Tethered Abstraction

Sage advice on how to think maturely about deep mathematical concepts from THE RISING SEA: Foundations of Algebraic Geometry by Ravi Vakil:

The purpose of this book is to give you a thorough foundation in these powerful ideas. Do not be seduced by the lotus-eaters into infatuation with untethered abstraction. Hold tight to your geometric motivation as you learn the formal structures which have proved to be so effective in studying fundamental questions. When introduced to a new idea, always ask why you should care. Do not expect an answer right away, but demand an answer eventually. Try at least to apply any new abstraction to some concrete example you can understand well.

Understanding algebraic geometry is often thought to be hard because it consists of large complicated pieces of machinery. In fact the opposite is true; to switch metaphors, rather than being narrow and deep, algebraic geometry is shallow but extremely broad. It is built out of a large number of very small parts, in keeping with Grothendieck's vision of mathematics. It is a challenge to hold the entire organic structure, with its messy interconnections, in your head.

(cf Think Cleanly (2022-03-01), Learn by Doing (2022-03-16), ...) - ^z - 2022-03-16