Not only for
the participants of Impanga, we collect here some thoughts, related to
how to carry
in mathematics, and - more generally - in science...
We shall know."
Aldo Andreotti:"Without optimism no mathematical theorem can be proved."
"If I feel unhappy, I do mathematics to become happy.
If I am happy, I do mathematics to keep happy."
Godfrey H. Hardy:"I am interested in mathematics only as a creative art".
[A reply to a question about how he got his expertise:]
"By studying the masters and not their pupils."
"Cleverness is not
J.- M. Hoene Wroński:
search of Truth is a testimony to the possibility of
the inscription on Wroński's tomb in Neuilly.)
night ... I realized that the DESIRE to know and the POWER
to know and to discover are one and the same thing."
from: A. Grothendieck Harvests and Sowings vol. I,
"Mathematics is written for mathematicians."
"I do not
know what I may appear to the world, but to myself I seem
to have been only like a boy playing on the sea-shore, and
diverting myself in now and then finding a smoother pebble
or a prettier shell than ordinary, whilst the great ocean
of truth lay
all undiscovered before me."
like I am somehow moving outer space. A particular idea
leads me to a nearby star on which I decide to land. Upon
my arrival I realize that somebody already lives there. Am
I disappointed? Of course not. The inhabitant and I are
cordially welcoming each other, and we are happy about our
Godfrey H. Hardy:
should prove theorems,
old men should write books."
Littlewood (about Ramanujan):
"Every positive integer was one of his personal friends."
In the 30. some excellent
are only three really great English mathematicians:
Hardy, Littlewood and Hardy-Littlewood.
1. When one wrote to the other, it was completely indifferent whether what they wrote was right or wrong.
2. When one received a letter from the other, he was under no obligation whatsoever to read it, let alone to answer it.
3. Although it did not really matter if they both thought about the same detail, still, it was preferable that they should not do so.
4. It was quite indifferent if one of them had not
contributed the least bit to the contents of a paper under
their common name.
*) Probably Harald Bohr.
From the collected works of Harald Bohr, quoted by Bela Bollobás in the foreword to Littlewood's Miscellany, Cambridge University Press, 1986.
in intuition and inspiration. Imagination is more
important than knowledge. For knowledge is limited,
whereas imagination embraces the entire world, stimulating
progress, giving birth to evolution. It is, strictly
speaking, a real factor in scientific research."
"A new idea
comes suddenly and in a rather intuitive way. But intuition
nothing but the outcome of earlier intellectual experience."
important thing is not to stop questioning; curiosity has
own reason for existing."
A thought of Jacob P. Murre about Grothendieck:
"He does not strive - at least in the first place -
for generality as such but for naturality."
"One can measure
the importance of a scientific work by
the number of earlier publications rendered superfluous by it."
Samuel Dickstein on Hoene-Wroński:
"His iron nature
required little sleep and food, he begins
work early in the morning and only after a couple of hours of work
he would have a meal saying: 'Now I have earned my day' . "
was gifted with the special capability of making many
mistakes, mostly in the right direction. I envied him for
this and tried to imitate him, but found it quite
difficult to make good mistakes."
Tjurin to PP in Paris in 2000:
"Mistakes, gaps in proofs, ... - all such things are in maths unimportant.
Only UNDERSTANDING is really important !"
"The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful."
principal aim of mathematical education is to develop
certain faculties of the mind, and among these intuition
is not the least precious."
"It is by
logic that we prove, but by intuition that we discover. To
know how to criticize is good, to know how to create is
"To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection."
“As in the sea between Scylla and Charybdis the helmsman is ever in danger, yet he will be thought shrewd and sagacious, if, keeping his ship on a straight course between the two, avoiding the rocks on the one side and the maelstrom on the other, he brings his ship safely to harbour:
in learning, the scholar is tossed between
difficulties and adversities; but he will be
worthy of praise and glory, if, directing his
mind and proper reason around them, likewise
avoiding any impediment or contention, he
penetrates without hindrance into the Truth he
nuova Accademia Peloritana detta de' Pericolanti, Messina, 1729)
combining two trivial things,
usually you obtain something nontrivial."
Lascoux to PP in 1978:
man: don't be afraid to think about trivialities!"
Corrado de Concini:
group acts on some mathematical object,
one has to use it."
science, we must be interested in things, not in persons."
are sadistic scientists who hurry to hunt down errors
instead of establishing the truth."
- " I try. I fail. I try again. I fail better."
consists of going from failure
to failure without loss of enthusiasm. "
"All great theorems were discovered after
"Good proofs are proofs that make us wiser."
"No book is ever
free from error or incapable of being improved."
"A lemma a
"But a mathematical theory cannot
thrive indefinitely on greater and greater
generality. A proper balance must ultimately be maintained between the generality
and the concreteness of the structure studied."
William Fulton to PP in 1997:
book should tell some story."
you will find the following fact concerning two Polish
Hoene-Wroński and Banach - interesting. In Lwów we had an edition of Wroński's work published in Paris and Banach showed me the page written by the philosopher which discussed the 'Highest Law' ; apparently Banach has proven to me that Wroński is not discussing messianic philosophy - the matter concerns expanding arbitrary functions into orthogonal ones."
A mathematician is someone who can find analogies among theorems;
a better one is someone able to see analogies among proofs, and still better
is one who perceives analogies among theories, and it is possible to imagine
one who sees analogies among analogies.
de Concini to PP in Pisa in 1993:
Giorgi gets stuck, he never consults a book or paper;
he always tries to find a solution by himself."
"Science is the
belief in the ignorance of experts."
"I don't know anything, but I do know that everything
is interesting if you go into it deeply enough."
for using formal mathematical manipulations,
without understanding how they worked:]
refuse a good dinner simply because
I do not understand the process of digestion?"
"The best way to
have a good idea is to have a lot of ideas."
"The opposite of
a correct statement is a false statement. The opposite
of a profound truth may well be another profound truth."
About Laurent Schwartz:
According to his teachers, Schwartz was an exceptional student. He was particularly gifted in Latin, Greek and mathematics. One of his teachers told his parents: "Beware, some will say your son has a gift for languages, but he is only interested in the scientific and mathematical aspect of languages: he should become a mathematician."
Cartier about Claude Chevalley:
"Chevalley was a
member of various avant-garde groups, both in politics and
in the art [...] Mathematics was the most important
part of his life, but he did not draw any boundary between
his mathematics and the rest of his life."
maths - contrary to art - it is not sufficient to find
something beautiful; it also must be true."
Shreeram Abhyankar in 2008 at RIMS, speaking to people presenting very general algorithms
to attack the desingularization in characteristic p>o (the desingularization is proved in dimensions
less than or equal to 3):
you prove the desingularization for any dimension, you
prove it, in particular, for dimension 4. So why do
not try to prove it first for dimension 4? This was
actually how Hironaka got his theorem in characteristic 0.
Ian G. Macdonald:
formulas - immediately change the subject!"
(communicated to PP by Corrado De Concini)
"We [he and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free , but when the chips are down we close the office door and compute with matrices like fury."
Andrey Tjurin to PP, in Moscow in 1986, about Igor Shafarevich:
"In our group of
all ideas go through Shafarevich."
Friedrich Hirzebruch about the Colloquium in Bonn in the 50. :
"Like catholics attend the Holy Mass every Sunday, mathematicians
should attend the Mathematical Colloquium every week."
Friedrich Hirzebruch introducing Jacques Tits as a speaker at the MPI in Bonn in the 90s.:
"Jacques Tits played an important role in the mathematical life of Bonn
in the 50s. ; it was he who taught people in Bonn to ask questions!"
"It is embarassing how many different meanings
the word 'explicit' in contemporary maths has !"
Dominique Foata to PP in Strasbourg in 1984:
mathematician is always alone."
Lascoux to PP:
of mind is what - above all - a mathematician needs in his
"All my talk was madness,
I wanted to know what, how and why.
I knocked on a door -
when it opened I found
I was knocking from the INSIDE!"
[this is a modified (by PP) version of the translation by Jonathan Star and Shahram Shiva's "A Garden Beyond Paradise, The Mystical Poetry of Rumi" (Bantam Books, New York, 1992)]
we want to prove a theorem that is a
conjecture. There are two radically
different ways of trying to do this. One is by brute force, the kind of thing we do when
we use a nutcracker to split a nutshell to get the nut inside. But there is another way.
We put the nut in a glass of softening fluid and wait patiently for some time. Then slight
finger pressure suffices for the nut to open by itself."
About LECTURING, there are several
thoughts. The first one is
often attributed to Friedrich Hirzebruch:
lecture should consist of three parts:
first part should be clear to everybody;
speaker should understand the second part;
then, there is the third part ... "
the favourite story of Niels Bohr:
"A small Jewish community, not far from Lublin, got one day a message that a famous Rabbi is supposed to visit Lublin soon to give a series of lectures. The community decided to send to Lublin a young man to follow the Rabbi's lectures. After coming back from Lublin, the young man reported: Rabbi gave three lectures:
- the first lecture was perfect, clear and simple: I understood everything;
- the second lecture was even better, deep and subtle: I understood some major ideas, but Rabbi understood all;
- but the
third lecture was just magnificent; it was an unforgettable intelectual experience: I did
understand nothing, and Rabbi understood not much."
nor the audience."
“In youth we learn;
in age we understand.”
Mies van der Rohe:
"Less means more."
"Whatever is worth saying,
can be stated in fifty words or less."
Now some thoughts about different domains of maths.
"What I don't like about measure theory is that you have to say
'almost everywhere' almost everywhere."
Egbert Brieskorn to PP in Bonn in 2006:
"There is NO singularity theory.
There are only singularities... "
Vladimir Voevodsky (ICM 2002):
"I am sorry that it is esoteric from the start,
but this is K-theory...''
"Later generations will regard Mengenlehre (set theory)
as a disease from which one has recovered."
"It is especially true in algebraic geometry that in this domain
the methods employed are at least as important as the results."
Grothendieck, asked why he developed the theory of
them, i.e., destroying them is an artificial 'intervention',
it obscures our vision and may led to pathologies."
[ collected by Piotr Pragacz (PP) ]