6 ms·

We don't even know if real numbers are "real."

by riskneutral 4y ago

labster 4y ago
That’s why I only use natural numbers. You can trust values from organic sources, like the number 5 and good old 23.
- joe__f 4y ago
  Yeah, all this modern stuff is too rational for me
- mayoff 4y ago
  Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. — Leopold Kronecker (“God made the integers; all else is the work of man.”)
chowells 4y ago
Actually, we know that the subset of R that can be used directly has measure zero. Everything else is a polite fiction to make some proofs work.
- avindroth 4y ago
  Any source? Curious as to what is meant by "used directly"
  - joe__f 4y ago
    Probably just that, all physically measurable quantities are rational, and the rational numbers have measure zero in the reals
    - ChrisLomont 4y ago
      >all physically measurable quantities are rational That's not even clear, unless you also assume you can measure exactly a unit length, which is not physically possible. Assuming you can measure something as a perfect rational value implies infinite precision, which is not possible. All physically measurable quantities have uncertainty is what I think you mean, but that doesn't say anything about possible cardinalities.
      - joe__f 4y ago
        I think you're saying that all physically measurable quantities are rational numbers with terminating decimal expansions. The set of rational numbers with terminating decimal expansions is a subset of the set of rational numbers, so my statement that all physically measurable quantities are rational is true. I also agree that there is some uncertainty in all measurements, I don't think this is at odds with what I said.
        ChrisLomont 4y ago
        > I think you're saying that all physically measurable quantities are rational numbers with terminating decimal expansions. No, this most certainly isn't true. For example, take a meter defined (as SI does) as a fraction of the speed of light. Now, for every length in the universe to be a rational fraction of this length is most certainly not true, because lengths, under relativity, form a continuum. Any speed is possible, so any length (via contraction) is possible. There'd be no physical reason all physical processes would be constrained to a subset of rational grid points - things can move freely. Heck, if all physical quantities are rational, are you claiming velocities only occur as rational numbers? That time is only rational numbers? It's far more likely that they'd be irrational, and our man made units are the weird things. Suppose two diagonal corners of a square with rational sides are somehow physically exact.... What is the physical diagonal length? Oh yeah, irrational. So no, physics isn't formed from rational numbers.
        joe__f 4y ago
        I'm making a statement about all possible experimentally measurable quantities, which as you pointed out have a finite precision and so can be represented as having terminating decimal expansions. This as far as I'm aware this is not a controversial statement. I'm not making any kind of statement about whether space-time is continuous or not. You are modelling spacetime as continuous and seem to be saying this is the only possible option. That's not true; there are a number of different models of how space-time behave at the smallest length scales. Try reading about loop quantum gravity for example; in that model spacetime is taken to be a discrete lattice at the smallest scales. In your model with continuous space-time then it's true that a square of side length 1 would have irrational diagonal of length root 2. In reality we don't currently have experimental evidence to say either way, and it could be that your square of side length 1 has a rational diagonal which approximates root 2.
  - bobbylarrybobby 4y ago
    I'm assuming they mean that for a number to be used, it must be computable.
  - chowells 4y ago
    Source: |R| > |N| = |any string of symbols in any notation that has existed in the past, exists now, or will exist in the future|
- riskneutral 4y ago
  > we know that the subset of R that can be used directly has measure zero The cardinality of the natural number set is the same as the cardinality of the rational number set. So, in some sense, you are saying that we know that the natural counting numbers are "real," which is a self-evident truth. In another sense, you are saying that fractions of an inch/cm/etc are "real" which is another self-evident truth. The question is whether uncountable infinity somehow exits in nature (in the case of real numbers, it is a question about the infinitesimally small scale). To say that it's a "polite fiction to make some proofs works" is too strong a statement, we do not know the answer to that question. At the same time, our measurement instrument will always be discrete and bounded, so the question is seemingly beyond science itself.
  - chowells 4y ago
    I'm not making a statement about the real world other than "it's impossible to communicate infinite information." I suppose that's a bit of a leap of faith, but I'm comfortable with it. Everything else I mean comes from math, with no connection to the real world.
    - otabdeveloper4 4y ago
      > it's impossible to communicate infinite information I think laws of conservation do not hold for information complexity. I don't believe you.
- ChrisLomont 4y ago
  We don't know that - some people posit that but it's far from being provable. All physically measured numbers have uncertainty, meaning the value obtained is not an actual number, but is a range, perhaps with some associated probability spread. This is not measure zero. We are also very capable "directly" using various things that may be continuums, such as energy, or time, or velocities, or many other physical quantities.
  - bobbylarrybobby 4y ago
    All the numbers we can use are computable (how can you use a number if you can't actually talk about it?) and there are only countably many of those.
    - ChrisLomont 4y ago
      >All the numbers we can use are computable That's not true :) A nice example is Chaitin's constant [1], which I can use in proof and books and define and on and on.... And it's explicitly and most definitely NOT computable :) There are lots of numbers in lots of areas of mathematics, even symbolic mathematics that are not computable in the sense you want them to be computable. Chaitin's constant is the tip of a very big iceberg. You're using circular logic by claiming the only numbers I can are are the computable ones then claiming all numbers I can use are computable. That's not true. It's a circular argument. [1] https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant
  - chowells 4y ago
    No, I mean something simpler. The subset of R that can be identified is countable, because language is countable. No matter what system you devise for describing numbers, it will be countable. And that means that it cannot describe approximately 100% of the numbers in R. They can't be uniquely described; they can't be used in computations. They're phantoms, at best. You know they're out there, but they will never be usable the way numbers you can actually describe are. The only thing they give you is the ability to declare R to be Cauchy complete in some proofs. They're a polite fiction.

We don't even know if real numbers are "real."

labster 4y ago

That’s why I only use natural numbers. You can trust values from organic sources, like the number 5 and good old 23.

joe__f 4y ago
Yeah, all this modern stuff is too rational for me
mayoff 4y ago
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. — Leopold Kronecker (“God made the integers; all else is the work of man.”)

chowells 4y ago

Actually, we know that the subset of R that can be used directly has measure zero. Everything else is a polite fiction to make some proofs work.

avindroth 4y ago
Any source? Curious as to what is meant by "used directly"
- joe__f 4y ago
  Probably just that, all physically measurable quantities are rational, and the rational numbers have measure zero in the reals
  - ChrisLomont 4y ago
    >all physically measurable quantities are rational That's not even clear, unless you also assume you can measure exactly a unit length, which is not physically possible. Assuming you can measure something as a perfect rational value implies infinite precision, which is not possible. All physically measurable quantities have uncertainty is what I think you mean, but that doesn't say anything about possible cardinalities.
    - joe__f 4y ago
      I think you're saying that all physically measurable quantities are rational numbers with terminating decimal expansions. The set of rational numbers with terminating decimal expansions is a subset of the set of rational numbers, so my statement that all physically measurable quantities are rational is true. I also agree that there is some uncertainty in all measurements, I don't think this is at odds with what I said.
      - ChrisLomont 4y ago
        > I think you're saying that all physically measurable quantities are rational numbers with terminating decimal expansions. No, this most certainly isn't true. For example, take a meter defined (as SI does) as a fraction of the speed of light. Now, for every length in the universe to be a rational fraction of this length is most certainly not true, because lengths, under relativity, form a continuum. Any speed is possible, so any length (via contraction) is possible. There'd be no physical reason all physical processes would be constrained to a subset of rational grid points - things can move freely. Heck, if all physical quantities are rational, are you claiming velocities only occur as rational numbers? That time is only rational numbers? It's far more likely that they'd be irrational, and our man made units are the weird things. Suppose two diagonal corners of a square with rational sides are somehow physically exact.... What is the physical diagonal length? Oh yeah, irrational. So no, physics isn't formed from rational numbers.
        joe__f 4y ago
        I'm making a statement about all possible experimentally measurable quantities, which as you pointed out have a finite precision and so can be represented as having terminating decimal expansions. This as far as I'm aware this is not a controversial statement. I'm not making any kind of statement about whether space-time is continuous or not. You are modelling spacetime as continuous and seem to be saying this is the only possible option. That's not true; there are a number of different models of how space-time behave at the smallest length scales. Try reading about loop quantum gravity for example; in that model spacetime is taken to be a discrete lattice at the smallest scales. In your model with continuous space-time then it's true that a square of side length 1 would have irrational diagonal of length root 2. In reality we don't currently have experimental evidence to say either way, and it could be that your square of side length 1 has a rational diagonal which approximates root 2.
- bobbylarrybobby 4y ago
  I'm assuming they mean that for a number to be used, it must be computable.
- chowells 4y ago
  Source: |R| > |N| = |any string of symbols in any notation that has existed in the past, exists now, or will exist in the future|
riskneutral 4y ago
> we know that the subset of R that can be used directly has measure zero The cardinality of the natural number set is the same as the cardinality of the rational number set. So, in some sense, you are saying that we know that the natural counting numbers are "real," which is a self-evident truth. In another sense, you are saying that fractions of an inch/cm/etc are "real" which is another self-evident truth. The question is whether uncountable infinity somehow exits in nature (in the case of real numbers, it is a question about the infinitesimally small scale). To say that it's a "polite fiction to make some proofs works" is too strong a statement, we do not know the answer to that question. At the same time, our measurement instrument will always be discrete and bounded, so the question is seemingly beyond science itself.
- chowells 4y ago
  I'm not making a statement about the real world other than "it's impossible to communicate infinite information." I suppose that's a bit of a leap of faith, but I'm comfortable with it. Everything else I mean comes from math, with no connection to the real world.
  - otabdeveloper4 4y ago
    > it's impossible to communicate infinite information I think laws of conservation do not hold for information complexity. I don't believe you.
ChrisLomont 4y ago
We don't know that - some people posit that but it's far from being provable. All physically measured numbers have uncertainty, meaning the value obtained is not an actual number, but is a range, perhaps with some associated probability spread. This is not measure zero. We are also very capable "directly" using various things that may be continuums, such as energy, or time, or velocities, or many other physical quantities.
- bobbylarrybobby 4y ago
  All the numbers we can use are computable (how can you use a number if you can't actually talk about it?) and there are only countably many of those.
  - ChrisLomont 4y ago
    >All the numbers we can use are computable That's not true :) A nice example is Chaitin's constant [1], which I can use in proof and books and define and on and on.... And it's explicitly and most definitely NOT computable :) There are lots of numbers in lots of areas of mathematics, even symbolic mathematics that are not computable in the sense you want them to be computable. Chaitin's constant is the tip of a very big iceberg. You're using circular logic by claiming the only numbers I can are are the computable ones then claiming all numbers I can use are computable. That's not true. It's a circular argument. [1] https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant
- chowells 4y ago
  No, I mean something simpler. The subset of R that can be identified is countable, because language is countable. No matter what system you devise for describing numbers, it will be countable. And that means that it cannot describe approximately 100% of the numbers in R. They can't be uniquely described; they can't be used in computations. They're phantoms, at best. You know they're out there, but they will never be usable the way numbers you can actually describe are. The only thing they give you is the ability to declare R to be Cauchy complete in some proofs. They're a polite fiction.