An essay on four dimensions
I was asked about how a mathematician understands multidimensional geometry. I decided to write my answer as a short essay.
Some links to resources related to math and education.

Two sites involved in providing free preuniversity textbooks:

I haven't yet examined either of these sites in detail, so I can only provide links; I can't recommend them based on content.

And of course there's WikibookS . That's probably the main site for free textbooks. I've found a few incomplete textbooks of some interest there a few years ago; they've probably been improved some since then.

I was asked for suggestions for serious reading about mathematics slightly above he high-school level. I'll probably add to this topic as time goes on. The list will be biased towards books I enjoyed long, long ago.
One, Two, Three, Infinity

I read this one in the 50's, and many times thereafter.

One Two Three Infinity covers set theory and some math, relativity, quantum nmechanics, genetics. This is preDNA stuff. They had figured out about genes, but hadn't got DNA yet. The cardinal number stuff in the math is both generally accepted and controversial. Generally accepted in the sense that most mathematicians have heard of this stuff, and more-or-less accept the way it's presented. Controversial, because the real experts disagree on a lot of it. Mostly not on what it is, and whether it's properly derived from the axioms of set theory, but on whether it shows that those axioms are deeply flawed. And you don't need a lot of that controversial stuff to do day-to-day mathematics, anyway.

Mathematician's Delight

One book I loved back in the 60's was Mathematician's Delight, by W.W. Sawyer. It didn't really teach much mathematics proper, but each chapter explained just what was the point about an area of mathematics. So somewhere in the middle there was a chapter on calculus, and it would explain what differentiation and integration were, but it wouldn't bother explaining how you actually go about the calculations.

This kind of thing was invaluable when I you finally got around to taking formal courses in the stuff. I already knew where the subject was heading, and what it was good for, before I was shown the technical details.


If you just want to find out what the calculus is about, read the aforementioned Mathematician's Delight by W.W. Sawyer's. It has a chapter on calculus. It's aimed at teachers, but nontheless it was a delight to read when I was in high school. It doesn't explain the details of the mathematics, the formulas, the symbolic manipulations on paper. It explains the point of the mathematics, what it's about. When I got around to learning the mechanics of differentiation and integration, I already understood what they were all about. That was an enormous help. Now for books that really teach calculus, with all the details.

One of the best books on traditional calculus I've actually read a lot of is Tom Apostol's Mathematical Analysis. But it didn't fare well on the market. It was too advanced for the elementary classes, and too elementary for advanced classes. So he wrote a new two-volume text, with the inspired titles Calculus I and Calculus II. These are better for the beginner. And they have wonderful problem sets.

And for a decidedly different take on the whole thing, which is possibly more accessible to beginners than Apostol's books, get a copy of Gabriel Stolzenberg's book on calculus. Be warned: this not exactly the same calculus you are llilkely to encounter in today's undergraduate Calculus courses (though I think this should change). It's a constructivist approach to calculus (in the mathematician's sense of constructive, not the educator's). This means, more or less, that it deals exclusively with entities that can actually be computed. It uses slightly different terminology from mainstream mathematics, and uses the same terminology in different ways. It's just as good for practical use, but the theory behind it is quite different.

Gabriel Stolzenbrg maintains that constructivism, and the corresponding subtle changes in terminology, makes the subject more accessible to students. I don't have any experience learning from the book, or teaching from it, but I have looked through a .pdf of a draft, and it looks solid.

See constructive mathematics for a short summary of what constructive mathematics is about, and how it differs from mainstream mathematics.