Hacker News Reader

One thing I cannot stand is the general lack of definitions. My mind is oriented to programming where anything outside of the language must be defined in scope. Or maps, which have a legend and define abstract symbols.

My first year linear algebra textbook is nothing like this. On page 1 (the very first page after the Roman numeral pages including the title page, table of contents, and forward) is a bright yellow box defining the set R^n.

The remainder of the book is chock full of these definition boxes. Additionally, there are blue boxes for remarks, green boxes for examples, mauve boxes for theorems, and red boxes for exercises.

It's really that straightforward. You could probably skip all of the prose text and just use these boxes to learn what you need to get through the exercises and pass the course. Other math texts I've read may not have this nice colour scheme, but they're still pretty easy to follow.

The exercises, on the other hand, will challenge you to think creatively and demand that you learn how to apply the definitions and theorems properly in your proofs.

I call this effect cultural mathematics. After college I avoided maths, went into programming, especially Functional Programming, which has a big math envy. With years I finally clicked on how mathematicians think and write. It's hyper cryptic when not explained or if you're just below the math level brainiac who does grok high level abstraction rapidly and thus will play that formula game with ease.

>You read the book. Start on page one. Eventually you get to a formula. You can scrape the text all you want, but there are characters without a definition.

If I remember correctly, this is much more common with more elementary math books which are presumably trying to be more informal and 'user friendly'. You might seek out a book pitched at a more advanced level which will be much more formal (and pedantic) if that is the kind of thing that's bothering you.

Also many books will have an 'index of notation' inserted at either the beginning or the end, so make sure to look for it - if it's there, the author might assume you know the convention and won't necessarily define everything in the exposition.

-----

One note about abstraction in mathematics vs computer programming. When model something in a program, you are often modeling a particular thing, or kind of thing, so it makes sense to give it a descriptive name. Contrast that with, say, a number: I have a small or medium or large pile of apples. I make the adjective more precise and say I have a 3 or 6 or 25 -pile of apples. Then i notice there's nothing special about apples, I can do the same thing for piles of rocks or tea bags. And I can do the same thing for non-piles, like links in a fence, or countries, or more abstract things like counting permutations or combinations or discrete transformations of whatever you can imagine.

The point being: in mathematics you are not working with a particular kind of thing, you are noticing patterns satisfied by many very very different kinds of things (hence the lack of descriptive names - there is often not much to describe). You stop asking questions about what it is, and learn to characterize things entirely by the logical properties they satisfy. Its a different thought process than what you might be accustomed to.

For me I need to see examples. The abstraction almost always just confuses me. Unfortunately most (advanced) mathematics texts are very thin on examples. It wasn't always like this. Mathematics used to (hundreds of years ago) be done with all examples and no theory! That would suit me just fine. It seems that some people are better at one way or the other. The present trend is towards the abstract side.

I've found I have to turn mathematics algorithmic to understand it, thinking through what each symbol does to the things passing "through" it. Attempting to understand equations per se does nothing for me. They're important for proofs, I get it, but aid me not at all figuring out what it all actually means. Yet equations are the primary tool and presentation method used in most math books.

I'm pretty sure this is why I've never had trouble with programming, despite feeling the way I imagine a dyslexic must when I try to read very "mathy" material even about things that I already understand. Programming's algorithm-first. Except the few languages that aren't, and I bounce off those, too.