Fail better: What formalized math can teach us about learning

TL;DR

This paper explores how formalized mathematics can inform better teaching and learning of logic by analyzing proof strategies, automation, and proof dynamics, and demonstrates a proof assistant as a proof of concept.

Contribution

It introduces a proof assistant that integrates existing tools and insights from formalized mathematics to enhance logic education and proof understanding.

Findings

Proof strategies and automation are crucial for effective learning.

Proofs should be viewed as dynamic objects, not static trees.

A prototype proof assistant demonstrates practical application of these ideas.

Abstract

Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in as much detail as our need for eliminating risk might require. For those who do not happen to have direct access to the realm of mathematical truths, the modern field of formalized mathematics has quite a few lessons to contribute, and one might pay heed to what it has to say, for instance, about: the importance of employing proof strategies; the fine control of automation in unraveling the structure of a certain proof object; reasoning forward from the givens and backward from the goals, in developing proof scripts; knowing when and how definitions and identities apply in a helpful way, and when they do not apply; seeing proofs [and refutations] as dynamical objects, not reflected by the static derivation trees that Proof Theory wants them to be. I believe that the great challenge for teachers and learners resides currently less on the availability of suitable generic tools than in combining them wisely in view of their preferred education paradigms and introducing them in a way that best fits their specific aims, possibly with the help of intelligent online interactive tutoring systems. As a proof of concept, a computerized proof assistant that makes use of several successful tools already freely available on the market and that takes into account some of the above findings about teaching and learning Logic is hereby introduced. To fully account for our informed intuitions on the subject it would seem that a little bit extra technology would still be inviting, but no major breakthrough is really needed: We are talking about tools that are already within our reach to develop, as the fruits of collaborative effort.