Reading Mathematics (2002) [pdf] | Hacker News Reader

Reading Mathematics (2002) [pdf]

math.cornell.edu

172 pointskercker20 comments

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.

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.

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.

At some point math becomes more about ideas and less about lower level details. Like you I don't write down my reasoning either and try to do exercises in my head or verbally. So long as I get the idea behind the proof correctly, I am not worried about the lower level implementations.

The best advice when learning math or anything is to "solve the smaller problem". Try to break the problem down into smaller pieces and work from there.

Trying to understand something complex as a whole can be daunting and intimidate people from even trying to understand it. But if break it apart into smaller parts and solve/understand those, many times, you find that it wasn't that complex in the first place.

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.

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

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

There's nothing special about mathematics in this regard.

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.

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

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

Proofs are fun when you study them on your own time. When you're time constrained, as you often are with school, they're way less fun and way more stressful.

From which book is this?

Looks to be a multivariate calculus text by the Hubbards:

http://matrixeditions.com/5thUnifiedApproach.html

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?"

The fifth edition has a similar section in chapter 0: http://matrixeditions.com/UnifiedApproach5thedSamples.html.

Was going to buy a PDF version, but after reading the requirements page [1] I decided not to buy anything from those authors. Also, the 5th edition of this book is not on Amazon.

[1] http://matrixeditions.com/Drumlin.html

Vector Calculus, Linear Algebra, and Differential Forms, A Unified Approach by John H. Hubbard and Barbara Burke Hubbard.

Great Thanks!

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

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