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:

• Praeclarum Theorema

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.

• Praeclarum Theorema : Proof by Case Analysis-Synthesis Theorem (CAST)