5 ms·

Roger Penrose gives a great explanation in The Road to Reality. We know some subset of the natural numbers are real because we can count things. And we therefor

Roger Penrose gives a great explanation in The Road to Reality. We know some subset of the natural numbers are real because we can count things. And we therefore know some subset of the rational numbers are real because we can cut things into pieces and count the pieces. But the reals? Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Mathematics is a tool we've developed to help us understand reality. But it itself is not reality.

chriswarbo 8y ago
> But the reals? Well they are invented purely so we can take the square root of two. I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc. There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'. PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!
SilasX 8y ago
I'm not following the logic there: >We know some subset of the natural numbers are real because we can count things. So he's saying: 1) X is real = there exists a one-to-one mapping between X and a consistent physical process. 2) Natural numbers map to counting. Okay, but then: >Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Not true: complex numbers also map to real-world dynamics, like oscillatory behavior. From a search for the physical significance of complex numbers: https://www.quora.com/What-is-the-physical-significance-of-complex-numbers https://www.quora.com/What-is-the-physical-significance-of-c... I would add that real numbers have physical significance in terms of having to arbitrarily operate on some fraction of another value. It's true that the universe might prohibit arbitrary precision, but when you don't know the depths of the permitted halvings, and you're permitted arbitrarily large units, it amounts to the same thing.
jVinc 8y ago
"But the reals? Well they are invented purely so we can take the square root of two. " To follow on you previous run: ... because we can cut things into pieces and count the pieces. We know real numbers are real because we can cut a square across the diagonal and see the relative length of the diagonal to the side.
- pauljurczak 8y ago
  We can't cut the square, because perfect square is an abstraction and what you are cutting is an approximation of the square. We can't cut across the diagonal, because all cuts are only an approximation of a straight line. We can't directly see the real number being the ratio of square diagonal to its side, because our eyes have finite number of photoreceptors, receiving information from a finite number of photons. It's natural numbers, all the way down...
vladTheInhaler 8y ago
My understanding is that many "intuitively obvious" statements about continuous functions, like the intermediate value theorem, are only true in the real numbers, not the rational numbers. So you might say that we defined the reals so that two intersecting lines actually have to touch somewhere. That seems like a reasonable motivation to me.
- acjohnson55 8y ago
  Aren't you presupposing that lines are themselves a continuum? It's a reasonable abstraction, but as far as we know, it's not physical real. If a line were just a collection of discrete points related by some formula, there's no reason two lines that aren't parallel should have to intersect, from the perspective of sharing a point.
  - vladTheInhaler 8y ago
    You're absolutely right that it might not reflect physical reality. I don't think it's known whether space is fundamentally discrete or not. But I can definitely say that continuous space meshes a lot more with my experience as a human being, and that because of that, the real numbers are motivated by more than just taking square roots. They model the human experience of space.
acjohnson55 8y ago
The rational numbers aren't physically real, because that would imply the ability to divide objects into exactly fractions, which generally isn't possible in a verifiable way, which also must be consistent with the operations we want to perform. Fractions are a useful abstraction. The reality is just relations between natural numbers. The real numbers aren't invented to capture the square root of two. You can deal with that using a simpler set. The real numbers are necessary for common approaches to calculus and other forms of math that depend on a continuum of quantity. I would describe math as a tool for modeling reality, rather than understanding it. Because we really don't know much at all. All we've got are rules that are pretty damn good at predicting things under everyday conditions.
- tree_of_item 8y ago
  The rational numbers are physically real because you can compute them. They just map on to the naturals.
  - acjohnson55 8y ago
    Depends on what you mean by physically real. I'm talking about in the sense of tangibility, not computability.
    - deleted 8y ago
      [deleted]

chriswarbo 8y ago

> But the reals? Well they are invented purely so we can take the square root of two. I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc. There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'. PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!

SilasX 8y ago

I'm not following the logic there: >We know some subset of the natural numbers are real because we can count things. So he's saying: 1) X is real = there exists a one-to-one mapping between X and a consistent physical process. 2) Natural numbers map to counting. Okay, but then: >Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Not true: complex numbers also map to real-world dynamics, like oscillatory behavior. From a search for the physical significance of complex numbers: https://www.quora.com/What-is-the-physical-significance-of-complex-numbers https://www.quora.com/What-is-the-physical-significance-of-c... I would add that real numbers have physical significance in terms of having to arbitrarily operate on some fraction of another value. It's true that the universe might prohibit arbitrary precision, but when you don't know the depths of the permitted halvings, and you're permitted arbitrarily large units, it amounts to the same thing.

jVinc 8y ago

"But the reals? Well they are invented purely so we can take the square root of two. " To follow on you previous run: ... because we can cut things into pieces and count the pieces. We know real numbers are real because we can cut a square across the diagonal and see the relative length of the diagonal to the side.

pauljurczak 8y ago
We can't cut the square, because perfect square is an abstraction and what you are cutting is an approximation of the square. We can't cut across the diagonal, because all cuts are only an approximation of a straight line. We can't directly see the real number being the ratio of square diagonal to its side, because our eyes have finite number of photoreceptors, receiving information from a finite number of photons. It's natural numbers, all the way down...

vladTheInhaler 8y ago

My understanding is that many "intuitively obvious" statements about continuous functions, like the intermediate value theorem, are only true in the real numbers, not the rational numbers. So you might say that we defined the reals so that two intersecting lines actually have to touch somewhere. That seems like a reasonable motivation to me.

acjohnson55 8y ago
Aren't you presupposing that lines are themselves a continuum? It's a reasonable abstraction, but as far as we know, it's not physical real. If a line were just a collection of discrete points related by some formula, there's no reason two lines that aren't parallel should have to intersect, from the perspective of sharing a point.
- vladTheInhaler 8y ago
  You're absolutely right that it might not reflect physical reality. I don't think it's known whether space is fundamentally discrete or not. But I can definitely say that continuous space meshes a lot more with my experience as a human being, and that because of that, the real numbers are motivated by more than just taking square roots. They model the human experience of space.

acjohnson55 8y ago

The rational numbers aren't physically real, because that would imply the ability to divide objects into exactly fractions, which generally isn't possible in a verifiable way, which also must be consistent with the operations we want to perform. Fractions are a useful abstraction. The reality is just relations between natural numbers. The real numbers aren't invented to capture the square root of two. You can deal with that using a simpler set. The real numbers are necessary for common approaches to calculus and other forms of math that depend on a continuum of quantity. I would describe math as a tool for modeling reality, rather than understanding it. Because we really don't know much at all. All we've got are rules that are pretty damn good at predicting things under everyday conditions.

tree_of_item 8y ago
The rational numbers are physically real because you can compute them. They just map on to the naturals.
- acjohnson55 8y ago
  Depends on what you mean by physically real. I'm talking about in the sense of tangibility, not computability.
  - deleted 8y ago
    [deleted]