[kannanvijayan]

I don't have an answer to your questions, but I think these thoughts are not uncommon for people who get into these topics. The relationship between the reals, including Pi, and the countables such as the naturals/integers/rationals is suggestive of some deeper truth.

The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe.

Make the areas between the circle and the square equal, and the infinite precision moves into the ratio between their lower order dimensional measures (circumfence, surface area, etc.).

You can't describe a system that expresses the one, in terms of a system that expresses the other, without requiring infinite precision (and thus infinite information).

Furthermore, it really seems like a bunch of the really fundamental reals (pi, e), have a pretty deep connection to algebras of rotations (both pi and e relate strongly to rotations)

What that seems to suggest to me is that if the universe is discrete, then the discreteness must be biased towards one of these modes or the other - i.e. it is natively one and approximates the other. You can have a discrete universe where you have natural rotational relationships, or natural linear relationships, but not both at the same time.

[schiffern]

  >The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe.

Easily fixed! I choose 1 dimension. :)

[kannanvijayan]

Hah, nice find :)

[schiffern]

Good show, and I appreciate your sentiment about the "messiness" of pi.

There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance.

Trig functions can be overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function.

That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway.

Still, I can dream!

[0] https://frinklang.org/

[1] https://en.wikipedia.org/wiki/Exact_trigonometric_values

[kannanvijayan]

I suppose you could have added root two as a fundamental as well. I suppose that's another problem with the irrationals: two irrationals that aren't linearly related by a rational are effectively two fundamentals from each others perspective.

It's a sad conclusion - though. Computation exists in the countable space. So there is no computationally representable symbolic model that can ever algebraically capture the reals.

The other thing that came to mind when you mentioned root-2 is a similar realization as with pi. That somehow a diagonal is not well defined in discrete terms with respect to two orthogonal vectors. So here once again, you have this weird impedance mismatch between orthogonality (a rotational concept) and diagonals (a linear concept).

I don't have the formalisms to explore these thoughts much further than this.. so it's hard to say whether this is just some trivial numerological-like observation or if there's something more to it. But it's kinda pleasant to think about sometimes.

[schiffern]

Yeah, once I got to "all I need to do is add a root for every prime! And cube roots! And..." I realized this is a path of madness. ;)

It could be done symbolically, by generalizing from their rational representation:

  X/Y

To

  (X/Y)^(A/B)

Again this is tantalizingly close to being workable in Frink -- it supports 'dangling' (unevaluated) rational exponents on units, but not simple numbers.

The problem of course is that I'm trying to twist a (powerful!) calculator into something like a computer algebra system. I really should just use an actual CAS.

But like you say, I'd be happy if I could "just" have an exact representation of (if not the reals because that's impossible, then at least) any number I can describe in finite terms with normal math operators.

Cheers and good day

[kcexn]

The universe, being a physical entity not bound by the rules of human logic doesn't have to be either.

Our logical approximation of the universe might need to be, assuming that we don't add more axioms to our system of logical reasoning.