Men of Mathematics (112 page)

To which the other side replies by a shrug of the shoulders and goes ahead with its great and fundamentally new task of reestablishing mathematics (particularly the foundations of analysis) on a firmer basis than any laid down by the men of the past 2500 years from Pythagoras to Weierstrass.

What will mathematics be like a generation hence when—we hope—these difficulties will have been cleared up? Only a prophet or the seventh son of a prophet sticks his head into the noose of prediction. But if there is any continuity at all in the evolution of mathematics—and the majority of dispassionate observers believe that there is—we shall find that the mathematics which is to come will be broader, firmer, and richer in content than that which we or our predecessors have known.

Already the controversies of the past third of a century have added new fields—including totally new logics—to the vast domain of mathematics, and the new is being rapidly consolidated and coordinated with the old. If we may rashly venture a prediction, what is to come will be fresher, younger in every respect, and closer to human thought and human needs—freer of appeal for its justification to extra-human “existences”—than what is now being vigorously refashioned. The spirit of mathematics is eternal youth. As Cantor said, “The essence of mathematics resides in its freedom”; the present “revolution” is but another assertion of that freedom.

*  *  *

Baffled and beaten back she works on still,

Weary and sick of soul she works the more,

Sustained by her indomitable will:

The hands shall fashion and the brain shall pore

And all her sorrow shall be turned to labour,

Till death the friend-foe piercing with his sabre

That mighty heart of hearts ends bitter war.

—J
AMES
T
HOMSON
.

I
. Quoted from R. E. Moritz'
Memorabilia Mathematica
, 1914. The original source is not accessible to me.

II
. L. Couturat,
Del ‘infini mathématique
, Paris, 1896,
p.49
. With the caution that much of this work is now hopelessly out of date, it can be recommended for its clarity to the general reader. An account of the elements of Cantorism by a leading Polish expert which is within the comprehension of anyone with a grade-school education and a taste for abstract reasoning is the
Leçons sur
les nombres transfinis, by Waclaw Sierpinski, Paris, 1928. The preface by Borel supplies the necessary danger signal. The above extract from Couturat is of some historical interest in connection with Hilbert's program. It anticipates by thirty years Hilbert's statement of his formalist creed.

E
RIC
T
EMPLE
B
ELL
was born in 1883 in Aberdeen, Scotland. His early education was obtained in England. Coming to the United States in 1902, he entered Stanford University and took his A.B. degree in 1904. In 1908 he was teaching fellow at the University of Washington, where he took his A.M. degree in 1909. In 1911 he entered Columbia University, where he took his Ph.D. degree in 1912. He returned to the University of Washington as instructor in mathematics and became full professor in 1921. During the summers of 1924-28 he taught at the University of Chicago, and in 1926 (first half) at Harvard University, when he was appointed Professor of Mathematics at the California Institute of Technology.

Dr. Bell was a former President of the Mathematical Association of America, a former Vice President of the American Mathematical Society and of the American Association for the Advancement of Science. He was on the editorial staffs of the
Transactions of the American Mathematical Society,
the
American Journal of Mathematics,
and the
Journal of the Philosophy of Science.
He belonged to The American Mathematical Society, the Mathematical Association of America, the Circolo Matematico di Palermo, the Calcutta Mathematical Society, Sigma Xi, and Phi Beta Kappa, and was a member of the National Academy of Sciences of the United States. He won the Bâcher Prize of the American Mathematical Society for his research work. His twelve published books include
The Purple Sapphire
(1924),
Algebraic Arithmetic
(1927),
Debunking Science,
and
Queen of the Sciences
(1931),
Numerology
(1933), and
The Search for Truth
(1934).

Dr. Bell died in December 1960, just before the publication of his latest book,
The Last Problem.

Abel, Niels Henrik,
3
,
164
,
167
,
223
,
229
–30,
260
,
270
–2,
295
, chap.
17
,
328
–9,
335
–7,
362
,
364
,
366
,
368
,
377
,
381
,
407
–8,
412
,
415
,
418
,
437
,
448
,
452
–3,
457
,
460
,
473
,
476
–8,
541

Abelian integral,
338
,
419

Adams, John Couch,
350

Airy, G. B.,
198
,
345
,
351
,
353

Alexander, J. W.,
268

Alexander the Great,
537

algebra,
13
,
21
,
40
,
52
–4,
61
–2,
64
–5,
97
,
123
,
140
,
149
,
163
–6,
168
,
199
,
208
,
211
,
213
,
222
–3,
225
,
227
,
231
–6,
238
,
253
–4,
260
–1,
272
,
282
–4,
287
,
309
–14,
317
,
321
–2,
324
,
328
,
330
,
334
–5,
337
–8,
347
–9,
354
–60,
364
,
368
,
372
,
376
,
386
,
388
–91,
394
–5,
398
–404,
408
,
437
–9,
442
–5,
448
,
450
–1,
453
,
456
–7,
459
–61,
463
–4,
470
,
472
–9,
482
,
489
,
493
,
511
,
517
–8,
527
–29,
540
–1,
564
–5,
568
–9

algebraic forms,
394
,
458

algebraic integers,
470
–2,
514
,
518
,
522
,
524

algebraic numbers,
462
,
464
,
469
–71,
474
,
477
–8,
482
,
523
–4,
562
,
564
–5,
567
–9

algebraic number field,
470
–4,
513
,
522
–4

algorithm,
140

Ampère, A. M.,
318

analysis,
13
,
16
,
22
–3,
54
,
64
,
70
,
87
,
117
–8,
139
–40,
144
,
150
–2,
154
–5,
161
–3,
166
,
168
–9,
171
,
174
,
176
,
183
,
187
,
198
,
201
–2,
207
,
213
,
222
–4,
236
–7,
248
–52,
268
,
270
–2,
275
,
286
,
311
–2,
316
,
319
,
322
,
327
,
334
–5,
338
–9,
346
,
364
,
377
,
400
,
406
–8,
413
–4,
416
–7,
419
,
421
–3,
429
,
446
,
451
,
456
–7,
460
–1,
469
,
474
–5,
477
–81,
488
–90,
494
,
510
,
513
,
519
,
521
–2,
526
–7,
537
–8,
541
–2,
544
,
547
,
556
–7,
561
,
567
,
571
,
575
,
579

analysis situs,
263
,
267
,
492
,
545

Antoinette, Marie,
166
,
170

Apollonius,
7
,
27
,
78
,
317
,
400

Appell, Paul,
454

Arago, F. J. D.,
139
,
150
,
190
,
193
,
205

Archimedes,
7
,
19
–20,
28
–34,
59
,
102
,
114
,
120
,
147
,
153
,
162
,
218
,
220
,
230
,
237
,
240
–1,
255
–6,
400
,
459
,
533

Archytas,
25

Aristotle,
20
,
25
,
78
,
240
,
278
,
569
,
575
,
577
–8

arithmetic,
21
,
64
,
118
,
152
,
161
–2,
168
,
185
,
208
,
221
,
223
,
225
–6,
227
–9,
231
–2,
234
–8,
240
–1,
248
,
252
–4,
256
,
260
–3,
267
,
271
–2,
276
,
284
,
356
,
368
,
408
,
419
,
438
,
446
,
451
,
455
–8,
470
–4,
477
–8,
481
,
485
,
488
,
500
,
510
,
513
,
517
–8,
522
–5,
527
–9,
541
,
551
,
561
,
574
–5,
577

arithmetical theory of forms,
356

Arnauld, A.,
83
,
128
–9

associative, associativity,
278
–9,
280
,
356

Ausonius,
39

Austen, Jane,
380

axioms,
20
–1,
305
–6,
333
,
419
,
503
,
576

Ayscough, Rev. Wm.,
91
–2

Babbage, Charles,
438

Bachet de Méziriac,
71

Baillet, A.,
39

Balzac, Honoré de,
548

Barrow, I.,
96
,
106
–7,
118

Bartels, Johann Martin,
222
–4

Bauer, Heinrich,
328

Beethoven, L. v.,
405

Berkeley, Bishop,
90
,
345

Bernoullis,
115
,
126
,
132
, chap.
8
,
143
–5,
155
–6

Berthollet, Count Claude-Louis,
183
–4,
189
–90,
193
–4,
196
,
273

Bertrand, J. L. F.,
453

Bessel, Friedrich Wilhelm,
245
,
248
,
250
–1,
331

Biot, J. B.,
181

Birkhoff, George David,
553

Bismarck, O. E. L., Prince von,
467

Blake, William,
10

Bliss, G. A.,
133
–4

Boeckh, P. A.,
328
–9

Bohr, N.,
19

Bolyai, Johann,
231

Bolyai, Wolfgang,
220
,
231
,
243
,
474
,
505

Boole, George,
118
,
121
,
124
,
213
,
354
,
389
,
390
,
406
, chap.
23
,
448
,
486

Boole, Mary,
446
–7

Borchardt, C. W.,
332
,
421
,
426
,
464
,
501

Borel, Émile,
454
,
566

boundary values,
339

Boutroux, Émile,
532