Science, Philosophy and Music English; French: Proceedings of the XXth International Congress of History of Science (Liège, 20-26 July 1997) Vol. XX 2503514146, 9782503514147

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SCIENCE, PHILOSOPHY AND MUSIC

DE DIVERSIS ARTIBUS COLLECTION DE TRA VAUX

COLLECTION OF STUDIES

DEL' ACADEMIE INTERN ATI ON ALE

FROM THE INTERNATIONAL ACADEMY

D'HISTOIRE DES SCIENCES

OF THE HISTORY OF SCIENCE

DIRECTION EDITORS

EMMANUEL

ROBERT

POULLE

HALLEUX

TOME 63 (N.S. 26)

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BREPOLS

PROCEEDINGS OF THE XXth INTERNATIONAL CONGRESS OF HISTORY OF SCIENCE (Liege, 20-26 July 1997) {~l.;!;JJQ

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VOLUMEXX

SCIENCE, PHILOSOPHY AND MUSIC Edited by

Erwin NEUENSCHWANDER and Laurence BOUQUIAUX

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BREPOLS

The XX1h International Congress of History of Science was organized by the Belgian National Committee for Logic, History and Philosophy of Science with the support of : ICSU Ministere de la Politique scientifique Academie Royale de Belgique Koninklijke Academie van Belgie FNRS FWO Communaute franc,;aise de Belgique Region Wallonne Service des Affaires culturelles de la Ville de Liege Service de l'Enseignement de la Ville de Liege Universite de Liege Comite Sluse asbl Federation du Tourisme de la Province de Liege College Saint-Louis Institut d'Enseignement superieur "Les Rivageois" Academic Press Agora-Beranger APRIL Banque Nationale de Belgique Carlson Wagonlit Travel Incentive Travel House

Chambre de Commerce et d'Industrie de la Ville de Liege Club liegeois des Exportateurs Cockerill Sambre Group Credit Communal Derouaux Ordina sprl Disteel Cold s.a. Etilux s.a. Fabrimetal Liege - Luxembourg Generale Bank n.v. Generale de Banque s.a. Interbrew L'Esperance Commerciale Maison de la Metallurgie et de l'Industrie de Liege Office des Produits wallons Peeters Peket de Houyeu Petrofina Rescolie Sabena SNCB Societe chimique Prayon Rupel SPE Zone Sud TEC Liege - Verviers Vulcain Industries

© 2002 Brepols Publishers n.v., Tumhout, Belgium

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. D/200210095159 ISBN 2-503-51414-6 Printed in the E.U. on acid-free paper

TABLE OF CONTENTS

Part one SCIENTIFIC MODELS FROM ANTIQUITY TO THE PRESENT

The mathematical model : epistemological tool or ideological notion ? .......................................................................................... 9 Martin ZERNER The discernment of ancient modeling efforts ............................................... 15 Don FAUST Ancient Egyptian cosmological, astronomical, and geometrical models ................................................................................. 21 Kurt LOCHER Grafische Umsetzung mentaler Modelle: als Problem-Losung " vorwissenschaftlicher " Technik mittels Experiment, graphischen und raumlichcn Modellen ......................................................... 27 Wolfgang von STROMER

Part two SCIENCE AND PHILOSOPHY

Aristotle as the first topologist ...................................................................... 45 Peeter MDDRSEPP Science and philosophy in the anthropometric stage .................................... 57 Bert MOSSELMANS and Ernest MATHUS La diffusion de Lucrece en Italie au xv1e siecle ........................................... 67 Amalia PERFETTI Bacon and eclecticism ................................................................................... 75 Stephen GAUKROGER Mechanicism, organicism, holism in Le Eon's concept of the crowd: a hermeneutical interpretation of paradigm combination ............................. 83 Oleg I. GUBIN Oswald Spengler : science, technology and the fate of civilization ............. 97 Antonello LA VERGATA Zur Geschichte der Zeitschrift " Erkenntnis " ( 1930-1940) im Lichte des Briefwechsels von Hans Reichenbach .................................................. 105 Hannelore BERNHARDT

TABLE OF CONTENTS

6

Kuhn's philosophical troubles with actual history of science .................... 113 Mario H. OTERO Essai sur la nature des theories scientifiques. Theories et structures ......... 119 Jean GADISSEUR Polarities within mathematics ...................................................................... 139 Roman DUDA The search for explanations in the methodology of mathematical physics .............................................................................. 149 Andres RIV AD ULLA La substance evanescente de la physique ................................................... 157 Yves GINGRAS De l'histoire des techniques Jean c. BAUDET

a la philosophie de la technologie ................ 165

The history of science and the ethics of science ........................................ 173 Martin COUNIHAN Vers une nouvelle modelisation des rapports entre science et politique .... 181 Sebastien BRUNET Concept of progress and geographical thought.. ......................................... 197 Witold J. WILCZYNSKI

Part three SCIENCE AND MUSIC

Aesthetic and technological aspects in Berio's Thema (Omaggio a Joyce) ....................................................................................... 207 Nicola SCALDAFERRI "Jeu de miroir Sus une fontayne " .............................................................. 217 Roberto DOA TI Re-synthesis of analogue electronic music compositions by computer : a teaching experience ................................................................ 227 Alvise VIDOLIN Riftessione sui sincretismi e l' evoluzione della musica

latinoamericana negli ultimi cinque secoli .................................................. 237 Carla MINELLI

Contributors .................................................................................................. 255

PART ONE

SCIENTIFIC MODELS FROM ANTIQUITY TO THE PRESENT

THE MATHEMATICAL MODEL: EPISTEMOLOGICAL TOOL OR IDEOLOGICAL NOTION ?

Martin

ZERNER

We sum up here the methodology and some of the conclusions of a research project which has been proceeding for a number of years. The methodology is twofold. One part is to make a historical study focusing on the appearance of the word " model " in scientific contexts. This makes it possible not to rely on a preconceived idea of what a mathematical model is and to detect a great variety of meanings. The other part is to make case studies of models in different fields in order to compare the ways in which they work. It is clear that the historical study cannot be exhaustive. On one hand every field and subfield cannot be explored. On the other hand, some occurrences certainly will escape the investigation. Nevertheless, if in a definite field no occurrence has been found, we can be reasonably sure that the word has been at most exceptionally used.

With this qualification, we first find the word " model " in the context of physics starting with Maxwell 1. It was taken up by several authors, including Boltzmann and Hertz 2 . Maxwell's first model was geometrical. However, most of these models of physical phenomena were themselves physical systems the function of which was to give the theory of the phenomenon of which they were a model. Rutherford's model of the atom (1911) is quite typical. Notice that there is no Bohr model, but, in his own words, a contribution to the theory of Rutherford's atomic model 3 . For this reason, I consider these models as physical, not mathematical. Whether they provided an explanation of the phys1. J.C. Maxwell," On Faradays' Lines of Force", Phil. Trans. Carnb. Soc., 10 (1857), part I. 2. See R. Muller, "Zur Geschichte des Modelldenkens und des Modellbegriffs ", in H. Stachowiak (ed.), Modelle-Konstruktion der Wirklichkeit, Miinchen, 1973. The author is grateful to Prof. Neuenschwander for this reference, the only one to his knowledge to pay attention to the actuel use of the word. Unfortunately, it does not cover the 20th century. 3. H. Bohr, "On the Constitution of Atoms and Molecules", Philosophical Magazine, VI, 26, (1913), 1-25.

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MARTIN ZERNER

ical phenomenon or a simple fiction which helped the mind to find out its mathematics was controversial4 . It is in the context of this controversy that Mach uses the phrase " mathematical model " for the atomic theory to emphasize his thesis according to which the atoms had no objective existence. Incidentally, this phrase was used in the same period with a quite different meaning, namely a concrete representation of a mathematical object, typically a plaster molding of a surface. The situation was quite different in the 1920s and 1930s. The word " model " is then very seldom found in physics. This may be due to the fact that the experimental evidence for the existence of atoms and subatomic particles was now overwhelming. Also with the advent of quantum physics a particle can no longer be conceived with the intuitions which we worked out in the macroscopic world and its very description relies now on fairly advanced mathematics. As a consequence the distinction between a physical and a mathematical model is blurred. This is e.g., the status of what has been first called the Thomas-Fermi theory or approximation and later increasingly often the Thomas-Fermi model. The year 1936 sees the permanent introduction of the word " model " in two fields: mathematical statistics5 and economics, more precisely econometrics6 . The first deserves a closer study. In economics the context of " model " is twofold. At the beginnings of articles or books, we find general statements about mathematical models. But when the word appears in specific situations, the context is statistical. This remains true until 1944, the year of the appearence of Haavelmo's famous manifesto7 which theorizes very clearly the necessity of a probabilistic mathematical formulation if an economic theory has to be checked against available data. It is impossible to sum up this article in a few lines, but let us try and give the main idea. The economist, contrary to e.g. the agricultural engineer, has no means whatsoever to test his theories against experimental situations in which he may choose various values of some of the parameters. He must rely on tests of hypotheses. Now these are meaningful only if there is a probabilistic theory to test (the phrases used by Havelmoo are "probabilistic model " and " theoretical model "). A completely different point of view is expressed in that same year 1944 in von Neumann and Morgenstern's still more famous Theory of Games and Eco4. E. Bellone, I Modelli e la Concezione def Mondo nella Fisica moderna da Laplace a Bohr, Milano, 1973 gives a detailed account of the main developments in 19th century physics, focusing on this controversy. 5. 1. Neyman, E. Pearson, "Contributions to the Theory of Testing Statistical Hypotheses'', Statistical Research Memoirs, vol. 1, London, 1936, 1-37. Reprinted in Joint Statistical Papers, Berkeley, 1967. 6. J. Tinbergen, "An Economic Policy for 1936 ",in J. Tinbergen (ed.), Selected Papers, Amsterdam, 1959. 7. T. Haavelmo, "The probability approach in econometrics", Econometrica, 12 (1944), (supplement).

THE MATHEMATICAL MODEL

11

nomic Behavior8. There appears the first statement of the conception of the mathematical model as the general method of the sciences which is now so widespread. In view of this, a special study was devoted to von Neumann9. Von Neumann very clearly expressed his conception of models, indeed a conception of the sciences in general, in a later article about physics : " To begin, we must emphasize a statement which I am sure you have heard before but which must be repeated again and again. It is that the sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of some verbal interpretations, describes observed phenomena " 10 • It must be added that in this conception " model " and " theory " are synonymous. The statement must be checked against the way in which von Neumann himself spoke about models in his specific scientific works. He uses the word in the context of three fields : logic, physics and economics. We need not insist on logic where it appears in one article of 1925 with its purely technical meaning11. The works of von Neumann in physics are clearly separated into two periods: 1927-1934 (quantum physics) and 1941-1959 (mostly fluid mechanics). In the first period, the word "model" is to be found thrice, in rather obscure sentences. In the second period, it appears in 5 papers out of 26 12 . It has in these papers a clear meaning. Some concrete situation is to be studied. Physics provides an unquestioned theory of this situation, only it is too complicated to be carried out to make concrete predictions. The model is a simpler situation in which this can be done and there are good reasons to believe that this simpler situation is an approximation of the genuine one. An exception is Burgers' model with which we cannot deal here. Contrary to" model"," theory " is often used in both periods and one word is opposed to the other on several occasions. So the physicist von Neumann is in fiat contradiction with von Neumann the essayist on physics. Von Neumann published two works on economics. The first has nothing to do with the notion of model except that the word appears in the misleading title 8. J. von Neumann, 0. Morgenstern, Theory of Games and Economic Behavior, Princeton,

1944. 9. M. Zerner, "Analogie et mode!es chez von Neumann", submitted to Rivista per la Storia delle Scienze. 10. J. von Neumann, "Method in the Physical Sciences", in L. Leary (ed.), The Unity of Knowledge, New York, 1955. All the works of von Neumann mentioned here except his books are reprinted in J. von Neumann, Collected Works, Oxford, Londres, New York, Paris, 1961-1963 (6 vols). 11. J. von Neumann, "Eine Axiomatisierung der Mengenlehre ", Journal fiir die reine und angewandte Mathematik, 154 (1925), 219-240. 12. One should may be add a paper with the title "The Point Source Model". Only the word does not appear elsewhere in the paper. This paper was only published after von Neumann's death in his Collected Works and the title may have been chosen by Taub who edited it. The name "point source model" had been adopted in stellar physics when the Collected Works appeared.

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MARTIN ZERNER

of the English translation (1945) 13 . The other is the joint book with Morgenstern. As has already been mentioned, von Neumann's conception of models and their function in the sciences is already expressed there, indeed repeatedly. It is put to work in two ways. In a general way, it enables the authors to justify the kind of mathematization they perform in economics by analogy with physics. The other way is quite specific, it justifies the use of a numerical utility (cardinal in the usual economic terminology) by a detailed comparison with the case of temperature. Both rely on the conception of the sciences of von Neumann the essayist. Within a few years after the publication of this book, the use of the word " model " became widespread in economics, usually with the same meaning as in the book. The next step was its being taken up in the so called behavioral sciences. This can be dated to 1951-1953 and it is quite clear that it was an importation from economics 14 . Operational research appears as the next field to have taken up the word. It was not used in the military (and secret) childhood of the field as can be seen in the book of Morse and Kimball 15 . The first volumes of the journal Operations Research, which started in 1952, contain occasional occurrences. It becomes of current use in the second half of the 1950s. Population dynamics is supposed to be a great user of models, but the word is to be found only in the late fifties and at that time in a statistical context only. In applied mathematics, the word " model " appears around 1965 and becomes usual in the 1970s in the United States and a few years later in France. This is conspicuously late in view of its later overwhelming presence in the literature of the field. It must be added that its meanings are manifold and it happens that the same author uses two different ones within a few lines (probably without realizing it). It is of interest to observe that in general papers about the methods of applied mathematics some authors illustrate their view of what a model is by examples taken from their practice with technical applications based on physics adding in fine " this is also the way it works in economics". In physics, the evolution is very complicated. From, very roughly, the middle of the century on, physicists speak about " models " again in various sub13. "A Model of General Equilibrium'', the original reference is: J. von Neumann, "Ueber ein rekonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixtpunktsatzes ",in K. Menger (ed.), Ergebnisse eines mathernatischen Kolloquiurns, Vienna, 1938. Notice that the object of the paper is not what economists call general equilibrium. 14. M. Zeruer, "Paul Lazarsfeld et la Notion de Modele mathematique ", in J. Lautman, B.P. Lecuyer (eds), Paul Lazarsfeld (1901-1976), Paris, 1998. 15. P. Morse, G. Kimball, Methods of Operational Research, Cambridge, New York, London, 1951.

THE MATHEMATICAL MODEL

13

fields and with different meanings. But the field is the very last to adopt the word " model " in the sense of von Neumann the essayist. Now a few words about the three case studies which have been carried out ; others would of course be highly desirable. They had obviously to be chosen in quite different fields. One of these fields had to be physics or some technology based on physics. The latter was deemed preferable as more typical and also, why not admit it, more familiar to the author. The oil reservoir simulations, as they appeared in the engineers' literature of the early 1970s were chosen for this case study. Another obvious choice was economics, mainstream economics of course. Now, very different things are called " models " in economics. To mention three, we have the supposedly basic theoretical models among which Arrow-Debreu is the best known, the big econometric models for prediction and a large number of intermediate models which rely on the former to study specific questions. These last make up the bulk of the research literature in economics ; one of them was selected for this and other reasons, namely a model of borrowing for a developing country 16 . To compare with a field quite different from both the preceding ones, the famous Lotka-Volterra systems (later called models) for population dynamics were studied. The study of each model has to be thorough enough to avoid superficial resemblances and it must make clear the general principles on which the models work in order to be able to make comparisons. This makes it impossible to report them here, even in summary. The conclusions they lead to confirm what appeared in the historical study, namely that models in physics and in economics rely on very different principles. The case of the Lotka-Volterra equations is different from both, moreover there is a big difference between the overly famous equations for a predator-prey system and the less well known but much more firmly grounded equations for two species in competition for food. It can be safely concluded that the use of any variant of the von Neumann conception of models can only lead to epistemological confusion. The ideological consequences of this confusion still need investigation. However, the contrast between a technological model based on physics and a mainstream economic model is clear. The first one relies on the principles of physics and controlable approximations to give indications about what can be gained in terms of efficiency (and ultimately profit). The second one is supposed, in the dominant view, to be justified by its logical coherence and a comparison with reality. In fact, when it comes to specific questions, the logical coherence is obtained at the cost of fantastic extra assumptions and the comparison with reality is, to put it mildly, very weak. In the specific case studied, there is none. And the historical context of the model shows that its function was to justify 16. R. Dornbusch, "Real Interest Rates, Home Goods and Optimal External Borrowing", Journal of Political Economy, 91 (1983), 141-153.

14

MARTIN ZERNER

the policies of macroeconomic adjustment to which the that time.

IMF

was switching at

THE DISCERNMENT OF ANCIENT MODELING EFFORTS

Don FAUST

In the history of science we see marches, many yet unfinished, toward a precise and complete description of Reality. To illustrate the great length of some of these marches, we consider three problem areas in the history of modeling m science. First, considering man's early struggles to model the human reproductive process leads to revisiting, from another perspective, an old conjecture concerning the 260 count of ancient American civilizations. Second, considering man's early attempts to model the moon's periodicity leads to a new conjecture concerning the prominence of sixty in the ancient numeration systems of the Near East. And third, considering man's attempts to model the concept of negation leads to some precise mathematical logic machinery and a theorem clarifying a connection between part of Aristotle's work 2000 years ago on privatives and part of recent work by Artificial Intelligence (AI) researchers on negation. This note is by necessity brief : for more detail, the reader may request the first reference in regard to problems one and two, the second for problem three. THE COUNT OF

260

The Tzolkin count of 260 of ancient American civilizations consists of consecutively forming pairings of two primitive numeral sets, one of 20 and another of 13. The first of one set is paired with the first of the other set, then the second of one with the second of the other, and so on, cycling through the sets repeatedly. The count is finished when all possible pairings have been formed. In this way one has a constructed numeral system, an ordered numeral set consisting of 260 unique pairs. This invention, for generating a counter set of larger cardinality from two given smaller counter sets, is certainly ingenious. It antedates the later placevalue system of base 20 in ancient America, and so can be seen to be a partial

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DON FAUST

solution to the problem of construction of a system of numeration allowing, as place-value systems do, the counting of sets of arbitrarily large cardinality. That such a counting system developed at all is, of course, interesting enough. But what is even more interesting is that at least one other ancient culture developed a system of similar design. The ancient Chinese developed a system based on such pairing as well, the Chia-tsu numeration system based on one set of 12 counters and another set of 10 counters, a sixty count of pairs. Let us look at the design involved in these two systems. For example, for the two counter sets with numerals a,b,c,d and 1,2,3,4,5,6 respectively, the counter set generated by pairing from them would consist of the following sequence of twelve numerals (each one a pair formed by the juxtaposition of one numeral from each of the two given counter sets) : al, b2, c3, d4, a5, b6, cl, d2, a3, b4, c5, d6. Note the uniqueness of each element (pair) of this derived counter set, giving a perfectly valid set of primitive numerals. In general, we may define such numeral sets as follows. The Pairing-Generated Counter Set generated by a counter set A with first element a and last element b and a counter set c with first element c and last element d is the counter set s such that : 1) the first element of s is ac ; and 2) if xy is ins then: xy is the last element of sin case x=b and y=d; otherwise xy has a successor in s, formed with the successor of x and the successor of y if x.7

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Tape 1.

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RE-SYNTHESIS OF ANALOGUE ELECTRONIC MUSIC

235

Time runs on the horizontal line and frequency on the vertical line. In this score we can read also the dynamic, in musical notation, and the elements label. The Construction score describes the construction of the each element, as shown in fig. 6. It is very precise and it gives a good documentation of the work made by Stockhausen. Making the re-synthesis of the score showed on fig. 5 we found some problems. Stockhausen gives only operational and not technical description of the electronic equipment. For instance, he describes the control panel of the filter but he does not give it pulse response which is the precise measure of the filter behaviour. The sound examples 11 and 12 present the original and the copy of this part of the score. Also in this piece the main difference is sound quality : the copy has a sound cleaner then the original. FIGURE

6.

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v,~,., (Cl• tut "" •ll lllot1" 1:1i1f11 .,.._oeui.n1, •., ,_ n, Of du loooUMl!tr- tlN