Advanced Math/Please help give some comment
I found this "note" on my school computer and would like to ask if you think it makes any sense, just a few remarks
will do. English is probably not the author's first language but I would like to have your opinion.
(How should one go about reading a mathematical paper)
A book can be seen as a collection of words, sentences or paragraphs arranged in a particular
sequence. In a math paper, we consider every word,sentence, paragraph and section as units of ideas in different
scale of details. Unlike a normal book, math ideas must be arrange in strict particular order otherwise it
will often make no sense. With this perspective, at least three things must be achieved to have a complete
grasp of the entire paper, naming, understand what every unit of ideas mean, understand how they are
connected by well-defined relations or on some intuitive level if they are non-trivial and why every unit
of ideas are arranged in a particular order. Ideally, one would like to hold highly connected collection of
ideas in his mind in contrast to ideas that are isolated and fragmentary pieces.
When one starts reading a paper, it helps to start from the general, skimming through the pages to get
a rough idea of what a paper is all about. With each subsequent attempts of readings one moved on to a
level of increased details, filling up gaps of understandings not achieved by previous readings.
On many occasions, one may find that he is getting diminishing return of progress in understanding
for every effort invested. If too severe, one should put the paper aside, switch it to a paper he finds
most interesting and come back to the initial paper after a few days with a more refreshed mind and
perspective. On many instances, what seems incomprehensible at first can suddenly reveal itself and
make complete sense. These kind of phenomena are well documented by psychologists.
(How one should go about understanding a mathematical paper)
When facing particular problems, think in terms of how this particular case fits into a larger
framework. Think in terms of the underlying principles at work. Think of particular cases as concrete
manifestation of a deeper abstract reality. The goal is to capture parts of this reality.
Realise that understanding something is fundamentally about connecting what you do not know
with what you already know. That is why analogies and metaphors sometimes help when we absolutely
know nothing about an idea. Beyond that we have to be precise, we have to pinpoint exactly
which idea is responsible for which consequences. That doesn't mean that we have to be absolutely
rigourous for all our thought process. When we try to make connections between ideas, it is really
a kind of discovery process, we make guesses, distinguish which guess is more true, make tentative
assumptions, test our own hypothesis, try out different configurations of ideas which we think may
yield the greatest amount of insights. We ask questions, lots and lots of them to ourselves and try
to answer them. We should ask not only what we should do to get from A to B, but also why we
do what we do. We know what we already know, but more importantly we try to figure out what we
still don't know. We try to achieve a level where we can "see" an idea and infer its consequences almost
immediately. Maximising connections allows us many different pathways to manuver in the landscape
I am afraid this 'note' is simply incoherent rambling from someone who has nothing to say but loves to scribble.
When you choose to read a mathematics paper you are already an expert in the field of the paper, or it will be totally incomprehensible to you. (as practically all of them are to me)