Wait, are you saying e and pi aren't "actual numbers"?
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.
I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no better than 113/355 meters long.
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 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 each in 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.
And, of course, the number has to have finite expression. Representing pi as a summation is finite. But presenting a numerical computation involving it has to be an approximation if it ever to finish.
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).
That was my point, thanks.
It’s like that for all numbers, not just the fancier ones! I have never experienced a 1.
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
There is a countable subset of the reals called "computable numbers". We only ever use a subset of computable numbers in real calculations.
But the concepts are useful, and the symbols have meaning. The article mentions normal words having different meanings in mathematical language, and that impacts even every-day use of maths.
. . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula.
Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π
But surely that's over simplification?
Is this magic by wizards?
It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.
It's another of those intuition challenging moments in math.