4 ms·

But is computation enough? The computable reals are a subset of the reals.

psychoslave 1y ago
This is an under specified question, until some observable goal is attached to "enough". Enough for what?
A_D_E_P_T 1y ago
"Reals" (tragically poorly named) can be interpreted as physical ratios. That is: Real numbers describe real, concrete relations. For e.g., saying that Jones weighs 180.255 pounds means there's a real, physical relationship -- a ratio -- between Jones' weight and the standard pound. Because both weights exist physically, their ratio also exists physically. Thus, from this viewpoint, real numbers can be viewed as ratios. In contrast, the common philosophical stance on numbers is that they are abstract concepts, detached from the actual physical process of measurement. Numbers are seen as external representations tied to real-world features through human conventions. This "representational" approach, influenced by the idea that numbers are abstract entities, became dominant in the 20th century. But the 20th century viewpoint is really just one interpretation (you could call it "Platonic"), and, just as it's impossible to measure ratios to infinite precision in the real world, absolutely nothing requires an incomputable continuum of reals. Physics least of all. In 20th and 21st century physics, things are discrete (quantized) and are very rarely measured to over 50 significant digits. Infinite precision is never allowed, and precision to 2000 significant digits is likewise impossible. The latter not only because quantum mechanics makes it impossible to attain great precision on very small scales. For e.g., imagine measuring the orbits of the planets and moons in the solar system: By the time you get to 50 significant digits, you will need to take into account the gravitational effects of the stars nearest to the sun; before you get to 100 significant digits, you'll need to model the entire Milky Way galaxy; the further you go in search of precision, the exponentially larger your mathematical canvas will need to grow, and at arbitrarily high sub-infinite precision you’d be required to model the whole of the observable universe -- which might itself be futile, as objects and phenomena outside observable space could affect your measurements, etc. So though everything is in principle simulatable, and precision has a set limit in a granular universe that can be described mathematically, measuring anything to arbitrarily high precision is beyond finite human efforts.
- anthk 1y ago
  Your whole comment can just be TL;DR Forth and the fixed point philosophy :) Rationals > irrationals on computing them. You can always approximate irrationals with rationals, even Scheme (Lisp, do'h) has a function to convert a rational to decimal and the reverse. decimal to rational.
- jiggawatts 1y ago
  Numbers interpretable as ratios are the Rational numbers, by definition, not the Reals. This entire discussion is about mathematical concepts, not physical ones! Sure, yes, in physics you never "need" to go past a certain number of digits, but that has nothing to do with mathematical abstractions such as the types numbers. They're very specifically and strictly defined, starting from certain axioms. Quantum mechanics and the measurability of particles has nothing to do with it! It's also an open question how much precision the Universe actually has, such as whether things occur at a higher precision than can be practically measured, or whether the ultimate limit of measurement capability is the precision that the Universe "keeps" in its microscopic states. For example, let's assume that physics occurs with some finite precision -- so not the infinite precision reals -- and that this precision is exactly the maximum possible measurable precision for any conceivable experiment. That is: Information is matter. Okay... which number space is this? Booleans? Integers? Rationals? In what space? A 3D grid? Waves in some phase space? Subdivided... how? Figure that out, and your Nobel prize awaits!
  - adrian_b 1y ago
    Rational numbers are ratios of integers. There are plenty of ratios that are ratios of other things than integers, so they are not rational numbers.
    - jiggawatts 1y ago
      Ratios of numbers that are not integers or Rationals are... the Reals. I mean sure, you could get pedantic and talk about ratios of complex integers or whatever, but that's missing the point: The Rationals are closed under division, which means the ratio of any two Rationals is a Rational. To "escape" the Rationals, the next step up is Irrational numbers. Square roots, and the like. The instant you mix in Pi or anything similar, you're firmly in the Reals and they're like a tarpit, there's no escape once you've stepped off the infinitesimal island of the Rationals.
somewhereoutth 1y ago
> But is computation enough? Of course not, but that would invalidate the entire project of some/many here to turn reality into something clockwork they can believe they understand. Reality is much more interesting than that.
112233 1y ago
I like to think about it like this: while real numbers in general are impossible to compute, write down or do anything else with them, many statements about real numbers can be expressed in a useful, computable form. It all gets mind-bendingly mind-bending really fast, especially with people like Harvey Friedman ( https://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/ https://u.osu.edu/friedman.8/foundational-adventures/fom-ema... ) or the author of this post trying to break the math by constructing an untractably large values with the simplest means possible (thus showing that you can encounter problems that do not fit in the universe even when working in a "simple" theory) (i saw the username and went to check audiomulch website, but it did not resolve :( )
btilly 1y ago
Are they? The idea that uncountable means more comes from a bad metaphor. See https://news.ycombinator.com/item?id=44271589 https://news.ycombinator.com/item?id=44271589 for my explanation of that. Accepting that uncountable means more forces us to debatable notions of existence. See https://news.ycombinator.com/item?id=44270383 https://news.ycombinator.com/item?id=44270383 for a debate over it. But, finally, there is this. Every chain of reasoning that we can ever come up with, can be represented on a computer. So even if you wish to believe in some extension of ZFC with extremely large sets, PA is capable of proving every possible conclusion from your chosen set of axioms. So yes, PA is enough. If you're not convinced, I recommend reading https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden....

This is an under specified question, until some observable goal is attached to "enough". Enough for what?

"Reals" (tragically poorly named) can be interpreted as physical ratios. That is: Real numbers describe real, concrete relations. For e.g., saying that Jones weighs 180.255 pounds means there's a real, physical relationship -- a ratio -- between Jones' weight and the standard pound. Because both weights exist physically, their ratio also exists physically. Thus, from this viewpoint, real numbers can be viewed as ratios. In contrast, the common philosophical stance on numbers is that they are abstract concepts, detached from the actual physical process of measurement. Numbers are seen as external representations tied to real-world features through human conventions. This "representational" approach, influenced by the idea that numbers are abstract entities, became dominant in the 20th century. But the 20th century viewpoint is really just one interpretation (you could call it "Platonic"), and, just as it's impossible to measure ratios to infinite precision in the real world, absolutely nothing requires an incomputable continuum of reals. Physics least of all. In 20th and 21st century physics, things are discrete (quantized) and are very rarely measured to over 50 significant digits. Infinite precision is never allowed, and precision to 2000 significant digits is likewise impossible. The latter not only because quantum mechanics makes it impossible to attain great precision on very small scales. For e.g., imagine measuring the orbits of the planets and moons in the solar system: By the time you get to 50 significant digits, you will need to take into account the gravitational effects of the stars nearest to the sun; before you get to 100 significant digits, you'll need to model the entire Milky Way galaxy; the further you go in search of precision, the exponentially larger your mathematical canvas will need to grow, and at arbitrarily high sub-infinite precision you’d be required to model the whole of the observable universe -- which might itself be futile, as objects and phenomena outside observable space could affect your measurements, etc. So though everything is in principle simulatable, and precision has a set limit in a granular universe that can be described mathematically, measuring anything to arbitrarily high precision is beyond finite human efforts.

anthk 1y ago
Your whole comment can just be TL;DR Forth and the fixed point philosophy :) Rationals > irrationals on computing them. You can always approximate irrationals with rationals, even Scheme (Lisp, do'h) has a function to convert a rational to decimal and the reverse. decimal to rational.
jiggawatts 1y ago
Numbers interpretable as ratios are the Rational numbers, by definition, not the Reals. This entire discussion is about mathematical concepts, not physical ones! Sure, yes, in physics you never "need" to go past a certain number of digits, but that has nothing to do with mathematical abstractions such as the types numbers. They're very specifically and strictly defined, starting from certain axioms. Quantum mechanics and the measurability of particles has nothing to do with it! It's also an open question how much precision the Universe actually has, such as whether things occur at a higher precision than can be practically measured, or whether the ultimate limit of measurement capability is the precision that the Universe "keeps" in its microscopic states. For example, let's assume that physics occurs with some finite precision -- so not the infinite precision reals -- and that this precision is exactly the maximum possible measurable precision for any conceivable experiment. That is: Information is matter. Okay... which number space is this? Booleans? Integers? Rationals? In what space? A 3D grid? Waves in some phase space? Subdivided... how? Figure that out, and your Nobel prize awaits!
- adrian_b 1y ago
  Rational numbers are ratios of integers. There are plenty of ratios that are ratios of other things than integers, so they are not rational numbers.
  - jiggawatts 1y ago
    Ratios of numbers that are not integers or Rationals are... the Reals. I mean sure, you could get pedantic and talk about ratios of complex integers or whatever, but that's missing the point: The Rationals are closed under division, which means the ratio of any two Rationals is a Rational. To "escape" the Rationals, the next step up is Irrational numbers. Square roots, and the like. The instant you mix in Pi or anything similar, you're firmly in the Reals and they're like a tarpit, there's no escape once you've stepped off the infinitesimal island of the Rationals.

somewhereoutth 1y ago

> But is computation enough? Of course not, but that would invalidate the entire project of some/many here to turn reality into something clockwork they can believe they understand. Reality is much more interesting than that.

112233 1y ago

I like to think about it like this: while real numbers in general are impossible to compute, write down or do anything else with them, many statements about real numbers can be expressed in a useful, computable form. It all gets mind-bendingly mind-bending really fast, especially with people like Harvey Friedman ( https://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/ https://u.osu.edu/friedman.8/foundational-adventures/fom-ema... ) or the author of this post trying to break the math by constructing an untractably large values with the simplest means possible (thus showing that you can encounter problems that do not fit in the universe even when working in a "simple" theory) (i saw the username and went to check audiomulch website, but it did not resolve :( )

btilly 1y ago

Are they? The idea that uncountable means more comes from a bad metaphor. See https://news.ycombinator.com/item?id=44271589 https://news.ycombinator.com/item?id=44271589 for my explanation of that. Accepting that uncountable means more forces us to debatable notions of existence. See https://news.ycombinator.com/item?id=44270383 https://news.ycombinator.com/item?id=44270383 for a debate over it. But, finally, there is this. Every chain of reasoning that we can ever come up with, can be represented on a computer. So even if you wish to believe in some extension of ZFC with extremely large sets, PA is capable of proving every possible conclusion from your chosen set of axioms. So yes, PA is enough. If you're not convinced, I recommend reading https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden....