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Hi Sir,

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.

Many Thanks!

(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

of ideas.

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)

Advanced Math

Answers by Expert:

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.**Education/Credentials**

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