Good Math/Bad Math

Sunday, April 23, 2006

Misc: Answering Questions and Sources

There's a bunch of questions that I've been asked about the group theory stuff I've been talking about lately. Since things are always slow on the weekend, I thought it would be a good time to go through and answer.

I also just want to take a moment and say thanks to all of the people who are reading the blog, and who've sent me kind words about how much they enjoy it. I can't tell you how happy I am that people are enjoying my writing. I'm having a great time writing this, and it really means a lot to me that people are enjoying reading it. So thank you all!
  1. Onto? What's that? When I was defining transformations of groups, I used a bunch of terms "onto", "total", "one-to-one", etc. Lots of people didn't know the terms - and they're right, I should have defined them. So here we go:
    1. Total/Partial: a function from A to B is total if f(a) is defined for all members of A. It's partial if there are members of A for which f(a) is not defined.
    2. Onto: a function from A to B is onto if for every member b of B, there is at least one member of A where f(a)=b. It's basically the reverse of total. An onto function is also called a surjection.
    3. One-to-one: a function from A to B is one-to-one if every member of A maps to one member of B; and no member of B is mapped to by more than one member of A. Note that in this definition of one-to-one, the function does not need to be onto. (Some definitions of "one-to-one" require that the function be onto; that's not the way I was taught.) By this definition, a one-to-one function is also called an injection.

  2. Symmetry means reflection around a mirror. Shouldn't you use a different term? A lot of people have asked this. The thing is, symmetry doesn't mean reflection around a mirror. That's the simplest common example of it, but the term symmetry does mean much more than just "unchanged by reflection". I know that most english dictionaries definitions are, unfortunately, the reflective definition, because that's the common usage. But the origin of the word in greek, and it's use in mathematics is broader than that. Mathematicians use the word in it's broader sense, and if I were to use a different term for it on my blog, all I'd do is end up confusing things, because every other mathematician in the world will continue to use the word symmetry.

  3. Why aren't you using MathML? After all of the discussion about whether or not to use MathML, I've ended up not using it, at least not for now. After experimenting with some tools for it, I've found that Blogger is very unfriendly to MathML. I really appreciate the fact that Blogger is free and all, but it's very frustrating at times. The only way I've found to make MathML work would require me to change a blogger setting which, in turn, would require me to go back and reformat all of the earlier posts to blog, or they'd become illegible.

  4. What are your sources? For the group theory stuff, I'm using my old class notes, Wolfram's Mathworld, and Wikipedia. In particular, Mathworld is a really fun source - there's a ton of great stuff up there. The writing quality varies from mediocre to fantastic; the group theory stuff tends towards the mediocre, but it's thorough, so it makes a good reference, if not a good place to go to learn the material.

    I've also been reading "The Equation that Couldn't be Solved" by Livio; lots of people think it's a great book. Personally, I find it a very frustrating read; Livio tries to write like Hofstadter or Gardner, with a lot of places where it goes off on semi-related tangents. When Hofstadter does that, I tend to enjoy it; his tangents are interesting and relevant. I find that Livio's tangents are often too unrelated to the basic material. I think it's mostly a matter of personal taste, but I find Livio a very difficult read; I can't get myself to read more than a few pages at a sitting. (Whereas when I read, say, Brian Greene, I read 50 pages at a sitting.) Given how many people seem to really love the book, if you're interested in this kind of stuff, it's probably worth getting a copy and reading; but it's not like "Godel, Escher, Bach" where I think it's an absolute essential to have in your library.

  5. When are you going to write about X? I'm amazed and flattered how many mails like this I've gotten. People have asked me to write about topography, recursive function theory, language families, rings, set theory, discrete vs continuous math, statistics, music theory, and more. My only answer is: give me a chance! Please do keep sending requests for what you'd be interested in seeing on the blog; I'll do my best to get to it all. The more requests I get for a topic, the sooner I'll get to it. But please be patient!

  6. Can I reprint articles from your blog? As long as it's not for profit, sure, as long as you put in an attribution identifying where you got it, and don't edit it other than for formatting. If you want to edit it, or you want to include it in something where you're going to charge for copies, get in touch with me by email, and we can talk about it.


  • I just want to chime in with another "what a cool blog!" comment. I studied group theory 20 years ago and your recent posts have reminded me of that time -- and having that special math experience of wonder and understanding -- of "getting" it.

    Keep up the good work.

    By Anonymous AndyS, at 2:10 AM  

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