griffzhowl 1y ago
Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies
- svnt 1y ago
  Energies, yes, but the concept of energy quanta is inverted to e.g. time and length in that we have a maximum, not a minimum, where our understanding/models are limited, right?