I think you're placing too much emphasis on the von Neumann architecture. It's
not essential to anything, and AFAIK was only intended as a short-term engineering
decision, pending redesign once we have more experience and understanding. This
separation of CPU and memory is now our biggest performance bottleneck; hence
why we shift so much on to GPUs, DSPs, etc. which aren't von Neumann.
Boole's big contribution was turning logic into an algebra: "and", "or", "not", etc. became mathematical operations for combining values ("true"/"false"), which we can quantify over, prove theorems about, etc. in a more abstract way than the sort of 'sentence templates' studied by the Greeks.
Important contributions were made by Shannon as well: in particular, his Masters thesis showed the equivalence of Boolean algebra, binary arithmetic, and digital circuits. Hence arithmetic and logic can be carried out by circuitry.
Regarding Turing machines: their definition is due to a philosophical argument, rather than mathematical (hence Church-Turing "thesis" rather than "theorem"). Essentially, Turing argues that any finite region of space can only have a finite set of distinguishable states; that any finite set can be enumerated and indexed; and therefore any physical system can be described by a (vast) transition table between these indexes. For input/output, a similar argument is made that only finitely-many possibilities can be distinguished between, that we can equivalentl... (read more)