Men of Mathematics (2 page)

23. COMPLETE INDEPENDENCE

B
OOLE
(1815-1864)

British mathematics. Damned at birth by snobbery. Boole's struggle for education. False diagnoses. Providence intervenes. Discovery of invariants. What is algebra? A philosopher attacks a mathematician. Frightful carnage. Boole's chance. “The Laws of Thought.” Symbolic logic. Its mathematical significance. Boolean algebra. Dead in his prime.

24. THE MAN, NOT THE METHOD

H
ERMITE
(1822-1901)

Old problems and new methods. Hermite's masterful mother. His detestation of examinations. Instructs himself. Higher mathematics sometimes easier than elementary. Educational disasters. Letters to Jacobi. A master at twenty one. Revenge on his examiners. Abelian functions. Pestered by Cauchy. Hermite's mysticism. Solution of the general quintic. Transcendental numbers. A hint to circle-squarers. Hermite's internationalism.

25. THE DOUBTER

K
RONECKER
(1823-1891)

Legend of an American saint. Lucky Kronecker. School triumphs. Great gifts. Algebraic numbers. Battles with Weierstrass. Kronecker's business career. Returns rich to mathematics. The Galois theory. Kronecker's lectures. His skepticism his most original contribution.

26. ANIMA CANDIDA

R
IEMANN
(1826-1866)

Poor but happy. Riemann's chronic shyness. Destined for the church. Saved. A famous hypothesis. Career at Göttingen. “A new mathematic.” Physical researches. Application of topology to analysis. Epoch-making essay on foundations of geometry. Gauss enthusiastic. The blessings of poverty. A root of tensor analysis. Quest for health.
Under a fig tree. Riemann's landmark in geometry. Curvature of space. Pathbreaking for relativity.

27. ARITHMETIC THE SECOND

K
UMMER
(1810-1893),
DEDEKIND
(1831-1916)

Aged in the wood. Napoleonic warp to Kummer's geniality. Equally gifted in the abstract and the concrete. What Fermat's Last Theorem started. Theory of ideal numbers. Kummer's invention comparable to Lobatchewsky's. Wave surface in four dimensions. Big of body, mind, and heart. Dedekind, last pupil of Gauss. First expositor of Galois. Early interest in science. Turns to mathematics. Dedekind's work on continuity. His creation of the theory of ideals.

28. THE LAST UNIVERSALIST

P
OINCARÉ
(1854-1912)

Poincaré's universality and methods. Childhood setbacks. Seized by mathematics. Keeps his sanity in Franco-Prussian war. Starts as mining engineer. First great work. Automorphic functions. “The keys of the algebraic cosmos.” The problem of
n
bodies. Is Finland civilized? Poincaré's new methods in celestial mechanics. Cosmogony. How mathematical discoveries are made. Poincaré's account. Forebodings and premature death.

29. PARADISE LOST?

C
ANTOR
(1845-1918)

Old foes with new faces. Rotting creeds. Cantor's artistic inheritance and father-fixation. Escape, but too late. His revolutionary work gets him nowhere. Academic pettiness. Disastrous consequences of “safety first.” An epochal result. Paradox or truth? Infinite existence of transcendentals. Aggressiveness advances, timidity retires. Further spectacular claims. Two types of mathematicians. Insane? Counter-revolution. The battle grows fiercer. Cursing the enemy. Universal loss of temper. Where stands mathematics today? And where will it stand tomorrow? Invictus.

INDEX

ABOUT THE AUTHOR

TO TOBY

W
ITHOUT A MASS OF FOOTNOTES
it would be impossible to cite authority for every statement of historical fact in the following pages. But little of the material consulted is available outside of large university libraries, and most of it is in foreign languages. For the principal dates and leading facts in the life of a particular man I have consulted the obituary notices (of the moderns); these are found in the proceedings of the learned societies of which the man in question was a member. Other details of interest are given in the correspondence between mathematicians and in their collected works. In addition to the few specific sources cited presently, bibliographies and references in the following have been especially helpful.

(1) The numerous historical notes and papers abstracted in the
Jahrbuch über die Fortschritte der Mathematik
(section on history of mathematics).

(2) The same in
Bibliotheca Mathematica.

Only three of the sources are sufficiently “private” to need explicit citation. The life of Galois is based on the classic account by P. Dupuy in the
Annales scientifiques de l' École normale supérieure
(3
^me
série, tome 13, 1896), and the edited notes by Jules Tannery. The correspondence between Weierstrass and Sonja Kowalewski was published by Mittag-Leffler in the
Acta Mathematica
(also partly in the
Comptes rendus du 2
^me
Congrès international des Mathématiciens
, Paris, 1902). Many of the details concerning Gauss are taken from the book by W. Sartorius von Waltershausen,
Gauss zum Gedächtniss
_y
Leipzig, 1856.

It would be rash to claim that every date or spelling of proper names in the book is correct. Dates are used chiefly with the purpose of orienting the reader as to a man's age when he made his most original inventions. As to spellings, I confess my helplessness in the face of such variants as Basle, Bale, Basel for one Swiss town, or Utzendorff,
Uitzisdorf for another, each preferred by some admittedly reputable authority. When it comes to choosing between James and Johann, or between Wolfgang and Farkas, I take the easier way and identify the man otherwise.

Most of the portraits are reproduced from those in the David Eugene Smith Collection, Columbia University. The portrait of Newton is from an original mezzotint loaned by Professor E. C. Watson. The drawings have been constructed accurately by Mr. Eugene Edwards.

As on a previous occasion (
The Search for Truth),
it gives me great pleasure to thank Doctor Edwin Hubble and his wife, Grace, for their invaluable assistance. While I alone am responsible for all statements in the book, nevertheless it was a great help to have scholarly criticism (even if I did not always profit by it) from two experts in fields in which I cannot claim to be expert, and I trust that their constructive criticisms have lightened my own deficiencies. Doctor Morgan Ward also has criticized certain of the chapters and has made many helpful suggestions on matters in which he is expert. Toby, as before, has contributed much; in acknowledgment for what she has given, I have dedicated the book to her (if she will have it)—it is as much hers as mine.

Last, I wish to thank the staffs of the various libraries which have generously helped with the loan of rare books and bibliographical material. In particular I should like to thank the librarians at Stanford University, the University of California, the University of Chicago, Harvard University, Brown University, Princeton University, Yale University, The John Crerar Library (Chicago), and the California Institute of Technology.

E. T. B
ELL

(Motto of Marischal College, Aberdeen)

The science of Pure Mathematics, in its modern developments, may claim to be the most original creation of the human spirit.—A. N. W
HITEHEAD
(Science and the Modern World,
1925)

A mathematical truth is neither simple nor complicated in itself, it is.—É
MILE LEMOINE

A mathematician who is not also something of a poet will never be a complete mathematician.—K
ARL
W
EIERSTRASS

I have heard myself accused of being an opponent, an enemy of mathematics, which no one can value more highly than I, for it accomplishes the very thing whose achievement has been denied me.—G
OETHE

Mathematicians are like lovers. . . . Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.—F
ONTENELLE

It is easier to square the circle than to get round a mathematician.—A
UGUSTUS DE
M
ORGAN

I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are . . . .—A. N. W
HITEHEAD
(
The Concept of Nature,
1920)

*  *  *

Number rules the universe.—T
HE
P
YTHAGOREANS

Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics.—C. F. G
AUSS

Thus number may be said to rule the whole world of quantity, and the four rules of arithmetic may be regarded as the complete equipment of the mathematician.—J
AMES
C
LERK
M
AXWELL

The different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.—T
HE
M
OCK
T
URTLE
(Alice in Wonderland)

God made the integers, all the rest is the work of man.—L
EOPOLD
K
RONECKER

[Arithmetic] is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths.—H. J. S. S
MITH

*  *  *

Plato's writings do not convince any mathematician that their author was strongly addicted to geometry. . . . We know that he encouraged mathematics. . . . But if—which nobody believes—the
[Let no man ignorant of geometry enter] of Tzetzes had been written over his gate, it would no more have indicated the geometry within than a warning not to forget to bring a packet of sandwiches would now give promise of a good dinner.—A
UGUSTUS
D
E
M
ORGAN

There is no royal road to geometry.—M
ENAECHMUS
(
to
A
LEXANDER THE
G
REAT
)

*  *  *

He studied and nearly mastered the six books of Euclid since he was a member of Congress.

He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.—A
BRAHAM
L
INCOLN
(Short Autobiography,
1860)

*  *  *

Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.—E
RNST
M
ACH

A
single curve, drawn in the manner of the curve of prices of cotton, describes all that the ear can possibly hear as the result of the most complicated musical performance. . . . That to my mind is a wonderful proof of the potency of mathematics.—L
ORD
K
ELVIN

*  *  *

The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe.—K
ARL
P
EARSON

Examples . . . which might be multiplied
ad libitum,
show how difficult it often is for an experimenter to interpret his results without the aid of mathematics.—L
ORD
R
AYLEIGH

But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.—A
LBERT
E
INSTEIN

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. For this reason a book on the new physics, if not purely descriptive of experimental
work, must be essentially mathematical.—P. A. M. D
IRAC
(Quantum Mechanics,
1930)

As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena [of electromagnetism] was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. I also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathematicians.—J
AMES
C
LERK
M
AXWELL
(A Treatise on Electricity and Magnetism,
1873)