[tialaramex]

Are you sure? Naively sure, you won't find sufficiently enormous circles and even if you could such a huge circle won't have the Pi ratio here, that's a property of Euclidean space and we don't live in a Euclidean space, ours is slightly off IIRC.

But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.

[Dylan16807]

The most precise situations I can think of involve impossibly perfect measurements of volume. And even there, okay a cubic meter is 10^105 cubic planck units and the visible universe is 10^186. Finite math can easily throw a million digits at any problem. How do you reach a point where you need reals to describe actual things?

[Nevermark]

Pi, i and e show up with apparent perfect "precision" in all kinds of physics.

Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.

Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory.

I think "precision" is the wrong way to look at what can mean something or not.

I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers.

From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.)

What is true about all these numbers is not precision, but that they emerge from a finite number of relationships.

They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.)

And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.

So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail.

Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.