Comment by omginternets | Hacker News Reader

Comment by omginternets | Hacker News Reader

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Broadly speaking, I've been approaching mathematics from two different angles: CS, and the mathematical background that subtends it.

On the CS front, I'm interested in two main subjects:

- Distributed Systems (especially peer-to-peer overlays)

- Programming Language Design (in particular: type theory, algebraic effects, and -- more recently -- optimal beta reduction)

On the pure math side, I've tried to focus my efforts on topics related to the above. I've tried working through "introductory" books on each of the following subjects:

- Combinatorics

- Graph Theory

- Category Theory

Prior to that, I tried working through some more general books. This is where most of my small progress emerged:

- Thompson's Calculus Made Easy

- Stewart and Tall's The Foundation of Mathematics (Set Theory)

In all cases, the biggest blocker for me is the proofs and exercises. I rapidly get stuck on a problem, which of course prevents me from understanding the subsequent chapters.

Proofs are the great 'leveling-up' of mathematics. At my former university they had a bridge class for students to take when they made the leap from more mechanistic mathematics to proofs. I didn't have the benefit of taking it when I was a math major (to my regret, it was painful to level-up), but it looks like the text they use is publicly available:

http://www2.math.umd.edu/~shalper/text.pdf

Many thanks! Proofs are something I've always struggled with, having never been taught any methodology. I've added this to my reading list.

The proofs in standard Euclidean high school plane geometry are a perfectly good introduction to how to write proofs. A good next step is just a college course in abstract algebra -- sets, groups, rings, fields, prime numbers, the fundamental theorem of arithmetic, the fundamental theorem of algebra, Galois theory (why with Euclidean construction it is impossible to construct a square with the same area as a given circle), why the square root of 2 is irrational, the Euclidean greatest common divisor algorithm, the Chinese remainder theorem, vector spaces, .... Here there are essentially no prerequisites.

The results proved in an abstract algebra course are often just embarrassingly childishly simple so that getting a solid proof is easy. For the issue of writing style for proofs, just pick that up from the text. E.g., usually say since instead of because. The proofs in abstract algebra and, really, the rest of math, are really essentially the same as in high school plane geometry but just written in a less rigid style.

If want a theorem proving course in calculus, first take calculus where theorem proving is not the main content, then take abstract algebra (where will use no calculus), and then take a theorem proving course in calculus, e.g., from W. Rudin, Principles of Mathematical Analysis. With this sequence, should never be without prerequisites or in doubt about the reason, point, value, or intuitive view of the material.

Yes, the Wiles quote is really good for describing research, but learning need be nowhere nearly that challenging.

The stuff I've done is 40 years ago now, but I've done some combinatorics and some graph theory. It's been so long that it's possible that I can't help you at all, but I might be able to talk in generalities, and maybe that's all you need.

My email is in my profile.

Thank you so much - you have no idea how much this means to me. I'll be sure to shoot you an email next time I get stuck, which will happen more or less as soon as I decide to try again :'D

Well, as I say, I may not be of any help, but at least you'll have someone with whom to share your confusion. I'll look for your email.

What Schaum's Outlines would be relevant here?

Just ordered Schaum's Outline on Graph Theory. Thank you for the recommendation!

I didn't find it, you did!

The Schaum's book of most interest to the original poster (Abigail Rabieh)is the following:

Schaum's 3000 Solved Problems in Linear Algebra by Seymour Lipschutz. Below is a precis:

"Master linear algebra with Schaum's--the high-performance solved-problem guide. It will help you cut study time, hone problem-solving skills, and achieve your personal best on exams! Students love Schaum's Solved Problem Guides because they produce results. Each year, thousands of students improve their test scores and final grades with these indispensable guides. Get the edge on your classmates. Use Schaum's! If you don't have a lot of time but want to excel in class, use this book to: Brush up before tests; Study quickly and more effectively; Learn the best strategies for solving tough problems in step-by-step detail; Get the big picture without spending hours pouring over long textbooks. Review what you've learned in class by solving thousands of relevant problems that test your skill. Compatible with any classroom text, Schaum's Solved Problem Guides let you practice at your own pace and remind you of all the important problem-solving techniques you need to remember--fast! And Schaum's are so complete, they're perfect for preparing for graduate or professional exams. Inside you will find: 3000 solved problems with complete solutions--the largest selection of solved problems yet published on linear algebra; A superb index to help you quickly locate the types of problems you want to solve; Problems like those you'll find on your exams; Techniques for choosing the correct approach to problems; Guidance on choosing the quickest, most efficient solution. If you want top grades and thorough understanding of linear algebra, this powerful study tool is the best tutor you can have! Chapters include: Vectors in R" and C." Matrix Algebra. Systems of Linear Equations. Square Matrices.Determinants. Algebraic Structures. Vector Spaces and Subspaces. Linear Dependence, Basis, Dimension. Mappings. Linear Mappings. Spaces of Linear Mappings. Matrices and Linear Mappings. Change of Basis, Similarity. Inner Product Spaces, Orthogonality. Polynomials over a Field. Eigenvalues and Eigenvectors, Diagonalization. Canonical Forms. Linear Functionals and the Dual Space. Bilinear, Quadratic, and Hermitian Forms. Linear Operators on Inner Product Spaces. Applications to Geometry and Calculus."

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