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

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.

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-et-un 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-and-one 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]

Pen-and-ink portrait of Mittag-Leffler by Albert Engström [8].

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]

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,

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. Gauthier-Villars, 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.

[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.