Re: R.J. Lipton • Surely You Are Joking?
Comment 1
Even at the mailroom entry point of propositional calculus, there is a qualitative difference between insight proofs and routine proofs. Human beings can do either sort, as a rule, but routinizing insight is notoriously difficult, so the clerical routines have always been the ones that lend themselves to the canonical brands of canned mechanical proofs.
Just by way of a very choice example, consider the Praeclarum Theorema (Splendid Theorem) noted by Leibniz, as presented in cactus syntax here:
I’ll discuss different ways of proving this in the comments that follow.
Comment 2
The proof given via the link above is the sort that a human, all too human was able to find without much trouble. You can see that it exhibits a capacity for global pattern recognition and analogical pattern matching — manifestly aided by the use of graphical syntax — that marks the human knack for finding proofs. When I first set to work developing a Simple Propositional Logic Engine (SPLE) those were the aptitudes I naturally sought to emulate. Alas, I lacked the metaptitude for that.
Comment 3
For my next proof of the Praeclarum Theorema I give an example of a routine proof, the sort of proof that a machine with all its blinkers on can be trained to derive simply by following its nose, demanding as little insight as possible and exploiting the barest modicum of tightly reigned-in look-ahead.
Re: “The proof given via the link above […] exhibits a capacity for global pattern recognition and analogical pattern matching — manifestly aided by the use of graphical syntax — that marks the human knack for finding proofs.” I’d say that chemical interactions, or what happens into a living cell, have the same, at least apparent, characteristics. Because molecules don’t need sorting and branch predictors.
I keep a constant eye out for natural analogues of logical structures and transformations, at every level of natural process. I call this question —