Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

understanding proofs 333an ordinary proof, we have assumed or established A, B, andC, and need toprove an assertion of the form ‘D and E’; we can do this by noting that ‘itsuffices to establish D, andthenE, in turn’ and then accomplishing each ofthese two tasks. We can also work forwards from hypotheses: for example,from A ∧ B, C ⇒ D we can conclude A, B, C ⇒ D. Inordinaryterms,ifwehave established or assumed ‘A and B’, we may use both A and B to deriveour conclusion. A branch of the tree is closed off when the conclusion ofa sequent matches one of the hypotheses, in which case the goal is clearlysatisfied.Things become more interesting when we try to take larger inferentialsteps, where the validity of the inference is not as transparent. Suppose, inan ordinary proof, we are trying to prove D, having established A, B, andC. In sequent form, this corresponds to the goal of verifying A, B, C ⇒ D.We write ‘Clearly, from A and B we have E’, thus reducing our task ofverifying A, B, C, E ⇒ D. But what is clear to us may not be clear to thecomputer; the assertion that E follows from A and B corresponds to the sequentA, B ⇒ E, and we would like the computer to fill in the details automatically.‘Understanding’ this step of the proof, in this context, means being able tojustify the corresponding inference.In a sense, verifying such an inference is no different from proving a theorem;a sequent of the form A, B ⇒ E can express anything from a trivial logicalimplication to a major conjecture like the Riemann hypothesis. Sometimes,brute-force calculation can be used to verify inferences that require a gooddeal of human effort. But what is more striking is that there is a large class ofinferences that require very little effort on our part, but are beyond the meansof current verification technology. In fact, most textbook inferences have thischaracter: it can take hours of painstaking work to get a proof assistant toverify a short proof that is routinely read and understood by any competentmathematician.The flip explanation as to why competent mathematicians succeed wherecomputers fail is simply that mathematicians understand, while computersdon’t. But we have set ourselves precisely the task of explaining how thisunderstanding works. Computers can search exhaustively for an axiomaticderivation of the desired inference, but a blind search does not get very far.The problem is that even when the inferences we are interested in can bejustified in a few steps, the space of possibilities grows exponentially.There are a number of ways this can happen. First, excessive case distinctionscan cause problems. Proving a sequent of the form A ∨ B, C, D ⇒ E reducesto showing that the conclusion follows from each disjunct; this results in twosubgoals, A, C, D ⇒ E and B, C, D ⇒ E. Each successive case distinction