# bogus comments on Book Review: Mathematics for Computer Science (Suggestion for MIRI Research Guide)

• More gen­er­ally, say we want to prove a the­o­rem that looks some­thing like “If A, then B has prop­erty C.” You start at A and, ap­peal­ing to the defi­ni­tion of C, show that B has it. There’s prob­a­bly some clev­er­ness in­volved in do­ing so, but you start at the ob­vi­ous place (A), end in the ob­vi­ous place (B satis­fies the defi­ni­tion of C), and don’t rely on any crazy, seem­ingly-un­re­lated in­sights. Let’s call this sort of proof mun­dane.

There is a virtue in mun­dane proofs: a smart reader can usu­ally gen­er­ate them af­ter they read the the­o­rem but be­fore they read its proof. Do­ing is benefi­cial, since proof-gen­er­at­ing makes the the­o­rem more mem­o­rable. It also gives the reader prac­tice build­ing in­tu­ition by play­ing around with the math­e­mat­i­cal ob­jects and helps them im­prove their proofwrit­ing by com­par­ing their out­put to a max­i­mally re­fined proof.

On the end of the spec­trum op­pos­ing mun­dane is witchcraft. Proofs that use witchcraft typ­i­cally have a step where you demon­strate you’re as in­ge­nious as Pas­cal by hav­ing a seem­ingly-un­re­lated in­sight that makes ev­ery­thing eas­ier. No­tice that, even if you are as in­ge­nious as Pas­cal, you won’t nec­es­sar­ily be able to gen­er­ate these in­sights quickly enough to get through the text at any rea­son­able pace.

I’m skep­ti­cal that this is a huge prob­lem in prac­tice,and my hunch is that this is mostly a mat­ter of the origi­nal proof be­ing badly con­veyed. In the case of Pas­cal in­genuious proof of the gam­bler’s ruin, what the ex­po­si­tion in MCS isn’t con­vey­ing clearly enough is that we’re es­sen­tially defin­ing a “game” (or more sub­tly, a val­u­a­tion) as f([gam­bler’s cap­i­tal]) that is fair in the ex­pec­ta­tion sense, and, more im­por­tantly, re­sults in “ruin” or “win” ex­actly when the origi­nal game does. If you start with a proof of this state­ment, you’re back in the fa­mil­iar pat­tern of “If A, then B has prop­erty C”. We’ve just in­serted a sim­ple step of “Ac­tu­ally, we’ll just need to show that B’ has prop­erty C, and then B does too”, which is mun­dane, be­cause it just fol­lows from what C is like!

The point though is that not all ex­po­si­tions make this ob­vi­ous enough; some just fo­cus on proofs that are “mun­dane” in your origi­nal sense, but this too is a kind of “dumb­ing down” and lack of math­e­mat­i­cal ma­tu­rity.