Experiments, Computability, and the Existence of Physical Functions

TL;DR

This paper demonstrates that experimental procedures can be modeled as algorithms producing physical functions, clarifying the relationship between experiment, computability, and measurement accuracy.

Contribution

It introduces a formal framework linking experimental science with computability theory, clarifying the existence and computability of physical functions under finite-precision measurements.

Findings

Experimental procedures can be viewed as algorithms producing physical functions.

Finite-precision measurements are compatible with the existence of these functions.

Protocol dependence and stochasticity specify the realized map and assumptions needed for protocol-independent quantities.

Abstract

Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the protocol, background conditions, and reporting rules are fixed. Assuming an explicit physical Church--Turing bridge principle, a reproducible experiment therefore computes a map from admissible inputs to outputs, and the corresponding function exists in the sense appropriate to those outputs. Furthermore, computable analysis allows us to explain why this conclusion is compatible with finite-precision measurement since in this case what matters is a systematic approximation to a requested accuracy, not the production of exact real numbers in a single step. Neither protocol dependence nor stochasticity undermines the existence claim. Rather, they specify which map is realized by a given protocol and what additional assumptions are required for stronger claims about a single protocol-independent quantity. The paper therefore separates three questions that are often conflated: whether the function exists, whether it is computable, and when results obtained under different protocols may be treated as measurements of the same quantity.