FACING THE PROBLEM OF MATHEMATICS
Many people are frightened of mathematics and think they cannot read it at all. … The problem is partly this. WE are not told, or not told early enough so that it sinks in, that mathematics is a language, and that we can learn it like any other, including our own. We have to learn our own language twice, first when we learn to speak it, second when we learn to read it. Fortunately, mathematics has to be learned only once, since it is almost wholly a written language.
Since mathematics is a language, it has its own vocabulary, grammar and syntax, and these have to be learned by the beginning reader. Certain symbols and relationships between symbols have to be memorized. The problem is different, because the language is different, but it is no more difficult, theoretically, than learning to read English or French or German. At the elementary level, in fact, it may even be easier. … There are no emotional connotations of mathematical terms, propositions, and equations when these are properly used.
HANDLING THE MATHEMATICS IN SCIENTIFIC BOOKS
We were observing that the presence of mathematics in scientific books is one of the main obstacles to reading them. There are a couple of things to say about that.
First, you can probably read at least elementary mathematics better than you think. We have already suggested that you should begin with Euclid, and we are confident that if you spent several evenings with the Elements you would overcome much of your fear of the subject.
That leads to the second point we want to make. If your intention is to read a mathematical book in and for itself, you must read it, of course, from beginning to end – and with a pencil in your hand, for writing in the margins and even on a scratch pad is more necessary here than in the case of any other kinds of books. But your intention may not be that, but instead to read a scientific work that has mathematics in it. In this case, skipping is often the better part of valor.
Take Newton’s Principia for example. The book contains many propositions, both construction problems and theorems, but it is not necessary to read all of them in detail, especially the first time through. Read the statement of the proposition, and glance down the proof to get an idea of how it is done; read the statements of the so-called lemmas and corollaries; and read the so-called scholiums, which are essentially discussions of the relations between propositions and of their relations to the work as a whole. You will begin to see that whole if you do this, and so to discover how the system that Newton is constructing is built – what comes first and what second, and how the parts fit together. Go through the whole work in this way, avoiding the diagrams if they trouble you (as they do many readers), merely glancing at much of the interstitial matter, but being sure to find and read the passages where Newton is making his main points. One of these comes at the very end of the work, at the close of Book III, which is titled “The System of the World.” This General Scholium, as Newton called it, not only sums up what has gone before but also states the great problem of almost all subsequent physics.
Keep in mind that your primary obligation is not to become competent in the subject matter but instead to understand the problem.
A NOTE ON POPULAR SCIENCE
In a sense, there is little more to say about reading scientific popularizations. By definition, these are words – either books or articles – written for a wide audience, not just for specialists. … they generally skirt or ovoid the two many problems that confront the reader of an original contribution in science. First, they contain relatively few descriptions of experiments (instead, they merely report the results of the experiments). Second, they contain relatively little mathematics (unless they are popular books about mathematics itself.)
Of course, these publications, no matter how good they are or how carefully and responsibly edited, pose the problem that was discussed at the end of the last chapter. In reading them, we are at the mercy of reporters who filter the information for us. If they are good reporters, we are fortunate. If they are not, we have almost no recourse.
Scientific popularizations are never easy reading in the sense that a story is or seems to be. … You cannot read it for understanding without keeping your mind awake. Thus, the requirement that you read actively is more important here than almost anywhere else. Identify the subject matter. Discover the relation between the whole and its parts. Come to terms and plot the propositions and arguments. Work at achieving understanding before you begin to criticize or to assess significance. These rules, by now, are all familiar. But they apply here with particular force.