This post is prompted by irritation and, dare I say, love – irritation at reading to understand yet another drab mathematical proof and love for the FORTH programming language as a thinking tool.
My problem with proofs in print media is their flatness. Each line is rendered in the same font in the same size. Newly defined symbols are scattered throughout the proof as needed often not even meriting their own paragraph or other typographical demarcation. The crux of the reasoning pattern is sometimes obscure or implicit but in any event just another line in the proof.
With all the computer presentation tools now extant and with more and more mathematics being published online, we ought to do better. That brings me to my favorite programming language FORTH. It may have changed over the thirty some years since I used it as a development tool but here is what I fondly remember. The structure of the language forced you to design top down and to build programs from the bottom up, testing as you went. The programs could be read from the top down since great care was taken naming subroutines (definitions) and factoring (separating) actions. And best of all there was a basic rule that each screen’s worth of program code including comments must stand alone and back then, screens were character-based and held much less information.
So my idea is for writing proofs is to write discrete blocks – one screen with large print – starting with a proof’s structure and essential reasoning pattern and then have blocks with successive refinements. Hyperlinks would be used extensively particularly for definitions and key background information.
What follows is my attempt without the proper tools ( I will be using Microsoft OneNote) to organize a particular proof in the way I mean. The proof I will rewrite is the main result of Polynomial Dynamics and a Proof of the Fermat Little Theorem by Valdimir Dragovic’ in The American Mathematical Monthly, February 2013.
This did not come out as well as I liked. Each block could have been designed better with different fonts and layout. The proof might have been factored differently. The principle is there but my execution reflects my lack of experience.
However, as I worked through this particular concept for presenting proofs other computer amenable methods flitted through my mind. Maybe proof as a Prezi. Or a proof as a series of stacked windows. Or color coding. Or trees. The intention is to suggest making mathematical proofs more accessible using modern tools and I’ll leave it at that.