Proof by Omission: Episode 0 — Why?

As a kid, I was rarely on good terms with my teachers. That probably had something to do with the fact that I asked a lot of questions, too many by most accounts. If you made a bar chart of every question I ever asked and grouped them by interrogative, the mode would be overwhelmingly why. Why is this the rule? Why does this formula work? Why does this follow from that?

The answer I received, more often than I can count, was some version of: just accept it. It's in the textbook. It will be on the exam. This bothered me as a child, and it kept bothering me through high school and university. There was always this sense that the most important part of every idea had been quietly removed before it reached me, either through incompetence, convenience, or both.

It was only when I took a real analysis course that I could finally assign a proper name to this frustration. In mathematics, Proof by Omission is a term coined satirically by the mathematician R.L. Goodstein for what happens when a lecturer cannot bridge the gap between two steps in an argument. They can get from their premises to statement A, and from statement B to their conclusion, but the connection between A and B is missing. So they write down A, and without hesitation write therefore B, trusting that if the theorem is complicated enough, nobody will question the leap. Goodstein called this part of Generalized Logic: the art of making incomplete arguments look complete. In practice it disguises itself with phrases you have almost certainly encountered before. The proof is left as an exercise to the reader. The remaining cases are entirely analogous. It can be trivially shown that. The details may be easily supplied.

What struck me when I first encountered this term is that it describes not just a classroom trick but an entire philosophy of education. Every formula handed down without its origin. Every rule stated without the problem that made it necessary. Every therefore written across a gap that was never acknowledged, let alone filled. I am a firm believer that Proof by Omission is deeply embedded in the way mathematics, statistics, and programming are taught today. I have always wanted to methodically and slowly build up the logic and first principles behind these subjects, starting not from the textbook but from the original problem that made each concept necessary in the first place.

This blog is that attempt. Not to replace the formulas, which are right and matter, but to restore what was left out. The struggles. The wrong turns. The humans who spent years failing before they spent one afternoon succeeding. The problems that made each concept necessary in the first place.

Every therefore deserves a because. This is where I try to find them.