Creativity
Carlo Tomasi
Bibliography
• J. Hadamard, The Psychology of Invention in the
Mathematical Field, Dover, New York, NY, 1954
(original 1945).
• N. Wiener, Invention -- The Care and Feeding of
Ideas, MIT Press, Cambridge, MA, 1993.
• J. L. Adams, Conceptual Blockbusting: A Guide to
Better Ideas, Perseus Books Group, New York, NY,
2001.
• J. E. Oliver, The Incomplete Guide to the Art of
Discovery, Columbia University Press, New York, NY,
1991.
• H. Simon, The Sciences of the Artificial, MIT Press,
Cambridge, MA, 1969.
Outline and a Warning
• Can we do without creativity in academia?
• Where does creativity come from in
mathematics, art, science?
• Poincaré’s introspections analyzed by
Hadamard
• But can we learn from this? Is CS a science?
• Simon’s Sciences of the Artificial
• Yes. Yes.
• There will be small fonts, as this talk is based on
readings
Not Quite Necessary
for Academic Success
• For every idea however wrong there are data for which the idea
will work
• For every paper however useless there is a venue that will
publish it
• For every talent however weak there is a school that will hire it
• The band-wagon method: Apply today’s method to anything at
hand
• The naked-emperor method: State it in a complicated way, and
few will argue
• The clique method: Find enough peers like you, and you have a
forum
• Resume padding and the LPU (Least Publishable-Unit)
• This is no fun!
Creativity and Discovery
• Novelty + usefulness
• Discovery is thrill, excitement, and euphoria.
Discovery is the difference between victory
and defeat, between satisfaction and
disappointment, between success and failure.
[Oliver]
• Discovery may be fickle and unpredictable to
the point of exasperation and frustration. It
may be elusive to the point of dismay or the
destruction of a career. It may be addictive to
the point of dereliction of duty. [Oliver]
Jules Henri Poincaré, 1854—1912
• Brother of Raymond Poincaré,
prime minister then president
of France in WWI
• Originator of algebraic topology,
theory of analytic functions of
several complex variables
• Major contributor to algebraic geometry,
number theory, physics, applied math
• Looked at his own thought processes in a
1908 lecture on Mathematical invention to the
Institute Général Psychologique in Paris
Poincaré’s Anecdotes
• Describes his discovery
of a theory of so-called
Fuchsian groups
• Not very important to know
what these are
• Started by thinking that
Fuchsian functions cannot
exist
A tessellation of the unit disc
with hyperbolically isometric
regions.
Tiles are related to each other
by Fuchsian transformations,
which form a group.
Drawing by Maurits Cornelis
(Another creative fellow) → Escher (1898-1972) .
I wanted to represent [Fuchsian] functions by the quotient of two
series; this idea was very conscious and deliberate; the
analogy with elliptic functions guided me. I asked myself what
properties these series must have if they existed, and
succeeded without difficulty in forming the series I have
called thetafuchsian.
Just at that time, I left Caen, where I was living, to go on a
geologic excursion under the auspices of the School of
Mines. The incidents of the travel made me forget my
mathematical work. Having reached Coutances, we entered
an omnibus to go some place or other. At the moment when I
put my foot on the step, the idea came to me, without
anything in my former thoughts seeming to have paved the
way for it, that the transformations I had used to define the
Fuchsian functions were identical with those on nonEuclidean geometry. I did not verify the idea; I should not
have had time, as, upon taking my seat in the omnibus, I
went on with a conversation already commenced, but I felt a
perfect certainty. On my return to Caen, for conscience’ sake,
I verified the results at my leisure.
[Poincaré]
Then I turned my attention to the study of some
arithmetical questions apparently without much
success and without a suspicion of any connection
with my preceding researches. Disgusted with my
failure, I went to spend a few days at the seaside and
thought of something else. One morning, walking on
the bluff, the idea came to me, with just the same
characteristic of brevity, suddenness, and immediate
certainty, that the arithmetic transformations of
indefinite ternary quadratic forms were identical with
those of non-Euclidean geometry. […]
Most striking at first is this appearance of sudden
illumination, a manifest sign of long, unconscious prior
work. The role of this unconscious work in
mathematical invention appears to me incontestable.
[Poincaré]
Jacques Hadamard,1865 – 1963
• French mathematician
• The number of primes
<n grows as fast as n /ln n
• Conjectured by Lebesgue
(1798) and Gauss (1849)
• Hadamard’s proof is based on his theory of
integral functions applied to the Riemann
zeta function The shortest path between two truths in the real domain
passes through the complex domain.
J. H.
Practical application is found by not looking for it, and one can say that
the whole progress of civilization rests on that principle. J. H.
Hadamard’s 1945 Book
• Introspection, his own, Poincaré’s, and
that of others
• Preparation, incubation, intimation,
illumination, formulation
• Follows a theory by English
social psychologist Graham
Wallas (1925, cited by J. H.)
Graham Wallas, 1858--1932
preparation, incubation, intimation,
illumination, formulation
• Conscious, long, hard work.
• Attack all questions “carrying all the outworks, one
after the other. There was one, however, that still
held out, whose fall would involve that of the whole
place. But all my efforts only served at first the better
to show me the difficulty, which indeed was
something.” [Poincaré]
• Errors, dead ends, frustration.
• Result: “After this shaking-up imposed upon them by
our will, these atoms do not return to their primitive
rest. They freely continue their dance.” [Poincaré]
preparation, incubation, intimation,
illumination, formulation
Hadamard’s (or Wallas’) proposal:
• Invention or discovery takes place by
combining ideas
• There are very many combinations, most
useless, few fruitful
• The mind constructs many possible
combinations, unconsciously and at random
• The conscious mind only chooses some that
could be fruitful
Un/subconscious creation of ideas as…
• Divergent, as opposed to
convergent thinking
(Joy Paul Guilford, 1897–1988,
US psychologist)
• Generate, as opposed to test
(Allen Newell, 1927—1992, H. Simon
Herbert Simon, 1916—2001,
fathers of AI, CMU)
• Unconscious or
subconscious?
J. P. Guilford
A. Newell
Unconscious or Subconscious?
• Fringe-subconscious
• Taine, French historian and psychologist, 1828—1893:
“You may compare the mind of a man to the stage of a theatre,
very narrow at the footlights but constantly broadening as it
goes back. At the footlights, there is hardly room for more than
one actor […] As one goes further and further away from the
footlights, there are other figures less and less distinct as they
are more distant from the lights. And beyond these groups, in
the wings and altogether in the background, are innumerable
obscure shapes that a sudden call may bring
forward and even within direct range of the
footlights. Undefined evolutions constantly
take place throughout this seething mass of
actors of all kinds, to furnish the chorus
leaders who in turn, as in a magic lantern
picture, pass before your eyes.”
Hippolyte Taine
preparation, incubation, intimation,
illumination, formulation
• “For some thinkers, while engaged in a creative work,
illumination may be preceded by a kind of warning by which
they are made aware that something of that nature is imminent
without knowing exactly what it will be.” [Hadamard]
• Paul Valéry, French Poet, 1871—1945: “Sometimes I have
observed this moment when a sensation arrives at the mind; it
is as a gleam of light, not so much illuminating as dazzling. This
arrival calls attention, points, rather than
illuminates, and in fine, is itself an enigma
which carries with it the assurance that it can
be postponed. You say `I see, and then
tomorrow I shall see more.’ There is an activity,
a special sensitization; soon you will go into
the dark-room, and the picture will be seen to
emerge.”
Paul Valéry
preparation, incubation, intimation,
illumination, formulation
• “Most striking at first is this appearance of sudden
illumination, a manifest sign of long, unconscious
prior work.” [Poincaré]
• Carl Friedrich Gauss, 1777—1855, on a theorem he
had tried to prove for years:
“Finally, two days ago, I succeeded, not
on account of my painful efforts, but by
the grace of God. Like a sudden flash
of lightning, the riddle happened to be
solved. I myself cannot say what was
the conducting thread which connected
what I previously knew with what made
my success possible.”
C. F. Gauss
Illumination:To Invent Is to Choose
• “It takes two to invent anything. The one makes up
combinations, the other one chooses, recognizes
what he wishes and what is important to him in the
mass of the things that the former has imparted to
him. What we call genius is much less the work of
the first one than the readiness of the second one to
grasp the value of what has been laid before him
and to choose it.” [Valéry]
• Divergent, then convergent thinking (Guilford)
• Generate, then test (Newell and Simon)
Illumination: Choice Criteria
The rules of choice “are extremely fine and delicate. It is almost
impossible to state them precisely; they are felt rather than
formulated. Under these conditions, how can we imagine a
sieve capable of applying them mechanically? The privileged
unconscious phenomena, those susceptible to becoming
conscious, are those which, directly or indirectly, affect most
profoundly our emotional sensibility.
It may be surprising to see emotional sensibility invoked à
propos of mathematical demonstrations which, it would seem,
can interest only the intellect. This would be to forget the feeling
of mathematical beauty, of the harmony of numbers and forms,
of geometric elegance. This is a true aesthetic feeling that all
true mathematicians know, and surely it belongs to emotional
sensibility.” [Poincaré]
preparation, incubation, intimation,
illumination, formulation
• Or formalization: express by writing
• Conscious, disciplined, focused, painstaking
• To verify. “The feeling of absolute certitude which
accompanies the inspiration generally corresponds to
reality; but it may happen that it has deceived us.”
[Hadamard]
• To “precise.” “It never happens that the unconscious
work gives us the results of a somewhat long
calculation already solved in its entirety.” [Poincaré]
• To form “relay results” so that “the outcome is not the
end of the research but one stage of it, so that we
think of utilizing it.” [Hadamard]
Summary of Hadamard
•
•
•
•
•
Preparation: No free lunch. Expect frustration.
Incubation: Random search. Give it time.
Intimation, Illumination: Eureka!
Formulation: Down to business.
Louis Pasteur, 1822—1895:
L. Pasteur
"Chance favors only the prepared
mind.“
• Thomas A. Edison, 1847—1931:
“Genius is one per cent inspiration
and ninety-nine per cent perspiration.” T. A. Edison
Reality Check
• Few scientists complete their careers
without making a discovery of some sort,
yet few make discoveries of major
significance. [Oliver]
• All scientists struggle through prolonged
intervals of absence of discovery and the
accompanying anxiety about a future that
threatens to be discovery free. [Oliver]
Math, Science, and CS
•
•
•
•
Do we care about all of this?
Is CS a science?
Is CS mathematics?
Frank Harary, 1921—2005,
UMich (math) and NMSU (CS),
graph theorist: “Any field that has the word
science in its name is guaranteed thereby
not to be a science.”
CS as a Science
• H. Simon, The Sciences of the Artificial, MIT Press, 1969
• Artificial: contingent to the goals or purposes
of their designer
• Sciences study an object (artificial or natural)
in relation to its environment:
– what function or purpose it serves (clock tells time),
– its interaction with the outer environment
(ship buffeting, moisture)
– how it achieves that purpose (inner environment:
gears or quartz)
• Human rationality in the artificial is akin to natural selection
in the natural: they both make objects adapt to purpose
• Study functions and interactions by simulation
• Simulation shows hard-to-infer consequences of well
understood laws and premises (e.g., weather prediction)
• Poorly understood systems can be simulated by abstracting
only the relevant
Abstract Study with the Computer
The more we are willing to abstract from the detail of a set of
phenomena, the easier it becomes to simulate the phenomena.
Moreover we do not have to know, or guess at, all the internal
structure of the system but only that part of it that is crucial to the
abstraction.
It is fortunate that this is so, for if it were not, the topdown strategy
that built the natural sciences over the past three centuries would
have been infeasible. We knew a great deal about the gross
physical and chemical behavior of matter before we had a
knowledge of molecules, a great deal about molecular chemistry
before we had an atomic theory, and a great deal about atoms
before we had any theory of elementary particles— if indeed we
have such a theory today.
This skyhook-skyscraper construction of science from the roof
down to the yet unconstructed foundations was possible because
the behavior of the system at each level depended on only a very
approximate, simplified, abstracted characterization of the system
at the level next beneath. This is lucky, else the safety of bridges
and airplanes might depend on the correctness of the “Eightfold
Way” of looking at elementary particles. [Simon]
Abstract Study of the Computer
• “No artifact devised by man is so convenient for this kind of
functional description as a digital computer. It is truly protean,
for almost the only ones of its properties that are detectable in
its behavior (when it is operating properly!) are the
organizational properties.” [Simon]
• “The computer is a member of an important family of artifacts
called symbol systems, or more explicitly, physical symbol
systems. Another important member of the family (some of us
think, anthropomorphically, it is the most important) is the
human mind and brain. [Simon]
• Symbol systems are almost the quintessential artifacts, for
adaptivity to an environment is their whole raison d’être. They
are goal-seeking, information-processing systems, usually
enlisted in the service of the larger systems in which they are
incorporated.” [Simon]
Empirical Study of the Computer
“The research that was done to design computer time-sharing
systems is a good example of the study of computer behavior as
an empirical phenomenon. Only fragments of theory were
available to guide the design of a time-sharing system or to
predict how a system of a specified design would actually behave
in an environment of users who placed their several demands
upon it. Most actual designs turned out initially to exhibit serious
deficiencies, and most predictions of performance were startlingly
inaccurate.
Under these circumstances the main route open to the
development and improvement of time-sharing systems was to
build them and see how they behaved. And this is what was done.
They were built, modified, and improved in successive stages.
Perhaps theory could have anticipated these experiments and
made them unnecessary. In fact it didn’t, and I don’t know anyone
intimately acquainted with these exceedingly complex systems
who has very specific ideas as to how it might have done so. To
understand them, the systems had to be constructed, and their
behavior observed.” [Simon]
Summary of Simon
• Science studies
– Internal environment (how it works)
– External environment (how it interacts)
– What function it serves
• Science is both mathematical and empirical, and
proceeds by abstraction
• Computers can be used to simulate by abstraction
• Computers can be studied mathematically and
empirically, often normatively (design) rather than
descriptively (discovery)
• Computer Science is a science
– As the science of symbol systems
– As the mathematical and empirical study of computing
systems
Creativity in CS
• Creativity is fundamental in computer science
when
– Posing a question
– Formulating an abstract model
– Conjecturing an answer
– “Solving” a model
– Designing a system
– Designing an experiment (about a natural or
artificial system)
– Writing a paper
• There is no free lunch: creativity is hard work,
courage, and intellectual generosity
Bibliography
• J. Hadamard, The Psychology of Invention in the
Mathematical Field, Dover, New York, NY, 1954
(original 1945).
• N. Wiener, Invention -- The Care and Feeding of
Ideas, MIT Press, Cambridge, MA, 1993.
• J. L. Adams, Conceptual Blockbusting: A Guide to
Better Ideas, Perseus Books Group, New York, NY,
2001.
• J. E. Oliver, The Incomplete Guide to the Art of
Discovery, Columbia University Press, New York, NY,
1991.
• H. Simon, The Sciences of the Artificial, MIT Press,
Cambridge, MA, 1969.
Be Obsessed!