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> 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 the

> 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.
  - andrewla 1y ago
    Yeah, we teach it. It ends up showing up again in measure theory, assuming that anyone still bothers to teach the mostly useless Lebesgue integral instead of the gauge integral. Measure theory shows up again in probability theory if you're not using Kolmogorov for some sadistic reason and you have to deal with countability. Otherwise it's pretty much a dead end unless you're in the weeds. You just mutter "almost everywhere" as a caveat once in a while and move on with your life. Nobody really cares about the immensely large group of numbers that by definition we cannot calculate or define or name except to kowtow to what is in retrospect a pretty bad theoretical underpinning for formal analysis.
- blueplanet200 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. Well, there are the same number. So, uh, sorry?
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?

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.
- andrewla 1y ago
  Yeah, we teach it. It ends up showing up again in measure theory, assuming that anyone still bothers to teach the mostly useless Lebesgue integral instead of the gauge integral. Measure theory shows up again in probability theory if you're not using Kolmogorov for some sadistic reason and you have to deal with countability. Otherwise it's pretty much a dead end unless you're in the weeds. You just mutter "almost everywhere" as a caveat once in a while and move on with your life. Nobody really cares about the immensely large group of numbers that by definition we cannot calculate or define or name except to kowtow to what is in retrospect a pretty bad theoretical underpinning for formal analysis.
blueplanet200 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. Well, there are the same number. So, uh, sorry?

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?