Document 229614

How to be a mathematician
It takes a long time to learn to live-by the time you learn your time
is gone. I spent most of a lifetime trying to be a mathematician-and
what did I learn? What does it take to be one? I think I know the
answer: you have to be born right, you must continually strive to
become perfect, you must love mathematics more than anything else,
you must work at it hard and without stop, and you must never give up.
Born right? Yes. To be a scholar of mathematics you must be born
with talent, insight, concentration, taste, luck, drive, and the ability to
visualize and guess. For teaching you must in addition understand
what kinds of obstacles learners are likely to place before themselves, and
you must have sympathy for your audience, dedicated selflessness,
verbal ability, clear style, and expository skill. To be able, finally, to
pull your weight in the profession with the essential clerical and administrative jobs, you must be responsible, conscientious, careful, and
organized-it helps if you also have some qualities of leadership and
You can't be perfect, but if you don't try, you won't be good enough.
To be a mathematician you must love mathematics more than family,
religion, money, comfort, pleasure, glory. I do not mean that you must
love it to the exclusion of family, religion, and the rest, and I do not
mean that if you do love it, you'll never have any doubts, you'll never
be discouraged, you'll never be ready to chuck it all and take up gardening instead. Doubts and discouragements are part of life. Great mathematicians have doubts and get discouraged, but usually they can't stop
doing mathematics anyway, and, when they do, they miss it very deeply.
" Mathematician" is, to be sure, an undefined term, and it is possible
that some people caned mathematicians nowadays (or ever) do not (or
did not) love mathematics all that much. A spouse unsympathetic to
mathematics demands equal time, a guilty parental conscience causes
you to play catch with your boy Saturday afternoon instead of beating
your head against the brick wall of that elusive problem-family, and
religion, and money, comfort, pleasure, glory, and other calls of life, deep
or trivial, exist for all of us to varying degrees, and I am not saying that
mathematicians always ignore all of them. I am not saying that the love
of mathematics is more important than the love of other things. What
I am saying is that to the extent that one's loves can be ordered, the
greatest love of a mathematician (the way I would like to use the term)
is mathematics. I have known many mathematicians, great and small,
and I feel sure that what I am saying is true about them. To mention
some famous names: I'd be very surprised if Marston Morse, and Andre
Weil, and Hermann Weyl, and Oscar Zariski didn't agree with me.
Mind you, I am not recommending or insisting that you love mathematics. I am not issuing an order: "If you want to be a mathematician,
start loving mathematics forthwith" - that would be absurd. What I am
saying is that the love of mathematics is a hypothesis without which the
conclusion doesn't follow. If you want to be a mathematician, look into
your soul and ask yourself how much you want to be one. If the wish
isn't very deep and very great, if it is not, in fact, maximal, if you have
another desire that takes precedence, or even more than one, then you
should not try to be a mathematician. The "should" is not a moral one;
it is a pragmatic one. I think that you would probably not succeed in
your attempt, and, in any event, you would probably feel frustrated and
As for working hard, I got my first hint of what that means when
Carmichael told me how long it took him to prepare a fifty-minute invited address. Fifty hours, he said; an hour of work for each minute of
the final presentation. When, many years later, six of us wrote our
"history" paper ("American mathematics from 1940... "), I calculated
that my share of the work took about 150 hours; I shudder to think
how many man-hours the whole group put in. A few of my hours went
toward preparing the lecture (as opposed to the paper). I talked it, the
whole thing, out loud, and then, I talked it again, the whole thing, into
a dictaphone. Then I listened to it, from beginning to end, six timesthree times for spots that needed polishing (and which I polished before
the next time), and three more times to get the timing right (and, in
particular, to get a feel for the timing of each part). Once all that was
behind me, and I had prepared the transparencies, I talked the whole
thing through one final rehearsal time (by myself-no audience). That's work.
Archimedes taught us that a small quantity added to itself often
enough becomes a large quantity (or, in proverbial terms, that every
little bit helps). When it comes 10 accomplishing the bulk of the world's
work, and, in particular, the work of a mathematician, whether it is
proving a theorem, writing a book, teaching a course, chairing a department, or editing a journal, I claim credit for the formulation of the
converse: Archimedes's way is the only way to get something done. Do
a small bit, steadily every day, with no exception, with no holiday. As
an example, I mention the first edition of my Hilbert Space Problem
Book, which had 199 problems. I wrote most of the first draft during
my Miami year, and I forced myself, compulsively, to write a problem
a day. That doesn't mean that it took 199 days to write the whole book
~the total came to about three times that many.
As for "never give up", that doesn't need explanation, and I have
been trying to illustrate it with anecdotal evidence all along, but here,
just for fun, is a pertinent little story. Somewhere around 1980 I was
invited to give a talk to a "general" audience, and, having given it, I
wrote it up and submitted it for publication in The Mathematics
Teacher. In due course I received reports from a couple of referees; they
read, in part, as follows ... The author apparently feels she/he is illustrating the thrill and power of abstraction. While his/her examples contain
the potential for doing that, I don't feel that the presentation creates
that effect. ... Much of the problem with the paper is that of a meandering style. The progression of ideas is not c1ear.... The mathematical
topics focused on are of only moderate interest." The paper was firmly
rejected. I didn't give up~ I just shrugged my shoulders, and submitted
the same article, word for word, to what was then called The Two-Year
College Mathematics Journal. It was accepted and printed, and, a year
after that, it received the P61ya award from the Association.
All these prescriptions and descriptions about how to be a mathematician arose, inevitably, from my own attempts to become one. Nobody can tell you what mathematicians should do, and I am not completely sure I know what in fact they do all I can really say is what
I did.
How close did I come'? What is my total mathematical contribution?
The first answers that occur to me are a small but pretty proof (the
monotone class theorem), a few reasonable theorems (mainly in my
ergodic papers" Approximation theories ... " and .• In general a measurepreserving theorem is mixing "), and a good idea in logic (polyadic algebras).
One thing that I am pretty good at is asking questions. Given a
mathematical problem that I understand the statement of, one whose
history I know something about and that I spent some time studying,
one for which I am moderately up to date with the standard techniques
...... given those circumstances, I have some talent for identifying and
formulating the central questions. If I look hard at a question for a
month, try to answer it, and fail, then I am confident that it is not
trivial. I am confident that a question like that is fit for mathematicians
much better than I; when one of them solves it, he is bound to feel at
least a small tingle of pride and pleasure. (Examples: the power inequality, and the Weyl-von Neumann theorem for normal operators.)
Related to the questions I have asked are the concepts I discovered
and introduced, notably subnormal and quasitriangular operators, and,
possibly, capacity in Banach algebras. Important theories have grown
out of such concepts; I think it's fair 10 call them contributions.
I wrote a few good surveys and some pretty good books. Perhaps the
best are Finite-Dimensional Vector Spaces and A Hilbert Space Problem
Book-but then my vote on such matters is probably the one that weighs
the least.
My most nearly immortal contributions are an abbreviation and a
typographical symbol. I invented "iff", for "if and only if" - but I could
never believe that I was really its first inventor. I am quite prepared to
believe that it existed before me, but I don't know that it did, and my
invention (re-invention'1) of it is what spread it through the mathematical
world. The symbol is definitely not my invention-it appeared in popular
magazines (not mathematical ones) before I adopted it, but, once again,
I seem to have introduced it into mathematics. It is the symbol that
sometimes looks like D, and is used to indicate an end, usually the end of
a proof. It is most frequently called the "tombstone ", but at least one
generous author referred to it as the "halmos".
That's it; that's a life, that's a career. I was, in I think decreasing
order of quality, a writer, an editor, a teacher, and a research mathematician.
What's next? The writing of this book took a lot out of me. It took
a year and a half of time and energy, during which I paid no attention
to research. It was a deliberate gamble. 1 wanted to write this book, I
wasn't at all sure that I could do it the way I dreamt, I wasn't sure I
could tell the audience I had in mind what I wanted to say. If it turns
out that I succeeded, I'll be happy; if not, I'll be sad. But, in either case,
I'm not ready to crawl into a hole and pull it in after me. I'd like to
write some more mathematics, to teach some more, and, who knows,
even to prove a theorem. I'm going to try, that's for sure. I thought, I
taught, I wrote, and I talked mathematics for fifty years, and I am glad
I did. I wanted to be a mathematician. I still do.
Index of Photographs
Ambrose, W. 48
Anosov, D. V. 302
Bishop, E. 162
Brown, H. A. 364
Collingwood, E. 292
Conway, J. B. 364
Day, M. M. 231
de Branges, L. 326
Doob, J. L. 52
Eberlein, W. F. 242
Eilenberg, S. 248
Everett, H. S. 133
Feller, W. 94
Fillmore, P. A. 364
Fomin, S. V. 306
Furniss, W. T. 345
Gamkrelidze, R. V. 295
Gelfand, L M. 306
Halmos, P. R.
1931 21
1947 242
1949 247
1970 368
1980 386
Indiana University, functional
analysts 364
Institute for Advanced Study,
Fuld Hall common room 242
Kaplan, S. 242
Kirillov, A. A. 308
Kolmogorov, A. N. 309
Krein, M. G. 313
Lefschetz, S., letter from 87
Leibler, R. A. 242
ios, J. 316
Mac Lane, S. 248
Martin, W. T. 109
Massera, J. 186
Mautner, F. I. 242
McKnight, J. D. 331
McLaughlin, J. E. 332
Milman, D. 386
Minakshisundaram, S. 242
Mishchenko, E. F. 296
Mookini, E. 344
Moore, R. L. 256
Naimark, M. A. 310
Nakata, J. 357
Room. T. G. 281
Rubin, H. 242
Samelson, H. 102
Sarason, D. E. 276
Segal, I. E. 242
Shields, A. L. 72
Silber, J. R. 346
Smithies, F. 389
Spanier, E. H. 242
Stampfli, J. G. 364
Steenrod, N. E. 391
Stone, M. H. 145
von Neumann, J. 92
Wallen, L. J. 347
Williams, J. P. 364
Zorn, M. A. 362
academic titles, call me mister 27
advice, how to give 267, 269
advice to students, don't stay put 52
Alexander, J. W.
and von Neumann 88
demotion 89
lectures 88
Alexandrov, P. S., German
gentleman 310
from Hazlett 45
from Shaw 32
gymnasium 14
high school, college, etc. 17,32
algebraic geometry
at Illinois 36
from Emch 49
algebraic logic
climax 211
flowers 212
representation theorem 212
all work and no play 53
almost periodic functions and Jessen
almost universal forms
versus Ramanujan 42
excitement of progress 57
Ambrose, W.
algebraic geometry exam 49
at Alabama 81
blase graduate student 50
first meeting? 29
music 29-30
pea pool 61
pins Doob down 69
politics 54
the light dawns 48
"American Mathematics from 1940 to the
Day Before Yesterday" 397
AMG-B, AMG-C, etc. 114
Board of Trustees 226
committee service 221
Council 226
Council voting procedure 226, 227
first meeting 78
functions of 226
in-group 224
class report 47
conversion to 56
from Steimley 32
analytic geometry
a revelation 24
mystery of rotation 25
solid 28
Ando, T., at Oberwolfach 386
Annals of Mathematics Studies 96
Anosov, D. V., Russian companion 301
anxiety attack 359
applied mathematician
consulting 116
reformulation 117
at Syracuse 108
converse of theorem 402
reasons better 205
area of ignorance 274
library problems 194
arithmetic in Hungarian II
arithmetic races 12
Aronszajn-Smith, invariant
subspaces 320
associate professor promotion 160
associate, vestigial rank 105
ASTP super course 111
Atiyah-Singer index theorem 398
Atomic Energy Commission
fellowships 217
authors suggest referees 383
automathography vii
axiology, the value of philosophy 39
Azumaya examination 271
bachelor's thesis with Harry Levy 33
Baer, R.
assistants 62
at Illinois 51
integral closure 62
Bagley, R., with McKnight 330
Banach, S.
in Fundamenta 65
book appears 49
book an addiction 78
book out of date 231
fixed point theorem 325
banquet in Russia 311-312
Barankin and Blakers on FDVS
Bartky talks about chairmanship 361
basic concepts of algebraic logic 390
basic minimum mathematics 60
Bateman, H., posthumous notes 294
Bear, H. S., Hawaii chairman 362
personal taste vii
from cranberries 23
Berens, H., linguistic control 387
Berg, I. D., same as Cantor set
theorem 325
Bergman, S., English 172
Berkson, E., helps found Wabash
seminar 373
Berlitz for Spanish 170
Bernoulli numbers in topology 72
Bernstein, J., plays trumpet 242
Bernstein-Robinson, invariant
subs paces 204, 320
Bessel's inequality, computational
proof 4
Beth in Daigneault's thesis 214
Bieberbach conjecture solved 329
Bing, R. H., student of Moore 255
Birkhoff, G. D.
conjectures ergodic case is general
ergodic theorem 68
prolific publisher 41
problem 103
Artin and Cantor 103
Bishop, E.
homework, a new theorem 137
thesis, first and second 161
Blackwell, D.
brilliant freshman 45
Doob's student 80
blitz in chess 85
Blyth, T. S., at St. Andrews 388
BMC, Dundee, 1965 290
Boas, R. P.
changes colors 381
Gaelic review 292
quashes review 120
Bocher's book is a mess 40
Bochner, S., as chairman 353
Bombieri, E., distinguished lecturer in
Bloomington 268
Bonsall, F. F., introduction 291
book editor, how to be 228
book production takes long 144
book review editor, how to be 375-376
book reviewers
where to find them 376
with a conscience 377
without a conscience 378
book reviews
on all of pure mathematics 223
what are they 375
books read, only two 64
Boolean algebra
free 206
as purified picture of mathematics 245
structure 47
article about 238
in Uruguay 185
Brahana, H. R.
metabelian groupie 56
oral exam 60
teaches algebra 40
teaches calculus 27
Bram, J., sits in the back 161
bribe offered 44
Britain in 1958 243
brothers study medicine 19
Brown, A. A., helpful in Princeton 85
Brown, H. A.
Brown-Halmos-Shields on Cesaro 319
Brown-Halmos on Toeplitz 319
turnabout jobs 161,362
undetermined Il 160
Brown, S., newsbreak 323
in the 1920's 5
revisited 284
the second time 315
Bulletin of the AMS, editorial
committee 372
Butzer, P., at Oberwolfach 385
a la Moore 258-259
not for freshmen 26
four step rule 26
Cambridge congress
Lefschetz 163
Schwartz and Hadamard 163
panel on measure theory 164
Campbell-Baker-Hausdorff formula 3
Campbell, Sam, offers loan 83
Cantor set
beautiful or nasty 63
in teaching 272
Cantor's proof not paradox 33
Caratheodory, a good book 58, 127
Carmichael, R. D.
elliptic functions 56
fifty hours for fifty minutes 401
oral exam 60
teaches differential equations 61
teaches number theory 40, 41-42
The Dean 42
Theory of Groups, read and
proofread 65
time for research 322
weak students 57
Castillo, H., Ph.D. exam 171
Cauchy integral formula, less beautiful
than group 3
Cauchy's theorem is "in" 49
Cayley just a rumor 35
CBMS regional conferences 369
Cesaro is normal 320
consult colleagues 351
how to select 349
length of service 352
foster contacts among faculty 354
pay attention to details 353
psychotherapist 361
"realist" 355
nibbles 239
offers 343
secondary job 356
good and bad 350
three greats 353
Chambana daylight time 20
change fields often 82, 156
change notation to understand it 69
Charlottesville meeting of the AMS 79
Chebyshev, P. L., what did he do? 65
Chelsea Publishing Co., in fur shop 159
chemical engineering career 18
equations balance 19
laboratory 23
chess by correspondence 38
Chevalley, C.
at Princeton 90
in Dundee 290, 293
on Council 58
college faculty 131
division faculty 130, I3!
farewell conference 253, 254
fired from 249
glamorous 50's 200
master's exams 145, 146
Math J in the college 131
mathematical advisors 332
outstanding students 152-155
permanent visitors 138
Ph.D. students 159-162
salaries 1959 249
students in the 1950' s 202
The College 122
top curriculum 144
Christmas meetings of the AMS 107
circle squarer, 1T = 4 335
circus in Moscow 299
citizen identification card 91
classical mechanics, proud title 101
Clebsch, R. F. A., in algebraic
geometry 49
Coble, A. B.
beginning negotiation 95
by the short hairs 104
chairman 43
Ph.D. supervisor 43
salary 43
Coburn, father and son 274
Cockcroft, W. R., dean 294
cocky. definition 17
Cohen, P., accomplishment 203, 204
Cohn, P., at St. Andrews 388
Ambrose and Kakutani 99
Ewing 397
Gustafson 397
Leibler 95, 97
Lumer and Schaffer 192
marriage principle 98
Moolgavkar 397
Savage 151
Sunder 371
Vaughan 78
von Neumann 100-101
Wheeler 397
Ziemer 397
college at Chicago 131
Collingwood, E.
offer of a drink 291
Sir 292
colloquium speaker, how to be 268
colloquium talks have a bad name 71
committees are inefficient 372
committees of one 372
bad ones 373
common room
at Chicago 129
in Fine Hall 85
not Commons 85
communication, art versus science 10
communist by allegation 218
commutators, Brown-Pearcy 191
competition in research 323
complete logic is large 207
complex numbers not known 17
computability, how define 244
computer in Uruguay 184
conceptual versus computational 3
Cambridge 162
Edinburgh 243
consistency proofs, fascinating and
admirable 203
consistent logic is small 207
constructive mathematics, a religion
continuity, definition, biggest single
step 205
continuous functions of Hermitian
operators 383
contributions of a life in
mathematics 402-403
Coolidge versus Veblen and Young 33
counterexamples are good examples 63
covering course material 260-262
Cramer's rule, not much fun 203
cranks and Wigner 334
crank papers submitted 382
Crathorne, A. R.
colloquium chairman 106
shaves moustache 37
0° = 7 135
American 16
early acquired 30
cylindric algebras, Tarski et al. 210
Daigneault, A., student via Church 214
Day, M. M., writes Ergebnisse book
de Branges-Rovnyak
Fillmore's gap 328
invariant subspaces claimed 326
many readers 327
press release 328
Research Announcement 327
revised reprint 328
Dedekind cuts are useless 204
Dedekind on Gauss 61
deduction theorem via generated
ideals 207
de la Vallee Poussin. real variable
wisdom 56
demanding high school students 335
democracy in the AMS? 226
departmental recommendations via
Jessie 358
department meetings at Quadrangle
Club 130
Descartes' great victory 25
desk calculators in primeval days 37
Diamond Head Circle poolside
seminar 362
Dickson's Moderrz Algebraic Theories,
badly written 41
dictionaries 394, 395
American Heritage and Webster II 10
Dieudonne, J.
committee member 392
recreational mathematics 325
dilation theory solves problems 157
dirty tricks in teaching 271
distinguished professor of the wrong
sex 367
distinguished visitor, how to be 268
Dogpatch is Oak Ridge 114
Doob. J. L.
arrives at Illinois 50
book as text 75
dedication like a monument 370
exploits ergodic theorem 68
first students 80
intellectual honesty or elegance 69
lecturer and teacher 51-52
martingale next to reprints 370
no research course 51
oral exam 60
suggests thesis direction 74
teaches topology 51, 61
thesis supervisor 61, 76
understands Oppenheimer 240
Douglas, R. G.
in Oberwolfach 384
reads de Branges-Rovnyak 327
Dowker, C. H., von Neumann's
assistant 93
draft mathematicians 105
Dresden, A., at Swarthmore 122
drinking during prohibition 23
driving in Scotland 291
duality, an instance of "same" 325
Dunford-Schwartz not yet in
existence 231
Dushnik. B., teaches Savage 136
Dutch is funny 59
Dyer, E., Proceedings editor 224
Eckhart Hall, neo-gothic 128
Edinburgh application, Scotch joke 25
Edinburgh congress, department
meeting 244
advertisement and advisor 228
judge 232
editorial work, part time, full time 222
Eells. J., at St. Andrews 388
Eilenberg, S.
thesis students 273
who is? 123
Einstein and Straus 141
Eisenbud, father and son 90
Eisenhart, father and son 85
elliptic functions, dreadful stuff 56
Emch, A.
oral exam 60
teaches algebraic geometry 49
Encyclopedia Britannica improves
writing 395
English in Hungary, sometimes 287
entropy in ergodic theory 236
Erdelyi, A.
in Dundee 290
in St. Andrews 388
on non-standard analysis 294
Erdos, P.
dimension of rational points 91
helps with measure theory 92
nojob 90
Erdos is not Erdos 292
Ergebnisse, how to edit 230
ergodic course at Chicago 137
ergodic theorem, von Neumann and
Birkhoff 68
ergodic theory, book and recent
progress 235
Ernest, J., job nibble 370
Ethiopians in Budapest 289
Euclidean geometry
in high school 18
advanced 34
Everett, H. S., brings Singer to
Chicago 133
Euler's proof in Latin 58
Everitt, W. N.
at Dundee 290
free dinner 291
Ewing, J. H.
likes et al. 399
topological informant 365
examples, typical and degenerate 62
excommunicated research
announcement 329
executive committee at Michigan doesn't
rock 351
existential quantification is an
operator 211
expository papers and surveys 390
at University of Montevideo 180
engineering 182, 183
humanities and sciences 180, 181
false exercise replaced 143
false theorems for use on tests 146
emigrates 14
lends money 83
offers college choice 18
teaches clock 10
FBI security checks, always
necessary? 217
almost rejected 96
second edition rewarding? 238
sold by Savage 150
fee versus honorarium 267
Feinstein, A., changes his mind 146
Feller, W., writes severe review 94
fellowship applicants a pitiful lot 335
Fermat's last theorem, often proved 335
Fibonacci sequence, rabbit story 18
fiction as lies 8
Fiji on way to Sydney 282
filling in manuscripts 76
Fillmore, P., finds gap 328
Fine Hall
Platonic 84
teas 85
finishing a book six times 143
finite simple groups versus Janko 282
fired from Chicago 249
first professors at Institute 84
Fisher, R. A., not a mathematician 65
Foia~, C.
and Krein 314
at Gelfand seminar 307
Fomin, S. V.
as translator 305
house, when painted? 306
Ford, L., father and son 112
foreign accent
bane of existence 16
how to say "what" 16
not eliminated 106
formal power series and convergence
meets ship 173
at Instituto 185
"Foundations of Probability"
corrected by Robbins 113, 114
exposition or survey 390
Fourier, learns something new 205
Fraenkel complains about review 223
free groups, like propositional
calculus 206
French exam, free translation 38
Friday evening discussion 54
Friedman, M., friend of Savage 149
Fubini, G., at Institute 89
Fuchs, L., Budapest host 285,286
Fundamenta, great journal 65
Furniss, W. T., Hawaii dean 345
Galler, B., inherited from Stone
Galois, just a rumor 35
Galuten, A., becomes a publisher 159
gambling systems can't win 78, 79
game theory is born 100
Gamkrelidze and Mishchenko
opera 300
salad bar 296
Garding, L.
English 172
Riesz's assistant 138
Gardner, M., columns 237
Gaul's three parts 31
Gauss as examiner 61
Gelfand, I. M.
always late 306
representation via Stone 245
three ring seminar 307
Gellert Hotel, no soap 286
general analysis at Chicago 122
generic means nothing in particular 49
geometry via Graustein 32
Gergen wants no refugees 80
course in college 22
learn at two 9
reading mathematics 58
Gillman, L., 1976 meeting at San
Antonio 397
"Glimpse into Hilbert Space" 390
Go is deep 86
goddam foreigner 15
Godel, K.
accomplishment 203, 204
at Institute 90
completeness theorem 212
computability 244
no stories 141
proof 212
Goldman, 0., elderly at 21 142
Gomory, R., in Sydney 281
Gonzalez Dominguez, A., interview 169
Gottschalk teaches philosophy 33-34
grades on languages 136
Graduate Club at Illinois 66
Granville, Smith, and Longley
the infamous 26
the light dawns 48
Green's theorem, a foggy mystery 32
Grundlehren is not Ergebnisse 230
Guggenheim fellowship
before Morrow 40
finally 140
GUM for shopping 299
Gustafson, W. H., algebraic therapist 365
curriculum 16
recitations 13
scary 12
Hadamard, J., dangerous? 163
Hadwin, D., at TCU 369
Hahn, H., among best sources 56, 127
halmos, tombstone 403
Halperin, I. S., square deal petition 216
Hamilton, N. T.
flunks Latin 142
genius manque 333
Hamlet in Spanish 198
handwriting Americanized 7
Harary, F., at St. Andrews 388
Hardy and Littlewood on
collaboration 98
Hardy Pure Mathematics versus
Titchmarsh 47
Harrison, D., calls Sammy Sammy 278
Harrison, K., as student 368
Hartman-Wintner versus Toeplitz 319
Hasse on "higher algebra" 229
Hatcher, A. G.
in the wrong picture 367
on theses 279
hats off to the ladies 30 I
Hausdorff, F.
beautiful and astonishing 57
just a rumor 35
not at Illinois 48
chairmanship offer 343
glamorous visitors 360, 361
hires 20 assistant professors 360
midnight in January 282
offers to Shields and Wallen 347
Hay, G. E.
makes Michigan offer 249
teaches Michigan procedures 254
Hazlett teaches algebra 45
healthy, becoming 240
Heisenberg uncertainty principle 191
Heller, A., takes three jobs 360
Hellinger, E.
a refugee 157
on multiplicity 157
Helson, H., on shifts 254
gigantic in Leningrad 300
white elephant 301
Hewitt, E., house, Porsche, and
secretary 245
high school
civics 17
geometry 18
history 17
in Chicago 15
Hilbert on Ph.D. supervision 272
Hilbert space
and logic 211
book, first as lecture notes 158
book, too short 159
not at Illinois 49
problem book, one a day 402
history of mathematics in Monthly 381
Hobson frustrating 65
Hochschild, G., learns logic 215
Hoersch, V., teaches 4 x 4 matrices 28
honorarium versus fee 267
Hopf, E., invariant measure 236
Hotelling's notes on statistics 65
Howie, J., at St. Andrews 388
Huff, G., spoilsport 43
Hughes, D., wangles Michigan offer
customs inspectors 285
in King's Cross 284
misinformation 318
vocabulary 16
Hungarians en masse 285
coffee during lecture 289
"Comrade Halmos" 289
democracy 316
English spoken 287
gallows humor 317
grumbles about Russian 287
how to reconfirm 288
interview 288
lecture in English 289
lighter after the war 286
middle class standard of living 287
New York Times is bad manners 287
polylogue 288
research institute versus university
scary 284
service is good and bad 317
Hutchins, M., invents the college
Hyde Park, a segregated island
iff, who invented? 403
Iliev, L. G., at Oberwolfach 386
Bachelor of Science 35
first registration 20
freshman curriculum 22
going home to 105
second registration 24
statistics 31, 36
Illinois Institute of Technology, minus
T 157
functional analysis seminar 363-365
goes down 363
invitation 362
law school and music school 365
less than Michigan 366
Ph.D. students, private seminar 367
indiscrete spaces 340
Infantozzi at Instituto 185
infinite regress, what's wrong with
it? 213
"Innovation in Mathematics", a
catastrophe 238
Ambrose fellowship 83
first six professors 84
housing project 142, 240
in 1947 140, 141
in 1957 241
"insignificant" people
office 141, 243
onerous administration
Instituto de Matematica
building 184
computer 184
founded by Laguardia
library 185
people 185
integrally closed subrings
f tt;
is not what you think 271
interpretation is homomorphism 208
interview about gambling 79
invariant measure problem 236
invariant subs paces
Aronszajn-Smith 320
Bernstein-Robinson 204, 320
de Branges-Rovnyak 326
Italian, learned in a day 82
Izumi, S.-I., non-theorem 156
Jacobs, K., photographs 385
Janko, Z., versus Hilbert space 282
Jerison, M., helps found Wabash
seminar 374
Jessen, B.
English 172
lectures on almost periodic
functions 138
Jessie Nakata
makes recommendations 358
the perfect secretary 357
job hunting
after the war 121
for no jobs 79
job pattern in U.S. 329-330
Jones, A.
at Instituto 186
in Brazil 188
jUdging students is easy 147
Julia's Theorie Quantique 79
Kakutani, S., on Haar measure 98
Kalmar, L., lectures in German 287
Kaplansky, I.
algebra and Mac Lane 248
at St. Andrews 384
teaches measure theory 145
Katok, A. B., in Sinai's seminar 313
Kazarinoff ruble exchange 304
Kelley, J. L.
the name is Kelley 370
topological terminology 339
Ketchum, P., teaches analysis 47
Khintchine, A.
is difficult 65
writes postcard 67
Kibbey, D.
Doob's student 80
Real Variable Club 66
Kirillov, A., translates 308
Kleene, S. C., at Institute 89
Klein, F., Elementary Mathematics as
culture 66
Kline, J. R., Secretary of AMS 79
Knopp, K., Theory of Functions
translated 56, 65
knowing students is important
know versus teach 133
Kolmogorov, A. N.
and discomfort 308-309
Grundbegriffe der
Wahrscheinlichkeitsrechnung 49, 65
the new testament 65
Kolmogorov-Sinai, solve conjugacy
problem 235
Krein, M. G.
says yes 313
and Foia§ 314
mailing reprints 314
Krein-Milman theorem 386
Kremlin theater for food 300
Kriegspiel, three board chess 85
Kriete-Trutt versus Cesaro 320
Laguardia, R.
arranges Punta del Este 192
at Cambridge congress 169
English 172
founds Instituto 184
meets ship 173
not on council 182
on time? 197
teaches engineers 183
Uruguay invitation 167
versus deans 183
versus Massera 187
Landau, E.
organization 233
steep number theory 57
versus Dedekind 61
Lane, E. P.
doesn't know Eilenberg 123
for mathematics 10
French and Fortran 9, 10
German, Latin, Hebrew 9
graded 136
Greek and Sanskirt 10
Larkin. P., libraries and bifocals 294
Lasker, E., plays Go 91
Lat is u.s.c., pleasant 384
law courses, here and there 18
law school at Indiana 365
Lawrence, T., at Rad. Lab. 115
Lax, P.
edits de Branges-Rovnyak 327
on shifts 254
English in Hungary 14
to read 6
to write 7
Le Blanc. L., via measure theory 214
lectures have a bad name 71
Lefschetz. S.
handwriting 86, 87
how to describe 395
left leaning affects social life 54
Leibler, R. A., plays pool 61
more beautiful 299
train 303
Levy and Roth needs amplifying 65
Levy, H., teaches geometry 33
Levy, P., incomprehensible 65,81
Liapounov's theorem
a la Kai Rander Buch 156
a la Lindenstrauss 156
as laboratory 355
problems in Argentina 194
technique from Harry Levy 33
Lieber, T. C. Mits books 59
Lima da Silva Dias, Candido, tries to
help 169
Lindemann and Littlewood 292
Linderholm. c., ergodic student and
author 221
Lindstrum, A., Real Variable Club 66
linear algebra
by Moore method 259
for two 106
from von Neumann 41
love-hate 40
not in analytic geometry 25
sold by Savage 150
linguistics is not philology 5
Liouville numbers in teaching 272
Littlewood, J. E.
and Lindemann 293
zero-infinity law 121
log cabin 271
log, not In 271
logarithmic law of A. Weil 123,349
and Hilbert space 211
books to learn from 206
course in Chicago 215
differs from mathematics 203
examines symbols 205
formal and symbolic 202
from Kubitz 31
mathematics dictionary 207
Ph.D. students 212
"recessional" 216
revolutionized 202
summer institute 215
syllogistics 31
Lomnicki and Ulam mistake 68
Lomonosov newsbreak 323
Lorenzen, P., invitation 242
Los, J., linguistics courage 315
Lumer, G.
at Oberwolfach 386
in Belgium 188
odd mathematics 187
swap problems 189
Lytle teaches Euclidean geometry
MacCluer, C., calls me mister 28
Mac Lane, S, chairman 248
Maharam review 120
Maker, P., Real Variable Club 66
Mandelbrot, B., and fractals 377
Markov, A. A., what did he do? 65
marriage problem, with Vaughan 78
Martin, R. S.
as teacher 66
comes to Illinois 51
on Tarski 66
Martin, W. T.
chairman at Syracuse 108
first appointments 108
leaves Syracuse 121
with Netzorg 46
martingale to Doob 370
Massera, J.
helps found Instituto 184
in parliament 188
leave of absence 187
mathematics and politics 186
master key for gentlemen 130
master's degree, consolation prize 42
master's exam in philosophy 39
Mate, A., large cardinals 316
mathematical daydreams 30
mathematical machines, theory of 244
Mathematical Reviews, how it works 120
mathematicians are peripatetic 167
as an art 391
made difficult 222
not deductive 321
Maxwell "explained" 238
Mayer, W., half a homology 88
McCarthy hysteria 219
McKnight, J. D., at Miami 330
McLaughlin, J. E.
on de Branges 328
shares linear algebra vice 333
measure spaces, all the same 100
Measure Theory
begins 128
false exercise 143
how written 143
last word 142
measure theory panel at Cambridge
congress 164
meetings of the AMS 106-107
memos from the chairman 361
Metallurgical Laboratory 114
method of successive approximations
Michal, A. D., peripatetic seminar 373
Michael, E., door buzzer and Saturday
lunch 154
almost Chicago 366
Executive Committee 351
mathematical advisors 333
Ph.D. students 278
terms of offer 250
Miles, H., teaches algebra 28
Miller, G. A., millionaire 56
Miller, Blichfeldt, and Dickson Linear
Groups 56
Miller, J. B., sends Ken Harrison
Milman, D., says they're crazy 387
minutes per week per class 269
mislead or lie 113
monotone class theorem, a proof 402
articles versus notes 382
editorial committee 373
editors 380
sneered at 379
Mookini, E.
Hawaii chairman 343
negotiations 345
Moore, E. H., dictum on
generalization 191
Moore, R. L.
and Lumer 187
"Challenge in the Classroom" 257
colloquium book 255
don't read, think 257
method and covering material 261
student interview 257
Texas, Socratic, discovery 257
versus empty set 255
Morgenstern, 0., and von Neumann
Morley triangle 34
Morrison, P., erudite 237
Morrow, G., villain and hero 38-40
Morse, M., loves math. 401
mathematicians 295
smells 299
visitors 314
musical taste
Haydn vii
Brahms and Bach 29
Nachbin, L
chair candidacy 178
editor of Summa 157
trying to arrange Brazil visit 169
Nagy, Sz. B.
at Oberwolfach 385
honorarium check 132
not in Edinburgh 244
Naimark, M. A., charming host 310
Naive Set Theory, on Hewitt's bed,
mistranslated 246
Nakata, J., perfect secretary 357
napkins, stochastic at Oberwolfach 385
negative three, no-minus three, yes 271
Nelson, E., as Chicago student 154
Netzorg, D.
explains linear algebra 46
formal power series 45
good Illinois Ph.D. 80
Newton, I., and the binomial theorem 31
New York Times in Budapest 287
Nikodym, 0., on Radon-Nikodym 79
Nikolskii, S. M., banquet 311
non-standard analysis for
mathematicians 204
Noriund, N. E., prolific publisher 41
Norse mythology from Flom 34
Northrop, E., at the College 122
Note about notes 229
number theory from Carmichael 40
isolated institute 384
stochastic napkins 385
Olmsted, J., helpful in Princeton 85
ontogeny recapitulates phylogeny 270
operator methods in classical
mechanics 101
Oppenheimer, J. R.
as director 240
in Hungarian 317
optional skipping 75
ordering of employers 165
Oregon State, first job 83
Orloff, D., first student 159
Ornstein, D.
and non-invariant measure 237
in courses 202
Parrott, S., breaks mug 278
passions, two are too many 53
application withdrawn 218
granted 219
patriotism in Hungary 8
Peano, G.
algebra 212
arithmetic 215
axioms 203
pencil sharpening 322
peripatetic seminar 373
permanent visitors at Chicago 138
personal tastes: dark beer, long walks,
TM vii
Ph.D. candidates
private seminar 279
versus advisor 274
Ph.D., do something else 278
Ph.D. rating system 66
Ph.D. supervision
brainstorming 276
competition 274
Hilbert 272
hoarding 274
regular meetings 276
weaning students 277
Ph.D. theses zigzag 275
philology is not linguistics 5
as major? 24
courses 38
of mathematics in Monthly? 381
photograph sources 381
as spectator sport 14
in high school 15
Pitcher, E.
on voting procedure 227
Pitt, H. R., in Nottingham 217
Plessner, A. I., on mUltiplicity 158
Poincare, H.
conjecture and Smale 398
just a rumor 35
pointless spaces 120
polyadic algebras
are mathematics 210
mirror mathematics 213
P6lya's dictum 77
P6lya-Szego and Savage 151
polynomially compact operators 320
pommy is English 281
Pontrjagin, L. S.
discovers duality 210
English 312
Topological Groups, a thriller 93, 200
positive approximation and
Dieudonne 324
Potapova, V. P., on shifts 254
prelims on basic knowledge 60
prepare for classes 135
matrix course 95
song 86
course bad at Illinois 34
is measure theory 325
Proceedings, AMS
division of labor 223
managing editor 224
Proceedings, NAS, in mathematics
library 106
proceedings of conferences are bad 369
professor promotion 160
Prohorov, Y. V., almost not met 311
prompt, plain, brief 334
proof, definition of 205
calculus, what is it? 206
function 208
logic, computational lecture 39
aim for perfection 265
are scholars 268
use medium 266
pseudo-metric, not semi- 339
Punta del Este conference
Establier arrangements 192, 193
going there 193
invitations 193
lectures. monitored 194
note taking 194
Pythagorean theorem classified 115
Quadrangle Club, department
meetings 130
quantification is cylindrification 210
quasitriangular, invariant subspaces? 320
questionnaires make bad theses 335
questions, open-ended 146
Radiation Laboratory invents radar 115
Radjavi-Rosenthal, operator
te.rminology 340
Radon-Nikodjim theorem and sufficient
statistics 151
Rainich, Y., thesis director 275
Ramsey's theorem enters logic 203
random alms in the real world 118
Rankin, R. A., reviews in Gaelic 292
R as a grade 153
Rasiowa and Sikorksi, algebraic logic 210
rating of mathematicians 305
rational points in Hilbert space 91
reading helps teaching 8
reading a paper 39
Real Variable Club at lIlinois 66
real variables at Indiana 366
"Recent Progress in Ergodic
Theory" 235
how to write 334
Institute applicant 337
not the whole truth 337
Ph.D. program 341
publisher 341
Singer 338
should compare 337
recreational mathematics, Dieudonne 325
recruitment committee constituency 352
recursive number theory, brilliant and
ugly 212
reductive algebra, not Hermitian 340
Boolean and prompt 121
how and why 119
referees, two at a time 383
Regellosigkeit 74
regional conferences 369
Reichenbach, H., mystic nonsense 68
rejected papers 321
reprint collection a good thing 67, 68
by the clock 324
in bed 324
in ergodic theory 112
needs tranquility 118
teachers must do it 322
via special cases 324
via writing 323
why do it? 322
Revesz, P., Budapest host 285
how and why 119
of books 223
of papers 120
Rey Pastor interview 169
rhetoric at Illinois 23
Riesz. F. and M. 138
Riesz, M., English 172
Riesz representation theorem, typical
special case 324
Rimoldi, H.
Chicago reminiscences 180
working conditions 18l
Robbins, H.
corrects "Foundations of
Probability" 113, 114
threatens measure theory 127
Robinson, A.
complains about review 223
sends preprint 204
Rogel's Thesaurus, how to use 395
Rohlin, V. A.
and Plessner, on multiplicity 158
as host 303
disagreement 112
Stanley Fry 20
AI Burton et al. 29
Room, T. G., and family 280
Rosenberg, A., editor of Monthly 399
Rosenthal, A., Landau's roommate 60
Rosenthal, P., finishes thesis 278
Rovnyak, J.
defective merchandise 329
with de Branges 326
royalties, authors' and editors' 239
Rudin, M. E., student of Moore 256
Ruibal, Uruguay computer 184
rules, learn them 224
Rumanian school 320
Russell and Whitehead
logic is attractive 202
Principia Mathematica 31
automobiles vandalized 296
bugged room? 297
camera 295
domestic passports 299
doors to hotels 299
hotel service 296
royalty accounts 303
service in dining room 297-298
street maps 298
tipping 297
exchange agreement 290
in Hungary 287
is not funny 59
Saaty, T.
suspicious of . 'Glimpse into Hilbert
Space" 390
St. Andrews colloquium. mathematical
holiday 387-390
St. Isaac's Cathedral is beautiful 30 I
Saks, S.
not enough 127
the old testament 65
"same" mathematics 325
Samelson, H.
arrives 102
birthday scam 103
Sammlung Goschen for straphangers 229
Sammy is Sammy 278
Santa Barbara invitation 370
Sarason, D. E.
dinner guest 277
introvert, lecturer 276
writing thesis 276
Savage, L. J.
joint course 75
learns from Dushnik 136
linear algebra 150
nystagmus 149
pemmican 151
P6lya-Szegii 151
sells FDYS 150
sufficient statistics 151
Wallis 152
SCFAS or Wabash west 375
Schaffer, J. J.
impressive 187
in Pittsburgh 188
swap problems 189
Schreiber, M., thesis surprise 202, 222
Schiffer, M., as committee member 392
Schwartz. L., dangerous') 163
Scientific American. can't enjoy 237
scotophilia 243
second editions, don't do them 238,239
as officialdom 253
help and hindrance 223
Seifert-Threlfall, good book 58
semantic completeness is
semisimplicity 207
semesters and quarters 269
seminar talks are not different n
serendipity in library 65
cop-out 220
departmental. professional,
university 221
for failed scholars 221
to the profession 220
sesquicentennial seminar at Indiana 363
Severi, F .. in algebraic geometry 49
Shabat, B. Y., pays royalties 304
Shangri La is Los Alamos 114
Shaw, J. B., courtly gentleman 32
Sherman. S .. in Fine Hall 93
Shields, A. L., two-man seminar 73
shifts, paper on 254
shift transformations and entropy 236
Shilov, G. E., not met 312
Shirokov, F. Y .. not met 312
shortest letter 337
should I be a mathematicianry 148
Silber, J.
blunt wholeheartedness 346
Texas negotiation 344, 345
simple topological spaces 340
simplify in research 77
Sinai, Y. G.
at circus 299
seminar in English 312
Singer, L M" and Scheidy Everett 132133
Smale. S.
and the Poincare conjecture 398
at Copacabana 93
Smithies, F. and N. at SI. Andrews 388
southern belles 256
Spanier. E., editor 327
at parties 172
Berlitz, Hugo, etc. 170
lecturing in 172
readings 171
through pictures 170
Spanish tutors. Castillo and
Chiaraviglio 171
special cases, source of research 324
spectral inclusion theorems 191, 275
spectra paper. the weakest 95
spouse. believer') 70
editorship 228
fellow editors 232
square roots with Lumer and
Schaffer 191
stable homotopy groups and Bernoulli
numbers 72
stage fright, always 44
Stallings, J. R., and the Poincare
conjecture 398
departments as they grew 36
from Czuber 65
Steenrod, N.
and his committee 39 j
and thesis advice 274
death 393
stochastic processes 236
Steimley teaches higher analysis 32, 40
Steinhaus, H.
in Fundamenta 65
on probability 68
Stokes' theorem, a foggy mystery 32
Stone. M. H.
as chairman 123
book appears 49
chapter on mUltiplicity 69
four theorems 353
graduate curriculum 144
invites Measure Theory 127
promotion news 160
representation via Gelfand 245
resented at Chicago 249
thesis topics 274
Straus, E .. and Einstein 141
Strunk and White, Steenrod's hope 392
study, how to 69
Sturm, G. C. F., who was he~ 60
subnormal operators here to stay 157
sutlicient statistics with Savage 151
Summer Institute 1957 215
Sunder, V. S.
at Indiana 370
at Santa Barbara 371
versus Korotkov and Bukhvalov 371
versus sexist language 371
sun worship 164
super course ASTP 110
surveys and expository papers 390
Swift, W" host to Wabash seminar 374
1964 280
faculty 282
lecture notes lost 283
syllabi, good things 262, 263
syllogisms are obvious 202
symbolic and formal logic 202
syntactic concepts in logic 207
colleagues 108, 109
grades 109
Master's candidates III
standards 109
students III
teaching load 112
Szabados, J., at Oberwolfach
Tamarkin, J. D., exam 271
Tarski, A.
cylindric algebras 210
in Fundamenta 65
on truth 66
secrets 210
teach versus know 133
teaching assignments at Chicago 130
begin with interesting facts 272
by challenge 271
do research 322
don '( lecture 273
freshman algebra 44
give advice 269
know the students 134
prepare 135
television course, expensive folly 359
as roller coaster 76
not lifetime contract 214
"they" say what you said 70
Thompson, J. G., square root of the
shift 202
Thrall, R, M.
as graduate student 41
at Rad. Lab. 115
Tihonov, A. N.
at banquet 311
theorem in marriage problem 78
Timoney, R., on Good Friday 113
tired liberal 55
Titchmarsh, E. C.
double course 272
Theory of Functions versus Hardy 47
Toeplitz matrices, defined 319
tombstone, who invented0 403
topological groups
course and students 200
exam 201
Tomier, E., mystic nonsense 68
topologies for measure-preserving
transformations 112
town versus gown 366
Townsend's Real Variables is "good" 56
train to Leningrad, baffling 303
tranquil Tuesday for research
Wednesday 118
translations into Russian 305
how to prove identities 24
too much 24
trigonometry (cont.)
vicarious 11
Tljitzinsky, W. J.
Baire functions and Borel sets 58
thesis director 274
Truesdell, C. A., review of error 120
truth tables are homomorphisms 207
Trutt, D., in de Branges coterie 326
Tsar Pushka in Kremlin 307
Tukey, J. W., general topology and
Zorn 36
Turing machines define computability 244
first experience 14
logic 31
two-person seminars 72
two referees at a time 383
typing 37
Ulam, S.
Cambridge congress 164
measure theory 127
product spaces 68
unbiased estimation 121
undecidable propositions, fascinating
and admirable 203
undetermined 1\ 160
unity of mathematical thought 211
University of Miami, Jimmy
McKnight 330
University of Washington, visit 245
unreasonable effectiveness of
mathematics 63
arrival 173
automobile horns 198
banquet 197
bus doesn't stop 198
cold 177
computer 184
decision to go 168-170
Hamlet 198
heavenly haven 170
joint appointments 180
lecture audience 190
Martinis 176
meals 174
pensiones 173
probability course 190
red tape 183
research 191
schizophrenia 189
shabby 199
spying 196
Stone's comments 196
strikes 189
student control 183
teaching 188
teas 175
timing (hora Uruguaya)
U.S. Embassy 195
vacation 183
weather 177
Vajda, S., at St. Andrews 388
valueless measures 120
van der Waerden, good book 58
Van Nostrand
and Introduction to Hilbert Space 159
Mathematical Studies 229
undergraduate series 228
a pronoun 209
different sort 215
what is it? 208
Veblen and Young versus Coolidge 33
Veblen, O.
lunches 243
operates 86
vector, what it is not 153
vector space over C 62, 64
Villegas, C.
in Canada 188
nojob 182
Vinogradov, I. M., pontificates 303
von Mises, R.
dicta about Kollektivs 65
mystic nonsense 68
Regel\osigkeit 74
von Neumann, J.
guess on conjugacy problem 236
invariant measures 98
lectures, high-powered 92
the writing of the legend 393
Wabash seminar
history 373
support it 266
Wald, A., a mathematics student 36
beginning 241
near Ann Arbor, and the record 331
Wallen, L. J.
comes to Hawaii 347
in ergodic theory 202
Wallis, A., writes about Savage 152
Wang, T., shows off 47
and the draft 105
exciting but remote 84
Weierstrass, K., just a rumor 35
at faculty meetings 130
L' integration. .. 200
logarithmic law 123, 349
loves math 401
monothetic groups 103
Ph.D. students 333
wants Grothendieck 130
WeIch, F.
on Moore 255
Real Variable Club 66
West, T. T., and haitch 293
Weyl, H.
dictates book 143
interview 89
loves math 401
Weyl-von Neumann theorem 403
white elephant in Hermitage 301
Whittaker and Watson, unlike
Townsend 56
Wielandt, H., on commutators 191
Wiener, N.
as teacher, 134
insecure 107
wife of forty years vii
Wigner, E.
and cranks 334
on applied mathematics 63
Wilder, R. L.
at 80 74
student of Moore 255
Wilf, H., computer science editor 380
Wilhelmina versus vectors 152
Wilks, S. S., a mathematics student 36
Williams, Tennessee, in Hungarian 317
Wintner, A.
on commutators 191
Spektraltheorie sweetens life 82
Artin yes 5
Courant no 5
Lefschetz no 5
versus Bessel's inequality 4
versus numbers 3
Zariski yes 5
Wos, L., waves his hand 155
writing as thinking 8
wrong solutions to trivial problems 120
yellow series, at least two 230
Zaanen, A., at Oberwolfach 386
Zariski, 0., loves math 401
Zeeman, E. C., and the Poincare
conjecture 398
Zermelo-Fraenkel axioms give set
theory 213
zero-infinity law 121
Zorn, M. A.
asks questions 363
exam 271
in German 387
lemma 36
Good Friday 114
Zygmund, A., at faculty meetings 130