# Politecnico di Milano 2015 MATHEMATICIANS ON CREATIVITY - Peter Liljedahl.

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Politecnico di Milano 2015 MATHEMATICIANS ON CREATIVITY - Peter Liljedahl

Politecnico di Milano 2015 INTRODUCTION Very few mathematicians leave a record of their emotional investment in their work. - Nitsa Movshovitz-Hadar

Politecnico di Milano 2015 INTRODUCTION Very few mathematicians leave a record of their emotional investment in their work. - Nitsa Movshovitz-Hadar

Politecnico di Milano 2015 INTRODUCTION 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

Politecnico di Milano 2015 INTRODUCTION 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

Politecnico di Milano 2015 INTRODUCTION The genesis of mathematical creation is a problem which should intensely interest the psychologist. - Henri Poincaré

Politecnico di Milano 2015 RESEARCH QUESTIONS 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?

Politecnico di Milano 2015 HISTORY (1908) 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 ".

Politecnico di Milano 2015 HISTORY (1902, 1904)  Édouard Claparède (1873-1940) and Théodore Flournoy (1854- 1920)  30 item survey  L'Enseignement Mathématique (1902, 1904)

Politecnico di Milano 2015 HISTORY (1945) The Psychology of Invention in the Mathematical Field Four Stages to the discovery process:  Initiation  Incubation  Illumination  Verification

Politecnico di Milano 2015 METHODOLOGY 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

Politecnico di Milano 2015 METHODOLOGY 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

Politecnico di Milano 2015 METHODOLOGY – Survey I 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]

Politecnico di Milano 2015 METHODOLOGY – Survey I 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]

Politecnico di Milano 2015 METHODOLOGY – Survey II 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.

Politecnico di Milano 2015 FIRST RESPONSE Dear Peter, … 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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 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

Politecnico di Milano 2015 EMERGENT THEMES 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 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 EMERGENT THEMES 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

Politecnico di Milano 2015 CONCLUSIONS 1.Mathematical creation is largely a creative endeavour. 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

Politecnico di Milano 2015 CONCLUSIONS 1.Mathematical creation is largely a creative endeavour. 2.The AHA! is an affective experience.

Politecnico di Milano 2015 MATHEMATICIANS ON CREATIVITY

Politecnico di Milano 2015 MATHEMATICIANS ON CREATIVITY 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

Politecnico di Milano 2015 MATHEMATICIANS ON CREATIVITY Chance will favour only those who are prepared. - Wendell Fleming Chance favours the prepared mind. - Louis Pasteur

Politecnico di Milano 2015 MATHEMATICIANS ON CREATIVITY 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. - 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

Politecnico di Milano 2015 THANK YOU! liljedahl@sfu.ca www.peterliljedahl.com/presentations

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