I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).
If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals.
This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.
You're taking an anomalously narrow view of the parent comment. Say the minimum distance is one inch.
You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.
dhosek is saying that in a quantized space, there are reals that cannot be demonstrated this way, and he is right, but the same thing is untrue of rationals.
("In a quantized space", by the way, just means that all measured quantities are necessarily integers. That causes all kinds of problems, but "lacking examples of arbitrary rational numbers" isn't one of them.)
> You can easily demonstrate it [half an inch] as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.
You seem to have your units confused. Half an inch is a distance, while
ratios, or comparisons of ratios, are all dimensionless scalars.
I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist.
But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.
In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. And any useful number can be approximated arbitrarily closely by rationals.
And for computable numbers there's even less of a gap. With rationals you can only approximate. But you can have a computable number that is exactly equal to the square root of 2, because a computable number is the algorithm by which you form arbitrarily close approximations. The square of that computable number is itself computable and is exactly equal to 2.
What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
And if you're worried that sticking to the rationals and the computable numbers is too much of a concession to "physical reality", rest assured -- the rationals are just as unphysical as the real numbers because they are continuous already, and physics does not give us the power to measure the difference between two sufficiently precise rational numbers just as it barfs when you throw "real" numbers at it.
Navel gazing is a physical phenomenon, so if you would like to know everything about physical phenomena, you have to be able to predict the navel gazers.
I don't think your position is silly, but this is not a great argument for it.
> But when we say things like "the rationals are discrete"
In the usual topology they are not?
> In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers.
This characterization captures neither the intuitive nor the formal definition of continuity. You are effectively saying that Q is dense in R, but this is insufficient to prove, for example, the intermediate value theorem.
> measure theory, a theory which yields almost nothing of value except endless paradoxes
Come on now. The usual definition of concepts as basic as areas is tethered to measure theory. We say it's "obvious" that the integral is the area under the curve (and it is: e.g. the Riemann integral is trivially the Peano-Jordan measure) but this only works because we're appealing to it.
You can route around it, but let's not pretend we're doing it for no reason.
I can see the elegance of a purely intuitionistic construction, but the "usual" real numbers are much closer to how we intuitively (no pun intended) work with numbers.
No, the rationals are not discrete in the usual topology. They end up being discrete when we consider continuous mappings from R->Q though. That is the "technical" sense that I refer to. The rationals, as you say, are dense in R but they are also dense in the computables.
The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techniques (Dedekind cuts or Cauchy sequences) effectively only yield the computable numbers, the real numbers outside of the computables are inherited from the diagonal argument rather than being foundational to the construction. I mean, this is trivially true because constructions are constructive.
I disagree that area is tethered to measure theory; I certainly learned about areas in geometry long before I ever heard of anything with measure theory. Measure theory exists to tie up some of the horrifying poorly behaved functions that increasingly wily mathematicians invented to break our notions of area and continuity. But we have better tools now for dealing with those that don't involve measure theory so there's no reason to ever hear the phrase "almost everywhere" or "subadditive" ever again.
To back it up to your closing and my main point -- the constructive numbers are way closer to the way we work with numbers because all numbers we ever deal with, even abstractly, fit this definition much better.
> What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
It buys you the rigor of doing calculus, which buys you a lot of results that, while could be computed without calculus, would also be very difficult without it.
Doing calculus with computable numbers is totally possible and you get all the continuity you need. You need to drop the Lebesgue formulation of the integeral and either use a Reimann integral or the gauge integral (Henstock–Kurzweil) if you need a well-behaved integral in the face of very poorly-behaved functions, but in physical reality these don't exist and in abstract mathematics they are rarely of interest and the gauge integral is as robust as Lebesgue without all the measure theory nonsense.
Intuitionalist analysis and calculus are very well established; the only thing you can't do with them is nonsense like showing that integrating over the characteristic function of the rationals is zero (who cares) or showing that you can break a three dimensional sphere up into three pieces are reassemble them after translations and rotations into a larger sphere (obviously not true).
Approximate relative to what? All actual measurement is implicitly or explicitly approximate such as L = x meters +/- epsilon. There is no infinite precision by which to discount rational measures as "approximate" and thus "invalid" in any way.
>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
You "buy" all of mathematics which operates on the assumption of "infinite" precision. It is an abstraction necessary to prove theorems and relationships of math. Abstracting from precision isn't a denial that it exists, it is the assumption that I can ignore it or leave it undefined. This is the assumption that distinguishes math from physics/engineering. Mathematicians deal with abstract e or pi but in the real world pi=3.14 if you are tiling your patio and 3.14159265... or whatever is necessary to get to the moon.
pi and e and sqrt(2) are real numbers and not rational, to be sure. But they are computable! Computable just means that they are arbitrarily approximable. "approximate relative to what" is that whatever criteria defines the number. You can't represent the "true" value of a non-rational number in the rationals, but you can prove that the error of an approximation is (rationally) bounded above and below, and you can have another approximation with a tighter bound.
Rational numbers are already infinitely precise relative to other representations -- finite decimals are another representation that is functionally equivalent to the rationals, but even a simple rational like 1/3 does not have a finite decimal value.
You can prove all the interesting theorems with computable numbers and rational/decimal numbers. You don't need the real numbers because you can't name a real number that exists and is not computable, BY DEFINITION! No mathematical construction can define a real number that is not constructible. These numbers are useless and there's no reason to continue even in abstract mathematics to pretend that they are useful because we have the formalisms to ignore them.
I'm on your side for most of what you say. This topic has been interesting to me for years. I've considered going back to school to build on my math degree, specifically because of this topic.
However, I thought things like Chaitin's Constants (you could make one per programming language) are real numbers you can name but not compute. I think you could do this from any undecidable problem.
Of course there only a countable number of those Reals. And they still don't seem useful for much more than naval gazing.
I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).
If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals.
This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.
You're taking an anomalously narrow view of the parent comment. Say the minimum distance is one inch.
You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.
dhosek is saying that in a quantized space, there are reals that cannot be demonstrated this way, and he is right, but the same thing is untrue of rationals.
("In a quantized space", by the way, just means that all measured quantities are necessarily integers. That causes all kinds of problems, but "lacking examples of arbitrary rational numbers" isn't one of them.)
> You can easily demonstrate it [half an inch] as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.
You seem to have your units confused. Half an inch is a distance, while ratios, or comparisons of ratios, are all dimensionless scalars.
Have you ever seen a map with a scale indicator?
I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist.
But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.
In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. And any useful number can be approximated arbitrarily closely by rationals.
And for computable numbers there's even less of a gap. With rationals you can only approximate. But you can have a computable number that is exactly equal to the square root of 2, because a computable number is the algorithm by which you form arbitrarily close approximations. The square of that computable number is itself computable and is exactly equal to 2.
What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
And if you're worried that sticking to the rationals and the computable numbers is too much of a concession to "physical reality", rest assured -- the rationals are just as unphysical as the real numbers because they are continuous already, and physics does not give us the power to measure the difference between two sufficiently precise rational numbers just as it barfs when you throw "real" numbers at it.
Navel gazing is a physical phenomenon, so if you would like to know everything about physical phenomena, you have to be able to predict the navel gazers.
I don't think your position is silly, but this is not a great argument for it.
> But when we say things like "the rationals are discrete"
In the usual topology they are not?
> In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers.
This characterization captures neither the intuitive nor the formal definition of continuity. You are effectively saying that Q is dense in R, but this is insufficient to prove, for example, the intermediate value theorem.
> measure theory, a theory which yields almost nothing of value except endless paradoxes
Come on now. The usual definition of concepts as basic as areas is tethered to measure theory. We say it's "obvious" that the integral is the area under the curve (and it is: e.g. the Riemann integral is trivially the Peano-Jordan measure) but this only works because we're appealing to it. You can route around it, but let's not pretend we're doing it for no reason.
I can see the elegance of a purely intuitionistic construction, but the "usual" real numbers are much closer to how we intuitively (no pun intended) work with numbers.
No, the rationals are not discrete in the usual topology. They end up being discrete when we consider continuous mappings from R->Q though. That is the "technical" sense that I refer to. The rationals, as you say, are dense in R but they are also dense in the computables.
The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techniques (Dedekind cuts or Cauchy sequences) effectively only yield the computable numbers, the real numbers outside of the computables are inherited from the diagonal argument rather than being foundational to the construction. I mean, this is trivially true because constructions are constructive.
I disagree that area is tethered to measure theory; I certainly learned about areas in geometry long before I ever heard of anything with measure theory. Measure theory exists to tie up some of the horrifying poorly behaved functions that increasingly wily mathematicians invented to break our notions of area and continuity. But we have better tools now for dealing with those that don't involve measure theory so there's no reason to ever hear the phrase "almost everywhere" or "subadditive" ever again.
To back it up to your closing and my main point -- the constructive numbers are way closer to the way we work with numbers because all numbers we ever deal with, even abstractly, fit this definition much better.
> What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
It buys you the rigor of doing calculus, which buys you a lot of results that, while could be computed without calculus, would also be very difficult without it.
Agree to disagree!
Doing calculus with computable numbers is totally possible and you get all the continuity you need. You need to drop the Lebesgue formulation of the integeral and either use a Reimann integral or the gauge integral (Henstock–Kurzweil) if you need a well-behaved integral in the face of very poorly-behaved functions, but in physical reality these don't exist and in abstract mathematics they are rarely of interest and the gauge integral is as robust as Lebesgue without all the measure theory nonsense.
Intuitionalist analysis and calculus are very well established; the only thing you can't do with them is nonsense like showing that integrating over the characteristic function of the rationals is zero (who cares) or showing that you can break a three dimensional sphere up into three pieces are reassemble them after translations and rotations into a larger sphere (obviously not true).
>With rationals you can only approximate.
Approximate relative to what? All actual measurement is implicitly or explicitly approximate such as L = x meters +/- epsilon. There is no infinite precision by which to discount rational measures as "approximate" and thus "invalid" in any way.
>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
You "buy" all of mathematics which operates on the assumption of "infinite" precision. It is an abstraction necessary to prove theorems and relationships of math. Abstracting from precision isn't a denial that it exists, it is the assumption that I can ignore it or leave it undefined. This is the assumption that distinguishes math from physics/engineering. Mathematicians deal with abstract e or pi but in the real world pi=3.14 if you are tiling your patio and 3.14159265... or whatever is necessary to get to the moon.
You are overestimating what real numbers buy you.
pi and e and sqrt(2) are real numbers and not rational, to be sure. But they are computable! Computable just means that they are arbitrarily approximable. "approximate relative to what" is that whatever criteria defines the number. You can't represent the "true" value of a non-rational number in the rationals, but you can prove that the error of an approximation is (rationally) bounded above and below, and you can have another approximation with a tighter bound.
Rational numbers are already infinitely precise relative to other representations -- finite decimals are another representation that is functionally equivalent to the rationals, but even a simple rational like 1/3 does not have a finite decimal value.
You can prove all the interesting theorems with computable numbers and rational/decimal numbers. You don't need the real numbers because you can't name a real number that exists and is not computable, BY DEFINITION! No mathematical construction can define a real number that is not constructible. These numbers are useless and there's no reason to continue even in abstract mathematics to pretend that they are useful because we have the formalisms to ignore them.
I'm on your side for most of what you say. This topic has been interesting to me for years. I've considered going back to school to build on my math degree, specifically because of this topic.
However, I thought things like Chaitin's Constants (you could make one per programming language) are real numbers you can name but not compute. I think you could do this from any undecidable problem.
Of course there only a countable number of those Reals. And they still don't seem useful for much more than naval gazing.
Similar fun at this Baez blog post from a decade ago: Surprises in Logic: https://math.ucr.edu/home/baez/surprises.html
How complex are complex numbers?
How integral are integers?