The Measure of Reality

At an alumni event I encountered a fellow member of my class of Reed College ‘06, Steven Shapin, who is now Franklin L. Ford Professor of the History of Science at Harvard University. After mentioning some books about the history of that had deeply affected me, I asked him to recommend some books on the subject that I might enjoy. He gave me a short reading list, and I will be reviewing these books as I get to them.

I find books about the history of science and particularly about the history of mathematics to be helpful to my teaching. The nineteenth century philosopher Herbert Spencer claimed that “If there be an order in which the human race has mastered its various kinds of knowledge, there will arise in every child an aptitude to acquire these kinds of knowledge in the same order.... Education is a repetition of civilization in little.” (Wikipedia) While I doubt that this is literally true, I have found that examining the long halting development of mathematical ideas by different cultures helps me understand the difficulty that students have in mastering them.

The first book from Steven Shapin’s list is The Measure of Reality: Quantification and Western Society, 1250 – 1600 by Alfred W. Crosby. This is one of those “big picture” books, like Guns, Germs and Steel that attempts (rather successfully) to explain the success of an entire civilization over a period of centuries.

The books’ argument is well described on the first page:

Western Europeans were among the first, if not the first, to invent mechanical clocks, geometrically precise maps, double-entry bookkeeping, exact algebraic and musical notations, and perspective painting. By the sixteenth century more people were thinking quantitatively in Western Europe than in any other part of the world. Thus, they became world leaders in science, technology, armaments, navigation, business practice, and bureaucracy, and created many of the greatest masterpieces of Western music and painting. [Emphasis added.]

The thesis of the book is contained in the word “thus”. That is, Crosby believes that the success of Western imperialism is due to the development of quantification. It would be pointless to argue whether this is true or not. Clearly, evidence can be found to either support or refute a thesis that is this broad. But what is fascinating to me are the examples that Crosby introduces, including a description of some of the high points of late medieval and renaissance culture: polyphony in music, perspective in art, the beginnings of modern physics and mathematics. I even gained an appreciation of the role of double-entry bookkeeping in enabling complex business arrangements.

What I found most interesting is trying to imagine the mentalité, or mind-set, of pre-quantitative people. How does one experience time, when one has never seen a clock? How does one picture a scene when viewing a picture of it that does not obey modern rules of perspective?

Despite its big picture, I found the most endearing feature of this book some of the details. For example, I have never realized that the invention of the staff to write music in the fourteenth century prefigured the Cartesian coordinate system. Note that the staff is a graph in which time is the horizontal axis and pitch the vertical. One wonders why it took centuries for the mathematicians to catch up.

Persistence in Solving Math Problems

I’m currently teaching a course to a middle-school math teacher in Teaching Mathematics through Problem Solving. I’ve taught this course a number of times before. One of the things I do is to ask students to work on problems which I think they will find difficult, but doable. My students – all teachers themselves – become frustrated if they encounter a problem that takes them more than a few minutes. They are so used to routine problems that will yield to a known method of attack, that they don’t know what they are capable of.

I think it may be sometimes better to assign one difficult problem rather than 10 routine ones. And we need to get students to commit to trying to solve a problem even if they have to put it away for a while, let it percolate in their unconscious, and come back to it later. At the risk of sounding like an old fogy, students today are very much used to expecting instant gratification. We need to teach them the rewards of persistence.

Math Humor

Here are a few math education jokes I heard recently. Two of them come from my Tai Chi teacher. I think they all contain a bit of wisdom as well as humor.

The teacher draws a right triangle on the board, labels the legs 3 and 4 and the hypotenuse x, and asks the student, “Can you find x?” The student rushes up to the board and pointing, says “Here it is!”

The teacher writes (a + b)² on the board, and asks if anyone can expand it. A student comes up and writes
( a____+____b )².

The teacher writes an equation on the board, and says “Suppose x is the solution to this equation” and starts to do some algebraic manipulation. A student waves his hand wildly. When the teacher calls on him, he says “But sir, suppose it isn’t?”

Follow up to problem E9 - Composite values of integer polynomials

I’ve been doing some more thinking about Problem E9, which asks the reader to prove that the range of a quadratic polynomial evaluated on the integers must include a composite number. In addition to the algebraic proof I linked to the original problem, my brother provided a proof based on congruences. I think both are interesting.

Each proof applies to all non-trivial polynomials, not just quadratics, and each actually shows (or can be extended to show) that the range of the polynomial must include an infinite number of composite numbers.

One question to which I do not know the answer is whether the range of a polynomial must contain an infinite number of prime numbers. Of course, the answer is “no” if the polynomial factors over the integers. If the polynomial is prime, however, I suspect the answer is “yes”.

For linear polynomials, the prime or composite nature of the ranges have been well studied. In 1837, Dirichlet proved that every sequence (an + b) contains an infinite number of prime numbers, iff (a, b) = 1, and moreover the fraction of all primes ≤ x that are in such a sequence approaches 1/φ(a) as x approaches infinity, where φ(a) is the number of natural numbers less than a that are relatively prime to a. As a corollary, this also shows that there are an infinite number of composite numbers in the range.

Dirichlet’s Theorem is famous for being the first that used complex analysis to solve a major problem in number theory. Recently (2004), Green and Tao (in an important and difficult paper) proved that there exist arithmetic sequences of arbitrary length that are all prime. The proof is non-constructive.

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Thoughts from Oliver Selfridge Memorial

I attended the memorial service for Oliver Selfridge in Cambridge yesterday (March 15). There were lots of people there from the AI community, as well as friends who knew him from his interests in education, madrigal singing, gardening, skiing, sailing, poetry writing, and a few others. He was considerably more than a dilettante in all of these fields, and was know for his prodigious memory, his sense of wonder, and his desire to know what made the world work.

One of the speakers, Marvin Minsky, said that when meeting an incredible intellect like Selfridge, von Neumann, or Nash, he (Minsky) always concentrated not just on what the person said, but on trying to figure out how they were able to arrive at it. He mentioned that Oliver had an uncanny sense of direction, so that (for example) he was able to determine which way was North when emerging from underground after a complex subway trip. Minsky finally realized that Oliver was frequently checking the position of the sun in the sky, and the directions of the shadows. It turned out that he was doing this unconsciously.

Someone mentioned also Oliver's belief in the primacy of learning. He was quoted as having said (approximate quote): "A mind without learning is scarcely a mind at all", and it was this belief which informed his researches in AI.