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Necessary prologue: I'd really like to become more fluent in the language of mathematics. I don't have a schedule that permits me taking a class and any on-line tutors that I find seem relatively sketchy. So, I'm looking to teach myself.

After reading quite a few books on the history of mathematics, it's become clear to me that geometry seems to be where mathematics all started. Not only that, but it seems to lead into other branches of mathematics quite well (for example, everything geometric can be discussed and represented algebraically, so I'll, by necessity, bump into algebra along the way).

My question: Given that I'm looking to teach myself, that I have no direct NEED for understanding a specific branch of mathematics, and that I intend to be studying casually, like, for the rest of my life off and on, is it fair to assume that geometry is a good starting place for a general mathematics education?

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If you are just looking to study for fun, then it's important to choose something you enjoy. For many people, geometry fits the bill nicely. Personally, I find problems in discrete mathematics more engaging, more readily apprehended, and their proofs more natural. I would suggest giving any undergraduate discrete mathematics textbook a try and see if you don't agree. –  Austin Mohr Sep 13 '11 at 20:25
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The main question is whether you enjoy it. Another nexus that comes to mind is number theory, which also attracts all fields of mathematics (including geometry), but going along with geometry is great as long as you enjoy what you study. Of course, "geometry" is a very (very!) broad term... –  Gadi A Sep 13 '11 at 20:27
    
To a large extent it's a matter of temperament. There are large areas of mathematics where geometry is not, strictly speaking, needed to understand the formal development, but where many mathematicians nevertheless feel that some geometric intuition is extremely helpful for finding and remembering results. You may or may not be among them. On the other hand, if by "geometry" you mean classic Euclidean compass-and-straightedge constructions without any numbers or coordinates, then you can safely skip that if it doesn't catch your interest. –  Henning Makholm Sep 13 '11 at 20:32
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Classical geometry is moribund. It does have live descendants, but they are very distant descendants. Unless geometry is of particular interest to you, I would suggest that other branches of mathematics may be more suitable. –  André Nicolas Sep 13 '11 at 20:41
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Historically, geometry was the subject in which the idea of mathematical proof was invented. More recently, many of us were first introduced to mathematical proof through geometry in high school (before the recent dumbing-down of the curriculum, but that's another story). It's also interesting because it involves a lot of interaction between the "left brain" (linear/logical reasoning) and "right brain" (spatial visualisation). –  Robert Israel Sep 14 '11 at 22:36
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First: I love this project. All the best to you. And I think you're going to have a great time.

Second: Mathematics is too big for this question to be able to be answered quite in the way you mean it. It is so big that not even a life of devoted study could lead any individual to an understanding of more than a small fraction of what's already well-understood by humanity as a whole. It is often said that while it used to be possible for the world's greatest mathematicians to have a birds-eye view on the subject as a whole, the last generation of mathematicians to have such a view was the generation of Poincare and Hilbert, 100 years ago. (Since you've read a lot of math history, you've probably heard this idea before.) The subject has sprouted in so many directions it's even difficult to keep track of them all. Ask a researcher in dynamical systems what the most important or fundamental math to know is and you'll get a totally different answer than if you ask a number theorist. So your choice of where to start is necessarily going to be a little bit arbitrary in relation to the question of "most central." What counts as the center depends on what part of the boundary you're looking at.

One piece of good news is that to quote math writers Bob and Ellen Kaplan, "anything leads to everything." Just as you're going to bump into algebra if you pick up geometry, you're going to bump into geometry if you pick up algebra. Basically you're going to bump into both if you start anywhere.

That said, I don't want to ignore your request for concrete advice about where to start. This request actually goes beyond the question of what field to start with. Does "geometry" mean Euclid? Or a high school geometry textbook? Or a book on projective or hyperbolic geometry, or the geometry of Einstein's theory of relativity? So let me suggest some actual books you might try. The three books I'm about to suggest have the advantages that they are a) aimed at educated laypeople, and b) broad in scope. (In fact, they each attempt to take a wide-angle view of mathematics as a whole. Of course a propos of the last paragraph, math is too big for this to be done in an objective way: that's why the 3 books are extremely different from each other.)

Here are the books:

What Is Mathematics? by Courant and Robbins

The Heart of Mathematics by Burger and Starbird

Mathematics and the Imagination by Kasner and Newman

The first and third are bona fide classics; the second hasn't been around more than a few years but it is wonderful.

All 3 books contain real, significant mathematical content, but What is Mathematics? and The Heart of Mathematics have the advantage that they contain exercises and problems, i.e. will actually get you doing math, not just reading about it, so they may be the better choice. My recommendation is you have a look at these two books online or in the bookstore, and buy one of them, and start working out of it.

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Thanks a bundle, Ben! This is, like, exactly what I needed to hear! I really appreciate the thorough response! –  Smovies Sep 13 '11 at 21:32
    
No sweat! Have a great time! –  Ben Blum-Smith Jan 13 '12 at 5:09
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Given that you have the freedom to choose your own path, you might as well study what you find interesting. Geometry is certainly a good place to start, if that interests you.

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I understand that, for sure. The problem is that it all interests me to the point that trying to figure out a starting place is too overwhelming. That's why I'm trying to think of it in terms of what would be the most practical. –  Smovies Sep 13 '11 at 20:27
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