Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.

SarahC comments on The Best Textbooks on Every Subject - Less Wrong

167 Post author: lukeprog 16 January 2011 08:30AM

You are viewing a comment permalink. View the original post to see all comments and the full post content.

Comments (327)

Sort By: Popular

You are viewing a single comment's thread.

Comment author: [deleted] 16 January 2011 05:30:02PM *  19 points [-]

Subject: Representation Theory

Recommendation: Group Theory and Physics by Shlomo Sternberg.

This is a remarkable book pedagogically. It is the most extremely, ridiculously concrete introduction to representation theory I've ever seen. To understand representations of finite groups you literally start with crystal structures. To understand vector bundles you think about vibrating molecules. When it's time to work out the details, you literally work out the details, concretely, by making character tables and so on. It's unique, so far as I've read, among math textbooks on any subject whatsoever, in its shameless willingness to draw pictures, offer physical motivation, and give examples with (gasp) literal numbers.

Math for dummies? Well, actually, it is rigorous, just not as general as it could potentially be. Also, many people's optimal learning style is quite concrete; I believe your first experience with a subject should be example-based, to fix ideas. After all, when you were a kid you played around with numbers long before you defined the integers. There's something to the old Dewey idea of "learning by doing." And I have only seen it tried once in advanced mathematics.

Fulton and Harris won't do this. The representation theory section in Lang's Algebra won't do this -- it starts about three levels of abstraction up and stays there. Weyl's classic The Theory of Groups and Quantum Mechanics isn't actually the best way to learn -- the group theory and the physics are in separate sections and both are a little compressed and archaic in terminology. Sternberg is really a different thing entirely: it's almost more like having a teacher than reading a textbook.

The treatment is really most relevant for physicists, but even if you're not a physicist (and I'm not), if you have general interest in math, and background up to a college abstract algebra course, you should check this out just to see what unusually clear, intuitive mathematical writing looks like. It will make you happy.

Comment author: Douglas_Knight 17 January 2011 09:38:56PM 3 points [-]

Fulton and Harris won't do this.

Won't do what? Almost everything you say about Sternberg seems to me to apply to Fulton & Harris. I have not looked at Sternberg, and it may well be better in all these ways, but your binary dismissal of F&H seems odd to me.

Comment author: realitygrill 17 January 2011 09:34:06PM 2 points [-]

Have you ever read Group Theory and Its Applications in Physics by Inui, Tanabe, Onodera? I have never been able to find this book and it's been recommended to me several times as the pedagogically best math/physics book they've ever read.

Comment author: komponisto 17 January 2011 06:45:59PM 2 points [-]

Also, many people's optimal learning style is quite concrete; .

I hasten to point out (well, actually I didn't hasten, I waited a day or two, but...) that while this is true for many people, it isn't true for all, and, in particular, it isn't true for me. (See here.)

I believe your first experience with a subject should be example-based, to fix ideas. After all, when you were a kid you played around with numbers long before you defined the integers

I don't think the way I learned mathematics as a young child (or indeed in school at any time, up to and including graduate school) was anywhere near optimal for the way my mind works.

The best way for me would have been to work through Bourbaki, chapter by chapter, book by book, in order. I'm dead serious. (If I were making an edition for my young self I would include plenty of colorful but abstract pictures/diagrams.)

Comment author: [deleted] 17 January 2011 07:38:43PM 2 points [-]

I assumed there were some folks like you but I'd never met one. Shame on me for making too many assumptions.

Comment author: komponisto 17 January 2011 11:02:05PM 5 points [-]

I assumed there were some folks like you but I'd never met one

It's not as stark as that. For example, Alicorn, whom I believe you've met, shares with me a psychological need for concepts to be presented in logical order.

In my case, if you're curious, I think the reason I'm the way I am comes down to efficient memory. To remember something reliably I have to be able to mentally connect it to something I already know, and ultimately to something inherently simple. The reason I can't stand ad-hoc presentations of mathematics is that remembering their contents (let alone being able to apply those contents to solve problems) is extremely cognitively burdensome. It requires me to create a new mental directory when I would prefer to file new material as a subdirectory under an existing directory. (I don't mind having lots of nested layers, but strongly prefer to minimize the number of directories at any given level; I like to expand my tree vertically rather than horizontally.)

This explains why it took me forever to learn the meaning of "k-algebra". The reason was that (for a long time) every time I encountered the term, the definition was always being presented in passing, on the way to explaining something else (usually some problem in algebraic geometry, no doubt), instead of being included among The Pantheon Of Algebraic Structures: Groups, Rings, Fields etc. -- so my brain didn't know where to store it.

Comment author: [deleted] 18 January 2011 12:58:43AM 2 points [-]

Well, I can see the need to have the concepts fit together. What I need on a first pass through a subject is something that can attach to the pre-abstract part of my brain. A picture, even a "real-world example." Something to keep in mind while I later fill in the structure.

The way I see it (which I realize is more of a metaphor than an explanation) human brains evolved to help us operate in large social groups of other primates. We're very good at understanding stories, socialization, human faces, sex and politics. I think for a lot of people (myself included) the farther we get from that core, the more help we need understanding concepts. I need to make concessions to human frailty by adding pictures and applications, if I want to learn as well as possible. (This is something that people in abstract fields rarely admit but I think LessWrong is a good place to be frank.)

Comment author: SilasBarta 18 January 2011 12:31:13AM *  2 points [-]

In my case, if you're curious, I think the reason I'm the way I am comes down to efficient memory. To remember something reliably I have to be able to mentally connect it to something I already know, and ultimately to something inherently simple.

Interesting. That sounds like my habit of making sure everything I learn plugs into my model for everything else, and how I'm bothered if it doesn't (literature and history class, I'm looking in your general direction here). Likewise, how I don't regard myself as understanding a subject until my model is working and plugged in (level 2 in my article).

This is why I've usually found it easy to explain "difficult" topics to people, at least in person: per my comment here, I just find the inferentially-nearest thing we both understand, and build out stepwise from there. And, in turn, why I'm bothered by those who can't likewise explain -- after all, what insights are they missing by having such a comparmentalized (level 1) understanding of the topic?

Comment author: steven0461 17 January 2011 11:51:33PM 2 points [-]

I find it takes much more effort to learn things when different sources don't coordinate well on definitions, notation, and the material's hierarchical structure. For example, if everyone agreed on how to present the Nine Great Laws of Information Theory, that would make them much easier for me to remember them. It's as if, instead of learning the overlap between different presentations, my brain shuts down and doesn't trust any of them. But it's hard to settle on such cognitive coordination equilibria.

Comment author: lukeprog 16 January 2011 05:53:27PM 1 point [-]

Thanks for all the detail! I've added it to the list above.

Comment author: jsumr 26 May 2015 04:01:09PM 0 points [-]

Has anyone been to OpenStax College?

http://openstaxcollege.org/books

If so, are their textbooks good?