6 ms·

If it turns our universe is discrete (at the Plank scale), that supports the possibility that all reality may be discrete. Which would mean integers are baked

If it turns our universe is discrete (at the Plank scale), that supports the possibility that all reality may be discrete. Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way. Reals are not real. (Roughly) Equivalently: There may be no perfect circles in nature.

threatofrain 1y ago
But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the reals naturally arises from the discrete.
- Nevermark 1y ago
  Integers are reals. But you can't claim something about reals because integers have that property. Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property. There are many reasons to doubt un-constructible numbers exist in nature. Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible. An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols! The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?) We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc. None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. (It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)
  - throwaway81523 1y ago
    It's hard to claim that an infinite (or anyway unbounded) collection of integers exists in nature either. If you accept the idea of an infinite collection, why not an infinite sequence? Write down a decimal point, then start flipping a coin, 1's and 0's forever: .011010010111... So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact.
    - Nevermark 1y ago
      > So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact. Well no, you will never have it. You can't start out with finite things, and built an infinite thing, even if you have infinite components to put together, and infinite time to do it. That is what countably infinite means. It is a very practical kind of infinity. And the concept comes directly, and inevitably, from the integers. Just like integers, and all the theorems/patterns we discover in them, reality may be countably infinite. Filled with an infinite number of structures of unbounded sizes, and infinitely large structures parameterized with finite constraints. The class of real numbers is not what our familiarity with its name makes us think it is. It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. Not by induction or other mathematical inevitability. Nowhere in the chain of numbers built up from the integers. With no hope of ever encountering a single concrete instance that isn't already in a smaller better defined subclass, that doesn't require the concept of reals. Mathematicians get to have their games. At a minimum they are useful as ways of stretching mathematical skill. Concepts that don't correlate with things that exist, can still be interesting, challenging, and spin off insights.
      - threatofrain 1y ago
        > It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.
griffzhowl 1y ago
It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. For one thing, they're speculative: the current theories that give the most precise and accurate predictions within their respective domains of applicability are general relativity and quantum field theory. These theories are based on continuous space and time, and no attempt to base them on discretized space and time has been successful (AFAIU both QFT and GR rely on Lorentz invariance, which means there's no absolute rest frame, hence no absolute unit of time and space, but a discretized spacetime would require absolute unit values for space and time, hence an absolute rest frame). Should we conclude then that the reals are real, because they're components of our best current physical theories? Maybe, maybe not: these are features of our current best models, but we don't know, and possibly will never know, the ultimate nature of physical space and time. For another thing, even if space and time are fundamentally discrete, there's still no doubt that the mathematical theories based on real numbers are effective in making predictions, and we would still like to use them. That's means they should have some logical foundation which can guarantee that reasoning using them is correct.
- NoahZuniga 1y ago
  > It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).
  - griffzhowl 1y ago
    You're right. The question is more about mathematical ontology than its logical foundations. I should have said something like "It seems wrong to me to base mathematical ontology on speculative theories about the nature of physical space and time."
    - deleted 1y ago
      [deleted]

threatofrain 1y ago

But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the reals naturally arises from the discrete.

Nevermark 1y ago
Integers are reals. But you can't claim something about reals because integers have that property. Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property. There are many reasons to doubt un-constructible numbers exist in nature. Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible. An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols! The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?) We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc. None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. (It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)
- throwaway81523 1y ago
  It's hard to claim that an infinite (or anyway unbounded) collection of integers exists in nature either. If you accept the idea of an infinite collection, why not an infinite sequence? Write down a decimal point, then start flipping a coin, 1's and 0's forever: .011010010111... So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact.
  - Nevermark 1y ago
    > So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact. Well no, you will never have it. You can't start out with finite things, and built an infinite thing, even if you have infinite components to put together, and infinite time to do it. That is what countably infinite means. It is a very practical kind of infinity. And the concept comes directly, and inevitably, from the integers. Just like integers, and all the theorems/patterns we discover in them, reality may be countably infinite. Filled with an infinite number of structures of unbounded sizes, and infinitely large structures parameterized with finite constraints. The class of real numbers is not what our familiarity with its name makes us think it is. It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. Not by induction or other mathematical inevitability. Nowhere in the chain of numbers built up from the integers. With no hope of ever encountering a single concrete instance that isn't already in a smaller better defined subclass, that doesn't require the concept of reals. Mathematicians get to have their games. At a minimum they are useful as ways of stretching mathematical skill. Concepts that don't correlate with things that exist, can still be interesting, challenging, and spin off insights.
    - threatofrain 1y ago
      > It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.

griffzhowl 1y ago

It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. For one thing, they're speculative: the current theories that give the most precise and accurate predictions within their respective domains of applicability are general relativity and quantum field theory. These theories are based on continuous space and time, and no attempt to base them on discretized space and time has been successful (AFAIU both QFT and GR rely on Lorentz invariance, which means there's no absolute rest frame, hence no absolute unit of time and space, but a discretized spacetime would require absolute unit values for space and time, hence an absolute rest frame). Should we conclude then that the reals are real, because they're components of our best current physical theories? Maybe, maybe not: these are features of our current best models, but we don't know, and possibly will never know, the ultimate nature of physical space and time. For another thing, even if space and time are fundamentally discrete, there's still no doubt that the mathematical theories based on real numbers are effective in making predictions, and we would still like to use them. That's means they should have some logical foundation which can guarantee that reasoning using them is correct.

NoahZuniga 1y ago
> It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).
- griffzhowl 1y ago
  You're right. The question is more about mathematical ontology than its logical foundations. I should have said something like "It seems wrong to me to base mathematical ontology on speculative theories about the nature of physical space and time."
  - deleted 1y ago
    [deleted]