DOGMA

Angèle Kremer Marietti

MEASUREMENT AND PRINCIPLES:
THE STRUCTURE 0F PHYSICAL THEORIES

Article revised in January 2002
Originally published in Revue Internationale de Philosophie 3/1992 pp.361-375

According to Duhem's analysis[1] , and as many philosophers of science, Einstein and the mathematician Robert D. Carmichael declared about this point, without the slightest hesitation I would be willing to affirm that there is nowhere a scientific theory which is perfect and ultimate: even though there may be such a theory, I think it is a good philosophical postulate to suppose that there is none.
Given as the result of a methodical process, being dependent on the history of inquiry and ever new findings, a theory in physics cannot be taken as an absolutely certain explanation of sensible appearances ; moreover, scientific activity[*] can be regarded as a special type of social activity which is culture regulated. By means of sensible appearances we refer to an underlying reality, the approach of which is only possible through scientific, but also historically determined research because epistemology becomes for us more and more historical. Theory cannot let our senses accede to the supposed underlying reality which is presumed to exist. From that specific point of view, scientific theories work only "as though" reality is exactly what they are asserting it to be. Therefore such a theory is always at first an hypothetical construction concerning the reality.
However, certainty of theories must be possible to scientists. That possibility is even an epistemological condition of their research, and a necessary belief without which most of them would not go on leading their own research. And Pierre Duhem, who was suspicious of the certainty of theories, nevertheless gave acoustical theories as an example of certainty[2].

I. MEASUREMENT

With the reservations above mentioned, we can recognize that Carl G. Hempel has exactly defined at what precise time, in the particular process of research, a theory may be considered as being effective. We may retain what he proclaimed justly : "Theories are usually introduced when previous study of a class of phenomena has revealed a system of uniformities that can be expressed in the form of empirical laws"[3]. Hempel's position is not different from Auguste Comte's position which stated, in the last century, that scientific discourse must be developed according to the "law" and not to the "cause", given that only the law can be expressed in mathematical terms. Ernst Mach required, like Comte, that every statement states relations between observable quantities. Also, concerning scientific law, we can make further distinctions and say that a law involves numerical relations allowing experimental verification by measurement. Carmichael expressed it very well when he compared law and principle in relation to measurement:
"By a law we shall mean a statement of phenomenal fact in terms involving numerical relations subject to experimental verification by measurement. By a principle we shall mean a statement of fact, relating to phenomena in such a form as to require a transformation by the use of other facts before one can arrive at numerical relations subject to experimental verification by measurement"[4].
If experiments have a generative role, what is clear and well underlined, in both of these cases enounced by Carmichael, is that the experimental function is the function of "measurement". And the function of measurement may be identified with Hempel's idea of uniformities, as much as with Comte's statements about quantitative laws. Indeed, Auguste Comte explained how, upon the three Keplerian laws, Newton could inversely determine a priori the planetary movements resulting from the dynamical, mathematical law of gravitation which proved the independence and the leading function of astronomy in relation to the other sciences.
With Carmichael's formulas we can better understand why one can speak, on the one hand, of the Principle of the Conservation of Energy, and on the other hand, of Newton's Law of Gravitation. Indeed, Newton's law of gravitation is like any law defined as above: "a statement of phenomenal facts involving numerical relations subject to experimental verification by measurement". Thus, the principle of conservation of energy, not subject to direct test, needs to be transformed in order to let the scientist arrive at those numerical relations subject to experimental verification by measurement, as above. Like any other principle, the principle of conservation states a fact in a form which requires a transformation by means of other scientific facts. And only the results obtained by means of a principle prove its value. We can see also that it is not easy to verify theoretical symbolism with respect to phenomenal reality, but it can be verified only punctually through another symbolism: that of numbers, and by means of a relative system of reference with respect to which measurement is possible.
Measurement begins to become effective through the regulated establishment of numbers: with two numbers, for instance, the position of a ship may be defined by its latitude and its longitude. But, for defining the position of an aeroplane, which requires a third dimension, the height above sea-level, measurements are accomplished with three numbers. In all these cases, measurements concern positions. But if we want to consider events, where there are not positions but things which happen, then we need not only a portion of space (three dimensions or three numbers) but also an interval of time: an extra dimension or number. Thus, in order to locate an event we need four numbers because of its four dimensions. Moreover, space and time are not independent in the theory of relativity for which fundamental physical things are events. Einstein's "basic concept", the concept of "event", corresponded at the beginning of Einstein's research to Mach's concept of "element". Applying the Lorentz transformation, Einstein discovered that events happening at different places cannot be absolutely simultaneous because of what was later called the "conical time order". Thus, we must think of two cones. The active present of an event is a four-dimensional cone stretching backward in space and time. Its active future is a similar cone stretching forward in space and time. And outside of both of these cones no part of the four-dimensional world can be either in the active past or in the active future of that event. But the time itself of an event was described in 1905 as being "given simultaneously with the event by a stationary clock located at the place of the event"[5]. And he went further : as that time has meaning "only when it connects with our consciousness through sense experience (that is, when it is subjected to measurement-in-principle by means of a clock present at that same place), also is the place, or space co-ordinate, of an event meaningful only if it enters our sensory experience while being subjected to measurement-in-principle (that is, by means of meter sticks present at the same time)"[6]. A re-reading of Einstein's paper with insight might let us distinguish between "reality" and "events" ; but, for Einstein : "the ‘time’ of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event"[7]. ‘Time’' is measured time. In other words: outside the fourth dimension is no event, that is no reality, observable to us.
The Restricted principle of Relativity holds in the general theory of relativity, and may be expressed as follows:
"If S1 and S2 are two systems of reference in free space having with respect to each other a uniform unaccelerated motion, then natural phenomena run their course with respect to S2 in accordance with precisely the same general laws as with respect to S1" [8]
The above statement means that if one system S1 is suitable, the other system S2 is equally suitable in as much the requisite conditions are satisfied. The required measurements may be brought into convenient relations by means of a Principle of Correspondence of Units. When the relation of the quantity L1 to the system S1is the same as the relation of the quantity L2 to the system S2, the units of S1 and S2 let us obtain the same numerical result by measuring a quantity L1 with the units of S1 and a quantity L2 with the units of S2. If we must understand the restricted principle of relativity in agreement with the correspondence of units, then it is in agreement with the facts of experiment, given that there is an intricate interdependence of experiment and theory.
What Einstein wanted to achieve was a unified field theory in which there would not be two separated "kinds of physical things"[9]: "measuring rods and clocks" and "all other things". Measurement would be included in the theory in as much as "the particles themselves would everywhere be described as singularity-free solutions of the complete field-equations"[10]. Classical mechanics describes phenomena as existing in themselves : a force is supposed to exist even if we cannot observe it directly. And even if Einstein relativist classical mechanics gave the "observer" a particularly important function, Einstein thought that the observed events existed in themselves : each observer would have a view of the objective reality. Hence it is possible to identify Einstein's philosophy with epistemological realism.
On the contrary, in quantum mechanics it cannot be so. Indeed, a measurement brings an answer to the question put by the experimenters. But a measurement model is described in quantum mechanics by coupling the system to be measured with a measuring apparatus. And another description of quantum-mechanical measurement is possible through the use of formalism. To be precise, as Samuel L. Braunstein and Carlton M. Caves pointed out, the formalism of Effects and Operations in the terms of a measurement model are all equivalent[11]. Obtaining the formalism of Effects and Operations is possible by applying the standard rules accepted in quantum mechanics to the apparatus observation in a measurement model. Formalism provides a convenient notation for a direct description of the measurement of the system state. In Effects and Operations, quantum state and type of apparatus, and quantum state and type of the interaction with the system are incorporated together. Barchielli, Lanz and Prosperi developed the formal description of a continuous measurement of position, for which they used Effects and Operations; yet, they did not specify the measurement model corresponding to the Effects and Operations they chose[12]. Since Bernard d'Espagnat found a discrepancy between what he thought to be a new theory conceived by Barchielli et al.[13] and the standard quantum mechanics, Braunstein and Caves have explored that discrepancy and may prove now, on the contrary, that there was another measurement model than the one which d'Espagnat supposed to be chosen by Barchielli et al. Braunstein and Caves are also able to conclude that Barchielli et al.. did not use a new theory of measurement. By the way, the formalism Effects and Operations remains valid in the examined case[14]. Braunstein and Caves underline the possibility of one Effect have more than one Operation: many model measurements accomplish the same measurement statistics ; however after measurement they lead to different system states. The measuring apparatus is an observer and it may be any macroscopic tool used by experimenters. But in that which concerns the very small, a macroscopic tool is being used to measure a microscopic system : under such conditions it may occur that the tool disturbs the measured system and give different system states.
Therefore, as Bernard d'Espagnat explains it[15], the result measurement is expressed by a number given by the tool through its temporary interaction with the system according to the rules direction for use of the measuring apparatus. But the number of the measured quantity does not necessarily show the real quantity supposed to be there before the measurement. Therefore Bohr used the term ‘phenomenon’ only in the proper meaning of observations including the measuring apparatus. When velocity is at first measured, then momentum may give the measurement of mass and permit the physicist to identify a particle.
All forces in nature are generated by the exchange of messenger particles between interacting particles : photons (quanta of electromagnetic energy with zero mass) may extend the influence in the case of the electromagnetic force to be identified.

 

II. PRINCIPLES

Since Auguste Comte's positivism, all scientists know that a theory lets them link phenomena to some principles, and that without a theory the isolated observations could not be combined, nor could the facts be noticed in the first place.
I began to speak about the essential function of measurement because physical laws are always verified through measurement without which they simply would not be laws. But in the meantime I have already referred to the notion of principle, necessary from the outset for measurement. A kind of circularity is operative between measurement and principles.
Indeed, in order to go all the way with measurement, scientists use the certainty of various principles: for instance, a principle of correspondence of units. But, before that principle, they suppose the use of systems of reference with respect to which measurement can be made and principles can be formulated. Principles are statements of facts relating to phenomena, which are to be transformed through other facts before being verified by measurement. By the term ‘facts’ Auguste Comte meant either "particular facts" observed on the condition of a theory, or "general facts" that are scientific laws or theories. Likewise, for Einstein, ‘facts’ were, besides "particular facts", either, for instance, the constancy of the velocity of light, which is a law, or the equality of gravitational and inertial mass, which is a principle called Einstein's principle of equivalence (E=mc2, where m is the relativistic mass, E the sum of kinetic energy: associated with the motion).
Whereas a law is subject to experimental verification by measurement, a principle is not subject to a direct test : it only proves its value through the results of measurement, after a transformation.
Thus, the principle of correspondence of units lets the physicist get a transformation allowing the continuation of measurement, with which he verifies another principle, the restricted principle of relativity. And so, we can see that measurement is in the very bosom of a physical theory, which includes laws the truth of which is dependent on measurement and principles, on the basis of systems of reference which imply also measurement and principles! Thus, any scientific theory is to be tested by experiment through measurement. There must be an unified theory which presents the relationship between theory and experiment.
As to the way of setting up a theory, Einstein has stated against Hume's induction that "there is no way from the experience to the setting up of a theory" [16]. What is needed at first is to have "sufficient strong formal conditions", such as principles or other theories ; afterwards is only needed a "little knowledge of facts"[17]. If he had been strictly getting along only with a "constructive theory" built up by induction of generalization, on the basis of a logical ladder departing from a set of individual observations, Einstein would never have created the theory of relativity. Einstein was not searching for a constructive theory, but, on the contrary, for a "principle theory"[18]. In order to discover the laws that he was looking for, Einstein applied another epistemological imperative, that is the reference to new universal principles. And he explained that first necessity of discovering "a universal formal principle (that) could lead us to assured results"[19], even though always "a theoretical system can claim completeness only when the relations of concepts and experiences facts are laid down firmly and unequivocally"[20]. Searching for "theories of principle", Einstein put forward his postulates and drew the logical deductions in order to point the deductions to experimental tests.
The fundamental start of the general theory of relativity is a four-dimensional manifold of space-time: in the general theory of relativity, the restricted principle of relativity is suitable under the condition that the phenomena involved occur in free space, that is in the absence of a gravitational field or of any disturbance. The restricted theory of relativity is also validly incorporated in the general theory of relativity, in small portions of space-time within the range of a vanishing difference due to macroscopic phenomena, since another of Einstein's principles is the principle of covariance.
Let us review the conditional principles of the theory of relativity: 1. The restricted principle of relativity; 2. the requirement that the first principle shall hold as a limiting condition in a gravitational field; 3. the requirement that, in an infinitesimal portion of space-time, the first principle shall hold in an indefinitely close approximation ; 4. the requirement to express the laws of nature in a form independent of the particular reference system used, because there are no co-ordinates in nature.
A few laws, a few concepts and a few principles could succeed in the most complete theory. But Einstein also had an operational approach to concepts which had meaning for him only when they could refer, like Kant claimed, to objects, and, besides, to the rules which were assigned to these objects. Einstein's ambition was to both generalize and unify as much as possible: thus, he obtained a logical simplicity which he thought was the essence of his theory of relativity. We must remember now that, though suspecting any scientific theory of not being perfect, Carmichael added: "But the theory of relativity comes nearer to this ideal than any of its rivals"[21]. Maybe the reason for that near perfection is that Einstein knew the way he thought himself: he claimed to his friend Solovine that it was necessary for the physicist to know the way he usually thought[22]. Einstein asked elsewhere : "What is thinking?"[23].
Therefore Einstein gave a theory of "physical" thinking: a kind of cognitive theory of his own. He established a reconstruction of the emergence of the conceptual framework that had been necessary to make possible his theory. Through a design with comments, Einstein showed that to the physicist is given the E (Experience), which is a multiplicity of immediate sense experiences : we philosophers must notice here the same beginning as Kant's in the Critique of Pure Reason, referring to sense experience or to what Kant called, in German, "Sinnlichkeit". But we need to let some A ("Axioms" or fundamental Principles) which are based upon these sense experiences, and from which consequences are going to be drawn, act upon them. Here, for Kant, were working together the pure principles of understanding and transcendental imagination. Now, for Einstein, between E and A, there would be no logical path, but only an intuitive connection which is, as Einstein said, "subject to revocation".
Thus, in Einstein's cognitive theory, the concept plays the part of a mental connection between sense experiences, but it "is not identical with the totality of sense impressions referred to"[24], and it "cannot be gained from material given to us by the senses"[25]. If the choice of concepts was seen by him as being free, Einstein also supposed working a theoretical structure of mind, permitting theoretical description not "directly dependent upon acts of empirical assertions"[26]. Einstein saw concepts and systems of concepts as a human creation. From the E (Experience) to the A (Axioms or Principles), there was for him a J (Jump); and, from A to E (again Experience but the final experience), there were the necessary consequences S, S', S", etc. Holton has recapitulated the complete cycle : E-J-A-S-E, as being Einstein's process of scientific theory construction, the criteria of which were an external validation along with an inner perfection[27].

 

III.QUESTIONS ABOUT INDUCTION AND EXPLANATION

In the same order of ideas about scientific imagination, Carl G. Hempel admits that there are "no generally applicable ‘rules of induction’"[28], by means of which both hypothesis and theory are mechanically inferred from empirical data. He recognizes the function of a creative imagination and particularly of the "method of hypothesis". Rules of induction and rules of deduction are to be understood as "canons of validation rather than of discovery"[29]. Therefore Hempel is right to reject the presumed ideal of scientific inquiry developed in four stages: "(1) Observation and recording of all facts, (2) analysis and classification of these facts, (3) inductive derivation of generalizations from them, and (4) further testing of the generalizations"[30]. Hempel calls that view "the narrow inductivist conception of scientific inquiry"[31]. However, he thinks that scientific inquiry may be said to be only "inductive in a wider sense"[32]. Hempel conceives correlatively the rules of induction as stating, like the rules of deduction, criteria for the soundest arguments. But we have seen that Einstein went much further with his own epistemology of the theory of relativity, when he "jumped" [*] to an intuition which did not necessarily follow the rules of induction. In order to be corroborated, theories must be already hypothetically imagined, since, without any theory at all, the physicist could not project a set of observations, for which he needs a basis of provisional hypotheses precisely belonging to an imaginary theory.
That was indeed the original idea of Comte in his Cours de philosophie positive (1830-1842). Already in 1825 - in Considérations sur les sciences et les savants - Comte claimed that "one cannot make observations without theories more than one can create a positive theory without observations"[33]. It is as though a special kind of circularity were needed : without any research theory there is no suitable observations, and, on the contrary, without suitable observations there will be no final theory at all. But this presumed circularity is only an appearance, because the first theory needed in order to make observations is but hypothetical, and the second theory justifying the effective observations has been, in principle, first corroborated. With the above reservations, there may be hypotheses without any inductive-evidential support, but they have some strong theoretical support, even if the first theory which elicited the observations was false.
The principle of following an inductive way was abandoned by Pierre Duhem in favor of holism: in his book concerning physical theory (1905), he gave preference to a theory conceived as a whole yielding a representation rather than an explanation. Duhem stated that no isolated hypothesis could be corroborated outside of the field of physics[34]. Auguste Comte held exactly the same opinion, when he rejected whatever observations and experimentations could be developed outside of a theory. And effectively he did so in 1854, when he rejected, probably with exaggeration, the astronomical prevision and prediction of the planet Neptune by Le Verrier in Paris (also claimed by John Couch Adams in Cambridge and verified by Galle in Berlin). Indeed, Comte asserted that it was an "astronomical illusion" to presume a deduction concerning the existence of a planet simply from calculation – while Le Verrier did what he had learned to do, that is: measuring angles and counting time[35].
We could now suggest that scientific explanation is often suspicious as induction. As did also Pierre Duhem, Carl G. Hempel justly emphasized the regularities and the system of uniformities which he recognized to stand at the very bottom of theories. Moreover, we must say that such an assertion might be right for corroborated theories. In the above quoted essay, Comte proved Adam Smith's History of Astronomy on the subject of regularities. In fact, Adam Smith evoked not exactly the regularities, but, the contrary, the "magnificent irregularities, whose grandeur cannot be overlooked" and which called forth the amazement of mankind in the first ages of society. According Adam Smith's observation, men ascribed all the irregular events of nature "to the agency power of their gods"[36] . Thus, explanations were looked for in the case of natural irregularities. Beginning with the remote ‘theologian’ age, explanations went on until the age of science.
Carl G. Hempel joined theories and regularities: "Theories then seek to explain (...) regularities"[37]. Measurement would then need an explanation! But explanation is not always what is necessarily procured by scientific theories. According to Comte, and despite the motto "ramener l'inconnu au connu", scientific work is essentially constituted by the fact of going out of the concrete reality in which we live, toward its abstract and relative representation. For Pierre Duhem, it is even more explicit: "A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, in order to represent as simply, as completely and as exactly as possible, a set of experimental laws"[38]. We find here the two principal characteristics of physical theory, that we emphasized : measurement and principles. Comte and Duhem thought like, later, Philipp Frank, that "scientific explanations" may often prove to be metaphysical. And Philipp Frank himself developed the idea that a particular type of equation being the only legitimate basis of the scientific explanation of physical phenomena, could not be justified by science unless the attempts of justification were based upon metaphysics.
From that point of view, the distinction between a description and a causal theory was for Philipp Frank purely metaphysical[39], For instance, to say that Einstein's theory of relativity is "descriptive", or that Newton's theory is "causal" and "explanatory", for both of these cases the similarity of the traditional attitude is noted by Frank : the difference is only metaphysical. What was supposed to explain in the sixteenth century was what could be derived from the Aristotelian philosophy ; in the nineteenth century, it was what could derived from Newton's philosophy. We can say that throughout time science follows practices which are regulated by culture. The reference of scientific theories to the different schools has been pointed out by Kuhn in 1962, but it has also been explicitly noted by Duhem in 1905. When Duhem distinguished four schools in succession :1) the Aristotelian, 2) the Newtonian, 3) the atomistic, 4) the Cartesian, he did not report these schools to a banished metaphysics, and he even thought that these philosophic schools had been also helpful in the creation of explanations in scientific theories considered as good. And we know that, in the Structure of Scientific Revolutions, Thomas S. Kuhn also referred to what he called the "paradigms" which came from such schools, and which helped the scientists at their own time : "the assimilation of Galileo’s and Newton's mechanics gave rise to a particular famous debates with Aristotelians, Cartesians, and Leibnizians about the standards legitimate to science"[40].
More recently, Hempel did not restrict the distinctive characteristics of a good scientific theory to the explanatory import nor the testability-in-principle because they were for him "only minimal necessary conditions that a scientific theory must satisfy"[41]. For explanation is to Hempel a necessary condition of scientific theory. Indeed, for Hempel, and according to the views of Comte, Duhem and Einstein, it seems that the best scientific theory would be the one which could best unify the greater number of diverse phenomena. For Hempel too, theory had to trace all the diverse phenomena back to the same underlying processes : there was then also a unification of views. Like Newton's theories of gravitation and of motion, like the kinetic theory of gases, and also like Bohr's theory of the hydrogen atom, for Hempel the best theory is the one which could best unify by saving the phenomena.
Therefore, Bas C. Van Fraassen has proposed a new way to a theory : "to specify a family of structures, its models ; and secondly to specify certain parts of those models (the empirical structures) as candidates for the direct representation of observable phenomena"[42]. Then, what is called ‘axioms of quantum energy’ in treatises on quantum mechanics is exactly what B. C. Van Fraassen calls a "family of models", with the indication of their "empirical substructures". In that presentation, what is actually becoming possible is to choose between two epistemic attitudes : either the assertion that the theory is "true" and deserves belief, or the assertion of its empirical adequacy with the consequence of its acceptance. Against any "metaphysical baggage" B. C. Van Fraassen thinks it necessary to advocate "empirical minimality", for pragmatic reasons. The assertion of empirical adequacy appears to B. C. Van Fraassen weaker than the assertion of truth. The scientist must save the phenomena by delivering himself from metaphysics, but we know that thematic analysis shows how the scientific discovery is really working : scientific imagination uses what Holton calls themata[43] persisting metaphysical issues in scientific change.

 

CONCLUSION

Theories in physics are based upon principles and measurement. Rather than the explanation of natural laws what the best theories are looking for is a greater unification at a higher level of axioms. There is a cognitive function of theories which has been well understood by Einstein, analyzing the way his proper mind proceeded in the process of learning about reality. We may now raise a question : is it possible be neither realist nor nominalist neither rationalist nor empiricist ? If any of these characteristics remains unavoidable then one could doubt that "science" or the human mind are able to avoid completely all metaphysics.

Université d’Amiens.


[1] Pierre Duhem, La théorie physique. Son objet. Sa structure, Deuxième édition revue et augmentée. Paris Marcel Rivière et Cie. 1914.
[*] My note in 2002 : I wrote: "science"; but I do not agree with cognitive relativism.
[2] Op.cit., p. 5.
[3] C. G. Hempel, Philosophy of Natural Science, Englewood-Cliffs, New Jersey: Prentice-Hall, 1966. See p.70.
[4] A Debate on the Theory of Relativity. With an Introduction by William Bryan. Favoring the Theory : Robert D Carmichael, Harold T. Davis. Opposing the Theory : William D. MacMillan, Mason E. Hufford. Chicago, London : The Open Court, 1927. See p. 6.
[5]Albert Einstein, « Zur Elektrodynamik bewegter Körper", Annalen der Physik, 17, 1905, p. 894.
[6] Gerald Holton, Thematic Origins of Scientific Thought. Kepler to Einstein (1973). Revised Edition, Harvard University Press, Cambridge, Massachusetts, 1988, p. 243.
[7] Quoted in G. Holton, op.cit., p. 243.
[8] R. D. Carmichael, op.cit., p. 15-16.
[9] A. Eintein, « Autobiographical notes », in Albert Einstein : Philosopher-Scientist. Paul A. Schilpp, Ed. Evanstons, Illinois : Open Court, 1949. See p. 59-61.
[10] Ibid.
[11] Samuel L. Braunstein and Carlton M. Caves, "Quantum rules : An effect can have more than one operation", in Foundations of Physics Letters.. Editor: Alwyn van der Merwe. FPLEET 1 (1), 1988, p. 3-12.
[12] A. Barchielli, L. Lanz and G. M. Prosperi, Nuevo Cimento 72 B, 79 (1982).
[13] Bernard d’Espagnat, Foundations of Physics, 16, 351, (1986).
[14] S. L. Braunstein and C. M. Caves, op.cit.
[15] Bernard d’Espagnat, Une incertaine réalité. Le monde quantique, la connaissance et la durée. Paris ; Gauthier-Villars, 1985, p. 130-131, note 1.
[16] A. Einstein. op. cit., p. 89, note 6.
[17] Ibid.
[18] A. Einstein, « What is the theory of relativity ? » London Times, November 28, 1919.
[19] A. Einstein, « Autobiographical notes », op. cit., p. 63.
[20] Quoted in F. Herneck, Forschungen und Fortschritte, 40, 1966, p. 133.
[21] R. D. Carmichael, op. cit., p. 6.
[22] Quoted in G. Holton, The Advancement of Science and its Burdens. Cambridge University Press, 1986, p. 29-33.
[23] Op. cit., p.33.
[24] Op. cit., p. 84.
[25] A. Einstein, Ideas and Opinions, New York, Crown, 1954, p. 307.
[26] A. Einstein, 'Reply to criticisms", in Albert Einstein Philosopher-Scientist, p. 674.
[27]G. Holton, 1986. See p. 34.
[28] C. G. Hempel, op. cit., p. 15.
[29] Ibid., p. 18.
[30] Ibid., p. 31.
[31] Ibid.
[32] Ibid., p. 18.
[*] My note in 2002 : In a way, this jump is, by itself, an inductive process.
[33] Auguste Comte, Ecrits de Jeunesse 1816-1828. Paris: Mouton, 1970. See p. 326.
[34] Pierre Duhem, op. Cit., p. 393.
[35] Auguste Comte, Correspondance générale, Tome VII, Paris : Éditions de l’École des Hautes Études en sciences Sociales, Librairie Philosophique J. Vrin, 1987. Textes établis par Paulo E de Berrêdo Carneiro, et présentés par A. Kremer Marietti. See my Introduction, p. XCII, and Comte’s letter of February, the 23rd 1854, pp. 191-192.
[36] Adam Smith, Essays Philosophical and Literary. London : Ward, Lock and Co. See p. 330.
[37] Carl G. Hempel, op. cit., p. 70.
[38] Pierre Duhem, op.cit., p.24.
[39] Philipp Frank, Foundations of Physics, in Foundations of the Unity of Science, Vol. I, 1955. Chicago and London : The University of Chicago Press. See p. 423-504.
[40] Thomas S. Kuhn, in Foundations of the Unity of Sciences, Vol.2, 1970. See p. 110.
[41] Carl G. Hempel, op. cit., p.75.
[42] Bas C. Van Fraassen, The Scientific Image, Oxford: Clarendon Press, 1980. See p. 64.
[43] See Gerald Holton, 1986 and 1988, also Angèle Kremer Marietti, Les racines philosophiques de la science moderne, Bruxelles, Mardaga, 1987.

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