
Reading Mathematics (2002) [pdf] - kercker
http://www.math.cornell.edu/~hubbard/readingmath.pdf
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johnbender
Beyond notation and ordering, I have had the best results reading and
comprehending complex concepts, including mathematics, by taking the following
suggestion to the extreme:

> Read with pencil and paper in hand, making up little examples for yourself
> as you go on.

I like to find a difficult question that I can answer with an understanding of
the material. This acts as a litmus test of my understanding and a forcing
function.

The question can be almost anything, but a general approach I use is to write
a "compiler" that maps some concept from the material to a concept I already
understand (this normally takes the form of a denotational semantics). Then
the question would be, "How can I interpret X as Y?" This technique has its
limits since the material can't be too far afield from something I already
know and the idea isn't novel but it has been effective for me. The critical
bit is forcing myself to write down a fairly comprehensive mapping function.
This gets me into the dark corners of my understanding very quickly and adds
new questions to answer.

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aisofteng
I do the exact opposite - I don't write anything down and try to visualize
what I'm reading entirely. I don't start writing anything down when doing
exercises until I have the entire solution in mind (unless it's something
computational). It's hard to get used to doing at first, but, after a couple
years, my abstract reasoning skills developed significantly past most or all
of my peers.

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logicchains
I've noticed something similar; it's harder at first, but once I learn to
think about something visually/intuitively, I can understand/use/reason with
it much better than if I'd started with pen and paper. It makes sense that
this approach is more effective because if the brain has internalized a
concept than it can reason with it subconsciously, but if external paper is
required for reasoning about it then such subconscious reasoning is not
possible.

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tw1010
Funny how utterly natural and subconsious this stuff becomes after a while. I
almost felt like commenting something snyde about how superfluous it is to
make it this explicit, but then I realized that it was only a few years ago
that it made my consious brain totally overwhelmed.

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opportune
Yeah at first I was thinking "is this not the only way of understanding
written mathematics?!!"

Then I realized that when I was a freshman undergrad, I understood much less
than half of the terminology that is considered basic mathematics. The
formalisms behind WOLOG and \forall \epsilon >0 \exists \delta > 0 s.t. yadda
yadda were completely lost on me. A lot of mathematics is simply familiarity,
and that's easy to forget

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trentmb
> A lot of mathematics is simply familiarity, and that's easy to forget

There's nothing special about mathematics in this regard.

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Improvotter
Right now I'm studying for my math final for computer engineering. This really
does speak to me as I often don't understand certain parts of the chapter and
just skip ahead to the exercises and then I try to reflect back to the theory.
Thank god that I don't have to learn a single proof though.

Alright, back to studying.

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jayshua
I'm studying computer science myself and very much enjoy writing proofs. Is
there any particular reason that you don't?

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cyberpunk0
Fellow computer scientist hear discrete mathematics was my worst subject. Not
for lack of enthusiasm I love combinatorics and various other related math
subjects. The concept of proofs is easy enough to understand but the problem
is the dense notation involved. Math papers are incredibly painful to read

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sergj
From which book is this?

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stablemap
Looks to be a multivariate calculus text by the Hubbards:

[http://matrixeditions.com/5thUnifiedApproach.html](http://matrixeditions.com/5thUnifiedApproach.html)

~~~
jostylr
That's correct (a later edition than the pdf given the date). It is a
beautiful book that blends the practical and abstract together to really give
a solid foundation in both linear algebra and multivariable calculus topics.
If someone asked me for exactly one book to read after learning single
variable calculus, this is the book I would recommend. It is long, but
extremely satisfying.

It covers in detail many topics often glossed over while keeping an eye on
what is actually applicable, including such things as (taken from their page):

> More big matrices! We included the Perron-Frobenius theorem, and its
> application to Google's PageRank algorithm More singular values! We included
> a detailed proof of the singular value decomposition, and show how it
> applies to facial recognition: "how does Facebook apply names to pictures?"

~~~
mcguire
The fifth edition has a similar section in chapter 0:
[http://matrixeditions.com/UnifiedApproach5thedSamples.html](http://matrixeditions.com/UnifiedApproach5thedSamples.html).

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josephhurtado
Useful, to the point, non controversial, and short. AWESOME.

