Memorizing Mathematical Definitions

[Avatrin]

Hi

Usually when learning math, understanding the theorems and ideas helps tremendously to remember math. I get that... I got through calculus, linear algebra and complex analysis easily.

The problem for me started with three branches in mathematics: Real analysis, measure theory and abstract algebra. The theorems are no problem. However, remembering the definitions is really hard:
Rings, fields, sigma-rings, algebras, sigma-algebras, integral domains, outer measure, Lebesgue measure, metric space, norm space... The list goes on... Not only are there many definitions; They are often very similar to each other. So, remembering the differences can be an art in itself.

I have to retake exams because I could not remember the definitions. What were the exact definitions of pointwise and uniform continuity? I remembered only vaguely and had to try to deduce parts of the definitions..

How can I remember the definitions more easily? I keep forgetting them..

 

[jedishrfu]

I'd look at the properties of each definition and see how one definition is a subset of another due to one or more added properties.

You might see a pattern that you can work with and own the subject.

 

[homeomorphic]

"It is impossible to understand an unmotivated definition, but this does not stop the criminal axiomitizers/algebraists." --V.I. Arnold

Forgetting the definitions is a symptom of the lack of motivation found in many math books and classes. One of the tricks to remembering things is to make them as meaningful as possible, and to see the point of the definition.

For algebra, you might try books by John Stillwell on algebra and number theory. Ideals are actually a meaningful concept, when viewed in the context of number theory, as Stillwell explains in the later chapters of his book. They are not an arbitrary definition from the sky, the way textbooks often present it. You could also think of an ideal as the appropriate notion for the kernel of a ring homomorphism or something, but that doesn't quite have the poetry of Dedekind's original theory of ideals.

For real analysis and measure theory, you might try the two volumes of A Radical Approach to Real Analysis or Understanding Analysis.

And generally, try to understand the definitions, not just memorize them. There's a clear picture that goes along with the epsilon-delta definition of continuity, for example. You should remember the picture and how to translate that into a definition, not the word-for-word, rote-memorized definition.

Possibly, memorizing word for word definitions could be useful for beginners in some cases, as a kind of crutch, while they are still developing their mathematical skills, but in general, I wouldn't recommend it.

One caveat, though: you might not want to spend excessive amounts of time searching for the best possible intuition behind every single definition. If you go further in math, all the way to doing research, you might find that in your research, you don't even need to know what an ideal is. In that case, you might want to cut some corners (choose which ones to cut carefully!) and leave the motivation to someone like me who can't leave it alone, to the point where I have abandoned a conventional mathematical career in favor of devoting all of my mathematical efforts to searching for and restoring the motivation and intuition.

 

[Borek]

Flash cards.