Men of Mathematics (6 page)

. . . the glory that was Greece

And the grandeur that was Rome.

—E. A. P
OE

To
APPRECIATE
our own Golden Age of mathematics we shall do well to have in mind a few of the great, simple guiding ideas of those whose genius prepared the way for us long ago, and we shall glance at the lives and works of three Greeks: Zeno (495–435
B.C
.), Eudoxus (408–355
B.C
.), and Archimedes (287–212
B.C
.). Euclid will be noticed much later, where his best work comes into its own.

Zeno and Eudoxus are representative of two vigorous opposing schools of mathematical thought which flourish today, the critical-destructive and the critical-constructive. Both had minds as penetratingly critical as their successors in the nineteenth and twentieth centuries. This statement can of course be inverted: Kronecker (18231891) and Brouwer (1881- ), the modern critics of mathematical analysis—the theories of the infinite and the continuous—are as ancient as Zeno; the creators of the modern theories of continuity and the infinite, Weierstrass (1815-1897), Dedekind (1831-1916), and Cantor (1845-1918) are intellectual contemporaries of Eudoxus.

Archimedes, the greatest intellect of antiquity, is modern to the core. He and Newton would have understood one another perfectly, and it is just possible that Archimedes, could he come to life long enough to take a post-graduate course in mathematics and physics, would understand Einstein, Bohr, Heisenberg, and Dirac better than they understand themselves. Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians permit themselves today with all the hard-won gains of twenty five centuries to smooth their way, for he alone of all the Greeks had sufficient stature and strength to stride clear
over the obstacles thrown in the path of mathematical progress by frightened geometers who had listened to the philosophers.

Any list of the three “greatest” mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton (1642-1727) and Gauss (1777-1855). Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first. Had the Greek mathematicians and scientists followed Archimedes rather than Euclid, Plato, and Aristotle, they might easily have anticipated the age of modern mathematics, which began with Descartes (1596-1650) and Newton in the seventeenth century, and the age of modern physical science inaugurated by Galileo (1564-1642) in the same century, by two thousand years.

*  *  *

Behind all three of these precursors of the modern age looms the half-mythical figure of Pythagoras (569?-500?
B.C
.), mystic, mathematician, investigator of nature to the best of his self-hobbled ability, “one tenth of him genius, nine-tenths sheer fudge.” His life has become a fable, rich with the incredible accretions of his prodigies; but only this much is of importance for the development of mathematics as distinguished from the bizarre number-mysticism in which he clothed his cosmic speculations: he travelled extensively in Egypt, learned much from the priests and believed more; visited Babylon and repeated his Egyptian experiences; founded a secret Brotherhood for high mathematical thinking and nonsensical physical, mental, moral, and ethical speculation at Croton in southern Italy; and, out of all this, made two of the greatest contributions to mathematics in its entire history. He died, according to one legend, in the flames of his own school fired by political and religious bigots who stirred up the masses to protest against the enlightenment which Pythagoras sought to bring them.
Sic transit gloria mundi.

Before Pythagoras it had not been clearly realized that
proof
must proceed from
assumptions.
Pythagoras, according to persistent tradition, was the first European to insist that the
axioms,
the
postulates,
be set down first in developing geometry and that the entire development thereafter shall proceed by applications of close deductive reasoning to the axioms. Following current practice we shall use “postulate,”
instead of “axiom” hereafter, as “axiom” has a pernicious historical association of “self-evident, necessary truth” which “postulate” does not have; a postulate is an arbitrary assumption laid down by the mathematician himself and not by God Almighty.

Pythagoras then imported
proof
into mathematics. This is his greatest achievement. Before him geometry had been largely a collection of rules of thumb empirically arrived at without any clear indication of the mutual connections of the rules, and without the slightest suspicion that all were deducible from a comparatively small number of postulates. Proof is now so commonly taken for granted as the very spirit of mathematics that we find it difficult to imagine the primitive thing which must have preceded mathematical reasoning.

Pythagoras' second outstanding mathematical contribution brings us abreast of living problems. This was the discovery, which humiliated and devastated him, that the common whole numbers 1,2,3, . . . are insufficient for the construction of mathematics even in the rudimentary form in which he knew it. Before this capital discovery he had preached like an inspired prophet that all nature, the entire universe in fact, physical, metaphysical, mental, moral, mathematical—
everything
—is built on the
discrete
pattern of the integers 1,2,3, . . . and is interpretable in terms of these God-given bricks alone; God, he declared indeed,
is
“number,” and by that he meant common whole number. A sublime conception, no doubt, and beautifully simple, but as unworkable as its echo in Plato—“God ever geometrizes,” or in Jacobi—“God ever arithmetizes,” or in Jeans—“The Great Architect of the Universe now begins to appear as a mathematician.” One obstinate mathematical discrepancy demolished Pythagoras' discrete philosophy, mathematics, and metaphysics. But, unlike some of his successors, he finally accepted defeat—after struggling unsuccessfully to suppress the discovery which abolished his creed.

This was what knocked his theory flat: it is impossible to find two whole numbers such that the square of one of them is equal to twice the square of the other. This can be proved by a simple argument
^I
within the reach of anyone who has had a few weeks of algebra,
or even by anyone who thoroughly understands elementary arithmetic. Actually Pythagoras found his stumbling-block in geometry: the ratio of the side of a square to one of its diagonals cannot be expressed as the ratio of any two whole numbers. This is equivalent to the statement above about squares of whole numbers. In another form we would say that the square root of 2 is
irrational,
that is, is not equal to any whole number or decimal fraction, or sum of the two, got by dividing one whole number by another. Thus even so simple a geometrical concept as that of the diagonal of a square defies the integers 1,2,3, . . . and negates the earlier Pythagorean philosophy. We can easily construct the diagonal
geometrically, but we cannot measure it in any finite number of steps.
This impossibility sharply and clearly brought irrational numbers and the infinite (non-terminating) processes which they seem to imply to the attention of mathematicians. Thus the square root of two can be calculated to any required
finite
number of decimal places by the process taught in school or by more powerful methods, but the decimal never “repeats” (as that for 1/7 does, for instance), nor does it ever terminate. In this discovery Pythagoras found the taproot of modern mathematical analysis.

Issues were raised by this simple problem which are not yet disposed of in a manner satisfactory to all mathematicians. These concern the mathematical concepts of the infinite (the unending, the uncountable), limits, and continuity, concepts which are at the root of modern analysis. Time after time the paradoxes and sophisms which crept into mathematics with these apparently indispensable concepts have been regarded as finally eliminated, only to reappear a generation or two later, changed but yet the same. We shall come across them, livelier than ever, in the mathematics of our time. The following is an extremely simple, intuitively obvious picture of the situation.

Consider a straight line two inches long, and imagine it to have been traced by the “continuous” “motion” of a “point.” The words in quotes are those which conceal the difficulties. Without analysing them we easily persuade ourselves that we picture what they signify. Now label the left-hand end of the line 0 and the right-hand end 2.
Half-way between 0 and 2 we naturally put 1; half-way between 0 and 1 we put ½; half-way between 0 and ½ we put ¼, and so on. Similarly, between 1 and 2 we mark the place 1½, between 1½ and 2, the place 1½ and so on. Having done this we may proceed in the same way to mark ⅓, ⅔, 1⅓, 1⅔, and then split each of the resulting segments into smaller equal segments. Finally, “in imagination,” we can conceive of this process having been carried out for
all
the common fractions and common mixed numbers which are greater than 0 and less than 2; the conceptual division-points give us
all the rational numbers between 0 and 2.
There are an infinity of them. Do they completely “cover” the line? No. To what point does the square root of 2 correspond? No point, because this square root is not obtainable by dividing
any
whole number by another. But the square root of 2 is obviously a “number” of some sort;
^II
its representative point lies somewhere between 1.41 and 1.42, and we can cage it down as closely as we please. To cover the line completely we are forced to imagine or to invent infinitely more “numbers” than the rationals. That is, if we accept the line as being
continuous,
and
postulate
that to each point of it corresponds one, and only one, “real number.” The same kind of imagining can be carried on to the entire plane, and farther, but this is sufficient for the moment.

Simple problems such as these soon lead to very serious difficulties. With regard to these difficulties the Greeks were divided, just as we are, into two irreconcilable factions; one stopped dead in its mathematical tracks and refused to go on to analysis—the integral calculus, at which we shall glance when we come to it; the other attempted to overcome the difficulties and succeeded in convincing itself that it had done so. Those who stopped committed but few mistakes and were comparatively sterile of truth no less than of error; those who went on discovered much of the highest interest to mathematics and rational thought in general, some of which may be open to destructive criticism, however, precisely as has happened in our own generation. From the earliest times we meet these two distinct and antagonistic types of mind: the justifiably cautious who hang back because the ground quakes under their feet, and the bolder pioneers who leap the chasm to find treasure and comparative safety on the other side. We shall look first at one of those who refused to leap. For penetrating
subtlety of thought we shall not meet his equal till we reach the twentieth century and encounter Brouwer.

Zeno of Elea (495–435
B.C
.) was a friend of the philosopher Parmenides, who, when he visited Athens with his patron, shocked the philosophers out of their complacency by inventing four innocent paradoxes which they could not dissipate in words. Zeno is said to have been a self-taught country boy. Without attempting to decide what was his purpose in inventing his paradoxes—authorities hold widely divergent opinions—we shall merely state them. With these before us it will be fairly obvious that Zeno would have objected to our “infinitely continued” division of that two-inch line a moment ago. This will appear from the first two of his paradoxes, the
Dichotomy
and the
Achilles.
The last two, however, show that he would have objected with equal vehemence to the
opposite
hypothesis, namely that the line is
not
“infinitely divisible” but is composed of a
discrete
set of points that can be counted off 1,2,3, . . . . All four together constitute an iron wall beyond which progress appears to be impossible.

First, the
Dichotomy.
Motion is impossible, because whatever moves must reach the middle of its course
before
it reaches the end; but
before
it has reached the middle it must have reached the quarter-mark, and so on,
indefinitely.
Hence the motion can never even start.

Second, the
Achilles.
Achilles running to overtake a crawling tortoise ahead of him can never overtake it, because he must first reach the place from which the tortoise started; when Achilles reaches that place, the tortoise has departed and so is still ahead. Repeating the argument we easily see that the tortoise will always be ahead.