These are some notes I wrote when I taught “intro to proofs” courses:
I wrote these notes to supplement a conventional book, because in my opinion typical intro to proofs courses are fundamentally questionable in their very nature. The premise of such a course is rather like that of My Fair Lady: uncivilised students must be taught to “talk the talk.” It’s the dress code and table etiquette of mathematics. In their fanatical devotion to the clinical, sterilised, Bourbaki way of doing mathematics, these courses have forgotten that precision elocution is pointless unless you have something to say. These courses deem actual mathematical content “too messy” and instead feed the student only fake theorems specifically concocted for the sake of being amenable to the desired mould of what a mathematical proof should look like.
Consider these notes, then, my pauper’s rebellion. Down with haut bourgeoisie snobbery for snobbery’s sake, down with fake perfume and powdered wigs. Let us have the courage to tackle real mathematics as it occurs in nature. Let us put meaning and purpose and exciting ideas first, and let us accept the airs and graces of the mathematical aristocracy only after they have proved their worth in this enterprise.