The scientist does not study nature because it is useful; he studies it because he
delights in it, and he delights in it because it is beautiful. If nature were not
beautiful, it would not be worth knowing, and if nature were not worth knowing,
life would not be worth living.
Henri Poincaré [11]
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more
permanent than theirs, it is because they are made with ideas. …The mathematician’s patterns,
like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must
fit together in a harmonious way. Beauty is the first test: there is no permanent place in the
world for ugly mathematics.
G. H. Hardy [5]
It is a fact that beautiful general concepts do not drop out of the sky. The truth is that, to begin
with, there are definite concrete problems, with all their undivided complexity, and these must be
conquered by individuals relying on brute force. Only then come the axiomatizers and conclude
that instead of straining to break in the door and bloodying one's hands one should have first
constructed a magic key of such and such shape and then the door would have opened quietly, as if
by itself. But they can construct the key only because the successful breakthrough enables them to
study the lock front and back, from the outside and from the inside. Before we can generalize,
formalize and axiomatize there must be mathematical substance. I think that mathematical
substance on which we have practiced formalization in the last few decades is near exhaustion and
I predict that the next generation will face in mathematics a tough time.
Hermann Weyl [14]
My dear Colleague: [4]
In the following, I am trying to answer in brief your questions as well as I am able.
I am not satisfied with those answers and I am willing to answer more questions if
you believe this could be of any advantage for the very interesting and difficult
work you have undertaken.
(A) The words or the language, as they are written or spoken, do not seem to play
any role in my mechanism of thought. The psychical entities which seem to serve
as elements in thought are certain signs and more or less clear images which can
be “voluntarily" reproduced and combined. There is, of course, a certain
connection between those elements and relevant logical concepts. It is also clear
that the desire to arrive finally at logically connected concepts is the emotional
basis of this rather vague play with the above mentioned elements. But taken
from a psychological viewpoint, this combinatory play seems to be the essential
feature in productive thought—before there is any connection with logical
construction in words or other kinds of signs which can be communicated to
others.
(B) The above mentioned elements are, in my case, of visual and some of
muscular type. Conventional words or other signs have to be sought for
laboriously only in a secondary stage, when the mentioned associative play is
sufficiently established and can be reproduced at will.
(C) According to what has been said, the play with the mentioned elements is
aimed to be analogous to certain logical connections one is searching for.
(D) Visual and motor. In a stage when words intervene at all, they are, in my
case, purely auditive, but they interfere only in a secondary stage already
mentioned.
(E) It seems to me that what you call full consciousness is a limit case which can
never be fully accomplished. This seems to me connected with the fact called the
narrowness of consciousness (Enge des Bewusstseins).
Remark: Professor Max Wertheimer has tried to investigate the distinction
between mere associating or combining of reproducible elements and between
understanding (organisches Begreifen); I cannot judge how far his psychological
analysis catches the essential point.
With kind regards…
Albert Einstein
A testimonial from Professor Einstein © The Hebrew University of
Jerusalem. Courtesy of the Einstein Archives Online [2]
My attitude towards mathematics is that most of it is lying out there, sometimes in hidden places,
like gems encased in a rock. You don’t see them on the surface, but you sense that they must be
there and you try to imagine where they are hidden. Suddenly, they gleam brightly in your face
and you don’t know how you stumbled upon them. Maybe they always were in plain view, and
we all are blind from time to time.
Enrico Bombieri
…I find it almost impossible to have a creative thought while sitting at a desk. To the extent I
“discover” things it is almost always while walking or pacing. Of course I can “work things out” at a
desk, or on the computer, but to really “turn things over in my mind” I have to walk around. Similarly
I think Hadamard mentions somewhere that he “thinks with his legs”.
Curtis McMullen
[S]’il y a une chose en mathématique qui (depuis toujours sans doute) me fascine plus que toute
autre, ce n’est ni “le nombre”, ni “la grandeur”, mais toujours la forme. Et parmi les mille-etun visages que choisit la forme pour se révéler à nous, celui qui m’a fasciné plus que tout autre et
continue à me fasciner, c’est la structure cachée dans les choses mathématiques.
[I]f there is one thing in mathematics that fascinates me more than anything else (and doubtless
always has), it is neither “number” nor “size”, but always form. And among the thousand-andone faces whereby form chooses to reveal itself to us, the one that fascinates me more than any
other and continues to fascinate me, is the structure hidden in mathematical things.
Alexandre Grothendieck [3]
At the Universität Bielefeld,
Grothendieck wrote this abstract into
the colloquium book after he spoke
there in 1971:
“Witch’s Kitchen 1971. Riemann-Roch
Theorem: The ‘dernier cri’: The
diagram [displayed] is commutative!
To give an approximate sense to the
statement about f: X  Y, I had to
abuse the listeners’ patience for almost
two hours. A gripping example of how
our thirst for knowledge and
discovery indulges itself more and
more in a logical delirium far removed
from life, while life itself is going to
Hell in a thousand ways—and is
under the threat of final extermination.
High time to change our course!” [6]
The mathematician's best work is art, a high
perfect art, as daring as the most secret dreams of
imagination, clear and limpid. Mathematical
genius and artistic genius touch one another.
Gösta Mittag-Leffler [12]
Pen-and-ink portrait of Mittag-Leffler by Albert
Engström [8].
PI
The admirable number pi:
three point one four one.
All the following digits are also initial,
five nine two because it never ends.
It can’t be comprehended six five three five at a glance,
eight nine by calculation,
seven nine or imagination,
not even three two three eight by wit, that is, by comparison
four six to anything else
two six four three in the world.
The longest snake on earth calls it quits at about forty feet.
Likewise, snakes of myth and legend, though they may hold out a
bit longer.
The pageant of digits comprising the number pi
doesn’t stop at the page's edge.
It goes on across the table, through the air,
over a wall, a leaf, a bird’s nest, clouds, straight into the sky,
through all the bottomless, bloated heavens.
Oh how brief—a mouse tail, a pigtail—is the tail of a comet!
How feeble the star’s ray, bent by bumping up against space!
While here we have two three fifteen three hundred nineteen
my phone number your shirt size the year
nineteen hundred and seventy-three the sixth floor
the number of inhabitants sixty-five cents
hip measurement two fingers a charade, a code,
in which we find hail to thee, blithe spirit, bird thou never wert
alongside ladies and gentlemen, no cause for alarm,
as well as heaven and earth shall pass away,
but not the number pi, oh no, nothing doing,
it keeps right on with its rather remarkable five,
its uncommonly fine eight,
its far from final seven,
nudging, always nudging a sluggish eternity
to continue.
Wislawa Szymborska [13]
FIGURES OF THOUGHT
To lay the logarithmic spiral on
Sea-shell and leaf alike, and see it fit,
To watch the same idea work itself out
In the fighter pilot’s steepening, tightening turn
Onto his target, setting up the kill,
And in the flight of certain wall-eyed bugs
Who cannot see to fly straight into death
But have to cast their sidelong glance at it
And come but cranking to the candle’s flame—
How secret that is, and how privileged
One feels to find the same necessity
Ciphered in forms diverse and otherwise
Without kinship—that is the beautiful
In Nature as in art, not obvious,
Not inaccessible, but just between.
It may diminish some our dry delight
To wonder if everything we are and do
Lies subject to some little law like that;
Hidden in nature, but not deeply so.
Howard Nemerov [10]
Together and holding hands they roam
to the limits of outer space.
Black hole and monopole exhaust
the secret of myths;
Fiber and connections weave to interlace
the roseate clouds.
Evolution equations describe solitons;
Dual curvatures defines instantons.
Surprisingly, Math has earned
its rightful place
for man and in sky;
Fondling flowers with a smile—just wish
nothing is said!
A poem written by Shiing Shen Chern. Provided courtesy of T. Y. Lam. [7]
Sculptures of Helaman Ferguson
Left: Marcel Duchamp’s Nude Descending a Staircase. 1912 and Right: Eliot Elisofon’s
Duchamp desciending a staircase, 1952.
A page from the draft of Riemann’s manuscript “Theorie der
Abel’schen Functionen”, 1857. Courtesy of the Göttingen
University Library [9].
Un des brouillons qu’Evariste Galois laissa sur sa table en partant pour son
duel. (One of the scribbles that Galois left on his table while leaving for his
duel.) [1]
References
[1] Bourgne, Robert and J. P. Azra. Écrits et Mémoires Mathématiques d’Évariste Galois. GauthierVillars, Paris, 1962.
[2] Einstein, Albert. A Testimonial for ‘An Essay on The Psychology of Invention in the
Mathematical Field’ by Jacques Hadamard. Courtesy of Einstein Archives Online.
http://www.alberteinstein.info/
[3] Grothendieck, Alexandre. Récoltes et Semailles: Réflexions et témoignages sur un passé de
mathématicien. Université des Sciences et Techniques du Languedoc, Montpellier, et Centre
National de la Recherche Scientifique, 1986.
[4] Hadamard, Jacques. Essay on the Psychology of Invention in the Mathematical Field. Dover
Publications, New York, NY, 1954.
[5] Hardy, G. H. A Mathematician’s Apology. Cambridge University Press, 1992.
[6] Jackson, Allyn. “Comme Appelé du Néant—As If Summoned from the Void: The Life of
Alexandre Grothendieck.” Notices of the American Mathematical Society 51, no. 10 (November
2004): 1196-1212.
[7] Jackson, Allyn. “Interview with Shiing Shen Chern.” Notices of the American Mathematical
Society 45, no. 7 (August 1998): 1050-1058.
[8] Jackson, Allyn. “The Dream of a Swedish Mathematician: The Mittag-Leffler Institute.” Notices
of the American Mathematical Society 46, no. 9 (October 1999): 860-865.
[9] Monastyrsky, Michael. Riemann, Topology, and Physics. Birkhäuser, Boston, MA, 1999.
[10] Nemerov, Howard. The Western Approaches. University of Chicago Press, Chicago, IL, 1975.
[11] Poincaré, Henri. The Foundations of Science: Science and Hypothesis, The Value of Science, Science and
Method. The Science Press, Lancaster, 1946.
[12] Rose, Nicholas J. Mathematical Maxims and Minims. Rome Press Inc., Raleigh, NC, 1988.
[13] Szymborska, Wislawa. Poems, New and Collected, 1957-1997. Harbourt Brace & Company, New
York, NY, 1998.
[14] Weyl, Hermann. “Part II. Topology and Abstract Algebra as Two Roads of Mathematical
Comprehension.” The American Mathematical Monthly 102, no. 7 (August-September 1995): 646-651.