topics: |  math | philosophy | biography

Very well-written memoirs of a leading mathematician, full of acidic
observations. 

Gian-Carlo Rota was born in Italy, where he went to school through the ninth
grade. He attended high school in Quito, Ecuador, and entered Princeton
University as a freshman in 1950. Three years later, he graduated summa cum
laude and went to Yale, where he received a Ph.D. in 1956 with a thesis in
functional analysis under the direction ofJ. T. Schwartz. After a position at
Harvard, in 1959 he moved to MIT, where he is now professor of mathematics
and philosophy. 

Excerpts

Beauty in mathematics

It is not uncommon for a definition to seem beautiful, especially
when it is new. However, mathematicians are reluctant to admit the
beauty of a definition; it would be interesting to investigate the reasons
for this reluctance. Even when not explicitly acknowledged as such,
beautiful definitions give themselves away by the success they meet.
A peculiarity of twentieth century mathematics is the appearance of
theories where the definitions far exceed the theorems in beauty.

The most common instance of beauty in mathematics is a brilliant
step in an otherwise undistinguished proof. Every budding
mathematician quickly becomes familiar with this instance of mathematical
beauty.

These instances of mathematical beauty are often independent of
each other. A beautiful theorem may not be blessed with an equally
beautiful proof; beautiful theorems with ugly proofs frequently occur.
When a beautiful theorem is missing a beautiful proof, attempts are
made by mathematicians to provide new proofs that will match the
beauty of the theorem, with varying success. It is however impossible
to find beautiful proofs of theorems that are not beautiful.

Alonzo Church=

		[this section became famous after it was referred to in the
		landmark graphic novel [doxiadis-2009-logicomix-an-epic|Logicomix], where the opening 
 		paragraph below is remarked to as a prelude for its theme 
		relating logic with madness.  

		this fulltext can be found at
		http://www.princeton.edu/~mudd/finding_aids/mathoral/pmcxrota.htm


It cannot be a complete coincidence that several outstanding logicians of the
twentieth century found shelter in asylums at some time in their lives:
Cantor, Zermelo, Godel, Peano, and Post are some. Alonzo Church was one of
the saner among them, though in some ways his behavior must be classified as
strange, even by mathematicians' standards.

Alonzo Church lecturing at Princeton. The smiling visage at right undermines some of the dark aspects in Rota's tale. sources: (princeton archives) (wikipedia)

He looked like a cross between a panda and a large owl. He spoke softly in complete paragraphs which seemed to have been read out of a book, evenly and slowly enunciated, as by a talking machine. When interrupted, he would pause for an uncomfortably long period to recover the thread of the argument. He never made casual remarks: they did not belong in the baggage of formal logic. For example, he would not say, "It is raining." Such a statement, taken in isolation, makes no sense. (Whether it is actually raining or not does not matter; what matters is consistence). He would say instead, "I must postpone my departure for Nassau Street, inasmuch as it is raining, a fact which I can verify by looking out the window/' (These were not his exact words). Gilbert Ryle has criticized philosophers for testing their theories of language with examples which are never used in ordinary speech. Church's discourse was precisely one such example.

He had unusual working habits. He could be seen in a corridor in Fine Hall at any time of day or night, rather like the Phantom of the Opera. Once, on Christmas day, I decided to go to the Fine Hall library (which was always open) to look up something. I met Church on the stairs. He greeted me without surprise.

He owned a sizable collection of science-fiction novels, most of which looked
well thumbed. Each volume was mysteriously marked either with a circle or
with a cross. Corrections to wrong page numberings in the table of contents
had been penciled into several volumes.  His one year course in mathematical
logic was one of Princeton University's great offerings. It attracted as many
as four students in 1951 (none of them were philosophy students, it must be
added, to philosophy's discredit). Every lecture began with a ten-minute ceremony of 
erasing the blackboard until it was absolutely spotless. We tried to save 
him the effort by erasing the board before his arrival, but to no avail. 
The ritual could not be disposed of; often it required water, soap, and 
brush, and was followed by another ten minutes of total silence while 
the blackboard was drying. Perhaps he was preparing the lecture while 
erasing; I don't think so. 

His lectures hardly needed any preparation. 
They were a literal repetition of the typewritten text he had written 
over a period of twenty years, a copy of which was to be found upstairs 
in the Fine Hall library. (The manuscript's pages had yellowed with the 
years, and smelled foul. Church's definitive treatise was not published 
for another five years).1 Occasionally, one of the sentences spoken in 
class would be at variance with the text upstairs, and he would warn us 
in advance of the discrepancy between oral and written presentation. 
For greater precision, everything he said (except some fascinating side 
excursions which he invariably prefixed by a sentence like, "I will now 
interrupt and make a meta-mathematical [sic] remark") was carefully 
written down on the blackboard, in large English-style handwriting, 
like that of a grade-school teacher, complete with punctuation and 
paragraphs. Occasionally, he carelessly skipped a letter in a word. At 
first we pointed out these oversights, but we quickly learned that they 
would create a slight panic, so we kept our mouths shut. Once he had 
to use a variant of a previously proved theorem, which differed only by 
a change of notation. After a moment of silence, he turned to the class 
and said, "I could simply say likewise,' but I'd better prove it again." 

It may be asked why anyone would bother to sit in a lecture which 
was the literal repetition of an available text. Such a question would 
betray an oversimplified view of what goes on in a classroom. What 
one really learns in class is what one does not know at the time one is 
learning. The person lecturing to us was logic incarnate. His pauses, 
hesitations, emphases, his betrayals of emotion (however rare) and 
sundry other nonverbal phenomena taught us a lot more logic than 
any written text could. We learned to think in unison with him as he 
spoke, as if following the demonstration of a calisthenics instructor. 
Church's course permanently improved the rigor of our reasoning. 

The course began with the axioms for the propositional calculus 
(those of Russell and Whitehead's Principia Mathematical I believe) 
that take material implication as the only primitive connective. The 
exercises at the end of the first chapter were mere translations of some 
identities of naive set theory in terms of material implication. It took 
me a tremendous effort to prove them, since I was unaware of the fact 
that one could start with an equivalent set of axioms using "and" and 
"or" (where the disjunctive normal form provides automatic proofs) 
and then translate each proof step by step in terms of implication. I 
went to see Church to discuss my difficulties, and far from giving away 
the easy solution, he spent hours with me devising direct proofs using 
implication only. Toward the end of the course I brought to him the 
sheaf of papers containing the solutions to the problems (all problems 
he assigned were optional, since they could not logically be made to 
fit into the formal text). He looked at them as if expecting them, and 
then pulled out of his drawer a note he had just published in Portugaliae 
Mathematical where similar problems were posed for "conditional  
disjunction," a ternary connective he had introduced. Now that I was 
properly trained, he wanted me to repeat the work with conditional 
disjunction as the primitive connective. His graduate students had 
declined a similar request, no doubt because they considered it to be 
beneath them. 

Mathematical logic has not been held in high regard at Princeton, 
then or now. Two minutes before the end of Church's lecture (the 
course met in the largest classroom in Fine Hall), Lefschetz would 
begin to peek through the door. He glared at me and the spotless text 
on the blackboard; sometimes he shook his head to make it clear that he 
considered me a lost cause. The following class was taught by Kodaira, 
at that time a recent arrival from Japan, whose work in geometry 
was revered by everyone in the Princeton main line. The classroom 
was packed during Kodaira's lecture. Even though his English was 
atrocious, his lectures were crystal clear. (Among other things, he 
stuttered. Because of deep-seated prejudices of some of its members, 
the mathematics department refused to appoint him full-time to the 
Princeton faculty). 

I was too young and too shy to have an opinion of my own about 
Church and mathematical logic. I was in love with the subject, and his 
course was my first graduate course. I sensed disapproval all around 
me; only Roger Lyndon (the inventor of spectral sequences), who had 
been my freshman advisor, encouraged me. Shortly afterward he  
himself was encouraged to move to Michigan. Fortunately, I had met 
one of Church's most flamboyant former students, John Kemeny who, 
having just finished his term as a mathematics instructor, was being 
eased — by Lefschetz's gentle hand — into the philosophy department. 
(The following year he left for Dartmouth, where he eventually became 
president). 

Kemeny's seminar in the philosophy of science (which that year 
attracted as many as six students, a record) was refreshing training in 
basic reasoning. Kemeny was not afraid to appear pedestrian, trivial, 
or stupid; what mattered was to respect the facts, to draw distinctions 
even when they clashed with our prejudices, and to avoid black-and- 
white oversimplifications.  Mathematicians have always found  
Kemeny 's common sense revolting. 

"There is no reason why a great mathematician should not also be a 
great bigot," he once said on concluding a discussion whose beginning 
I have by now forgotten. "Look at your teachers in Fine Hall, at how 
they treat one of the greatest living mathematicians, Alonzo Church." 

I left literally speechless. What? These demi-gods of Fine Hall were 
not perfect beings? I had learned from Kemeny a basic lesson: a good 
mathematician is not necessarily a "nice guy" 

[biographical note:
Alonzo Church (1903-1995): He invented (discovered?) The Lambda Calculus,
proved that Peano Arithmetic was undecidable, and articulated what is now
called the Church-Turing Thesis. ]

 

Solomon Lefschetz

No one who talked to Lefschetz failed to be struck by his rudeness. 

He was rude to everyone, even to people who doled out funds in 
Washington and to mathematicians who were his equals. I recall Lefschetz
meeting Zariski, probably in 1957 (while Hironaka was already  
working on the proof of the resolution of singularities for algebraic 
varieties). After exchanging with Zariski warm and loud Jewish  
greetings (in Russian), he proceeded to proclaim loudly (in English) his 
skepticism on the possibility of resolving singularities for all algebraic 
varieties. "Ninety percent proved is zero percent proved!" he retorted 
to Zariski's protestations, as a conversation stopper. He had reacted 
similarly to several other previous attempts that he had to shoot down. 
Two years later he was proved wrong. However, he had the satisfaction 
of having been wrong only once. 

Solomon Lefschetz was an electrical engineer trained at the Ecole 
Centrale, one of the lesser of the French grandes ecoles. He came to 
America probably because, as a Russian-Jewish refugee, he had trouble 
finding work in France. A few years after arriving in America, an 
accident deprived him of the use of both hands. 

	At the age of 23, while working as a member of the engineering staff
	of the Westinghouse Electric and Manufacturing Company in Pittsburgh,
	he tragically lost both his hands and forearms due to a transformer
	explosion. He was fitted with artificial hands worn inside a pair of
	shiny gloves. Later when he taught, a student would push a piece of
	chalk into his hand at the beginning of class and remove it at the
	end.
		from http://www.robertnowlan.com/pdfs/Lefschetz,%20Solomon.pdf

He went back to school and got a quick Ph.D. in mathematics at Clark
University (which at that time had a livelier graduate school than it has
now). He then accepted instructorships at the Universities of Nebraska and
Kansas, the only means he had to survive. For a few harrowing years he
worked night and day, publishing several substantial papers a year in
topology and algebraic geometry. Most of the ideas of present-day algebraic
topology were either invented or developed (following Poincare's lead) by
Lefschetz in these papers; his discovery that the work of the Italian
algebraic geometers could be recast in topological terms is only slightly
less dramatic.

His colleagues must have been surprised when Lefschetz himself 
started to develop anti-Semitic feelings which were still lingering when 
I was there. One of the first questions he asked me after I met him 
was whether I was Jewish. In the late thirties and forties, he refused to 
admit any Jewish graduate students in mathematics. He claimed that, 
because of the Depression, it was too difficult to get them jobs after 
they earned their Ph.D.'s. He liked and favored red-blooded American 
boyish Wasp types ... 

He despised mathematicians who spent their time giving rigorous or elegant
proofs for arguments which he considered obvious. Once, Spencer and Kodaira,
still associate professors, proudly explained to him a clever new proof they
had found of one of Lefschetz's deeper theorems. "Don't come to me with your
pretty proofs! We don't bother with that baby stuff around here!" was his
reaction. Nonetheless, from that moment on he held Spencer and Kodaira in
high esteem.

He liked to repeat, as an example of mathematical pedantry, the story of one
of E. H. Moore's visits to Princeton, when Moore started a lecture by saying,
"Let a be a point and let b be a point." "But why don't you just say, 'Let a
and b be points!'" asked Lefschetz. "Because a may equal b," answered
Moore. Lefschetz got up and left the lecture room.

Lefschetz was a purely intuitive mathematician. It was said of him 
that he had never given a completely correct proof, but had never made 
a wrong guess either. 

When he was forced to relinquish the chairmanship of the Princeton 
mathematics department for reasons of age, he decided to promote 
Mexican mathematics. His love /hate of the Mexicans got him into 
trouble. Once, in a Mexican train station, he spotted a charro dressed 
in full regalia, complete with a pair of pistols and rows of cartridges 
across his chest. He started making fun of the charro's attire, adding 
some deliberate slurs in his excellent Spanish. His companions feared 
that the charro might react the way Mexicans traditionally react to 
insult. The charro eventually stood up and reached for his pistols. 
Lefschetz looked at him straight in the face and did not back off. There 
were a few seconds of tense silence. "Gringo loco!" said the charro 
finally, and walked away. When Lefschetz decided to leave Mexico and 
come back to the United States, the Mexicans awarded him the Order 
of the Aztec Eagle. 

During Lefschetz's tenure as chairman of the mathematics  
department, Princeton became the world center of mathematics. He had an 
uncanny instinct for sizing up mathematicians' abilities, and he was 
invariably right when sizing up someone in a field where he knew next 
to nothing. In topology, however, his judgment would slip, probably 
because he became partial to work that he half understood. 

His standards of accomplishment in mathematics were so high that 
they spread by contagion to his successors, who maintain them to this 
day. When addressing an entering class of twelve graduate students, 
he told them in no uncertain terms, "Since you have been carefully 
chosen among the most promising undergraduates in mathematics 
in the country, I expect that you will all receive your Ph.D.'s rather 
sooner than later. Maybe one or two of you will go on to become 
mathematicians." 

from history of math at st-andrews.ac.uk

Lefschetz had two artificial hands over which he always wore a shiny black
glove. First thing every morning a graduate student had to push a piece of
chalk into his hand and remove it at the end of the day. The students at
Princeton made up a ditty about Lefschetz:- 

	Here's to Lefschetz, Solomon L.
	Irrepressible as hell
	When he's at last beneath the sod
	He'll then begin to heckle God.

For Lefschetz, independent thinking and originality were what mattered in
mathematical research. Unlike most mathematicians he had no respect for
elegance and if something was to him clearly true, he would consider it at
best a waste of time producing a rigorous argument to verify it. When a
student proudly showed him a clever argument that he had produced to give a
short proof of one of Lefschetz's theorems, rather than compliment the
student, he is claimed to have retorted:-

    Don't come to me with your pretty proofs. We don't bother with that baby
    stuff around here.

Even if there is little truth in a joke which circulated about Lefschetz,
namely that he never wrote a correct proof or stated an incorrect theorem,
there is an underlying truth in it reflecting on his style of mathematics.

Sylvia Nasar gives this vivid description of the impact Lefschetz had on
Princeton [17]:- 

     Entrepreneurial and energetic, Lefschetz was the supercharged human
     locomotive that ... pulled the Princeton department out of genteel
     mediocrity right to the top. He recruited mathematicians with only one
     criterion in mind: research. His high-handed and idiosyncratic editorial
     policies made the Annals of Mathematics, Princeton's once-tired monthly,
     into the most revered mathematical journal in the world. He was
     sometimes accused of caving in to anti-Semitism for refusing to admit
     many Jewish students (his rationale being that nobody would hire them
     when they completed their degrees), but no one denies that he had
     brilliant snap judgement. He exhorted, bossed, and bullied, but with the
     aim of making the department great and turning his students into real
     mathematicians, tough like himself.

He was the editor of the Annals of Mathematics from 1928 to 1958, bringing it
up to the standard of one of the very best world class journals.