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The psychogenesis of an idea
"From the epistemological point of view we see here how the construction of a new theory is far from being reducible to the accumulation of data, but necessitates an extremely complex structuring of interpretative ideas which are linked to the facts and which enrich them by framing them in context". Jean Piaget, foreword to H.E. Gruber's "Darwin on Man".
Agenda: List the persons, the places, the letters, the voyages, the drafts, the papers, the conferences, etc. Make a graph with all relevant relations among them. Make a comparison with the personal story of some important discovery (example: James Watson The Double Helix, 1969). Follow the different paths leading to other microdiscoveries in your scientific life. And try to arrive at some cognitive and epistemological conclusions.
I start with a list of 27 persons and the places where I have met them, grouped in 6 classes by the kind of help and expertise (general or specific) they have kindly offered me during my research on eye movements and fractals:
1 Piaget J., Geneva, genetic epistemology
Fraisse P., Paris, experimental psychology
Noizet G., Paris, experimental psychology
Bresson P., Paris, cognitive psychology
Inhelder B., Geneva, developmental psychology
Vonèche J., Geneva, developmental psychology
Vidal F., Buenos Aires, Boston/Cambridge, Geneva, history of psychology
Lynch K., Boston/Cambridge, urban planning
2 Lejeune J., Paris, genetics
Mosovich A., Buenos Aires, neurology
3 Reggini H., Buenos Aires, computer sciences, education
Minsky M., Boston/Cambridge, computer sciences
Papert S., Boston/Cambridge, computer sciences, education
4 Mandelbrot, B., Boston/Cambridge, mathematics
González Domínguez A., Buenos Aires, mathematics
Ortiz E., London, mathematics
5 Vuillemin J., Paris, philosophy
Bunge M., Buenos Aires, Kathmandu, philosophy
Coffa A., Buenos Aires, philosophy
6 Eye movements
Vinh Bang, Geneva, developmental psychology
Lévy-Schoen A., Paris, experimental psychology
Kauffman J., Geneva, experimental psychology
Bullinger A., Geneva, experimental psychology
Mehler J., Boston/Cambridge, Paris, cognitive psychology
Mackworth N., Boston/Cambridge, experimental psychology
Guirao, M., Buenos Aires, psychophysics
Garrahan J. P., Buenos Aires, theoretical physics
It is interesting to sum up the different personal meeting places related with my research: Buenos Aires (7), Rio de Janeiro (1), Geneva (7), Paris (7), Boston/Cambridge (5), London (1), Kathmandu (1). In all, South and North America, Europe and Asia! It somehow represents the different stages of my own scientific life, M.D. in Buenos Aires, Ph.D. in Paris, member of the Centre d'Epistémologie Génetique in Geneva, visiting professor in several universities of Brazil, Fulbright Fellow at MIT in Cambridge, Mass. This, I believe, is also a good example of the spread of scientific activities in the modern world. Even in a very limited microdiscovery as this one, the tendency is toward increasing scientific integration and collaboration. This is an important result to emphasize. Once again, in support of my thesis, the similarities between micro and macrodiscoveries are clear, also in the international context.
A sociologist of science will note the predominance of male scientists in my records. Only three women scientists, Inhelder, Lévy-Schoen and Guirao, are in my list, but their contribution was decisive in many aspects. Inhelder offered me continuous support, Lévy-Schoen provided me with reliable computer records of ocular movements and Guirao helped me with detailed calculations (with the assistance of two women collaborators Paula Harris and Andrea Fernández, that add to the list) and, last but not least, accepted my article for publication. During this long story five dear masters and colleagues died, Piaget, Noizet, González Domínguez, Coffa and Lejeune. Life is broken everywhere at any time, and so are many common and cherished projects.This is a sad fact that must be explicit in every scientific memoir made of flesh and bones. The richer our human network the more the broken threads around us.
As for the different documents I conserve on saccadic eye movements and fractals I count 2 manuscripts and 8 papers in different versions (from 1979 to 1994):
Manuscript 1 Eye movements as fractals (English, Date: November 1, 1979, finished at 8.15 pm; Geneva, Restaurant Hôtel de Ville) 4 pages in a small notebook format. It is a very concise and technical summary of my findings. It contains a definition of saccades as "random variables of length A". I later abandoned the "random" character of saccades but the rest remained, in my opinion, correct. This manuscript is the origin of versions 1, 2, 3 and 4.
Manuscript 2 La temperatura de la mirada (Spanish, Date: December 7, 1979; Rome, Fiumicino Airport - New Delhi, Imperial Hotel) has 26 pages (I have also a typed copy of it, with small changes). This is a long essay about my research and the details of my journeys. That manuscript became the seed of this fractal story (15 years later!) and is at the origin of versions 5 and 5bis.
Version 1. Eye movements as fractals (English, November 1979, Paris; Laboratoire de Psychologie Expérimentale). Sent to E. Ortiz to the Imperial College, London. In this version the hyperbolic function was expressed with exponent -1/D. I also mentioned the saccadic trees and applied Zipf's interpetation of the principle of least effort.
Version 2 Eye movements as fractals (English, November 21, 1979, Paris,; Laboratoire de Psychologie Expérimentale). Sent to B. Mandelbrot to Harvard. I changed the exponent to -D (and I reserved 1/D to express, instead, the temperature of sight).
Version 3 Saccadic eye movements as fractals (English, March 18, Buenos Aires, 1980; Centro de Investigaciones Filosóficas). Sent to publication to the first journal (6 pages, 3 figures). I eliminated the references to the saccadic tree in the text, I quoted it in a footnote.
Refused, July 1980.
Version 4 Saccadic eye movements as fractals (English, Buenos Aires, 1980; Centro de Investigaciones Filosóficas). Sent to publication to the second journal (4 pages, 3 figures). No mention of the saccadic tree. Refused, January 1981.
Version 5 La température du regard. Réflexions sur une démarche scientifique (French, Buenos Aires, 1981; Centro de Investigaciones Filosóficas) Sent to M. Bunge and F. Vidal (21 pages, 5 figures).
Version 5 bis with slight corrections of style. La température du regard. Réflexions sur une démarche scientifique (French, Buenos Aires, 1981; Centro de Investigaciones Filosóficas). Sent to publication to the third journal (21 pages, 5 figures). Refused, June, 1982.
Version 6 La temperatura de la mirada. Esbozo de un modelo fractal de los movimientos sacádicos. (Spanish, Buenos Aires, November 1993, Centro de Investigaciones Filosóficas). Procesos Sensoriales y Cognitivos. (Laboratorio de Investigaciones Sensoriales. CONICET, Buenos Aires, Dunken, 1998).
My list of the letters exchanged with Mandelbrot during this period is the following. He wrote me three letters: 2/19/80,1/3/83, 1/21/83 and a short note: 8/12/88. I kept only the copies of two letters I wrote to him (11/21/79, 3/18/80). I visited Mandelbrot at Harvard in April 1986, as an Eisenhower Fellow, a most extraordinary fellowship that gave me the occasion to meet some of the top scientists in my fields of interest (cognitive and developmental psychology, education for the talented and handicapped, computer networking). But this is another story (V). If I add the different fellowships during the long span of my research on eye movements, scale and fractals, I must count at least the following five: Boursier du Gouvernement Français (Paris 1957-58), Becario externo de la Universidad de Buenos Aires (Paris 1959-60), Guggenheim Fellow (Geneva, New York 1968), Fulbright Fellow (Boston/Cambridge 1976), Eisenhower Fellow (USA 1986). This represents quite an investment on my education and research made by the generosity of public and private institutions of several countries. The appointment as Directeur Associé à l'Ecole Pratique des Hautes Etudes (Laboratorie de Pyschologie Expérimentale et Comparée, Université de Paris-Descartes) was central to my investigation on fractal theory. The analogy with some major discoveries as reported by their authors is striking. Scholarships and fellowships were of paramount importance in many of them. See for instance James Watson's dealing, and troubles, with American and British support during his search in Cambridge for the double helix.
The list of places related to this research includes Buenos Aires, Curitiba, Rio de Janeiro, Cambridge, Boston, Paris, Geneva, London, Rome, New Delhi, Kathmandu. The universities where I studied the problem, met relevant people or performed specific research on fractal eye movements were: in the United States, Harvard and the Massachusetts Institute of Technology, in Europe the Université de Paris and the Université de Genève, in Asia the Tribhuvan University of Kathmandu (who sponsored the Unesco's meeting). This is a long list for a "standard" scientific research and it might be compared with some other lists provided by outstanding and mobile scientists during a "major" discovery. Once again there is a striking analogy between the micro and the macrodiscoveries concerning the number and importance of places visited and revisited.
And now let us try to analyse "l'ordre des raisons". I have found that a substantial part of my research was guided by analogies. As a student of Thomistic philosophy I have read much about the metaphysical importance of analogy. As an epistemologist I was aware of some recent contributions like Mary Hesse's book Models and analogies in sciences (1966). But now, as a scientist I was confronted by my "own" insights. My whole research, in fact, was triggered by the analogy between the highly irregular fractal paths discussed by Mandelbrot and saccadic eye movements. Later I developed a kind of mimetic stance with Mandelbrot's formalism, in search of a mathematical expression of the fractal nature of eye movements. I will name the first kind as Image Analogy IA, and the second as Formal Analogy FA. I propose the following sequence on my way to the fractal model:
IA1 between self-similarity obtained by the change of scale in fractal natural objects (the measurement of the coast of Britain) and in the construction of maquettes of large or small open places.
IA2 between Mandelbrot's mathematical fractal paths (Perrin, Cauchy flights, etc) and eye movements during a visual search (Yarbus).
FA1 between some mathematical properties of particular fractal sets (like the hyperbolic distribution, Zipf and Pareto laws and the isotropy of space and independence of Cauchy flights).
FA2 between the fractal dimension D of a text and the fractal dimension D of a sample of saccadic movements.
FA3 between the informational temperature of a literary text 1/D and the temperature of a sample of saccades 1/D (the temperature of sight)
FA4 between Mandelbrot's self-similarity of lexicographic trees and the self-similarity of "saccadic trees".
FA5 between Perrin's self-similarity of Brownian movements in time and the computer simulation of saccadic movements in short and long runs.
Also I must underline the central role played by chance in my whole research. I propose three main categories: Chance encounters with relevant information (CI), chance meetings with persons (CP) and chance findings of experimental data (CE).
I can order them as:
CI1 my first reading about self-similarity and Mandelbrot's fractal objects in Scientific American (IA1). I used this information in my research about the image of the city.
CI2 my first contact with Mandelbrot's fractal flights in the pictures of his book Fractals 1977 (IA2).
CP1 my unexpected meeting with the great mathematician González Domínguez on the same plane where I was reading Fractals, and the chance to discuss with him the analogy between fractal paths and saccades. He told me: "Mandelbrot has found a client!".
CP2 Fraisse's invitation to spend some months in Paris as Directeur Associé at his Lab and my desire to explore the fractal world of saccades.
CI3 Mandelbrot's pre-fractal analysis of the Zipf law in a volume of the Etudes d'Epistémologie Génétique and my link to Piaget and his Center.
CE2 the starlike graphs of saccades that were registered for other purposes and confirmed the isotropy of the directions of the eye in the visual space.
But I will not emphasize "le hasard et l'analogie" in the process of ideas as a transposition of Jacques Monod's "le hasard et la necessité" in the process of life. I am very conscious of the role played by both chance and analogy in my findings, but it was clearly a teleonomic search from the start. I prefer instead Inhelder's description in his famous article "If you want to get ahead get a theory" (with A. Karmiloff-Smith, 1975). I played quite freely with my fractal ideas but I had taken Mandelbrot's theory as a rigorous model to be tested, and not a different one. My frantic search of the hyperbolic distribution of the amplitude of saccades, first, and the isotropic distribution of the directions of eye movements, later, was a painful and difficult path to follow. My motivation was to find reliable data to fit into the model, but the model was previously there, it was Mandelbrot's model not mine's. And now I feel relieved and satisfied because I can give a new fractal intepretation of the highly irregular path of saccadic eye movements. One parameter, the exponent D of a simple power function suffices. The fractal model "got a new client" in the field of visual perception: saccades are a natural fractal object of D>1.
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