mathematical creativity

7/28/2019 Mathematical Creativity

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The scientist does not study nature because it is useful; he studies it because hedelights in it, and he delights in it because it is beautiful. If nature were notbeautiful, 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 theworld 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 beginwith, there are definite concrete problems, with all their undivided complexity, and these must beconquered by individuals relying on brute force. Only then come the axiomatizers and concludethat instead of straining to break in the door and bloodying one's hands one should have firstconstructed 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 mathematicalsubstance on which we have practiced formalization in the last few decades is near exhaustion andI 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 playany 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 thenarrowness 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 bethere and you try to imagine where they are hidden. Suddenly, they gleam brightly in your faceand 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 adesk, or on the computer, but to really “turn things over in my mind” I have to walk around. SimilarlyI 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 touteautre, 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 etcontinue 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 doubtlessalways 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 anyother and continues to fascinate me, is the structure hidden in mathematical things.

Alexandre Grothendieck [3]



At the Universität Bielefeld,Grothendieck wrote this abstract intothe colloquium book after he spokethere in 1971:

“Witch‟s Kitchen 1971. Riemann-RochTheorem: The „dernier cri‟: Thediagram [displayed] is commutative!To give an approximate sense to the

statement about f: X

Y , I had toabuse the listeners‟ patience for almosttwo hours. A gripping example of howour thirst for knowledge anddiscovery indulges itself more andmore in a logical delirium far removedfrom life, while life itself is going toHell in a thousand ways—and isunder the threat of final extermination.

High time to change our course!” [6]



Pen-and-ink portrait of Mittag-Leffler by AlbertEngström [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,eight nine by calculation,seven nine or imagination,not even three two three eight by wit, that is, by comparison

four six to anything elsetwo 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 pidoesn‟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 nineteenmy phone number your shirt size the year nineteen hundred and seventy-three the sixth floor the number of inhabitants sixty-five centship 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 eternityto 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 outIn the fighter pilot‟s steepening, tightening turn Onto his target, setting up the kill,And in the flight of certain wall-eyed bugsWho cannot see to fly straight into deathBut have to cast their sidelong glance at itAnd come but cranking to the candle‟s flame—

How secret that is, and how privilegedOne feels to find the same necessityCiphered in forms diverse and otherwiseWithout kinship—that is the beautifulIn 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 doLies subject to some little law like that;Hidden in nature, but not deeply so.

Howard Nemerov [10]



Together and holding hands they roamto the limits of outer space.

Black hole and monopole exhaustthe secret of myths;

Fiber and connections weave to interlacethe roseate clouds.

Evolution equations describe solitons;Dual curvatures defines instantons.Surprisingly, Math has earned

its rightful placefor man and in sky;

Fondling flowers with a smile— just wishnothing 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‟sDuchamp desciending a staircase, 1952.



A page from the draft of Riemann‟s manuscript “Theorie derAbel‟schen Functionen”, 1857. Courtesy of the GöttingenUniversity Library [9].



Un des brouillons qu‟Evariste Galois laissa sur sa table en partant pour sonduel. (One of the scribbles that Galois left on his table while leaving for hisduel.) [1]



References

[1] Bourgne, Robert and J. P. Azra. É crits et M émoires Mathématiques d’É variste Galois. Gauthier-

Villars, Paris, 1962.

[2] Einstein, Albert. A Testimonial for „An Essay on The Psychology of Invention in theMathematical Field‟ by Jacques Hadamard. Courtesy of Einstein Archives Online.http://www.alberteinstein.info/

[3] Grothendieck, Alexandre. Récoltes et Semailles: Ré flexions et témoignages sur un passé demathématicien. Université des Sciences et Techniques du Languedoc, Montpellier, et CentreNational de la Recherche Scientifique, 1986.

[4] Hadamard, Jacques. Essay on the Psychology of Invention in the Mathematical Field. DoverPublications, New York, NY, 1954.

[5] Hardy, G. H. A Mathematician’ s Apology. Cambridge University Press, 1992.

[6] Jackson, Allyn. “Comme Appelé du N éant—As If Summoned from the Void: The Life ofAlexandre Grothendieck.” Notices of the American Mathematical Society 51, no. 10 (November2004): 1196-1212.

[7] Jackson, Allyn. “Interview with Shiing Shen Chern.” Notices of the American MathematicalSociety 45, no. 7 (August 1998): 1050-1058.

[8] Jackson, Allyn. “The Dream of a Swedish Mathematician: The Mittag-Leffler Institute.” Noticesof the American Mathematical Society 46, no. 9 (October 1999): 860-865.

http://www.alberteinstein.info/

http://www.alberteinstein.info/



[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, NewYork, NY, 1998.

[14] Weyl, Hermann. “Part II. Topology and Abstract Algebra as Two Roads of MathematicalComprehension.” The American Mathematical Monthly 102, no. 7 (August-September 1995): 646-651.

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