Proving all the theorems in a book or completely recreating a field for yourself is incredibly time-intensive, however, and so I think it ought to be used sparingly. Trying to prove theorems while reading a book before looking at the proof is a good idea. Striving to understand material to the point that definitions, proof steps, and even results seem intuitive is an excellent idea as well. I frequently find myself at a blackboard for hours playing with some idea to achieve this end. This has been my experience with lectures, textbooks, course notes, research monographs, and papers.

This sort of thing is also good practice for reading research works, which often contain no exercises, can skip many steps, and sometimes, sadly, fail to motivate or to explain adequately their ideas. Making a new idea seem obvious for yourself, sketching your own examples, and developing your own conjectures are all incredibly powerful aids to understanding.

]]>As it turns out, almost everyone understands mathematics by examples, simplifications, examples, and familiarity, not dry, logical, abstract symbols. Similarly, progress in mathematics usually begins with examples. People usually develop the theory to match the examples, not the other way around. John Conway has some wisdom here:

“To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples…”

Halmos also emphasized the importance of examples. The following quote, which is probably so well known in part thanks to the blog Annoying Precision, demonstrates this fact:

“A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.”

The mathematics educational system, at least in the United States, does a poor job of emphasizing these features of mathematics, especially at the advanced undergraduate and beginning graduate level. The goal is absolutely rigorous theorem proving, no more, no less. In undergraduate mathematics, you need to learn how to prove theorems and deal with abstract, formal mathematics. In beginning graduate school—where lectures are so often too fast, with too few examples, and with too few deviations from the definition-theorem-proof format Vladimir Arnold so hated—you simply practice applying this new skill to new, more sophisticated example.

We often fail to emphasize adequately the roles of intuition, example, time, and play. Students maximize the number of theorems they learn and prove in a small period of time, but they do so at great cost.

]]>