I am not a Platonist, or any sort of theist.

e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world.

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature.

Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.

I probably confused matters with my undefined expression "actual numbers". e and pi are as actual as i or 0.

I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their existence is not just countable, but finite and not very large. You have to have expressed a representation of it at least once. We can only express a small number of numbers over the course of a life. Computers can do more of them. You might claim all those that programs you wrote have expressed.

How is "The circumferance of an idealized circle divided by its diameter" not a finite expression of π? Saying something cannot be expressed finitely in an integer-based numeral system, and saying that it admits no finite representation are two radically different statements.

Despite it being a non-starter from a pragmatic standpoint, we could for instance easily imagine a novel numeral type that encodes the set S = {a + b·π where a and b are integers} (we can encode integers quite easily and all we need to reposesent such a number in silico is to encode a and b). Using such a numeral type, we are able to do exact arithmetic if our operations are restricted to addition and subtraction (and if we are content with fractional representation of numbers as being considered "exact", we can also do division and multiplication although we would have to work within the larger set S' = { (a + b·π) / (c + d·π) where, a, b, c, and d are integers and c·d ≠ 0} rather than within S).

> Despite it being a non-starter from a pragmatic standpoint

That was my point, thanks.

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature.

It’s like that for all numbers, not just the fancier ones! I have never experienced a 1.

Even spiders and newborn babies count; it is a natural animal behavior. Axiomatic systems, not so much.

I don't think you have to be a Platonist to allow that numbers that do not exactly correspond to humanly measurable quantities can yet be part of a calculation.

Another example would be Binet's formula for Fibonacci n[1], which though it involves Phi, sqrt(5), etc., always evaluates to an integer, given integer n.

The existence of these irrationals in Platonic heaven doesn't follow from the formula, which afaik depends only on the usual rules of algebra and logic, though I admit I'm not sure what constructivists make of this type of calculation.

1: see https://r-knott.surrey.ac.uk/Fibonacci/fibFormula.html or https://mathworld.wolfram.com/BinetsFibonacciNumberFormula.h... for example