Politecnico di Milano 2015
MATHEMATICIANS ON CREATIVITY
- Peter Liljedahl
INTRODUCTION
Politecnico di Milano 2015
Very few mathematicians leave a record of their
emotional investment in their work.
- Nitsa Movshovitz-Hadar
INTRODUCTION
Politecnico di Milano 2015
Very few mathematicians leave a record of their
emotional investment in their work.
- Nitsa Movshovitz-Hadar
INTRODUCTION
Politecnico di Milano 2015
In working on this problem and in general, mathematicians
wander in a fog not knowing what approach or idea will work, or
if indeed any idea will, until by good luck, perhaps some novel
ideas, perhaps some old approaches, conquer the problem.
Mathematicians, in short, are typically somewhat lost and
bewildered most of the time that they are working on a problem.
Once they find solutions, they also have the task of checking that
their ideas really work, and that of writing them up, but these are
routine, unless (as often happens) they uncover minor errors and
imperfections that produce more fog and require more work.
What mathematicians write, however, bears little resemblance to
what they do: they are like people lost in mazes who only describe
their escape routes never their travails inside.
- Dan J. Kleitman
INTRODUCTION
Politecnico di Milano 2015
In working on this problem and in general, mathematicians
wander in a fog not knowing what approach or idea will work, or
if indeed any idea will, until by good luck, perhaps some novel
ideas, perhaps some old approaches, conquer the problem.
Mathematicians, in short, are typically somewhat lost and
bewildered most of the time that they are working on a problem.
Once they find solutions, they also have the task of checking that
their ideas really work, and that of writing them up, but these are
routine, unless (as often happens) they uncover minor errors and
imperfections that produce more fog and require more work.
What mathematicians write, however, bears little resemblance to
what they do: they are like people lost in mazes who only describe
their escape routes never their travails inside.
- Dan J. Kleitman
- Henri Poincaré
INTRODUCTION
Politecnico di Milano 2015
The genesis of mathematical creation is a problem
which should intensely interest the psychologist.
RESEARCH QUESTIONS
Politecnico di Milano 2015
1. In general, what is the genesis of
mathematical creation among research
mathematicians (travails inside)?
2. In particular, what is the role of
illumination (AHA!) in the genesis of
mathematical creation?
HISTORY (1908)
Politecnico di Milano 2015
Mathematical Creation/ Mathematical
Discovery
Creativity involves a period of
conscious work, followed by a
period of unconscious work.
The experience of having an idea
come to mind with "characteristics of
brevity, suddenness, and immediate
certainty".
HISTORY (1902, 1904)
Politecnico di Milano 2015
 Édouard Claparède (1873-1940)
and Théodore Flournoy (18541920)
 30 item survey
 L'Enseignement Mathématique
(1902, 1904)
2.
Towards branch of mathematical science did you feel especially attracted?
3.
Are you more interested in mathematical science per se or in its application to natural phenomena?
4.
Have you a distinct recollection of your manner of working while you were pursuing your studies, when the goal was rather to assimilate the results of others than to indulge in personal research? Have you any interesting information to offer on that point?
5.
After having completed the regular course of mathematical studies (which, for instance, corresponds to the program of the Licence mathematique or of two Licences or of the Aggregation) in what direction did you consider it expedient to continue your studies? Did you
endeavor, in the first place, to obtain a general and extensive knowledge of several parts of science before writing or publishing anything of consequence? Did you, on the contrary, at first try to penetrate rather deeply into a special subject, studying almost exclusively what was
strictly requisite for that purpose, and only afterwards extending your studies little by little? If you have used other methods, can you indicate them briefly? Which one do you prefer?
6.
Among the truths which you have discovered, have you attempted to determine the genesis of those you consider the most valuable?
7.
What, in your estimate, is the role played by chance or inspiration in mathematical discoveries? Is this role always as great as it appears to be?
8.
Have you noticed that, occasionally, discoveries or solutions on a subject entirely foreign to the one you are dealing with occur to you and that these relate to previous unsuccessful research efforts of your?
9.
Would you say that your principal discoveries have been the result of deliberate endeavor in a definite direction, or have they arisen, so to speak, in your mind?
10. Have you ever worked in your sleep or have you found in dreams the answers to problems? Or, when you waken in the morning, do solutions which you had vainly sought the night before, or even days before, or quite unexpected discoveries, present themselves ready-made
to your mind?
11. When you have arrived at a conclusion about something you are investigating with a view to the publication of your findings, do you immediately write down the part of your work to which your discovery applies; or do you let your conclusions accumulate in the form of notes
and begin the redaction of the work only when its contents are important enough?
12. Generally speaking, how much importance do you attach to reading for mathematical research? What advice in this respect would you give to a young mathematician who has had the usual classical education?
13. Before beginning a piece of research work, do you first attempt to assimilate what has already been written on that subject?
14. Or do you prefer to leave your mind free to work unbiased and do you only afterwards verify by reading about the subject so as to ascertain just what is your personal contribution to the conclusions reached?
15. As far as method is concerned, do you make any distinction between invention and redacting?
16. Does it seem to you that your habits of work are appreciably the same as they were before you had completed your studies?
17. When you take up a question, do you try to make as general study as possible of the more or less specific problems which occur to you? Do you usually prefer, first to study special cases or a more inclusive one, and then to generalize progressively?
18. In your principal research studies, have you followed the same line of thought steadily and uninterruptedly to the end, or have you laid it aside at times and subsequently taken it up again?
19. What is, in your opinion, the minimum number of hours during the day, week, or the year, which a mathematician who has other demands on his time should devote to mathematics so as to study profitably certain branches of these same mathematics? Do you believe that
one should, if one can, study a little every day, say for one hour at least?
20. Do artistic and literary occupations, especially those of music and poetry, seem to you likely to hamper mathematical invention, or do you think they help it by giving the mind temporary rest?
21. What are your favorite hobbies, pursuits, or chief interests, aside from mathematics, in your leisure time? Do metaphysical, ethical, or religious questions attract or repel you?
22. If you are absorbed by professional duties, how do you fit these into you personal studies?
23. What council, in brief, would you offer to a young man studying mathematics? To a young mathematician who has finished the usual course of study and desires to follow a scientific career?
Questions about daily habits
24. Do you believe that it is beneficial to mathematician to observe a few special rules of hygiene such as diet, regular meals, time for rest, etc.?
25. What do you consider the normal amount of sleep necessary?
26. Would you say that a mathematician's work should be interrupted by other occupations or by physical exercises which are suited to the individual's age and strength?
27. Or, on the contrary, do you think one should devote the whole day to one's work and not allow anything to interfere with it; and, when it is finished, take several days of complete rest? Do you experience definite periods of inspiration and enthusiasm succeeded by periods of
depression and incapacity to work? Have you noticed whether these intervals alternate regularly and, if so, how many days, approximately, does the period of activity last and also the period of inertia? Do physical or meteorological conditions (i.e. temperature, light, darkness,
the season of the year, etc.) exert an appreciable influence on your ability to work?
28. What physical exercise do you do, or have you done as relaxation form mental work? Which do you prefer?
29. Would you rather work in the morning or in the evening?
30. If you take a vacation, do you spend it studying mathematics (if so, to what extent?) or do you devote the entire time to rest and relaxation?
31. Does one work better standing, seated or lying down? Does one work better standing at the blackboard or on paper? To what extent is one disturbed by outside noises? Can one pursue a problem while walking or in a train? How do stimulants or sedatives (tobacco, coffee,
alcohol, etc.) affect the quality and quantity of one’s work?
32. It would be very helpful for the purpose of psychological investigation to know what internal or mental images, what kind of “internal words” mathematicians make use of; whether they are motor, auditory, visual, or mixed, depending on the subject which they are studying.
33. Especially in research thought, do the mental pictures or internal words present themselves in the full consciousness or in the fringe-consciousness? The same question is asked concerning the arguments which these mental pictures or words may symbolize?
HISTORY (1945)
Politecnico di Milano 2015
1.
At what time, as well as you can remember, and under what circumstances did you begin to be interested in mathematical sciences? Have you inherited your liking for mathematical sciences? Were any of your immediate ancestors or members of your family (brothers,
sisters, uncles, cousins, etc.) particularly good at mathematics? Was their influence or example to any extent responsible for your propensity for mathematics?
1. Initiation
2. Incubation
3. Illumination
4. Verification
HISTORY (1945)
Politecnico di Milano 2015
The Psychology of Invention in the
Mathematical Field
Four Stages to the discovery process:
METHODOLOGY
Politecnico di Milano 2015
And yet, the task is inherently difficult. The absence of
sufficient knowledge on this topic is not a matter of a
mere negligence on the part of researchers. There are at
least two reasons why collecting direct observational
data on the AHA! seems like an impossible mission.
First, being a private phenomenon, it is directly
accessible only to the experiencing subject. Second,
being defined as an experience that happens suddenly
and "without warning", it cannot be captured just when
the observer has time and means to observe.
- Anna Sfard
METHODOLOGY
Politecnico di Milano 2015
Resurrected Hadamard's survey - 5 questions (Survey I)
150 prominent mathematicians
 Field’s Medallists
 American Society of Arts & Sciences
 Royal Society
 Academie des Sciences
25 responses to Survey I (16.7%)
Second survey of an additional 2 questions dealing directly with
the AHA! (Survey II)
25 responses to Survey II (100%)
Sorted data according to questions and emergent themes
(analytic induction).
NO PSEUDONYMS
METHODOLOGY – Survey I
Politecnico di Milano 2015
1. Would you say that your principle discoveries
have been the result of deliberate endeavour in a
definite direction, or have they arisen, so to speak,
spontaneously? Have you a specific anecdote of a
moment of insight/inspiration/illumination that
would demonstrate this? [Hadamard # 9]
2. How much of mathematical creation do you
attribute to chance, insight, inspiration, or
illumination? Have you come to rely on this in any
way? [Hadamard# 7]
METHODOLOGY – Survey I
Politecnico di Milano 2015
3. Could you comment on the differences in the
manner in which you work when you are trying to
assimilate the results of others (learning
mathematics) as compared to when you are
indulging in personal research (creating
mathematics)? [Hadamard # 4]
4. Have your methods of learning and creating
mathematics changed since you were a student?
How so? [Hadamard # 16]
5. Among your greatest works have you ever
attempted to discern the origin of the ideas that
lead you to your discoveries? Could you comment
on the creative processes that led you to your
discoveries? [Hadamard # 6]
METHODOLOGY – Survey II
Politecnico di Milano 2015
6. How do you know that you have had an AHA!
experience? That is, what qualities and elements
about the experience set it apart from other
experiences.
7. What qualities and elements of the AHA!
experience serve to regulate the intensity of the
experience? This is assuming that you have had
more than one such experience and they have
been of different intensities.
… 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.
Sincerely yours,
Enrico Bombieri
FIRST RESPONSE
Politecnico di Milano 2015
Dear Peter,
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
Obviously you work like hell and once in while you notice
something really unexpected.
- David L. Donoho
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
I assimilate the work of others best through personal
contact and being able to question them directly.
- Jerry Marsden
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
I am impatient when reading work by others and usually
try to work things out myself.
- Dick Askey
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
Understanding others is often a painful process until one
suddenly goes beyond the details and sees the whole of
what's going on.
- Solomon Feferman
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
I'm convinced that I do my best work while asleep. The
evidence for this is that I often wake up with the solution
to a problem, or at least with a clear idea of how to
proceed to solve it.
- Charles Peskin
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
Rather, I stop and ask myself: What did I really find?
What is next? Sometimes this is the first step for real
progress.
- Enrico Bombieri
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
What I think you mean by an AHA! experience comes at
the moment when something mathematically significant
falls into place.
- Wendell Fleming
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
I can be talking to a colleague or my wife or eating
breakfast and suddenly, like a voice from the blue, I get
told what to do.
- Jerry Marsden
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
You must try and fail by deliberate efforts, and then rely
on a sudden inspiration or intuition or if you prefer to call
it luck.
- Dan J. Kleitman
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
Do I experience feelings of illumination? Rarely, except
in connection with chance, whose offerings I treasure.
- Benoit B. Mandelbrot
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Role of significance
Value of talking
Context of illumination
Importance of re-creating
Failures and wrong ideas
De-emphasis of details
Contribution of chance
Work while sleeping
Use of metaphors
Metacognition
Gaps in the field
My approach to research consists in looking to the mathematical
landscape, taking notice of the things I like and judge interesting and
of those I don't care about, and then trying to imagine what should be
next. If you see a bridge across a river, you try to imagine what lies
on the other shore. If you see a mountain pass between two high
mountains, you try to imagine what is in the valley you don't see yet
but secretly know must be there.
- Enrico Bombieri
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
In my discoveries the general direction or topic has been
planned, of course. However after that one has to search the
terrain until one finds an opening (or gives up), and where that
is cannot be planned.
- Gerd Faltings
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
Extend Hadamard's Work
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
Role of significance
Context of illumination
Failures and wrong ideas
Contribution of chance
Use of metaphors
Gaps in the field
Non-Obvious Pedagogical Implication
EMERGENT THEMES
Politecnico di Milano 2015
Persistent work
Value of talking
Importance of re-creating
De-emphasis of details
Work while sleeping
Metacognition
It is the activity in which the human mind seems to
take the least from the outside world, in which it acts
or seems to act only of itself and on itself.
- Poincaré
CONCLUSIONS
Politecnico di Milano 2015
1. Mathematical creation is largely a creative
endeavour.
CONCLUSIONS
Politecnico di Milano 2015
1. Mathematical creation is largely a creative
endeavour.
2. The AHA! is an affective experience.
Politecnico di Milano 2015
MATHEMATICIANS ON CREATIVITY
MATHEMATICIANS ON CREATIVITY
Politecnico di Milano 2015
This description fits Poincaré's scenario so perfectly
that I worry that I have embroidered the tale and
rearranged it to make it fit. But I'm pretty sure that it
really did happen the way I've just told you.
- Ian Stewart
Chance favours the prepared mind.
- Louis Pasteur
MATHEMATICIANS ON CREATIVITY
Politecnico di Milano 2015
Chance will favour only those who are prepared.
- Wendell Fleming
- Andrew Wiles, 1997
It is like going into an unfamiliar hotel room late at night without
knowing even where to switch on the light. You stumble around in
the dark room, perceive confused black masses, feel one or the
other piece of furniture as you are groping for the switch. Then,
having found it, you turn on the light and everything becomes
clear.
- George Pólya, 1965
MATHEMATICIANS ON CREATIVITY
Politecnico di Milano 2015
Perhaps I could best describe my experience of doing mathematics
in terms of entering a dark mansion. One goes into the first room,
and it's dark, completely dark. One stumbles around bumping into
the furniture, and gradually, you learn where each piece of
furniture is, and finally, after six months or so, you find the light
switch. You turn it on, and suddenly, it's all illuminated.
liljedahl@sfu.ca
www.peterliljedahl.com/presentations
Politecnico di Milano 2015
THANK YOU!