Men of Mathematics (5 page)

The last—the conception of
continuity,
“no nextness”—when developed in the manner of Newton, Leibniz, and their successors leads out into the boundless domain of
the calculus
and its innumerable applications to science and technology, and to all that is today called
mathematical analysis.
The other, the
discrete
pattern based on 1,2,3, . . . , is the domain of algebra, the theory of numbers, and symbolic logic. Geometry partakes of both the continuous and the discrete.

A major task of mathematics today is to harmonize the continuous
and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.

*  *  *

It may be doing our predecessors an injustice to emphasize modern mathematical thought with but little reference to the pioneers who took the first and possibly the most difficult steps. But nearly everything useful that was done in mathematics before the seventeenth century has suffered one of two fates: either it has been so greatly simplified that it is now part of every regular school course, or it was long since absorbed as a detail in work of greater generality.

Things that now seem as simple as common sense—our way of writing numbers, for instance, with its “place system” of value and the introduction of a symbol for zero, which put the essential finishing touch to the place system—cost incredible labor to invent. Even simpler things, containing the very essence of mathematical thought
—abstractness and generality,
must have cost centuries of struggle to devise; yet their originators have vanished leaving not a trace of their lives and personalities. For example, as Bertrand Russell observed, “It must have taken many ages to discover that a brace of pheasants and a couple of days were both instances of the number two.” And it took some twenty five centuries of
civilization
to evolve Russell's own logical definition of “two” or of any cardinal number (reported in the concluding chapter).

Again, the conception of a point, which we (erroneously) think we fully understand when we begin school geometry must have come very late in man's career as an artistic, cave-painting animal. Horace Lamb, an English mathematical physicist, would “erect a monument to the unknown mathematical inventor of the mathematical point as the supreme type of that abstraction which has been a necessary condition of scientific work from the beginning.”

Who, by the way,
did
invent the mathematical point? In one sense Lamb's forgotten man; in another, Euclid with his definition “a point is that which has no parts and which has no magnitude”; in yet a third sense Descartes with his invention of the “coordinates of a point”: until finally in geometry as experts practise it today the mysterious “point” has joined the forgotten man and all his gods in everlasting oblivion, to be replaced by something more usable—
a set of numbers written in a definite order.

The last is a modern instance of the abstractness and precision toward which mathematics strives constantly, only to realize when abstractness and precision are attained that a higher degree of abstractness and a sharper precision are demanded for clear understanding. Our own conception of a “point” will no doubt evolve into something yet more abstract. Indeed the “numbers” in terms of which points are described today dissolved about the beginning of this century into the shimmering blue of pure logic, which in its turn seems about to vanish in something rarer and even less substantial.

It is not necessarily true then that a step-by-step following of our predecessors is the sure way to understand either their conception of mathematics or our own. Such a retracing of the path that has led up to our present outlook would undoubtedly be of great interest in itself. But it is quicker to glance back over the terrain from the hilltop on which we now stand. The false steps, the crooked trails, and the roads that led nowhere fade out in the distance, and only the broad highways are seen leading straight back to the past, where we lose them in the mists of uncertainty and conjecture. Neither space nor number, nor even time, have the same significance for us that they had for the men whose great figures appear dimly through the mist.

A Pythagorean of the sixth century before Christ could intone “Bless us, divine Number, thou who generatest gods and men”; a Kantian of the nineteenth century could refer confidently to “space” as a form of “pure intuition”; a mathematical astronomer could announce a decade ago that the Great Architect of the Universe is a pure mathematician. The most remarkable thing about all of these profound utterances is that human beings no stupider than ourselves once thought they made sense.

To a modern mathematician such all-embracing generalities mean less than nothing. Yet in parting with its claim to be the universal generator of gods and men mathematics has gained something more substantial, a faith in itself and in its ability to create human values.

Our point of view has changed—and is still changing. To Descartes' “Give me space and motion and I will give you a world,” Einstein today might retort that altogether too much is being asked, and that the demand is in fact meaningless: without a “world”—matter—there is neither “space” nor “motion.” And to quell the turbulent, muddled mysticism of Leibniz in the seventeenth century, over the mysterious
: “The Divine Spirit found a sublime outlet
in that wonder of analysis, the portent of the ideal, that mean between being and not-being, which we call the imaginary [square] root of negative unity,” Hamilton in the 1840's constructed a number-couple which any intelligent child can understand and manipulate, and which does for mathematics and science all that the misnamed “imaginary” ever did. The mystical “not-being” of the seventeenth century Leibniz is seen to have a “being” as simple as ABC.

Is this a loss? Or does a modern mathematician lose anything of value when he seeks through the postulational method to track down that elusive “feeling” described by Heinrich Hertz, the discoverer of wireless waves: “One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them”?

Any competent mathematician will understand Hertz' feeling, but he will also incline to the belief that whereas continents and wireless waves are discovered, dynamos and mathematics are invented and do what we make them do. We can still dream but we need not deliberately court nightmares. If it is true, as Charles Darwin asserted, that “Mathematics seems to endow one with something like a new sense,” that sense is the sublimated common sense which the physicist and engineer Lord Kelvin declared mathematics to be.

Is it not closer to our own habits of thought to agree temporarily with Galileo that “Nature's great book is written in mathematical symbols” and let it go at that, than to assert with Plato that “God ever geometrizes,” or with Jacobi that “God ever arithmetizes”? If we care to inspect the symbols in nature's great book through the critical eyes of modern science we soon perceive that we ourselves did the writing, and that we used the particular script we did because we invented it to fit our own understanding. Some day we may find a more expressive shorthand than mathematics for correlating our experiences of the physical universe—unless we accept the creed of the scientific mystics that everything
is
mathematics and is not merely
described
for our convenience in mathematical language.
If
“Number rules the universe” as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.

When a modern mathematician turns aside for a moment from his symbols to communicate to others the feeling that mathematics in
spires in him, he does not echo Pythagoras and Jeans, but he may quote what Bertrand Russell said about a quarter of a century ago: “Mathematics, rightly viewed, possesses not only truth but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”

Another, familiar with what has happened to our conception of mathematical “truth” in the years since Russell praised the beauty of mathematics, might refer to the “iron endurance” which some acquire from their attempt to understand what mathematics means, and quote James Thomson's lines (which close this book) in description of Dürer's
Melencolia
(the frontispiece). And if some devotee is reproached for spending his life on what to many may seem the selfish pursuit of a beauty having no immediate reflection in the lives of his fellowmen, he may repeat Poincaré's “Mathematics for mathematics' sake. People have been shocked by this formula and yet it is as good as life for life's sake, if life is but misery.”

*  *  *

To form an estimate of what modern mathematics compared to ancient has accomplished, we may first look at the mere bulk of the work in the period after 1800 compared to that before 1800. The most extensive history of mathematics is that of Moritz Cantor,
Geschichte der Mathematik,
in three large closely printed volumes (a fourth, by collaborators, supplements the three). The four volumes total about 3600 pages. Only the outline of the development is given by Cantor; there is no attempt to go into details concerning the contributions described, nor are technical terms explained so that an outsider could understand what the whole story is about, and biography is cut to the bone; the history is addressed to those who have some technical training. This history
ends with the year 1799
—just before modern mathematics began to feel its freedom. What if the
outline
history of mathematics in the nineteenth century alone were attempted on a similar scale? It has been estimated that nineteen or twenty volumes the size of Cantor's would be required to tell the story, say about 17,000 pages. The nineteenth century, on this scale, contributed to mathematical knowledge about
five times as much
as was done in the whole of preceding history.

The beginningless period before 1800 breaks quite sharply into two. The break occurs about the year 1700, and is due mainly to Isaac Newton (1642-1727). Newton's greatest rival in mathematics was Leibniz (1646-1716). According to Leibniz, of all mathematics up to the time of Newton, the more important half is due to Newton. This estimate refers to the power of Newton's general methods rather than to the bulk of his work; the
Principia
is still rated as the most massive addition to scientific thought ever made by one man.

Continuing back into time beyond 1700 we find nothing comparable till we reach the Golden Age of Greece—a step of nearly 2000 years. Farther back than 600
B.C
. we quickly pass into the shadows, coming out into the light again for a moment in ancient Egypt. Finally we arrive at the first great age of mathematics, about 2000
B.C
., in the Euphrates Valley.

The descendants of the Sumerians in Babylon appear to have been the first “moderns” in mathematics; certainly their attack on algebraic equations is more in the spirit of the algebra we know than anything done by the Greeks in their Golden Age. More important than the technical algebra of these ancient Babylonians is their recognition—as shown by their work—of the necessity for
proof
in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions. This was one of the most important steps ever taken by human beings. Unfortunately it was taken so long ago that it led nowhere in particular so far as our own civilization is concerned—unless the Greeks followed consciously, which they may well have done. They were not particularly generous to their predecessors.

Mathematics then has had four great ages: the Babylonian, the Greek, the Newtonian (to give the period around 1700 a name), and the recent, beginning about 1800 and continuing to the present day. Competent judges have called the last the Golden Age of Mathematics.

Today mathematical invention (discovery, if you prefer) is going forward more vigorously than ever. The only thing, apparently, that can stop its progress is a general collapse of what we have been pleased to call civilization. If that comes, mathematics may go underground for centuries, as it did after the decline of Babylon; but if history repeats itself, as it is said to do, we may count on the spring bursting forth again, fresher and clearer than ever, long after we and all our stupidities shall have been forgotten.