Men of Mathematics (82 page)

Thanks to Mittag-Leffler, Madame Kowalewski at last obtained a position where she could do herself justice; in the autumn of
1884
she was lecturing at the University of Stockholm, where she was to be appointed (in
1889)
as professor for life. A little later she suffered a rather embarrassing setback when the Italian mathematician Vito Volterra pointed out a serious mistake in her work on the refraction of light in crystalline media. This oversight had escaped Weierstrass, who at the time was so overwhelmed with official duties that outside of them he had “time only for eating, drinking, and sleeping. . . . In short,” he says, “I am what the doctors call brain-weary.” He was now nearly seventy. But as his bodily ills increased his intellect remained as powerful as ever.

The master's seventieth birthday was made the occasion for public honors and a gathering of his disciples and former pupils from all over Europe. Thereafter he lectured publicly less and less often, and for ten years received a few of his students at his own house. When they saw that he was tired out they avoided mathematics and talked of other things, or listened eagerly while the companionable old man reminisced of his student pranks and the dreary years of his isolation from all scientific friends. His eightieth birthday was celebrated by an even more impressive jubilee than his seventieth and he became in some degree a national hero of the German people.

One of the greatest joys Weierstrass experienced in his declining years was the recognition won at last by his favorite pupil. On Christmas Eve,
1888,
Sonja received in person the Bordin Prize of the French
Academy of Sciences for her memoir
On the rotation of a solid body about a fixed point.

As is the rule in competition for such prizes, the memoir had been submitted anonymously (the author's name being in a sealed envelope bearing on the outside the same motto as that inscribed on the memoir, the envelope to be opened only if the competing work won the prize), so there was no opportunity for jealous rivals to hint at undue influence. In the opinion of the judges the memoir was of such exceptional merit that they raised the value of the prize from the previously announced 3000 francs to 5000. The monetary value, however, was the least part of the prize.

Weierstrass was overjoyed. “I do not need to tell you,” he writes, “how much your success has gladdened the hearts of myself and my sisters, also of your friends here. I particularly experienced a true satisfaction; competent judges have now delivered their verdict that my 'faithful pupil/ my 'weakness' is indeed not a 'frivolous humbug.' ”

We may leave the friends in their moment of triumph. Two years later (February 10, 1891) Sonja died in Stockholm at the age of forty one after a brief attack of influenza which at the time was epidemic. Weierstrass outlived her six years, dying peacefully in his eighty second year on February 19, 1897, at his home in Berlin after a long illness followed by influenza. His last wish was that the priest say nothing in his praise at the funeral but restrict the services to the customary prayers.

Sonja is buried in Stockholm, Weierstrass with his two sisters in a Catholic cemetery in Berlin. Sonja also was of the Catholic faith, belonging to the Greek Church.

*  *  *

We shall now give some intimation of two of the basic ideas on which Weierstrass founded his work in analysis. Details or an exact description are out of the question here, but may be found in the earlier chapters of any competently written book on the theory of functions.

A
power series
is an expression of the form

a
₀
+ a
₁
z
a
₂
z
+
²
+ . . . +
a
_n
z
ⁿ
+ . . . ,

in which the coefficients
a
₀
, a
₁
a
₂
, . . . , a
_n
, . . .
are constant numbers and
z
is a variable number; the numbers concerned may be real or complex.

The sums of 1, 2, 3, . . . terms of the series, namely
a
₀
, a
₀
+
a
₁
Z, a
₂
Z
²
, . . . are called the
partial sums.
If for some particular value of
z
these partial sums give a sequence of numbers which converge to a definite limit, the power series is said to converge to the same limit for that value of
z.

All the values of
z
for which the power series converges to a limit constitute the
domain of convergence
of the series; for any value of the variable
z
in this domain the series
converges;
for other values of
z
it
diverges.

If the series converges for some value of
z,
its value can be calculated to any desired degree of approximation, for that value, by taking a sufficiently large number of terms.

Now, in the majority of mathematical problems which have applications to science, the “answer” is indicated as the solution in series of a differential equation (or system of such equations), and this solution is only rarely obtainable as a finite expression in terms of mathematical functions which have been tabulated (for instance logarithms, trigonometric functions, elliptic functions, etc.). In such problems it then becomes necessary to do two things: prove that the series converges, if it does; calculate its numerical value to the required accuracy.

If the series does not converge it is usually a sign that the problem has been either incorrectly stated or wrongly solved. The multitude of functions which present themselves in pure mathematics are treated in the same way, whether they are ever likely to have scientific applications or not, and finally a general theory of convergence has been elaborated to cover vast tracts of all this, so that the individual examination of a particular series is often referred to more inclusive investigations already carried out.

Finally, all this (both pure and applied) is extended to power series in
2, 3, 4,
 . . . variables instead of the single variable
z
above; for example, in two variables,

a
+
b
₀
z
+
b
₁
w
+
c
₀
z
²
+ C
₁
zw
+
c
₂
w
²
+ . . ..

It may be said that without the theory of power series most of mathematical physics (including much of astronomy and astro-physics) as we know it would not exist.

Difficulties arising with the concepts of limits, continuity, and convergence
drove Weierstrass to the creation of his theory of irrational numbers.

Suppose we extract the square root of 2 as we did in school, carrying the computation to a large number of decimal places. We get as successive approximations to the required square root the
sequence
of numbers 1, 1.4, 1.41, 1.412, . . . . With sufficient labor, proceeding by well-defined steps according to the usual rule, we could if necessary exhibit the first thousand, or the first million, of the
rational
numbers 1, 1.4, . . . constituting this sequence of approximations. Examining this sequence we see that when we have gone far enough we have determined a perfectly definite rational number containing as many decimal places as we please (say 1000), and that
this
rational number differs from any of the
succeeding
rational numbers in the sequence by a number (decimal), such as .000 . . . .000 . . . , in which a correspondingly large number of zeros occur
before
another digit (1, 2, . . . or 9) appears.

This illustrates what is meant by a
convergent sequence
of numbers: the
rationals
1, 1.4, . . . constituting the sequence give us ever closer approximations to the “irrational number” which we
call
the square root of 2, and which we conceive of as having been
defined
by the
convergent sequence of rationals,
this definition being in the sense that a method has been indicated (the usual school one) of calculating
any particular member of the sequence in a finite number of steps.

Although it is impossible actually to exhibit the whole sequence, as it does not stop at any finite number of terms, nevertheless we regard the
process
for constructing
any
member of the sequence as a sufficiently clear conception of the whole sequence as a single definite object which we can reason about. Doing so, we have a workable method for
using
the square root of 2 and similarly for any irrational number, in mathematical analysis.

As has been indicated it is impossible to make this precise in an account like the present, but even a careful statement might disclose some of the logical objections glaringly apparent in the above description—objections which inspired Kronecker and others to attack Weierstrass' “sequential” definition of irrationals.

Nevertheless, right or wrong, Weierstrass and his school made the theory
work.
The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its applications, by any competent judge in his
right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.

After all, why should mathematicians, who are human beings like the rest of us, always be so pedantically exact and so inhumanly perfect? As Weierstrass said, “It is true that a mathematician who is not also something of a poet will never be a perfect mathematician.” That is the answer: a perfect mathematician, by the very fact of his poetic perfection, would be a mathematical impossibility.

Pure Mathematics was discovered by Boole in a work which he called
The Laws of Thought.—B
ERTRAND
R
USSELL

“O
H, WE NEVER READ ANYTHING
the English mathematicians do.” This characteristically continental remark was the reply of a distinguished European mathematician when he was asked whether he had seen some recent work of one of the leading English mathematicians. The “we” of his frank superiority included Continental mathematicians in general.

This is not the sort of story that mathematicians like to tell on themselves, but as it illustrates admirably that characteristic of British mathematicians—insular originality—which has been the chief claim to distinction of the British school, it is an ideal introduction to the life and work of one of the most insularly original mathematicians England has produced, George Boole. The fact is that British mathematicians have often serenely gone their own way, doing the things that interested them personally as if they were playing cricket for their own amusement only, with a self-satisfied disregard for what others, shouting at the top of their scientific lungs, have assured the world is of supreme importance. Sometimes, as in the prolonged idolatry of Newton's methods, indifference to the leading fashions of the moment has cost the British school dearly, but in the long run the take-it-or-leave-it attitude of this school has added more new fields to mathematics than a slavish imitation of the continental masters could ever have done. The theory of invariance is a case in point; Maxwell's electrodynamic field theory is another.

Although the British school has had its share of powerful developers of work started elsewhere, its greater contribution to the progress of mathematics has been in the direction of originality. Boole's work is a striking illustration of this. When first put out it was ignored
as
mathematics,
except by a few, chiefly Boole's own more unorthodox countrymen, who recognized that here was the germ of something of supreme interest for all mathematics. Today the natural development of what Boole started is rapidly becoming one of the major divisions of pure mathematics, with scores of workers in practically all countries extending it to all fields of mathematics where attempts are being made to consolidate our gains on firmer foundations. As Bertrand Russell remarked some years ago, pure mathematics was
discovered
by George Boole in his work
The Laws of Thought
published in
1854.
This may be an exaggeration, but it gives a measure of the importance in which mathematical logic and its ramifications are held today. Others before Boole, notably Leibniz and De Morgan, had dreamed of adding logic itself to the domain of algebra; Boole did it.

George Boole was not, like some of the other originators in mathematics, born into the lowest economic stratum of society. His fate was much harder. He was born on November
2, 1815,
at Lincoln, England, and was the son of a petty shopkeeper. If we can credit the picture drawn by English writers themselves of those hearty old days—
1815
was the year of Waterloo—to be the son of a small tradesman at that time was to be damned by foreordination.

The whole class to which Boole's father belonged was treated with a contempt a trifle more contemptuous than that reserved for enslaved scullery maids and despised second footmen. The “lower classes,” into whose ranks Boole had been born, simply did not exist in the eyes of the “upper classes”—including the more prosperous wine merchants and moneylenders. It was taken for granted that a child in Boole's station should dutifully and gratefully master the shorter catechism and so live as never to transgress the strict limits of obedience imposed by that remarkable testimonial to human conceit and class-conscious snobbery.