Now we’re talking! I found the second installment of the program much more satisfying than the first.
True, a lot of the mathematical ideas are implied rather than stated, and several times I was left with the feeling that the program stopped just short of something really interesting, but there’s a lot of good here. If something in this program doesn’t get your mind going, then you and I probably aren’t going to get along.
The program made much of the distinction between rational and irrational numbers, and the connotations of the latter term. I wish it had made a little more explicit this connection — numbers are called rational because they are ratios.
Still, the connection between numbers that don’t have the highly-special form that the Pythagoreans hoped all numbers had and madness is more than a coincidence. There has always been a (more than a little closed-minded) tendency to deprecate new types of numbers (new in the sense of “more general than had previously been understoof”) by an association with madness. Thus the distinction between real and imaginary numbers, say, terms born of the tumultuous time when the complex number system was extremely controversial.
I cannot resist mentioning in this context the surd field. This set is made up of all the real numbers you can make out of integers, addition, subtraction, multiplication, division, and square roots (but not other kinds of roots). Numbers in this field are sometimes called surds. These numbers are important because, if you start with a line segment and take that as your unit of length, then you can use a compass and straightedge to make segments with any surd length, but lengths that are not surds. The reason for this term? Just as a set-up line for calling the lengths you can’t construct absurd. *wince* Nowadays, this field is more sensibly called the field of constructible numbers, but it’s still a good trivia fact for days when you really want to bust out a silly-sounding word in the middle of a serious mathematical conversation.
(Amazing how often I find myself in precisely that situation.)
The brief mention of the distinction between a discrete view of numbers (what you mean by “number” when you’re counting your children) and a continuous view of numbers (what you mean by “number” when you’re weighing your children) is actually the tip of a very large iceberg. Both senses of number are important to modern mathematics. Analytic number theory, where much of my own mathematical research lives, is built on unexpected interactions between the discrete and continuous worlds.
The story of the man who used numbers, date, and the search for pattern and meaning as a survival mechanism in captivity is a fascinating one in many ways, and the following insights about the human hunger for patterns in number are valuable and well-stated. This search for patterns is a large part of the heart of mathematics. Just as mathematics is about much that does not have number-nature, the human impulse to study patterns and networks of ideas does not apply only to numbers. Much of geometry is highly non-numeric, as is symbolic algebra, and the kinship algebra mentioned in the program (and isn’t that interesting?).
If spending time calculating combinations of types of kinship relations seems arbitrary and dull to you, consider sudoku. I think the sudoku craze is another manifestation of this human impulse. (Sudoku seem to be about numbesr, of course, they are often called number puzzles, but a moment’s thought reveals that any symbols would be just as good. It really burns me up when a sudoku magazine, in an attempt to seem non-threatening, describes suduko as a number puzzle, but don’t worry, it doesn’t require any math! This is wrong on both counts; sudoku is one of the most deeply mathematical puzzle tpyes I know of, but it’s not really a number puzzle.)
And my heart goes out to anyone who was as inspired by Benoit Mandelbrot as the creator of this program evidently was. It is impossible to deny the role that Mandelbrot and his discoveries played in bringing interesting and beautiful mathematics into the public eye. I know I was. In French class in high school, I adopted the nom Benoit. I remember how exciting it was for me, at a conference a few years back, to meet and speak with Dr. Mandelbrot.
Perhaps my favorite part of the program (in large part because it is an anecdote which I hadn’t heard) was Mandelbrot’s story of the difference between computing at Harvard and computing at IBM. In a nutshell, computing at Harvard was sometimes more illuminating because the computers were slower — it was actually possible for a human . I can relate to this story. There is often more going on in a computation than the final answer. I can think of many times in my own mathematical life when I have used a computer to get answers to 5000 instances of a problem, but it was only after I did 5 or 6 by hand that I had the mathematical insight I had been looking for.
The final installment will air next Wednesday and discuss connections between numbers and music, architecture, and space. I’m looking forward to it.
One of the mathematicians interviewed in the program uses the phrase “well-ordered” to describe the set of counting numbers. This, it turns out, is a technical term, though no mention of what it actually means is made in the program. I’ll be discussing it in an upcoming article though , so I couldn’t resist bringing some attention to it.