The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
andrewla · 1h ago
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.
NoahZuniga · 13m 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?
blueplanet200 · 41m ago
>I'm an enthusiastic Cantor skeptic
A skeptic in what way? He said a lot.
andrewla · 14m 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.
alphazard · 21m 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.
chasd00 · 25m 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 · 22m 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.
NooneAtAll3 · 46m 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"
empath75 · 1h 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 · 47m 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 · 37m 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 · 35m 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 · 11m 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.
Eddy_Viscosity2 · 35m 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 · 5m 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.
tialaramex · 26m ago
> They're unphysical, and yet the very physical human mind can work with them just fine
Nah, you're likely think 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.
shkkmo · 40m 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.)
LegionMammal978 · 13m ago
Personally, I like to split the difference: the physical continuum definitely exists, to whatever extent any physical thing exists, but the real number line (and indeed the completed inductive set of integers) may just be a human-constructed fiction. The physical continuum is not necessarily identical to the real continuum; the latter is just a very useful model that lets us do human things like calculus.
athrowaway3z · 26m ago
Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago.
Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing).
The integers also create infinite gaps. The primes.
So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.
The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
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.
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?
A skeptic in what way? He said a lot.
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.
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.
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.
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".
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.
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.
it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
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.
Nah, you're likely think 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.
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.)
Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes.
So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.