Think Cleanly

From the Preface to THE RISING SEA: Foundations of Algebraic Geometry by Ravi Vakil:

Because our goal is to be comprehensive, and to understand everything one should know after a first course, it will necessarily take longer to get to interesting sample applications. You may be misled into thinking that one has to work this hard to get to these applications – it is not true! You should deliberately keep an eye out for examples you would have cared about before. This will take some time and patience.

As you learn algebraic geometry, you should pay attention to crucial stepping stones. Of course, the steps get bigger the farther you go.

Chapter 1. Category theory is only language, but it is language with an embedded logic. Category theory is much easier once you realize that it is designed to formalize and abstract things you already know. The initial chapter on category theory prepares you to think cleanly. For example, when someone names something a "cokernel" or a "product", you should want to know why it deserves that name, and what the name really should mean. The conceptual advantages of thinking this way will gradually become apparent over time. Yoneda's Lemma – and more generally, the idea of understanding an object through the maps to it – will play an important role.

and

Pathological examples are useful to know. On mountain highways, there are tall sticks on the sides of the road designed for bad weather. In winter, you cannot see the road clearly, and the sticks serve as warning signs: if you cross this line, you will die! Pathologies and (counter)examples serve a similar goal. They also serve as a reality check, when confronting a new statement, theorem, or conjecture, whose veracity you may doubt.

... more quotes and notes to follow!

(cf Ultimate Abstraction (2017-08-24), If You Need a Theorem (2018-11-08), Macro vs Micro (2019-02-03), Why Care about Category Theory (2019-03-03), Conceptual Building Blocks (2019-11-05), ...) - ^z - 2022-03-01