Not only for
the participants of Impanga, we collect here some thoughts, related to

how to carry
research

in mathematics, and - more generally - in science...

David
Hilbert:

We shall know."

Aldo Andreotti:

"Without optimism no mathematical theorem can be proved."

Alfréd Rényi:

* "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."

Euripides:

"Cleverness is not
wisdom."

J.- M. Hoene Wroński:

"The
search of Truth is a testimony to the possibility of
finding it."

(Translation of
the inscription on Wroński's tomb in Neuilly.)

Alexandre Grothendieck:

"... one
night ... I realized that the DESIRE to know and the POWER

to know and to discover are one and the same thing."

(Translation
from: A. Grothendieck Harvests and Sowings vol. I,
p.94.)

Nicolaus
Copernicus:

"Mathematics is written for mathematicians."

(*De
Revolutionibus)*

Isaac Newton:

"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."

Issai
Schur:

"I feel
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
common discovery."

Godfrey H. Hardy:

"Young men
should prove theorems,

old men should write
books."

J. E.
Littlewood (about Ramanujan):

"Every positive integer was one of his personal friends."

In the 30. some excellent
mathematician* said:

Nowadays, there
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 toLittlewood's Miscellany, Cambridge University Press, 1986.

Albert
Einstein:

"I believe
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
is

nothing but the outcome of earlier intellectual experience."

"The
important thing is not to stop questioning; curiosity has
its

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."

David
Hibert:

"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' . "

Goro Shimura:

"Taniyama
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."

Andrey
Tjurin to PP in Paris in 2000:

"Mistakes,
gaps in proofs, ... - all such things are in maths
unimportant.

Only UNDERSTANDING is really important !"

Henri Poincaré:

"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."

"The
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
better."

"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:*

*So
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
seeks.”*

*Dichiarazione dell'Impresa generale della*

* nuova Accademia
Peloritana detta de' Pericolanti*, Messina,
1729)

Igor Dolgachev:

"By
combining two trivial things,

usually you obtain something nontrivial."

Alain
Lascoux to PP in 1978:

"Young
man: don't be afraid to think about trivialities!"

Corrado de Concini:

"If a
group acts on some mathematical object,

one has to use it."

Maria Skłodowska-Curie:

"In
science, we must be interested in things, not in persons."

"There
are sadistic scientists who hurry to hunt down errors

instead of establishing the truth."

Samuel
Beckett:

- " I try. I fail. I try again. I fail better."

Winston
Churchill:

"Success
consists of going from failure

to failure without loss of enthusiasm. "

Adrian Mathesis:

*"All great theorems were discovered after
midnight."*

Yuri Manin:

"Good proofs are proofs that make us wiser."

Joseph Silverman:

"No book is ever
free from error or incapable of being improved."

Piotr Blass about Oscar
Zariski:

"Zariski was
serious and professional about mathematics and that was
picked up by all his students. He claimed to be 'slow',
which forced people to really explain things. His approach
was to do something almost every day. A lemma a day... (he
used to say)."

William Fulton to PP in 1997:

"A mathematical
book should tell some story."

Hugo
Steinhaus:

"Maybe
you will find the following fact concerning two Polish
mathematicians -

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."

Stefan Banach:

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.

Corrado
de Concini to PP in Pisa in 1993:

"When de
Giorgi gets stuck, he never consults a book or paper;

he always tries
to find a solution by himself."

Richard
Feynman:

"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."

Oliver Heaviside:

[Criticized
for using formal mathematical manipulations,

without understanding
how they worked:]

"Should I
refuse a good dinner simply because

I do not understand the process of digestion?"

Linus
Pauling:

"The best way to
have a good idea is to have a lot of ideas."

Niels Bohr:

"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."

Pierre
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."

Ofer Gabber:

"In
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):

"... If
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)

Irving Kaplansky:

"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
algebraic geometry,

all ideas go through Shafarevich."

Nicolas Bourbaki:

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!"

PP:

"It
is embarassing how many different meanings

the word 'explicit' in contemporary maths has !"

Dominique Foata to PP in Strasbourg in 1984:

"A
mathematician is always alone."

Alain
Lascoux to PP:

"A peace
of mind is what - above all - a mathematician needs in his
work."

** Rumi:**

*"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)]

**Alexandre Grothendieck:**

*"Suppose
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:

"A good
lecture should consist of three parts:

- the
first part should be clear to everybody;

- the
speaker should understand the second part;

- and
then, there is the third part ... "

Here is
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."

Winston
Churchill:

"Exhaust neither
the topic

nor the audience."

“In youth we learn;

in age we understand.”

Mies van der Rohe:

"Less means more."

Stanisław Ulam:

"Whatever is worth saying,

can be stated in fifty words or less."

Now some thoughts about different domains of maths.

Kurt Friedrichs:

"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...''

Henri Poincaré:

*"Later generations will regard Mengenlehre (set theory)*

as a disease from which one has recovered."as a disease from which one has recovered."

Oscar Zariski:

"It is especially true in algebraic geometry that in this domain

the methods employed are at least as important as the results."

Alexandre
Grothendieck, asked why he developed the theory of
schemes:

them, i.e., destroying them is an artificial 'intervention',

it obscures our vision and may led to pathologies."

***

(collected by Piotr Pragacz)