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This is the hardest of the three because it is about the beauty of abstraction. It's difficult to pin down that allure with words. What is intrinsically appealing is that it doesn't need words, that it eludes simple description, even though it is so simple.

Math was always fun. It was not secret, like reading and spelling obviously were; all of the rules were easy to understand. There were no exceptions. No silent e's or g's. No doubled consonants, long or short vowels. Of course, what they called "math" was really arithmetic. Math came later and was a lot more than just fun.

Arithmetic was the simplest rules imaginable. In a day you could understand nearly all of it. Once you learned to add numbers of three or four places you could add the largest numbers you could imagine. The understanding was entirely extensible across the other operations. I remember in sixth grade racing with two friends to see whether we could finish the extra credit long division homework before the class ended, so we could hand it in without even taking it home.

That's great, and the teachers spend a few years making sure that you are comfortable traversing the landscape of large numbers, fractions, decimals and negative integers. Nothing breathtaking, but it was necessary to develop a certain familiarity with the questions and their answers.

At first, word problems are similarly simple. Integers being added or multiplied. But to me they were so banal, just pieces fitted together to make the mute answer with the units tacked on: Sally has six dollars left. Rudyard walked eleven miles to school. Anne will be on fire for forty-five minutes.

Then I was introduced to x. First, x made it possible to look at more complex word problems. That was great. Sally had twice as much money as she did when she started, and it is the same number of quarters as she used to have dimes. Well, that was complicated enough to be interesting. Now we had a conversational place holder, a way to easily discuss something which we did not yet have a value for.

Symbols which could represent any number were great. Locking them into relationships with one another until they represented a definite number was even better. It had the pleasant feel of a polished piece of wood sliding into a Chinese puzzle. But we put aside these interesting pieces and took out some graph paper. It was nice to draw in math class. It became a calm spot in the chaos of my seventh grade days, a time to sit down and take out my fine point markers, my compass, and my clear, gridded ruler.

The first day the teacher wrote a simple equation on the board and drew one line on the permanent graph etched up there: x = 2. There were other things that day, that week, but that was the best of them. That contained so much of what was great about symbology, abstraction and the visual representation of a small piece of truth. The line was an idea, not a line. The line as drawn had a width, but the truth is that the line was conceptual. Everywhere on the line, x was equal to 2. It stretched off the page with arrows at each end and it went on forever. It was infinite, and the area to either side, where x did not equal two, was also infinite. It was a ball bearing which rolled along in a groove in the wood floor, pleasingly steadily supported and guided, it hummed with certainty.

I read Stephen Hawking's book "A Brief History of Time." In the preface he mentions having to include a few formulae, and that a colleague had warned him that every time he included a formula he would lose half his audience. That seemed so sad to me, because what could be more interesting and beautiful than an equation which explains some part of the random, natural universe? The first equations I saw weren't tied to anything concrete, they were just mathematical doodles and were still beautiful:

x = 2y

That was a steep line. Too steep a roof to walk up. But so fascinating. And we started to draw more than one equation at a time so we could look at the intersections:

x = 2y
y = 5

It wasn't difficult to figure that x was 10, but you could draw the two lines and see the answer as the pulsating point where the two lines met. Maybe the pulsating was only happening in my head, but it was there. And a pair of equations which had two possible solutions would graph as two lines (at least one of them a curve) which met at two different points. The truth, drawn on a cartesian plane.

It was great, and it got better and better as the equations became more complex. Trigonometry offered the curves of sine and cosine waves. The next course of geometry delivered parabolas and the quadratic equation. And then calculus, a world of beauty all its own, where complex questions can be reduced to simple answers. Oddly recursive questions like, "A snake is eating his tail, when will he feel full?" worked out to very satisfying answers like, "In five minutes." (In truth, the most fascinating question I remember answering in calculus was about high rise buildings. If it cost two thousand dollars per floor to build, plus another thousand dollars every floor higher, and given the rent per floor collected, what was the ideal height to build? Calculus delivered the answer so easily while trial and error would have taken so long and, really, would never be certain the way the calculus was.)

Eventually, though, it was clear that math was static. That is, the answers were there, and you just needed to find them. You peered down the narrow tube of

4(ab) + c = xy

and the answer was there, a bright spark where the tube was pierced by

x = y^2 + (ac)/b

There was some drama in the revelation, but as I began to reach the limits of what I was able to grasp (somewhere beyond trigonometric identities and the simple parts of topology) my continuing curiosity was not enough to animate the subject matter for me. It's difficult to bring to life that which you do not fully understand.

(As an aside, I did sit through a thesis presentation for a Bachelors in Mathematics. The discussion was about the Hairy Ball Theorem, which posits that if you have a sphere covered with hairs evenly spaced over its surface it is impossible to comb them all in a single direction without leaving one standing straight up. Puzzle over that for a moment or two.)

Fortunately, computers came along at the right time for me. A generation earlier math would have slipped away from me, unused and atrophied over several years, the beauty of abstraction would be lost to me. But starting in seventh grade there were computers in every school I attended, available with different degrees of effort, but they were there. I still remember typing

if x > 80 then vx = -vx

in 1978. That meant that if the little blip I was steering reached the right edge of the screen it should bounce off of it, changing direction but not losing any of its velocity. That was written in BASIC on a TRS-80, a very simple little computer from Radio Shack. Later it was referred to as a Trash-80. They had very limited memory and the only way to store the programs you wrote was to save them to a cassette tape. I rarely did that, I just typed them in again. Simple games where you tried to steer your block over to touch a blinking block. Or you ran from the blinking block. One cold winter afternoon I stood in the Radio Shack at twenty-third and seventh avenue and typed in a hundred or so lines so that my brother and sister could play my latest little game.

The computer was a compatriot. Someone I could explain the pure abstraction to and who would agree, expound, and draw what I meant. It was fantastic. With a few lines

x = 7
for y = 1 to 20 ; gset x,y ; next y

the computer drew the same plot I had on my page. It was not infinite, but in some ways the areas where it insisted on being discrete made it even more satisfying. It was an exploration of the abstraction, a peering into a virtual world with a concrete display of what was in there.

For the next decade I would struggle, torn between the easy construction of truth in the virtual world and the bloody, sweaty work of construction in the real world. Ultimately, I found a way for one to help with the other, and in the times that the frustration of physical building was highest I was able to fall back to construction things out of assertions, little lines of code and abstraction which illuminates truth with almost as much satisfaction as a wall reaching for the sky.