How to Read a Book: The Classic Guide to Intelligent Reading (34 page)

It is attained by frank confession of it.

The leading terms in a scientific work are usually expressed by uncommon or technical words. They are relatively easy to spot, and through them you can readily grasp the propositions. The main propositions are always general ones.

Science thus is not chronotopic. Just the opposite; a scientist, unlike a historian, tries to get away from locality in time and place. He tries to say how things are generally, how things generally behave.

There are likely to be two main difficulties in reading a scientific book. One is with respect to the arguments. Science is primarily inductive; that is, its primary arguments are those that establish a general proposition by reference to observable evidence-a single case created by an experiment, or a vast array of cases collected by patient investigation. There are other arguments, of the sort that are called deductive. These are arguments in which a proposition is proved by other propositions already somehow established. So far as proof is concerned, science does not differ much from philosophy. But the inductive argument is characteristic of science.

This first difficulty arises because, in order to understand the inductive arguments in a scientific book, you must be able to follow the evidence that the scientist reports as their basis.

Unfortunately, this is not always possible with nothing but the book in hand. If the book itself fails to enlighten him, the reader has only one recourse, which is to get the necessary special experience for himself at first hand. He may have to witness a laboratory demonstration. He may have to look at and handle pieces of apparatus similar to those referred to in the book. He may have to go to a museum and observe specimens or models.

Anyone who desires to acquire an understanding of the history of science must not only read the classical texts, but must also become acquainted, through direct experience, with the crucial experiments in that history. There are classical experiments as well as classical books. The scientific classics become more intelligible to those who have seen with their own eyes and done with their own hands what a great scientist describes as the procedure by which he reached his insights.

This does not mean that you cannot make a start without going through all the steps described. Take a book like Lavoisier's Elements of Chemistry, for instance. Published in 1789, the work is no longer considered to be useful as a textbook in chemistry, and indeed a student would be unwise to study it for the purpose of passing even a high school examination in the subject. Nevertheless, its method was revolutionary at the time, and its conception of a chemical element is still, on the whole, the one that we have in modern times. Now the point is that you do not have to read the book through, and in detail, to receive these impressions of it. The Preface, for example, with its emphasis on the importance of method in science, is enlightening. "Every branch of physical science," wrote Lavoisier, must consist of three things: the series of facts which are the objects of the science, the ideas which represent these facts, and the words by which these facts are expressed. . . . And, as ideas are preserved and communicated by means of words, it necessarily follows that we cannot improve the language of any science without at the same time improving the science itself; neither can we, on the other hand, improve a science without improving the language or nomenclature which belongs to it.

This was exactly what Lavoisier did. He improved chemistry by improving its language, just as Newton, a century before, had improved physics by systematizing and ordering its language-in the process, as you may recall, developing the differential and integral calculus.

Mention of the calculus leads us to consider the second main difficulty in reading scientific books. And that is the problem of mathematics.

Facing the Problem of Mathematics

Many people are frightened of mathematics and think they cannot read it at all. No one is quite sure why this is so.

Some psychologists think there is such a thing as "symbol blindness"-the inability to set aside one's dependence on the concrete and to follow the controlled shifting of symbols. There may be something to this, except, of course, that words shift, too, and their shifts, being more or less uncontrolled, are perhaps even more difficult to follow. Others believe that the trouble lies in the teaching of mathematics. If so, we can be gratified that much recent research has been devoted to the question of how to teach it better.

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.

As we have already observed, learning a new written language always involves us in problems of elementary reading.

When we underwent our initial reading instruction in elementary school, our problem was to learn to recognize certain arbitrary symbols when they appeared on a page, and to memorize certain relations among these symbols. Even the best readers continue to read, at least occasionally, at the elementary level: for example, whenever we come upon a word that we do not know and have to look up in the dictionary. If we are puzzled by the syntax of a sentence, we are also working at the elementary level. Only when we have solved these problems can we go on to read at higher levels.

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.

Any language is a medium of communication among men on subjects that the communicants can mutually comprehend.

The subjects of ordinary discourse are mainly emotional facts and relations. Such subjects are not entirely comprehensible by any two different persons. But two different persons can comprehend a third thing that is outside of and emotionally separated from both of them, such as an electrical circuit, an isosceles triangle, or a syllogism. It is mainly when we invest these things with emotional connotations that we have trouble understanding them. Mathematics allows us to avoid this.

There are no emotional connotations of mathematical terms, propositions, and equations when these are properly used.

We are also not told, at least not early enough, how beautiful and how intellectually satisfying mathematics can be. It is probably not too late for anyone to see this if he will go to a little trouble. You might start with Euclid, whose Elements of Geometry is one of the most lucid and beautiful works of any kind that has ever been written.

Let us consider, for example, the first five propositions in Book I of the Elements. (If a copy of the book is available, you should look at it.) Propositions in elementary geometry are of two kinds: (1) the statement of problems in the construction of figures, and (2) theorems about the relations between figures or their parts. Construction problems require that something be done, theorems require that something be proved. At the end of a Euclidean construction problem, you will find the letters Q.E.F., which stand for Quod erat faciendum, "(Being) what it was required to do." At the end of a Euclidean theorem, you will find the letters Q.E.D., which stand for Quod erat demonstrandum, " (Being) what it was required to prove."

The first three propositions in Book I of the Elements are all problems of construction. Why is this? One answer is that the constructions are needed in the proofs of the theorems.

This is not apparent in the first four propositions, but we can see it in the fifth proposition, which is a theorem. It states that in an isosceles triangle (a triangle with two equal sides) the base angles are equal. This involves the use of Proposition 3, for a shorter line is cut off from a longer line. Since Proposition 3, in turn, depends on the use of the construction in Proposition 2, while Proposition 2 involves Proposition 1, we see that these three constructions are needed for the sake of Proposition 5.

Constructions can also be interpreted as serving another purpose. They bear an obvious similarity to postulates; both constructions and postulates assert that geometrical operations can be performed. In the case of the postulates, the possibility is assumed; in the case of the propositions, it is proved. The proof, of course, involves the use of the postulates. Thus, we might wonder, for example, whether there is really any such thing as an equilateral triangle, which is defined in Definition 20. Without troubling ourselves here about the thorny question of the existence of mathematical objects, we can at least see that Proposition 1 shows that, from the assumption that there are such things as straight lines and circles, it follows that there are such things as equilateral triangles.

Let us return to Proposition 5, the theorem about the equality of the base angles of an isosceles triangle. When the conclusion has been reached, in a series of steps involving reference to previous propositions and to the postulates, the proposition has been proved. It has then been shown that if something is true (namely, the hypothesis that we have an isosceles triangle) , and if some additional things are valid (the definitions, postulates, and prior propositions) , then something else is also true, namely, the conclusion. The proposition asserts this if-then relationship. It does not assert the truth of the hypothesis, nor does it assert the truth of the conclusion, except when the hypothesis is true. Nor is this connection between hypothesis and conclusion seen to be true until the proposition is proved. It is precisely the truth of this connection that is proved, and nothing else.

Is it an exaggeration to say that this is beautiful? We do not think so. What we have here is a really logical exposition of a really limited problem. There is something very attractive about both the clarity of the exposition and the limited nature of the problem. Ordinary discourse, even very good philosophical discourse, finds it difficult to limit its problems in this way. And the use of logic in the case of philosophical problems is hardly ever as clear as this.

Consider the difference between the argument of Proposition 5, as outlined here, and even the simplest of syllogisms, such as the following:

All animals are mortal;

All men are animals;

Therefore, all men are mortal.

There is something satisfying about that, too. We can treat it as though it were a piece of mathematical reasoning. Assuming that there are such things as animals and men, and that animals are mortal, then the conclusion follows with the same certainty as the one about the angles of the triangle. But the trouble is that there really are animals and men; we are assuming something about real things, something that may or may not be true. We have to examine our assumptions in a way that we do not have to do in mathematics. Euclid's proposition does not suffer from this. It does not really matter to him whether there are such things as isosceles triangles. If there are, he is saying, and if they are defined in such and such a way, then it follows absolutely that their base angles are equal. There can be no doubt about this whatever-now and forever.

Handling the Mathematics in Scientific Books

This digression on Euclid has led us a little out of our way. 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. Having done some work on Euclid, you might proceed to glance at the works of other classical Greek mathematicians-Archimedes, Apollonius, Nicomachus. They are not really very difficult, and besides, you can skip.

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 an 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.

Newton's Optics is another scientific classic that you might want to try to read. There is actually very little mathematics in it, although at first glance that does not appear to be · so because the pages are sprinkled with diagrams. But these diagrams are merely illustrations describing Newton's experiments with holes for the sun to shine through into a dark room, with prisms to intercept the sunbeam, and with pieces of white paper placed so that the various colors of the beam can shine on them. You can quite easily repeat some of these experiments yourself, and this is fun to do, for the colors are beautiful, and the descriptions are eminently clear. You will want to read, in addition to the descriptions of the experiments, the statements of the various theorems or propositions, and the discussions that occur at the end of each of the three Books, where Newton sums up his discoveries and suggests their consequences. The end of Book Ill is famous, for it contains some statements by Newton about the scientific enterprise itself that are well worth reading.