On Proof and Progress in Mathematics
đź“ś Abstract
This essay contains some thoughts about how we use our brains and our methods to acquire understanding. Its particular focus is on the roles of proof, reasoning, and rigor in mathematics. I draw on some ideas from issues of "The Notices" in the past two years, which continually spark new thoughts that make me wish to communicate. It is a view from one particular angle, and I would like to make it clear that I realize that other angles and outlooks are valid ultimately because the way that people think and come to understand forms the creator of the mathematical reality we perceive.
✨ Summary
William P. Thurston’s essay, “On Proof and Progress in Mathematics,” delves into the cognitive processes involved in mathematical understanding, focusing on proof, reasoning, and rigor. The work does not contain an abstract but rather a preface outlining his thoughts on mathematical cognition. The paper discusses how mathematicians perceive mathematical reality and emphasizes the diverse ways in which individuals achieve understanding, contrasting the classical view of mathematical proof as the sole arbiter of mathematical truth. While the work itself has been primarily significant in philosophical discussions within the mathematics community, it does not appear to have directly influenced other tangible research or industry innovations. The abstract nature of its topic limits its direct applications, focusing rather on reshaping perspectives within the field. This paper can be primarily seen as contributing to the discourse on the philosophy of mathematics rather than directly inspiring new technological or academic advancements.