ACCORDING to tradition that strange mystic, philosopher, and mathematician, Pythagoras, spent over a quarter of a century in study and travel in Egypt before founding his own great Brotherhood and school at Croton, a Greek colony in Southern Italy. With his peculiar brand of mysticism we have no concern here, except to recall that it was a particularly wild one based on numbers, and that it infected most of his thinking. But in spite of his flights into the clouds of pure verbalism, Pythagoras did three things of the first magnitude, any one of which is probably sufficient to ensure his remembrance for as long as human beings can remember anything. These were: the first definitely recorded physical experiment in history; the invention of irrational numbers, on which. the whole vast structure of modern mathematical analysis rests; and the first definitely recorded insistence upon proof for statements about numbers and geometrical figures. It is the last only of these which we need to discuss in our pursuit of "truth" through the mazes of deductive reasoning. Pythagoras, dates (doubtful) are B.C. 569–496.

Let us go back for a moment to Egypt where, Pythagoras tells us, he learned much from the wise priests.

We saw that the Egyptian who found a consistently usable formula for the volume of a pyramid either used abstract reasoning subconsciously or was such a phenomenally good guesser that he needed no reasoning. It was suggested that civilized human beings sooner or later must resort to abstract reasoning if their civilization is not to slip backward.

To bring out the point about the "common agreement" which we spoke of in connection with land surveying, let us restate an extremely simple problem in that practical science in a form which would have appealed to a Greek, in particular to Pythagoras.

Here is the problem: how many square yards are there in a rectangular field which is 100 yards long and 50 yards broad? This is not quite hard enough for Pythagoras, so we generalize it: how many square yards are there in a field which is L yards long and B yards broad, where L and B are *any* numbers?

If you answer 5000 to the first problem, you are right. That answer leads to consistency with other problems of the same kind. But this is not enough for Pythagoras. "Prove it," he demands, just like one rude little boy to another who has used an offensive epithet. That seems easy: "Oh, you get the area of a rectangle by multiplying the length by the breadth—." Pythagoras interrupts: "Prove it." A very stupid person would multiply 100 by 50 and proudly exhibit the correct answer 5000.

We have all done such things when we were at school and got stuck in arithmetic; we looked up the answer at the end of the book. But Pythagoras is not interested in the answer; what he wants to know is *how do you know that it is right when you get it?*

When we understand what he is driving at we make a fussy attempt to recall what we were taught, get all hot and bothered, and finally fling back at him the Baconian answer: "Go and measure your beastly field. Cut it up with stretched strings or vermicelli or anything you like into square yards and count them." "Do that for B and L," Pythagoras grins, and possibly you tell him to go to L himself, for he has caught you in a trap from which you cannot escape by measurement, that is, by *experiment*, no matter if you have a million years to try. So here is a strikingly simple problem completely beyond the reach of the "operational" method. The element of *generality*, or *universality*
in the "any" of the problem puts it in a realm which is inaccessible to concrete experiment.

Pythagoras relents. Seeing that *any* B and *any* L are too hard, he draws a simple figure in the sand: a square with one of its diagonals. By an easy construction he then makes a rectangle with length equal to the diagonal of the square and breadth equal to one side of the square. "Measure it," he says, handing you a thin thread. "If the thread is too coarse, you may use a spider's string. The breadth of that rectangle is 12 inches. You measure the area and tell me how many square inches there are in the rectangle. When you give me the right answer, I'll give you all the gold in Greece. But if you give me a wrong answer, I'll have you sewed up in a sack and pitched off that cliff at high tide. Something like that happened to one of our Brotherhood only last Monday."

Of course nobody would accept such odds as that from a wild-eyed mystic like Pythagoras. He might sew the gambler up anyway, win or lose.

Men who cannot give Pythagoras the answer to his childishly simple-looking problem can nevertheless tell us how long the universe will last, when time began and when it will end, and what God has in store for us to pass away the ages through eternity. When we thoroughly understand what Pythagoras was talking about, and when we see into and through the machinery invented by Pythagoras and his immediate successors for disposing of his problem, we shall be able to laugh in the prophets' faces. His problem is simply to find the area of that rectangle he drew on the sand by measurement, that is, by experiment.

It cannot be done, and Pythagoras knew that it could not. The fact that the numerical measure of the diagonal of the square is "irrational" (not obtainable by dividing one whole number by another) is the disconcerting fact which caused Pythagoras to abandon his sublime extrapolation that nature and (possibly) reason are based on the simple pattern of the whole numbers 1, 2, 3,.... The "universe of geometrical lines" is, in this sense, *not* "rational;" "irrational" lengths can be humanly constructed.

But common sense tells us that the rectangle does have a definite area; we can see it with our eyes, if not with our minds. A rough approximation is 1.4 square feet, a little better 1.41, and so on, indefinitely, that is, interminably. There is no end to the process. And if there is no end, is it likely that we shall be able to And an exact answer by experiment? In this simple problem we have again stumbled across the infinite. Can any performable experiment continue without end? It begins to look as if that "common agreement" desired by all sane farmers on the way of measuring a field is less simple than it seemed. If one man says that the correct answer is 1.41, and another 1.412,
how is it to be decided which, if either, is right, or which has a closer approximation to the right answer *if there is one*? How do we know that there is a "right" answer to the problem? It is fairly obvious that we do not, until we agree upon some set of *conventions*.

Now, all this is as old as the hills to anyone who has ever been through a carefully presented course in elementary school geometry. But mere familiarity is not enough to prevent some devout believer in an abstract and eternal "truth," over and above our human conventions by which we reach agreements in geometry, as in everything else, from appealing to this "everlasting truth" for the right answer. "*There must be one answer which is right, and we can find that one because it is true*."

They can call upon "truth" till they are out of breath and blue in the face; they will get no answer. That "still, small voice" which they expect to hear has nothing whatever to say about the area of a rectangle. This may be hard doctrine, but unless our habits were completely perverted before we were seventeen by traditional teaching we shall see as we go on that it is saner doctrine than the other. There is nothing new in this, and most of us have known it ever since we began thinking at all, however much some of us may have rebelled when we first realized it.

Pythagoras insisted upon *proof*. Although we cannot say what "truth" is in Pilate's sense, simply because it has no meaning, we can say with rigid exactness what proof is in a deductive system of reasoning. That kind of system is the one which is relevant for the problem of the rectangle.

First, we lay down certain outright *assumptions* which we agree to accept without further argument. These are called our *postulates* (sometimes *axioms*). For example in geometry one postulate is: "The whole is greater than any one of its parts." The postulates agreed upon may or may not have been suggested by experience, or by induction from a large number of experiments. However they may have been suggested is wholly irrelevant in this question of proof.

To dispose of a possible objection here, the postulates are not always "obvious," or such that all sane beings could agree upon as being sensible. This objection harks back to the subconscious belief in that mysterious "truth." To give this dogmatic denial some shadow of backing, I may state that the trivial example above about the whole and its parts was chosen deliberately. It works admirably when we reason about a *finite* collection of things, say all the stars in all the nebulae reached by telescopes, or.all the human beings who have ever lived. But it does not work when we try to reason about an *infinity* of things, say all the points on a straight line. There is nothing "obvious" about it, nor does it "necessarily" apply to such a simple thing as the entire universe. It was indicated in an earlier chapter that the "whole-part" axiom fails for the infinite collection of *all* the common whole numbers. To repeat, because the point really is important for everything that is to follow, postulates are out-and-out assumptions.

Having seen the necessity for agreeing upon a set of postulates before undertaking to *prove* anything, Pythagoras and his successors next laid down the *completely arbitrary rule* that a statement shall be said to be *proved when*, and *only when*, the *statement follows from the postulates by an application of the rules of logic*. Nothing but these rules is to be injected into the process of "proof." When the statement does follow as just described, we say that it has been *deduced* from the postulates. The process is called deduction, and the type of reasoning by which it proceeds is the abstract, deductive reasoning already mentioned in connection with the Egyptians.

There are no historical grounds for saying indisputably that Pythagoras himself ever got as far as this cold, clear conception of proof. Indeed many of his speculations seem to show that he did not recognize the complete arbitrariness of the postulates, particularly in geometry. He seems to have been still in the mystic stage, and he reasoned subconsciously. A musician does not have to understand the theory of music in order to compose; harmonizing and the rest come after the composition has been conceived.

It is even doubtful whether any of the Greeks ever got as far as the above conception of proof. That conception seems to have been clearly grasped only in the Nineteenth Century. The Twentieth has gone considerably ahead of this, as regards logic, but that part of the story does not come until almost the end. Pythagoras however undoubtedly was the first human being on record who recognized the necessity for *proof* in order to enable men to reach common conclusions everywhere and always from the same set of data concerning numbers or geometrical figures. The suggestion he made was so simple, so rational, that it seems inevitable. We are so used to it that we take it for granted, forgetting the chaos which reigned before Pythagoras lived.

If we are inclined to underestimate what he did, we need but think of the situation with regard to human problems—social, ethical, moral, economic, religious—at the present time. Is there any sign of an agreement in sight? Is anyone ever likely to devise a set of rules for procedure in human problems on which almost all sane men can agree everywhere and in all times? Ask the Egyptians of 4241 B.C. To them the simpler task—simpler perhaps only because it has been done—of formulating such a set of rules for the material problems of their day may have looked as hopeless as our own task looks to us.

Any man who can get the rest of mankind to accept a treaty on anything must have quite unusual powers of persuasion. Pythagoras seems to be the only man on record who almost succeeded. His actual success however was so great that for long it completely overshadowed his partial failure. That partial failure is the significant item in the evolution—or history—of abstract thinking. The trouble began when the pyramid cropped up again to bother the Greeks. But we must first glance at the rules of logic which have been mentioned several times but not yet sufficiently discussed.

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