ajassat said:
When studying advanced mathematics and physics do you find that your brain is quickly able to understand new concepts and complicated looking math equations on the page?
Is anyone here actually able to pick up a book, say "Linear Algebra" and read through it understanding it easily like a fiction book without prior experience in the topic?
Do these new mathematical concepts come quickly to you, or do you have to stare at the page for hours before you understand?
Regards,
Adam
I'm not nearly as qualified as some people here, but I have done a lot of reading, so I feel I could say something meaningful on this subject. What follows simply reflects my experience in essentially reading on my own.
Reading mathematical books and articles (whether they are in pure math or some field of applied math) is a skill in and of itself.
First, mathematics almost has a language of its own. I'm not talking about the symbols here, but the English (or another language, of course; I'll talk about English here since I've the most experience with it). There are a variety of phrases that have technical meanings. At first, one has to consciously translate these phrases and expand them to arrive at their true meaning; eventually, one is able to do this subconsciously. As a simple example, I used to have to mentally map out "if and only if" every time I saw it, being extra careful to make sure I got both directions; now I've seen the phrase so much I'm somehow significantly faster at processing its meaning in a given context.
The growth of this skill has coincided with the growth of my ability to write proofs. I honestly believe that one of the first hurdles that one must overcome when first learning to write proofs is simply learning how to precisely control one's language and how to appropriately use phrases like "if and only if" and other such technical uses of language. I think one of the reasons for this is that people are often first exposed to proofs in calculus classes of some sort (not real analysis but introductory courses), in which the proofs are mostly calculations, not verbal arguments.
Once one has mastered this skill, I don't think one should have any (or at least not many) troubles reading mathematical articles and books if one has all the prerequisites filled. And this is a bigger point than it sounds like. Nearly without fail I've discovered that when I have to stare out a sentence or paragraph for more than a few minutes it's because I was lacking a piece of knowledge or understanding that the author had assumed.
In mathematics (and I'm sure it's the same in applied math) it's very easy for this to happen. As a basic example, the implicit function theorem is something that can be interpreted in a variety of ways and stated in a variety of ways. It's easy to learn only one rigid way of thinking of the theorem and then be caught off guard when an author interprets the theorem differently. Or another example could be linear maps between two vector spaces. In linear algebra, it's basically okay to think of it as a black box that takes a vector in the first space and produces a new one in the second space. But in multilinear algebra and related fields, linear maps can interact with other objects in a wide variety of ways, and the sort of rigid interpretation of linear maps promoted by classical linear algebra can be very unwieldy in tensor analysis and differential geometry and multilinear algebra in general.
As just a little side note, a lot of proofs from textbooks are carefully crafted to be as minimal as possible in order to keep the logic of the proof as crisp and appealing as possible. This usually makes manifest some simple "tricks" that make the proof work. So when reading a textbook I usually try to extract these "tricks" and remember them, which means I'll often re-read proofs several times even if I understood them on the first reading just trying to distill the abstract strategy used by the author.
