PlanetPhysics/Computer 2

Any automaton $$\mathcal{C}$$ which is capable of either executing a set of logical instructions $$\mathbb{I}$$ (that is called a program, $$\mathbb{P}$$) or whose operation is defined either by an algorithm/ set of algorithms $$\A$$ or a recursive function $$\mathcal{F}_R$$ is called a computer.

Remarks
Occasionally, and incompletely, a computer is simply being defined as ""a machine that manipulates data according to a list of instructions. . First of all, implicit in the latter description is the concept of sequential machine or automaton that has a precise mathematical definition, and is not simply just any `machine'. Secondly, the vague term of "list of instructions" needs actually be replaced by a "set of {\mathbf logical} instructions", which is precisely defined, for example by algorithms or recursive functions as in the top definition of the computer term.

Notably, and contrary to widespread misconceptions in old-age philosophy ( e.g. Descartes, John von Neumann, etc.), AI and the computer community, complex, living systems and the human brain cannot be adequately described or represented by any computer, computer model, or classical automaton; this is, in essence, because the latter cannot be adequately modelled by any recursive function, finitary algorithm or (computer) program. Furthermore, any computer can be encoded with a categorical commutative diagram. On the other hand, most organisms-- that possess variable topology and varying transition functions $$\delta_v$$ (viz. entry on automata)-- may only be encoded by the unique limit of a sequence of non-commutative categorical diagrams which is not necessarily finite, and that cannot be recursively computed.