B.5.MPC:10.2 - What the sources contribute to the synthesis

Rodin’s Axiomatic Architecture of Scientific Theories develops a constructive account of axiomatization in which object-forming activity matters alongside propositions. That supports asking how the needed object is obtained and which operations its theory permits. In :4.3–4.4, this becomes recovery of an actual construction. Physical interpretation still requires its subject account; the mathematical construction does not establish that a proposed physical interaction occurs. See Rodin, 2020, §§4.2.2–4.2.3.

Fong and Spivak make preservation under composition explicit: a functor preserves identities and composition between categories. The use here is the comparison of corresponding operations, supplied by C.29.1. It explains why relabelling objects alone cannot establish a transfer. A physical approximation may instead need a bound or another qualified relation; the coordinating Method does not require every connection to be a functor. See Seven Sketches in Compositionality, §3.3.2.

Horsman, Stepney, Wagner and Kendon distinguish the representation of a physical system from the preparation and interpretation needed to use a physical process for computation. Their account supports :4.5’s two routes and the distinction between refining an abstract description and realizing it. It also permits input preparation and result reading to differ. The present Method adapts that comparison to a receiving physical question and to useful bounds, rather than adopting their account as a universal definition of all computation. See When Does a Physical System Compute?, 2014, §§VI–VIII.

Turing’s 1936 construction of a universal computing machine provides a historical demonstration that an executor can interpret an encoded description of another machine’s procedure. That helps distinguish the rule description, its interpreter and the realized operation. The finite interpreter in C.29.2:5.1 provides a small entry to that distinction. Use C.29.2 for the computation-specific cost account when a resource limit matters; :4.4 carries that cost and the computation’s meaning into the joint use. See Turing, On Computable Numbers, §6.

These contributions answer different construction questions. The synthesis is to hold their input and result meanings together for a physical use, inspect their joint and alternative dependencies, and let a failed connection determine the next contribution. It preserves mathematical, empirical and realization grounds at the points where they are needed.