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THE DEVELOPMENT OF MATHEMATICS
By Eric Temple Bell Dover Publications, Inc.
Copyright © 1972 Taine T. Bell
All rights reserved.
ISBN: 978-0-486-15228-8
CHAPTER 1
General Prospectus
In all historic times all civilized peoples have striven toward mathematics. The prehistoric origins are as irrecoverable as those of language and art, and even the civilized beginnings can only be conjectured from the behavior of primitive peoples today. Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seventeenth century these two united, forming the ever-broadening river of mathematical analysis. We shall look back in the following chapters on this great river of intellectual progress and, in the diminishing perspective of time, endeavor to see the more outstanding of those elements in the general advance from the past to the present which have endured.
'Form,' it may be noted here to prevent a possible misapprehension at the outset, has long been understood mathematically in a sense more general than that associated with the shapes of plane figures and solid bodies. The older, geometrical meaning is still pertinent. The newer refers to the structure of mathematical relations and theories. It developed, not from a study of spacial form as such, but from an analysis of the proofs occurring in geometry, algebra, and other divisions of mathematics.
Awareness of number and spacial form is not an exclusively human privilege. Several of the higher animals exhibit a rudimentary sense of number, while others approach genius in their mastery of form. Thus a certain cat made no objection when she was relieved of two of her six kittens, but was plainly distressed when she was deprived of three. She was relatively as advanced arithmetically as the savages of an Amazon tribe who can count up to two, but who confuse all greater numbers in a nebulous 'manly.'
Again, the intellectual rats that find their way through the mazes devised by psychologists are passing difficult examinations in topology. At the human level, a classic puzzle which usually suffices to show the highly intelligent the limitations of their spacial intuition is that of constructing a surface with only one side and one boundary.
Although human beings and the other animals thus meet on a common ground of mathematical sense, mathematics as it has been understood for at least twenty-five centuries is on a far higher plane of intelligence.
Necessity for proof; emergence of mathematics
Between the workable empiricism of the early land measurers who parceled out the fields of ancient Egypt and the geometry of the Greeks in the sixth century before Christ there is a great chasm. On the remoter side lies what preceded mathematics, on the nearer, mathematics; and the chasm is bridged by deductive reasoning applied consciously and deliberately to the practical inductions of daily life. Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist. This does not deny that intuition, experiment, induction, and plain guessing are important elements in mathematical invention. It merely states the criterion by which the final product of all the guessing, by whatever name it be dignified, is judged to be or not to be mathematics. Thus, for example, the useful rule, known to the ancient Babylonians, that the area of a rectangular field can be computed by 'length times breadth,' may agree with experience to the utmost refinement of physical measurement; but the rule is not a part of mathematics until it has been deduced from explicit assumptions.
It may be significant to record that this sharp distinction between mathematics and other sciences began to blur slightly under the sudden impact of a greatly accelerated applied mathematics, so called, in the second world war. Semiempirical procedures of calculation, certified by their pragmatic utility in war, were accorded full mathematical prestige. This relaxation of traditional demands brought the resulting techniques closer in both method and spirit to engineering and the physical sciences. It was acclaimed by some of its practitioners as a long-overdue democratization of the most aristocratic of the sciences. Others, of a more conservative persuasion, deplored the passing of the ideal of strict deduction, as a profitless confusion of a simple issue which had at last been clarified after several centuries of futile disputation. One fact, however, emerged from the difference of opinion: It is difficult, in modern warfare, to wreck, to maim, or to kill efficiently without a considerable expenditure of mathematics, much of which was designed originally for the development of those sciences and arts which create and conserve rather than destroy and waste.
It is not known where or when the distinction between inductive inference-the summation of raw experience—and deductive proof from a set of postulates was first made, but it was sharply recognized by the Greek mathematicians as early as 550 B.C. As will appear later, there may be some grounds for believing that the Egyptians and the Babylonians of about 2000 B.C. had recognized the necessity for deductive proof. For proof in even the rough and unready calculations of daily life is indeed a necessity, as may be seen from the mensuration of rectangles.
If a rectangle is 2 feet broad and 3 long, an easy proof sustains the verdict of experience, founded on direct measurement, that the area is 6 square feet. But if the breadth is [square root of 2] and the length [square root of 3] feet, the area cannot be determined as before by cutting the rectangle into unit squares; and it is a profoundly difficult problem to prove that the area is [square root of 6] feet, or even to give intelligible, usable meanings to [square root of 2], [square root of 3], [square root of 6] and 'area.' By taking smaller and smaller squares as unit areas, closer and closer approximations to the area are obtained, but a barrier is soon reached beyond which direct measurement cannot proceed. This raises a question of cardinal importance for a just understanding of the development of all mathematics, both pure and applied.
Continuing with the [square root of 2] × [square root of 3] rectangle, we shall suppose that refined measurement has given 2.4494897 as the area. This is correct to the seventh decimal, but it is not right, because [square root of 6], the exact area, is not expressible as a terminated decimal fraction. If seven-place accuracy is the utmost demanded, the area has been found. This degree of precision suffices for many practical applications, including precise surveying. But it is inadequate for others, such as some in the physical sciences and modern statistics. And before the seven-place approximation can be used intelligently, its order of error must be ascertained. Direct measurement cannot enlighten us; for after a certain limit, quickly passed, all measurements blur in a common uncertainty. Some universal agreement on what is meant by the exact area must be reached before progress is possible. Experience, both practical and theoretical, has shown that a consistent and useful mensuration of rectangles is obtained when the rule 'length times breadth' is deduced from postulates abstracted from a lower level of experience and accepted as valid. The last is the methodology of all mathematics.
Mathematicians insist on deductive proof for practically workable rules obtained inductively because they know that analogies between phenomena at different levels of experience are not to be accepted at their face value. Deductive reasoning is the only means yet devised for isolating and examining hidden assumptions, and for following the subtle implications of hypotheses which may be less factual than they seem. In its modern technical uses of the deductive method, mathematics employs much sharper tools than those of the traditional logic inherited from ancient and medieval times.
Proof is insisted upon for another eminently practical reason. The difficult technology of today is likely to become the easy routine of tomorrow; and a vague guess about the order of magnitude of an unavoidable error in measurement is worthless in the technological precision demanded by modern civilization. Working technologists cannot be skilled mathematicians. But unless the rules these men apply in their technologies have been certified mathematically and scientifically by competent experts, they are too dangerous for use.
There is still another important social reason for insistence on mathematical demonstration, as may be seen again from the early history of surveying. In ancient Egypt, the primitive theory of land measurement, without which the practice would have been more crudely wasteful than it actually was, sufficed for the economy of the time. Crude both practically and theoretically though this surveying was, it taxed the intelligence of the Egyptian mathematicians. Today the routine of precise surveying can be mastered by a boy of seventeen; and those applications of the trigonometry that evolved from primitive surveying and astronomy which are of greatest significance in our own civilization have no connection with surveying. Some concern mechanics and electrical technology, others, the most advanced parts of the physical sciences from which the industries of twenty or a hundred years hence may evolve.
Now, contrary to what might be supposed, modern trigonometry did not develop in response to any practical need. Modern trigonometry is impossible without the calculus and the mathematics of [square root of -1]. To cite but one of the commoner applications, over a century and a half elapsed before this trigonometry became indispensable in the theory and practice of alternating currents. Long before anyone had dreamed of an electric dynamo, the necessary mathematics of dynamo design was available. It had developed largely because the analysts of the eighteenth century sought to understand mathematically the somewhat meager legacy of trigonometry bequeathed them by the astronomers of ancient Greece, the Hindus, and the mathematicians of Islam. Neither astronomy nor any other science of the eighteenth century suggested the introduction of [square root of 1], which completed trigonometry, as no such science ever made any use of the finished product.
The importance of mathematics, from Babylon and Egypt to the present, as the primary source of workable approximations to the complexites of daily life is generally appreciated. In fact, a mathematician might believe it is almost too generally appreciated. It has been preached at the public, in school and out, by socially conscious educators until almost anyone may be pardoned for believing that the rule of life is rule of thumb. Because routine surveying, say, requires only mediocre intelligence, and because surveying is a minor department of applied mathematics, therefore only that mathematics which can be manipulated by rather ordinary people is of any social value. But no growing economy can be sustained by rule of thumb. If new applications of a furiously expanding science are to be possible, difficult and abstruse mathematical theories far beyond the college level must continue to be developed by those having the requisite talents. In this living mathematics it is imagination and rigorous proof which count, not the numerical accuracy of the machine shop or the computing laboratory.
A familiar example from common things will show the necessity for mathematics as distinguished from calculation. A nautical almanac is one of the indispensables of modern navigation and hence of commerce. Machines are now commonly used for the heavy labor of computing. Ultimately the computations depend upon the motions of the planets, and these are calculated from the infinite (non-terminating) series of numbers given by the Newtonian theory of gravitation. For the actual work of computation a machine is superior to any human brain; but no machine yet invented has had brains enough to reject nonsense fed into it. From a grotesquely absurd set of data the best of machines will return a final computation that looks as reasonable as any other. Unless the series used in dynamical astronomy converge to definite limiting numbers (asymptotic series also are used, but not properly divergent), it is futile to calculate by means of them. A table computed by properly divergent series would be indistinguishable to the untrained eye from any other; but the aviator trusting it for a flight from Boston to New York might arrive at the North Pole. Despite its inerrant accuracy and attractive appearance, even the most highly polished mechanism is no substitute for brains. The research mathematician and the scientific engineer supply the brains; the machine does the rest.
Nobody with a grain of common sense would demand a strict proof for every tentative application of complicated mathematics to new situations. Occasionally in problems of excessive difficulty, like some of those in nuclear physics, calculations are performed blindly without reference to mathematical validity; but even the boldest calculator trusts that his temerity will some day be certified rationally. This is a task for the mathematicians, not for the scientists. And if science is to be more than a midden of uncorrelated facts, the task must be carried through.
Necessity for abstractness
With the recognition that strict deductive reasoning has both practical and aesthetic values, mathematics began to emerge some six centuries before the Christian era. The emergence was complete when human beings realized that common experience is too complex for accurate description.
Again it is not known when or where this conclusion was first reached, but the Greek geometers of the fourth century B.C. at latest had accepted it, as is shown by their work. Thus Euclid in that century stated the familiar definition: "A circle is a plane figure contained by one line, called the circumference, and is such that all straight lines drawn from a certain point, called the center, within the figure to the circumference are equal."
There is no record of any such figure as Euclid's circle ever having been observed by any human being. Yet Euclid's ideal circle is not only that of school geometry, but is also the circle of the handbooks used by engineers in calculating the performance of machines. Euclid's mathematical circle is the outcome of a deliberate simplification and abstraction of observed disks, like the full moon's, which appear 'circular' to unaided vision.
This abstracting of common experience is one of the principal sources of the utility of mathematics and the secret of its scientific power. The world that impinges on the senses of all but introverted solipsists is too intricate for any exact description yet imagined by human beings. By abstracting and simplifying the evidence of the senses, mathematics brings the worlds of science and daily life into focus with our myopic comprehension, and makes possible a rational description of our experiences which accords remarkably well with observation.
Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.
History and proof
In any account of the development of mathematics there is a peculiar difficulty, exemplified in the two following assertions, about many statements concerning proof.
(A) It is proved in Proposition 47, Book 1, of Euclid's Elements, that the square on the longest side of a right-angled triangle is equal to the sum of the squares on the other two sides (the so-called Pythagorean theorem).
(B) Euclid proved the Pythagorean theorem in Proposition 47 of Book I of his Elements.
In ordinary discourse, (A), (B) would usually be considered equivalent—both true or both false. Here (A) is false and (B) true. For a clear understanding of the development of mathematics it is important to see that this distinction is not a quibble. It is also essential to recognize that comprehension here is more important than knowing the date (c. 330-320 B.C.) at which the Elements were written, or any other detail of equal antiquarian interest. In short, the crux of the matter is mathematics, which is at least as important as history, even in histories of mathematics.
(Continues...)
Excerpted from THE DEVELOPMENT OF MATHEMATICS by Eric Temple Bell. Copyright © 1972 Taine T. Bell. Excerpted by permission of Dover Publications, Inc..
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