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 must know.
    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".

 Niels H. Abel:

[A reply to a question about how he got his expertise:]

"By studying the masters and not their pupils."


"Cleverness is not wisdom."

Leonardo da Vinci:

"Simplicity is the ultimate sophistication."

                                                            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."

"The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics."

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.

 Four principles of collaboration of Hardy and 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.


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 -
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."

       (quoted after the  web page of D. Zeilberger)

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."

Carlo Vitali:

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.”

  (excerpt, translated from the 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."

Oscar Zariski:

"A lemma a day... "

"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:

"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."

     (letter to Zofia Pawlikowska-Brożek, 28.06.1969).

                                                                                     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
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. "

           (PP was in the audience.)

Ian G. Macdonald:

"If, in your work, you come to more and more complicated
 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:
"Structures are the weapons of a mathematician."

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."

                           (communicated to PP by Egbert Brieskorn)

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 was in the audience.)                                                     


"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."

Andre Weil:

"God exists since mathematics is consistent,
and the Devil exists since we cannot prove it."


"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)]

Ludwig Mies van der Rohe:

"God is in the details."

"It is better to be good than original."

Alexander 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."

Kenzo Tange:

"Only beautiful is functional."

"Tradition can - to be sure - participate in a creation, but it can no longer
be creative itself."

                                                            Frank Lloyd Wright:

"Love of an idea is the love of God."

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."


Maria von Ebner-Aschenbach:

“In youth we learn;
in age we understand.”

Ludwig Mies van der Rohe:

"Less is 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."

Oscar Zariski:

"It is especially true in algebraic geometry that in this domain
the methods employed are at least as important as the results."

Jean Dieudonne:

"We have not begun to understand the relationship
between combinatorics and conceptual mathematics."

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

"Nilpotent elements are in algebraic geometry by nature! Neglecting
them, i.e., destroying them is an artificial 'intervention',
it obscures our vision and may led to pathologies."


[ collected by Piotr Pragacz  (PP) ]