There's an interesting paper[1] that argues that real numbers aren't physical, for the precise reasons you stated. That is, only the subset of real numbers that contain a finite amount of information (like constructive real numbers) are physically meaningful.

A consequence of this is that classical physics is not really deterministic: this is because, in general (ie. including chaotic systems), the evolution of a system depends on a set of initial conditions that are specified by full real numbers, impossible to measure with finite precision. So, the use of real numbers is hiding the indeterminism in the initial conditions, much like the function in this article encodes a dataset in a single parameter.

[1]: https://arxiv.org/abs/1803.06824

I think the precise definition of what exactly is physically meaningful is more interesting a bit, because pi is just as 'real' as 1 in some sense. You can't any more precisely measure 1m than pi meters. A perfect square doesn't exist any more or less precisely than a perfect circle.

Basically, the universe can just as well be approximated in two different ways. One, it's all straight lines, and anything that looks like a circle/curve is actually a piece of a veeery many sided polygon if you look closely enough at it. Two, it's all curves of some curvature, and anything that looks like a straight line is actually a segment of a veeery large circle. The first perspective corresponds to taking the rational numbers as physical. The second one corresponds to taking pi as physical (and then it's unclear whether e is also physical, or which other transcendental numbers are, but 1 is definitely un-physical then).

Probably the constructive framework is the best way to express this concept, just starting from whatever constant we chose to define as the unit.

Note that π, like practically any constant or number normally found in mathemathics is constructive and contains a finite amount of information. What the author consider unphysical are the non-constructive reals: unfathomable numbers that can't be defined or written in any finite amount of words or paper. It's hard to argue they play some role in our physical world: it would basically amount to metaphysics.

However, even if rarely used outside of proofs, they are the vast majority of the reals, in fact there are only countably infinite constructive/computable numbers.

information here, ie log of a probability, is a continuous notion -- it is real-valued, as not least, log is a transcendental fn --

i specifically said with a /discrete/ conception , geometry is infinitely dense (of discrete states)

In all physical theories, any finite system has a finite number of distinguishable states. So it is not infinitely informationally dense, especially when working with discrete bits of information.

Not to mention, the finer the distinctions between two states of a system, the more energy you need to distinguish them. So, the less impact these differences can have, unless the system is extraordinarily energetic (and even then, you end up in fundamental limits of energy per volume, like the Schwarzschild radius).

So again, there is no sense in which a finite part of the universe is universally dense.

Even worse for your argument, all currently known laws of physics use computable functions (the randomness in QM notwithstanding). So, by definition, all known laws of physics can be simulated by an ideal Turing machine (again, give or take some randomness in QM, depending on the interpretation you chose to believe in and on how you chose to simulate the QM system).

QM is even linear!