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Towards a logical reconstruction of revolutionary change: The case of Ohm as an example
MICHAEL
HEIDELBERGER *
TOWARDS A LOGICAL RECONSTRUCTION OF REVOLUTIONARY CHANGE: THE CASE OF OHM AS AN EXAMPLE I
ALL FORMALIZATIONSof historical scientific theories, as yet established, are confined to a small number of mathematized theories. In addition, the relation existing between actual historical theories has rarely been treated. There has been, it seems, no explicit investigation of a theory-pair in which one theory is separated from another by a scientific revolution. However, the claim has been made that by formalizing theories and reducing them to each other even the last rationality gap within a logical reconstruction of scientific change may be closed.’ In this paper I attempt to show in detail on how many levels an actual revolutionary change may take place and how, if at all, formalization can serve for the reconstruction of that change on these levels. As my historical example I choose the change in the field of electric research which occurred in Germany around 1830.’ As the method of formalization I take the ‘structuralist approach’ first advocated by Suppes, Adams, McKinsey and further developed by Sneed, Stegmiiller, and others.3 For several reasons, the transition to Ohm’s theory seems to be a good start for systematic reflections on the change of theories. On the one hand, this event happened not very long ago. Thus, the problems seem to us more familiar and easier to evaluate than those which arose, e.g. in the transition from Aristotelian to Galilean physics. On the other hand, the science of electricity forms part of the ‘Baconian Sciences’ (theory of heat, magnetism, chemistry. . .) which unlike the classical sciences (statics, mechanics,
*Seminar ftir Philosophie, Logik & Wissenschaftstheorie der Universitat Mtinchen, Ludwigstrasse 3 1, 8000 Mtinchen 22, West Germany. ‘See e.g. W. Stegmtiller, Theorie and Erfahrung (Berlin, 1973), p. 249; English edition: The Structure and Dynamics of Theories (New York, 1976). ‘Compare also the very interesting article of Kenneth L. Caneva, ‘From Galvanism to Electrodynamics: The Transformation of German Physics and its Social Context’, Hist. Stud. Phys. Sci. 9 (1978), 63 - 159. My paper has been written independently of Caneva’s work. ‘See J. D. Sneed, The Logical Structure of Mathematical Physics (Dordrecht, 1971) and W. Stegmtiller, The Structuralist View of Theories (New York, 1979) for the relevant literature.
Stud. Hist. Phil. Sci., Vol. 11 (1980), No. 2, pp. 103 - 121. Pergamon
Press Ltd.,
Printed
in Great
Britain. 103
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astronomy. . .) have as yet been greatly neglected by philosophy of science.4 Moreover, the episode of Ohm may be regarded as one of the typical cases for the change in the whole of natural science between 1830 and 1850. Recently, Bijhme has advocated the view that the investigation of scientific controversies could result in a similar function for the philosophy and sociology of science as twin-research does for biology and medicine.5 Thus, the history of the origin and reception of Ohm’s law is very suitable for throwing light upon the period in question; for the history of Ohm’s law is one of the most celebrated examples of resistance of scientists to scientific innovation. In the following paragraphs I should like to characterize the nine most essential levels on which, in my opinion, the change of electrical research took place. After that, some of the theories in question will be formalized and the question posed on which of these levels some formalizing of theory-elements might preferably be of use for deeper analysis. We shall see that the structuralist view proves especially valuable to illuminate the relation of a theory to its range of applications and at the same time the theoretical presuppositions on which a theory rests. Our main result will be that the revolutionary change brought about by Ohm’s theory can only partly be explicated in formal terms. 1. Level
of the world view: from
‘physics of essences’
to theoretical
physics
In Ohm’s time there were two main trends in electrical research, atomism and the dynamism of the ‘Naturphilosophie’. The atomists tried to explain electrical phenomena by means of the conception that electricity was an imponderable matter of finest atoms, whereas the dynamists rejected this thesis, explaining electricity as the sole effect of one or more forces. The atomists on their part were divided into two parties with regard to their attempts at explaining the origin of electricity in the voltaic pile. Some who followed Volta explained the pile by the so called ‘contact-theory’, others by chemical conceptions. Controversial as the discussions between atomists and dynamists, contact-theorists and chemists might have been, one fundamental attitude united them which was only superseded by the beginning of theoretical physics. For the schools mentioned the aim of physics consisted in explaining the cause, the essence and the nature of electricity. This attitude did not even change after it had been realized that the Baconian hope of ‘distilling’ electricity in ‘pure form’ in the laboratory, already long in coming, was fading ‘classical sciences’ vs ‘Baconian sciences’ is used as in Thomas Kuhn, Experimental Traditions in the Development of Physical Science’, J. Interdisciplinary Hkt. 7 (1976), l-31. German translation in: Th. Kuhn, Die Entstehung des Neuen (Frankfurt/Main, 1977). pp. 84 - 124. W. Bbhme, ‘Cognitive Norms, Knowledge-Interests and the Constitution of the Scientific Object’, E. Mendelsohn er al. (eds.) The Social Producfion Scientific Knowledge, (Dordrecht, 1977), p. 132. ‘The dichotomy ‘Mathematical vs.
of
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into the distance. The enormous quantity of accumulated facts demanded explanations which could bring some order into this chaos. Dynamism and atomism developed into theories which hypothetically anticipated a representation of the essence of electricity and which attempted to organize the abundant material by the immediate and pure intuition of this essence. By contrast, theoretical physics initially keeps to a positivistic program which Fourier was one of the first to formulate in his ThPorie analytique de la chaleur of 1822. With this theory Ohm viewed the flow of electric current by analogy with the flow of heat in an abstract way. Ohm still regards himself as a contact-theorist, but the importance of this preconception gradually fades, and his theory, properly speaking, shows all the signs of the phenomenological model, after which it was moulded. In its final form the theory is consistent with the hypothesis of the contact-theorists as well as with that of the chemists, as set forth by F. C. Daniell, Ch. Wheatstone and the Royal Society.” In consequence of this development questions of essence are no longer asked. In the end, the adoption of Ohm’s law means also the adoption of new principles of explanation which greatly differ from those of atomism and dynamism. from the intuitive (anschaulich) 2. Standards of theory formation: explanation to mathematical description The new way of theorizing in physics also influenced the criteria which had to be followed by physicists in theory formation. For dynamism and atomism, a theory had to be anschaulich, i.e. intuitive, in the peculiar sense of idealistic and romantic Naturphilosophie. Even mathematics had to submit to intuition. A contemporary of Ohm writes that the physicist needs mathematics only ‘where the intuitive concepts fade away in the contemplation of the infinite in nature’.’ For many scientists physics and mathematics even excluded each other completely. The achievements of mechanics and geometrical optics were considered to be a genuine part of mathematics and were institutionalised as ‘applied mathematics’. By contrast, Ohm maintains (and with him theoretical physicists in general) that, as he put it himself ‘every theory built on facts concerning a class of natural phenomena is imperfect if it does not bear the form of detailed mathematical representation’.8 In his main work Die galvanische Kette, mathematisch bearbeitet (1827) he tries to comply with this claim by stipulating an axiomatic theory of the electric circuit.
‘F. C. Daniell, Phil. Trans. R. Sot. 132 (1842). 269- 287; Ch. Wheatstone. hi/. Trans. R. Sm. 133 (1843), 303 -327 and Proc. R. Sot. 4 (1843) 336. ‘G. W. Muncke, Handbuch der Nuturlehre, Teil 1 (Heidelberg, 1829). p. 6. *G. S. Ohm, ‘Die galvanische Kette mathematisch bearbeitet’, Gesummelfe Abhandhmgen, E. Lommel (ed.) (Leipzig, 1892). p. 106.
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106
3. Attitude
towards experiment: from ‘ars inveniendi’
At the time of Ohm the conception change.
For
activity:
an ‘ars inveniendi’.
discovering
in History and Philosophy of Science
atomists
and
dynamists,
to ‘ars demonstrandi’
of the function of experiment began to experimenting was still a creative
They tried to expand
all sorts of new effects and by finding
the set of known new cross-linkages
facts by to other
areas of knowledge. As an atomist one even hoped to be able some day to isolate the ‘electric matter’ in an unmixed and uncompounded way in the testtube. Through this ‘experimentum crucis’ the aim of electric science would have been reached and the final insight into the essence of electricity would have been won. On the other hand, theoretical physics views experiment as an attempt to confirm a theory which has been formulated beforehand. In his Massbestimmungen iiber die galvanische Kette of 183 1, Gustav Th. Fechner sets out to confirm Ohm’s law by a thoroughly systematic and detailed investigation. He follows Biot’s method of counting the oscillations of a magnetic needle making use of Voltaic piles which were still rather variable at that time. By his work, Fechner set unprecedented standards for any further experimentation in the field of German electrical science. With Ohm and Fechner electrical experimentation lost its Baconian excitement and turned into a more sober undertaking - an ‘ars demonstrandi’.
4. View of technical utilization: from its irrelevance to its use for confirmation Atomists technical scientific
and dynamists
frequently
applicability of a scientific value, i.e. its contribution
way, e.g. Auguste
acknowledged
the commonplace
that the
theory says only little or nothing about its to clear up the problem of essence. In this
De la Rive, alluding
to Ohm and Fechner,
pronounced
the
judgement that the mere and restricted study of electrical conductivity, practical results excepted, cannot contribute much to progress in electrical research.s Like Ampere and many other contemporaries De la Rive could imagine
progress
only in a comprehensive
explanation
that would
phenomena of electricity, heat, light and chemistry together conception. After Ohm this situation changed. A series of physicists particularly Lenz and Wheatstone, adopting and defending because of its applicability for constructing telegraph electromagnetic engines though they had to admit that they did
unify
the
in one single emerged,
e.g.
Ohm’s theory lines and not altogether
*A. De la Rive, ‘Coup d’oeil SW I’Ctat actuel de nos connaissances en klectricit?, Archs. de I’&ectricitP (Suppkment g la Bibliotbkque universelle de Genkve) 1 (1841). 18: ‘Je ne prtsume pas, du reste, que I’ttude pure et simple de la conductibilitt klectrique doive, sauf en ce qui concerne les rtsultats pratiques, contribuer beaucoup aux progrks de la science de I’klectricit?.
Logical Reconstruction
of Revolutionary
107
Change
understand the fundamental terms of Ohm’s theory. Increasingly a new commonplace occurs in literature, converting the search for essence into its opposite: technical utilization now functions as a criterion for the value of a theory. Between 1860 and 1890, electrical research became the first branch of natural science which was industrially applicable in a manner worth mentioning if one leaves out the organic chemical colour industry of the seventies.‘O 5. Model-formation: heat conduction
from
the model
of mechanical
friction
to the model
of
We now come to more internal elements in investigating the change of a scientific theory. In the 18th century F.U.Th. Aepinus was the first to pave the way for a Newtonian conception of electricity and magnetism resorting to attractive and repulsive forces. The effluvial and vortex theories produced by a Cartesian spirit were now considered to be refuted and outdated. Coulomb, pursuing this path, discovered his force-law which became paradigmatic for electrostatics. His conception of the flow of electricity in conductors became exemplary, too. He explained leakage of electricity through imperfectly isolating conductors analogous to friction of a body moving on a surface, thus by mechanical resistance of displacement. Coulomb’s interest in friction dates back to his experience of twelve years as a civil engineer. In 1781 he found the law of mechanical friction, equally named after him, the law which puts the force of friction R proportional to the normal force N.” How could this idea be transferred into the flow of electricity? If a body on a plane base is pushed or pulled with a constant force K and with the consequent appearance of a friction R, then for the resulting force follows K+ = K - R, in case K> R; otherwise, K+ = 0. If Coulomb’s law of friction is applied to R, we get: K+ = K - p . N
in which lo is a coefficient depending on the condition of the surface. Coulomb also calls the force of friction a ‘coercitive force’ which unlike the active force of gravitation can only be measured by the maximum of its resistance.12 In 1785 Coulomb transfers these conceptions to charged spheres leaking electricity along imperfectly conducting suspenders. The force F by which the “See Th. Kuhn, Die Enfsfehung des Neuen (Frankfurt/Main, 1977), p. 210 and C. C. Gillispie, ‘Science and Technology’, The New Cumbridge Modern History, Vol. 9 (Cambridge, 1965), p. 145: ‘The ensuing development of the electrical industry was in all the long history of science the first truly portentous application of a major piece of basic research (and not just of rational method) to the occasions of industry’. “See C. S. Gillmor, Coulomb and the Evolution of Physics and Engineering (Princeton, 1971). “Ch. A. Coulomb, ‘36me memoire sur I’electricite et magnetisme’, Histoire et Mkmoires de I’Acadkmie Royale (Paris, 1785). p. 636.
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108
density D of the sphere’s charge affects the suspending wires is set analogous to the mechanical force K (Fig. 1). F’
F
The force B of the resistance by which a unit of length of the thin suspender opposes the flow of electricity, matches the normal mechanical force N. Thus the total resistance of a wire equals its length s times the ‘normal resistance’ B of the material, if its cross-section is negligibly small. The resulting force, i.e. F - B - s, corresponds to the force F+ by which the direct neighbourhood of the conductor’s end acts upon the charge of the end itself: F’
=
F
-
B - s
in case F > B s; otherwise, F’ = 0. In this way we have found an electrical analogue to the law of mechanical friction. In Coulomb’s time one still believed in the possibility of perfect conduction. The formula allows for this: if B = 0 then the resistance of the whole conductor is 0 and the force which exists on the surface of the sphere does not experience any diminution if measured at the end of the suspender. According to his electrostatic force law, Coulomb proceeds by converting F into B/s2 (more exactly: Dd/(s + r)?, where r is the radius of the sphere) and F’ to ddd/&. He believes that the effect of F can be neglected and, after a somewhat obscure derivation, he obtains the formula d’ = 0’
- B * s.
From this Coulomb concludes: The lengths of different silk threads or of any imperfectly conducting ideoelectric suspenders are proportionate to the squares of the densities as soon as these suspenders isolate completely. And this we have found in accordance with experiment.”
The model of friction can also be transferred to closed circuits. Here, Humphrey Davy’s experiments were of special influence.laO For him, the quantity of electricity which affected the closing wire is considered to be “Ibid., p. 635. ““Phil. Trans. R. Sot. 111(1821). German translation: Gilb. Ann. 71 (1822) 225-261.
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Logical Reconstruction of Revolutionary Change
dependent on the intensity of the pile diminished by the resistance which the current meets in the closing wire. As already mentioned, Ohm replaces this model by a different one: Fourier’s model of heat conduction. The one-dimensional transition q of heat between two body-elements is here proportional to the difference in temperature du: qx = - kw dt
where k is the inner heat conductibility, From this, Ohm derives: dQ =
0” ox
w the cross section and x the distance.
kw (a’-u) s
dt
k is the specific conductivity; U, u ’ the electroscopic force at the beginning and end of a conductor with length s and cross-section w. The flow of electricity is now interpreted in a manner completely different from the mechanical analogue. After a long and complicated derivation Ohm at long last obtains his law:
wherein S is the ‘size of current’, A the ‘sum of all tensions’ and L the ‘sum of all reduced lengths’ in the circuit. To the ‘pre-theoretical’ physicists of Ohm’s time this model did not seem intuitive enough, by contrast with the frictional model through the aid of which they thought they were able to imagine something more concrete.14 6. Classificational level of the sciences: from electrostatics/-dynamics and chemistry to ‘kinematics of the constant movement of electricity’ Before Ohm the phenomena of electric conduction through different
“Only in the forties of the 19th century, that is some two decades after the establishment of Ohm’s law, did Louis Poisseulle find a hydrodynamical law which says that the following quantity flows through capillaries per unit time:
Q=k
HD’ L
where k is a coefficient depending on material and temperature, H height of fall, D diameter and L the length. If this law had been known before Ohm it might very well have served as an anschauliches model, and a better analogue for Ohm’s law than the model of heat conduction.
110
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materials were tackled either by electrostatics or electrodynamics or chemistry: Some researchers were of the opinion that electrical conduction should be viewed as leakage of charged conductors and be explained in some way or other, by Coulomb’s law. On the other hand, one tried to explain higher degrees of conductibility by the strength of the current, an electrodynamical category. In both cases, electricity was thought of as composed of smallest particles, endowed with a force which was inversely proportional to the distance, ideally to the square of the distance. That meant, at least for the atomists, that the way of explaining the microstructure of electrical phenomena was mechanical. Apart from the dynamist, there existed a chemical alternative to this mechanical view. The chemists tried to explain conduction of electricity either as a chemical process or at least as the result of such a process. Ohm did not take sides with any of these conceptions. As a contact-theorist he could not follow any chemical explanation. But equally, he explicitly rejected electrical action at a distance in favour of only short-range forces when (analytically) treating the phenomena of the galvanic circuit. He found a theory which is independent of all former theories treating the so-far disparate relations between intensity, electroscopic force and resistance in the electric circuit. A connection with electrostatics could only be established after Kirchhoff in 1849 had interpreted for the first time the electroscopic force as potential difference.15 Equally, the relation to electrodynamics is not unambiguous. The electromotive force E of the pile is a static category. And Ohm’s theory claims that current intensity Z, a dynamical category, can be brought in connection with the terminal voltage U,, a static category. But since Ampere the existence of terminal voltage had been denied for, or seen incompatible with, the dynamical case. In modern formulation this connection would be: U, = E - Ri * I; where Ri is the internal resistance which usually was of a very considerable size in the piles used in Ohm’s time. We can say now that Ohm’s theory, whilst separated from the established theories, is at the same time also a mixture of disparate theoretical elements taken from very different theories. Thus Ohm’s theory did not live up to any established way of classifying the electric phenomena. Instead of doing ‘normal science’, i.e. normal electrostatics, -dynamics, or chemistry, Ohm declares the problems a class of its own. In this he calls for a completely new classification of the relevant sciences. 7. Organization circuit.
of the scope of theories: from
The applications
the Voltaic pile to the electric
of Ohm’s theory are not only demarcated
‘sPogg. Anna/n. 78 (l&19), 506-513
in a new and
Logical Reconstruction
of Revolutionary Change
111
different way, but also internally organized by a completely new principle. Physicists before and after Ohm see their experiments in different ways: applications which had to appear very heterogeneous when compared with each other are now seen as similar or identical. This falls under the Wittgenstein and Hanson category of ‘seeing as’ and of the ‘gestalt-switch’. In a Voltaic pile closed with a wire atomists and dynamists see a device which produces special electrical phenomena in the closing wire by virtue of its chemical composition and various effects inside the pile. According to this view, the conductivity of the wire, for instance, can depend on the type of the current source used (Voltaic pile, thermo-electric source, friction device. . .), the chemical composition as well as the pile’s ‘intensity’. By a law of conduction was meant a law that integrates all the physical properties or the chemical composition of a body together with all the modifications of conductivity, mentioned above. In addition, the phenomena experienced with the open pile and the closed pile were strictly separated. The classification induced thereby can be found in all the textbooks of the time. Since Ohm’s theory, especially in Wheatstone’s version,” the nature of the current source is irrelevant. It is a ‘black box’, only of interest for its current supply. The same arrangement of experiment which before Ohm was explained solely by the qualities of the pile is now seen as an ‘electric circuit’, an abstract conception under which also the discharge of statically charged bodies may be classified. In the same way, any electric source whatsoever, whether closed or not, can be subsumed under the concept of a circuit. The measurement of tension (‘electroscopic force’) on the open and closed pile can be formulated through one and the same formula: in modern terms, for the closed pile Uk = E - Ri *I,for the open pile E = Uk + Ri =I.In the same way the behaviour of a galvanometer, for instance, is similarly explained by using the conception of the circuit. Between all these applications and numerous other phenomena a similarity relation could be stated which was impossible before Ohm. With the construction of this similarity the road was freed for the investigation of general structural properties of circuits. The formulation of Kirchhoff’s laws” even became a land-mark for graph theory. Possibly one may say that in the changing configuration from ‘Voltaic pile’ to ‘electric circuit’, ‘substance’ is replaced’by ‘function’ in the way that Cassirer thought characteristic of the development of modern thought.”
‘ePhil. Trans. R. Sot. 133 (1843). 303 - 327 and in German in Pogg. Annaln 62 (1844). 499-543. “Pogg. Anna/n 72 (1847), 497 - 508. See N. L. Biggs et ui., Graph Theory 1736- 1936 (Oxford, 1976), p. 131ff. “E. Cassirer, Substanzbegriff und Funktionsbegriff. (Berlin, 1910).
Studies in History and Philosophy of Science
112 8. Level
of the underlying
theory of measurement:
new foundations
for
electric measurement
Ohm’s theory also had drastic effects on the methods of measurement in electrical science. After the general acknowledgement of Ohm’s law all newly proposed methods and instruments for measuring intensity, tension and resistance of a circuit presupposed the validity of Ohm’s law in one way or other. The measurement of the (inner and outer) resistance of a galvanic circuit is a direct derivation from Ohm’s law in all the proposed methods such as Ohm’s himself, Fechner’s, Poggendorff’s and W. Weber’s.” For Wheatstone’s method one even has to presuppose the validity of Kirchhoff’s laws for branched circuits. For the measurement of intensity by a galvanometer, one has to take into account the resistance of the instrument itself. This can only be justified by reference to Ohm’s law. At least until Kohlrausch’s experiments in 1849” the measurement of the ‘electroscopic force’ of the open and closed pile was done exclusively by presupposing Ohm’s law. Here also, the methods of Ohm, Fechner, Wheatstone and Poggendorff are basic. There are at least two reasons why the tension phenomena of a closed circuit were not measured directly by a voltage meter for a very long time: no piles were available which could produce a constant current for a considerable time, and the accuracy of the electrometer was not enough to arrive at an independent confirmation. But even if one measured the tension directly, in the end one again had to take into account the resistance of the measuring instrument itself, thus presupposing Ohm’s concepts. It was only in 1856 that, for the first time, Wilhelm Weber and Rudolf Kohlrausch were able to give a satisfying theory for the absolute measurement of the electromotive force and current intensity without assuming the validity of Ohm’s law beforehand. At the same time, they were the first to express these magnitudes in mechanical units.2’ What was the change then which Ohm’s theory brought about in measurement? First of all, it serves as a theory for measuring resistance, thereby differing from all former theories. Thus, whenever measuring ‘resistance’ one has to presuppose that there already exists a successful application of Ohm’s law. Therefore, ‘resistance’ is a theoretical term relative to Ohm’s theory.22 “See G. Wiedemann, Die Lehre von der Elektrizitiit, and 676ff. and D. Rutenberg, ‘The Early History of Measurement’, Ann. Sci. 4 (1939), 212 -243. _ “Pogg. Annaln 78 (1849), 1 - 21. “Pogg. Annaln 99 (1856). 10 -25. 221n the structuralist view ‘T-theoretical’ is defined theoretical in Tiff from the description of measuring n always that Thas at least one successful application. See l), p. 45ff. and A. Kamlah, ‘An Improved Version
Erkenntnis 10 (1976), 349 - 359.
Band 4, (Braunschweig, 1898). p. 4OOff. the Potentiometer Svstem of Electrical
in the following way: a function n is in a given application of T there follows Sneed (note 3), p. 31ff.. Stegmiiller (note of “Theoretical in a given Theory”‘.
Logical Reconstruction of Revolutionary Change Until
Weber
electromotive
and Kohlrausch, force
It was not
until
Coulombian
the same holds for the measurement
and the terminal
force law as the measuring
theory,
Kirchhoff’s
way of measuring
113
voltage:
instead
one presupposed
analysis
in 1847 that
electrostatic
of taking
the validity
density
one could
of the
Coulomb’s
of Ohm’s law. identify
the
of charge and the Ohmian
way of measuring the ‘electroscopic force’ as the same ‘potential difference’. But this is an empirical hypothesis which could be regarded analytically true only with Weber’s and Kohlrausch’s unifying theory of measurement. Thus, until Weber and Kohlrausch, ‘electromotive force’ exclusively served as a theoretical
term relative
to Ohm’s
theory
and not to any preceding
one.
9. Definition
of the basic terms: from ‘quantity’, ‘intensity’, and ‘conductivity’ to ‘size of current ‘, ‘electroscopic force’ and ‘reduced length’ Everything fundamental
has now been prepared for properly terms at the time of Ohm. Previous
understanding the change of to Ohm three basic concepts
were in use: quantity, intensity and conductivity. Intensity is measured with the electrometer, being only available at the open pile. It was then regarded as a mere static phenomenon which disappeared when discharged. When the Voltaic pile is closed such a static discharge is believed to take place. Thus, the intensity only acts at the moment of closure, in order to overcome the resistance of the closing wire. In a similar way a body slides ‘by itself’ on an inclined plane if its static friction is overcome (kinetic [sliding] friction is slightly less than static friction). The quantity of electricity flowing through the closing wire is indicated and measured by the galvanometer. For its part, it is dependent on the conductivity of those parts of the arrangement which do not excite conception pile: a pile resistances
electricity. Here, it is essential to realize that according to this the quantity per time is also dependent on the initial intensity of the having ever so much quantity, but not intensity enough to overcome will not bring about current flow.
Compared very strange. (‘electroscopic
with these concepts,
Ohm’s new definitions
had at best to appear
His new concepts are called elektroskopische Kraft force’), Stromgr@e (‘size of current’) and reduzierte Liinge
(‘reduced length’). By the latter he understood the length of a standard wire which has exactly the same conductivity as the part of the circuit under investigation. By Leitungsgiite (‘conductive quality’ or specific conductivity) of the standard wire Ohm means the quantity of electricity which flows from one place to another in unit time, but not the velocity of flow, as was often done before him. The size of current S is defined as the quantity of electricity flowing through the cross-section w of a conductor per unit time:
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114
S is measured The quantity
by the Multiplikator Q is defined
(galvanometer).
as the ‘sum of the electroscopic
manifestations’.
This would mean that the quantity might have become measurable by means of the electroscope which appeared unthinkable before Ohm. By the ‘electroscopic force’ at a point in the circuit Ohm understands the density of electricity at that point - a conception which has no longer anything to do with the old concept of intensity. This term proved to be the most difficult one to accept for Ohm’s contemporaries. In Ohm’s theory, the Spannung (‘tension’) which is present at the endpoints of a current carrying conductor is the difference of the electroscopic forces at the endpoints. As an axiom, Ohm stipulates this difference to be constant: a = i4 - U’ = constant. For Ohm, this supposition was the most important one of his theory. At first, Ohm stated his law for current carrying wires of finite length. Later, by adding the hypotheses of voltaic contact theory he tried to give a general theory of electric circuits.
II Having sketched the essential levels at which the change in electrical science had taken place I will now try and put forward the set-theoretical predicates of the theories in question. First of all the basic predicate
for Coulomb’s
theory:
P 1 : X is a Coulomb force system iff there is D, d, a, F, such that (1) X = < D, d, a, F>. (2) D is a non-empty finite set of electrically charged bodies. (3) d : D - R +(density of charge of a body). (4) a : D x D + R + (distance between two bodies). * (5) F : D x D + W + (force acting between two bodies). (6) for all p, q E D, where p Z q : F@,q)
=
d(P) * d(q) . eP>q)
This is the basic predicate
of Coulomb’s
theory.
By
f :D + R +we mean a
Logical Reconstruction of Revolutionary Change
115
function from D into the positive real numbers and the asterisk (*) indicates the theoretical term of the predicate. Now, any entity to which this predicate can be attributed is a model of the theory. d and a are T-non-theoretical terms, whereas F is T-theoretical. Coulomb expanded this predicate in order to take into account the case of electric conduction. The resistance or retarding force B a of the wire of length a appears in the same way as the loss of force in the case of mechanical friction. In doing this Coulomb makes use of two new T-theoretical terms: the concept of specific resistance and the concept of the molecular force F’ with which a charged point on a charged body is effected by the charge of its immediate neighborhood: P 2 : X is a Coulomb conducting system iff there is D, d, a, F, F’, B, such that (1) X = < D, d, a, F, P, B > . (2) - (6)asinPl. * (7) F+ : D - W + (molecular force). * (8) B : D x D + R +(specific resistance of the material betweenp E D and q E D). (9) p E D, q E D, p # q, d(q) Q d@). Then:
B@,q)*d.nq) = F@,q) - F+(q) or equivalently: (9’) . . .
B07,q) a@,q)
= d*@)-d*(q).
Humphry Davy transferred this idea to the arrangement which is formed by a closed pile. His theory is not very elaborated and so the following predicate has to be taken with some care. As T-non-theoretical concepts Davy uses the quantity of electricity, the cross-section of the wire, and its temperature, in addition to Coulombian concepts: P 3 : X is a Davy conducting system iff there is D, d, a, F, P B, Q, d, I?, such that (1) X =. (2) - (8) as in P 2 . (9) Q : D + R? + (quantity of the pile). (lO)d: D- W + (cross section of wire). (ll)~? : D - R + (temperature of wire).
Studies in History and Philosophy of Science
116
(12)~ E 0, q E D, d @) 2 d (q), p # q. Then:
If F(p,q)-F+(q)=0
then B@,q) a(q) * 19(4)
= Q @)
d(q)
In 1825 and 1826, Ohm put forward two new theories which are intermediate between the Coulomb conception and his eventual theory.” Though very interesting, we have to omit these theories and reserve the treatment to a later date. In the following predicate representing Ohm’s second article of 1826 the concept of the Ohmian current carrying conductor is introduced. ‘Specific conductivity’ and ‘electroscopic .force’ are the T-theoretical terms of the theory. P 4 : Z is an Ohmian conductor iff there is D, I, w, X, k, u, u 0 such that (1) Z = < D, I, w, X, k, u, u’ >. (2) D is a homogeneous current carrying prismatic conductor. (3) 1 : D + R +(length of the conductor). (4) w:D-‘W +(cross-section of conductor). (5) X : D +W + (strength of electric current). * (6) k : D -*R + (specific conductivity). * (7) u : D +R D(electroscopic force at the beginning of conductor). * (8) u’ : D - R o (electroscopic force at the end of conductor). (9)
X(D) =
(u(D)-
u ‘(D))
. k(D)
. w(D)
IV’)
The next predicate corresponds to Ohm’s theory of 1827, put forward in his Die galvanische Kette, mathematisch bearbeitet, his main work: P 5 : X is an Ohmic circuit iff there is D, E, g, E, T, I, q, a, u, such that (1) X = < D, E, g, =, T, I, g, u, u >. (2) D is a nonempty finite set of homogeneous prismatic conductors. (3) E is a set of points, the endpoints of the conductors. (4) g : D +_E x E; and 2 & E x E is an equivalence relation such that < D, E, g, i > induces a single closed circuit with no branches and no loops, and for any e E E, there is an e’ c E which is equivalent to it (i.e. is in direct contact with it). z’Schweigg. J. Chem. Phys. 44 (1825). llO137 - 166.
118 and Schweigg.
J. Chem.
Phys.
46 (1826),
Logical Reconstruction (5)
of Revolutionary
T is an interval
117
Change
of R D(time-interval).
(6) I : D + R + (length). (7) q : D + R (8) Q : D +W * (9) u : D - R * (10)~ : E -R * (1l)for
+ (cross-section). D(quantity of electricity). + (specific resistance). o (electroscopic
all tE
T, for all eE
du(e) = 0 (condition
force). E:
for stationary
current).
dt (12)for
all e, e’ E E, e # e’ and e 3
1 u(e) - u(e’)l>
0
(contact-theoretical (13)for
e’:
hypothesis).
all t E T, for all s, s’ E D:
dQ(s) q(s)dt
-
dQ(s ‘) (condition
of continuity).
q(s’)dt
(14) be s,E D. Then
for all t E T: dQ(so) .
e=e’
I u(e)-@‘)I
= dt
4s) 4s).q(s)
c sED
e% e’ e,e’EE or: In 1841 the Royal
u Society
decided
=I to regard
.R Ohm’s
law as independent
of
any chemical or contact-theoretical hypothesis.24 After this, Ohm’s law quickly turned into the form which can be taken as still valid today. It was Fechner’s and Wheatstone’s work which was mainly responsible for this development: P 6 : X is an Ohm-Fechner- Wheatstone circuit iff there is LL D2, I, 1, w, k, E, R,, Ri, such that (1) X = < D,, Dz, I, I, w, k, E, R,, Ri >. “i’roc. R. Sot. 4 (184lL 336.
Studies in History and Philosophy of Science
118
(2) D, is the outer (closing) part of the circuit. (3) D2 is a source of current. (4) I : D, x D* - R o (strength of current). (5) I : D, -, R + (length of outer part). (6) w : D, ‘R + (cross-section). (7) k : D, +R + (specific resistance). * (8) E : D2 +R D(electromotoric force). *(9) R,:D,+R + (outer resistance). * (10)R): D, + 1R D(inner resistance). (ll)E(Dz) = (R.(D,) + R,(D,) ) * I(D, Dz). (12)R,(D,)
= k(Q) I(D,) -. ~(DI )
In this predicate we have three different T-theoretical terms: specific conductivity, electromotive force and the inner resistance. P 6 as such is only applicable to one circuit at a time and does not state any connections between different circuits. But a very important part of the lawstatements made by Ohm’s law in pruxi concern sets of circuits. We can account for this by introducing three additional constraints which intuitively say the following: The resistance of any finite part of a circuit remains constant, if this part is included as a proper part in any other electric circuit. And equally the e.m.f. of a source of current remains the same, no matter what circuit it is a proper part of . More specifically: Let o, o’ be Ohm-Fechner-Wheatstone circuits. Then let D,(O) be the external part of the circuit o and D*(w) be the internal part thereof. Then for all Ohm-Fechner-Wheatstone circuits the following constraints hold: If D,(W) = D,(o’), (2) If D*(o) = Dz(o’), (3) If D&J) = D&I’), (I)
then R.(Dj(w)) = RJD(w’)) then Ri(D4w)) = Ri(Dz(o ’ 1) then EU&(w)) = E@(Q)‘))
This concludes the formal part of this paper. III
Having formalized the relevant historical theories in informal set-theoretical terms we go back to the initial question: what can formalization, and especially the ‘structuralist’ one, contribute to the logical reconstruction of the Ohm-episode in the history of electricity? The most important advantage of
Logical Reconstruction the structuralist of the theory grasp
of Revolutionary
approach, in question.
the theory
119
Change
it seems to me, is its ability By a set-theoretical
at one glance.
predicate,
This advantage
to produce
an overview
we can, so to speak,
is not merely
an educational
one. Wittgenstein tried to show that if we want to see how language functions we need an overview of the different uses of words:
really
Our grammar is lacking in this sort of perspicuity. A perspicuous representation produces just that understanding which consists in ‘seeing connections’. . ..The concept of a perspicuous representation is of fundamental significance for us. (Investigations, section 122). I think it is also true in the case of scientific theories that we need a perspicuous mode of representing their essential features in order to understand them and their place in history, i.e. the relations of their internal parts to each other and their connections and relations to other theories. In set-theoretical terms we can give a clear definition
of an application
of a
theory and we can display the theory’s structure in a very precise and unmetaphorical way. By an application of a theory there is meant any entity that can be described by the conceptual apparatus of the theory. We can distinguish two different levels in the structure of the application of a theory: the T-non-theoretical level and the T-theoretical level. At the T-nontheoretical level we can describe what the structure of an entity looks like before one applies the theory to it, i.e. before one has available the special the application of a information supplied by the theory. 25 For example, Coulomb theory (P 1) at the T-non-theoretical charged bodies, their distance from each
level is formed by the set of two other and the density of their
electrical charge. Analogously, for an Ohm conductor (P4) an application at the T-non-theoretical level consists of a conductor, its length, cross-section, and the size of current it carries. And for an Ohmic circuit (P 5) an application consists of a set of conductors which forms a closed cross-sections and the current in the circuit. The respective T-theoretical level of the applications adding
the T-theoretical
functions
circuit,
their lengths,
can be obtained
to the list. The applications
by
of Ohm’s and
Coulomb’s theories already differ at their T-non-theoretical level. The gestaltswitch required when changing from one theory to the other also consists in this difference. Thus, a set-theoretical formalization shows us how a theory organizes its scope and how this organization changed from Coulomb to Ohm. In this way we have elucidated the change which we tried to describe in part II as level 7 (organization of the scope of theories). The formalization also shows us how the meaning of a theoretical term is established. When comparing a Coulomb-type-theory with an Ohm-typeY&x also note 22.
120
theory
Studies in History and Philosophy of Science we can
immediately
see a difference
in the meaning
between
the
Coulomb-term of electrostatic force and the Ohm-term of electroscopic force (resp. tension). Although in both cases an electrometer is involved in measuring
the respective
for the measurement
magnitudes,
differ
from
the theoretical each other.
Thus,
presuppositions by formalizing
contribute to the explication of the meaning of the T-theoretical contributing to level 9 (definition of the basic terms).
necessary we can
terms, thereby
So far we have dealt with metatheoretical predicates which are one-place: ‘organization of the scope of a theory’ and ‘definition of the basic terms of a theory’. Now we will turn to metatheoretical predicates with more than one place, representing intertheoretical relations. First, we have the predicate of a ‘theory being the model of another theory’. Let us assume, that the theories of mechanical friction and of heat-conduction had been formalized respectively. Then the change in model-formation (described at level 5) can now be expressed by stating that the structure of the Coulomb-predicate (P 2) is isomorphic to the structure of the predicate describing the theory of mechanical friction, but not the theory of heat-conduction; and that the structure of the Ohm-predicate (P 5) is isomorphic to the predicate expressing the theory of heat-conduction but not the theory of friction. Another intertheoretical relation would be the classification of a range of phenomena by a theory in relation to a set of other theories. A classification induced by a theory can be seen as in agreement with that of another theory if the non-theoretical terms of the theories are identical. And the classification would be a new one in relation to another theory if the theories differ in at least one non-theoretical term. Let us, in addition, call a new classification by a theory T revolutionary in relation to T' if T introduces such a new nontheoretical term which appears in another theory T", not identical with T or
T'. In this sense, Ohm’s classification is new in relation to Coulomb’s theory, since he does not use ‘density of charge’ as a non-theoretical predicate when describing one and the same phenomenon and uses ‘current’ as a new basic predicate instead. If we take the term ‘electroscopic force’ as a non-theoretical term (which can be done after Kirchhoff, Kohlrausch and Weber) Ohm’s classification is even revolutionary both in relation to Ampere’s and Coulomb’s theory. In the first case, Ohm’s non-theoretical vocabulary differs from Ampere’s in the concept of electroscopic force which is a non-theoretical predicate in another theory, namely Coulomb’s. And in the second case, Ohm uses ‘electroscopic force’ in agreement with Coulomb’s ‘density of charge’ but he includes ‘current’ which is a basic predicate in another theory, namely Ampere’s. As we can easily see, such a change in classification (described on level 6) is most clearly represented by the structure, which is provided by the set-theoretical predicate of a theory.
Logical Reconstruction
of Revolutionary
Change
121
A complete set-theoretical description of a theory T would have to include the description of all those theories on which the measurement of the nontheoretical terms of T is founded. Thus, we can introduce the intertheoretical predicate of a ‘theory T resting on one or more theories of measurement’. Here, we cannot reconstruct all the theories of measurement relevant to our historical episode. But I think it is clear that such a specification of theories of measurement and their change over time from Coulomb to Weber could very well be described in set-theoretical terms. Thus, the ‘structuralist view’ can serve as a reconstruction of level 8 (underlying theory of measurement). If there appear different theories of measurement over time for one and the same non-theoretical term, it can also point out how the meaning of that term has changed (level 9: meaning-change of basic terms). So far, we have tried to show that by formalizing theories in set-theoretical terms one can reconstruct levels 5 - 9. But what about levels 1 - 4? It seems obvious to me that the set-theoretical view cannot contribute anything in reconstructing the change in a theory’s world view (level l), in its standards of theory formation (level 2), in its attitude towards experiment (level 3) and in its view of technical utilization (level 4). If giving a definition of the general term ‘scientific revolution’ would necessarily include change on these and/or similar levels, as I think it does, then it is obvious that revolutionary change could only partly be explicated in ‘structuralist’ terms. The result seems to force upon us the suspicion that there is a general difference in category between levels 1 - 4 and 5 - 9. I think it amounts to the following: levels 5 - 9 describe scientific change at the object-level and levels 1 -4 describe scientific change at the meta-level. At the object-level we describe how the special theories have changed, their concepts, their relations to other theories, the structure of their applications, etc. But at the meta-level we describe traits of a completely new category. We try to find an explication of general terms like explanation, theory, confirmation, law, experiment, usefulness, etc. And I think that the idea of revolutionary change in science did not so much prove itself revolutionary for the philosophy of science by pointing out the historical change in scientific concepts, perceptions, and methods, but by showing that there cannot be given a general explication of metascientific terms which could be adequate for history of science in all its stages. This limitation, I think, cannot be overcome by any philosophy of science which claims to be adequate in reconstructing historical scientific theories, whether using formal means or not.
HEIDELBERGER *
TOWARDS A LOGICAL RECONSTRUCTION OF REVOLUTIONARY CHANGE: THE CASE OF OHM AS AN EXAMPLE I
ALL FORMALIZATIONSof historical scientific theories, as yet established, are confined to a small number of mathematized theories. In addition, the relation existing between actual historical theories has rarely been treated. There has been, it seems, no explicit investigation of a theory-pair in which one theory is separated from another by a scientific revolution. However, the claim has been made that by formalizing theories and reducing them to each other even the last rationality gap within a logical reconstruction of scientific change may be closed.’ In this paper I attempt to show in detail on how many levels an actual revolutionary change may take place and how, if at all, formalization can serve for the reconstruction of that change on these levels. As my historical example I choose the change in the field of electric research which occurred in Germany around 1830.’ As the method of formalization I take the ‘structuralist approach’ first advocated by Suppes, Adams, McKinsey and further developed by Sneed, Stegmiiller, and others.3 For several reasons, the transition to Ohm’s theory seems to be a good start for systematic reflections on the change of theories. On the one hand, this event happened not very long ago. Thus, the problems seem to us more familiar and easier to evaluate than those which arose, e.g. in the transition from Aristotelian to Galilean physics. On the other hand, the science of electricity forms part of the ‘Baconian Sciences’ (theory of heat, magnetism, chemistry. . .) which unlike the classical sciences (statics, mechanics,
*Seminar ftir Philosophie, Logik & Wissenschaftstheorie der Universitat Mtinchen, Ludwigstrasse 3 1, 8000 Mtinchen 22, West Germany. ‘See e.g. W. Stegmtiller, Theorie and Erfahrung (Berlin, 1973), p. 249; English edition: The Structure and Dynamics of Theories (New York, 1976). ‘Compare also the very interesting article of Kenneth L. Caneva, ‘From Galvanism to Electrodynamics: The Transformation of German Physics and its Social Context’, Hist. Stud. Phys. Sci. 9 (1978), 63 - 159. My paper has been written independently of Caneva’s work. ‘See J. D. Sneed, The Logical Structure of Mathematical Physics (Dordrecht, 1971) and W. Stegmtiller, The Structuralist View of Theories (New York, 1979) for the relevant literature.
Stud. Hist. Phil. Sci., Vol. 11 (1980), No. 2, pp. 103 - 121. Pergamon
Press Ltd.,
Printed
in Great
Britain. 103
104
Studies in History and Philosophy of Science
astronomy. . .) have as yet been greatly neglected by philosophy of science.4 Moreover, the episode of Ohm may be regarded as one of the typical cases for the change in the whole of natural science between 1830 and 1850. Recently, Bijhme has advocated the view that the investigation of scientific controversies could result in a similar function for the philosophy and sociology of science as twin-research does for biology and medicine.5 Thus, the history of the origin and reception of Ohm’s law is very suitable for throwing light upon the period in question; for the history of Ohm’s law is one of the most celebrated examples of resistance of scientists to scientific innovation. In the following paragraphs I should like to characterize the nine most essential levels on which, in my opinion, the change of electrical research took place. After that, some of the theories in question will be formalized and the question posed on which of these levels some formalizing of theory-elements might preferably be of use for deeper analysis. We shall see that the structuralist view proves especially valuable to illuminate the relation of a theory to its range of applications and at the same time the theoretical presuppositions on which a theory rests. Our main result will be that the revolutionary change brought about by Ohm’s theory can only partly be explicated in formal terms. 1. Level
of the world view: from
‘physics of essences’
to theoretical
physics
In Ohm’s time there were two main trends in electrical research, atomism and the dynamism of the ‘Naturphilosophie’. The atomists tried to explain electrical phenomena by means of the conception that electricity was an imponderable matter of finest atoms, whereas the dynamists rejected this thesis, explaining electricity as the sole effect of one or more forces. The atomists on their part were divided into two parties with regard to their attempts at explaining the origin of electricity in the voltaic pile. Some who followed Volta explained the pile by the so called ‘contact-theory’, others by chemical conceptions. Controversial as the discussions between atomists and dynamists, contact-theorists and chemists might have been, one fundamental attitude united them which was only superseded by the beginning of theoretical physics. For the schools mentioned the aim of physics consisted in explaining the cause, the essence and the nature of electricity. This attitude did not even change after it had been realized that the Baconian hope of ‘distilling’ electricity in ‘pure form’ in the laboratory, already long in coming, was fading ‘classical sciences’ vs ‘Baconian sciences’ is used as in Thomas Kuhn, Experimental Traditions in the Development of Physical Science’, J. Interdisciplinary Hkt. 7 (1976), l-31. German translation in: Th. Kuhn, Die Entstehung des Neuen (Frankfurt/Main, 1977). pp. 84 - 124. W. Bbhme, ‘Cognitive Norms, Knowledge-Interests and the Constitution of the Scientific Object’, E. Mendelsohn er al. (eds.) The Social Producfion Scientific Knowledge, (Dordrecht, 1977), p. 132. ‘The dichotomy ‘Mathematical vs.
of
Logical Reconstruction of Revolutionary Change
105
into the distance. The enormous quantity of accumulated facts demanded explanations which could bring some order into this chaos. Dynamism and atomism developed into theories which hypothetically anticipated a representation of the essence of electricity and which attempted to organize the abundant material by the immediate and pure intuition of this essence. By contrast, theoretical physics initially keeps to a positivistic program which Fourier was one of the first to formulate in his ThPorie analytique de la chaleur of 1822. With this theory Ohm viewed the flow of electric current by analogy with the flow of heat in an abstract way. Ohm still regards himself as a contact-theorist, but the importance of this preconception gradually fades, and his theory, properly speaking, shows all the signs of the phenomenological model, after which it was moulded. In its final form the theory is consistent with the hypothesis of the contact-theorists as well as with that of the chemists, as set forth by F. C. Daniell, Ch. Wheatstone and the Royal Society.” In consequence of this development questions of essence are no longer asked. In the end, the adoption of Ohm’s law means also the adoption of new principles of explanation which greatly differ from those of atomism and dynamism. from the intuitive (anschaulich) 2. Standards of theory formation: explanation to mathematical description The new way of theorizing in physics also influenced the criteria which had to be followed by physicists in theory formation. For dynamism and atomism, a theory had to be anschaulich, i.e. intuitive, in the peculiar sense of idealistic and romantic Naturphilosophie. Even mathematics had to submit to intuition. A contemporary of Ohm writes that the physicist needs mathematics only ‘where the intuitive concepts fade away in the contemplation of the infinite in nature’.’ For many scientists physics and mathematics even excluded each other completely. The achievements of mechanics and geometrical optics were considered to be a genuine part of mathematics and were institutionalised as ‘applied mathematics’. By contrast, Ohm maintains (and with him theoretical physicists in general) that, as he put it himself ‘every theory built on facts concerning a class of natural phenomena is imperfect if it does not bear the form of detailed mathematical representation’.8 In his main work Die galvanische Kette, mathematisch bearbeitet (1827) he tries to comply with this claim by stipulating an axiomatic theory of the electric circuit.
‘F. C. Daniell, Phil. Trans. R. Sot. 132 (1842). 269- 287; Ch. Wheatstone. hi/. Trans. R. Sm. 133 (1843), 303 -327 and Proc. R. Sot. 4 (1843) 336. ‘G. W. Muncke, Handbuch der Nuturlehre, Teil 1 (Heidelberg, 1829). p. 6. *G. S. Ohm, ‘Die galvanische Kette mathematisch bearbeitet’, Gesummelfe Abhandhmgen, E. Lommel (ed.) (Leipzig, 1892). p. 106.
Studies
106
3. Attitude
towards experiment: from ‘ars inveniendi’
At the time of Ohm the conception change.
For
activity:
an ‘ars inveniendi’.
discovering
in History and Philosophy of Science
atomists
and
dynamists,
to ‘ars demonstrandi’
of the function of experiment began to experimenting was still a creative
They tried to expand
all sorts of new effects and by finding
the set of known new cross-linkages
facts by to other
areas of knowledge. As an atomist one even hoped to be able some day to isolate the ‘electric matter’ in an unmixed and uncompounded way in the testtube. Through this ‘experimentum crucis’ the aim of electric science would have been reached and the final insight into the essence of electricity would have been won. On the other hand, theoretical physics views experiment as an attempt to confirm a theory which has been formulated beforehand. In his Massbestimmungen iiber die galvanische Kette of 183 1, Gustav Th. Fechner sets out to confirm Ohm’s law by a thoroughly systematic and detailed investigation. He follows Biot’s method of counting the oscillations of a magnetic needle making use of Voltaic piles which were still rather variable at that time. By his work, Fechner set unprecedented standards for any further experimentation in the field of German electrical science. With Ohm and Fechner electrical experimentation lost its Baconian excitement and turned into a more sober undertaking - an ‘ars demonstrandi’.
4. View of technical utilization: from its irrelevance to its use for confirmation Atomists technical scientific
and dynamists
frequently
applicability of a scientific value, i.e. its contribution
way, e.g. Auguste
acknowledged
the commonplace
that the
theory says only little or nothing about its to clear up the problem of essence. In this
De la Rive, alluding
to Ohm and Fechner,
pronounced
the
judgement that the mere and restricted study of electrical conductivity, practical results excepted, cannot contribute much to progress in electrical research.s Like Ampere and many other contemporaries De la Rive could imagine
progress
only in a comprehensive
explanation
that would
phenomena of electricity, heat, light and chemistry together conception. After Ohm this situation changed. A series of physicists particularly Lenz and Wheatstone, adopting and defending because of its applicability for constructing telegraph electromagnetic engines though they had to admit that they did
unify
the
in one single emerged,
e.g.
Ohm’s theory lines and not altogether
*A. De la Rive, ‘Coup d’oeil SW I’Ctat actuel de nos connaissances en klectricit?, Archs. de I’&ectricitP (Suppkment g la Bibliotbkque universelle de Genkve) 1 (1841). 18: ‘Je ne prtsume pas, du reste, que I’ttude pure et simple de la conductibilitt klectrique doive, sauf en ce qui concerne les rtsultats pratiques, contribuer beaucoup aux progrks de la science de I’klectricit?.
Logical Reconstruction
of Revolutionary
107
Change
understand the fundamental terms of Ohm’s theory. Increasingly a new commonplace occurs in literature, converting the search for essence into its opposite: technical utilization now functions as a criterion for the value of a theory. Between 1860 and 1890, electrical research became the first branch of natural science which was industrially applicable in a manner worth mentioning if one leaves out the organic chemical colour industry of the seventies.‘O 5. Model-formation: heat conduction
from
the model
of mechanical
friction
to the model
of
We now come to more internal elements in investigating the change of a scientific theory. In the 18th century F.U.Th. Aepinus was the first to pave the way for a Newtonian conception of electricity and magnetism resorting to attractive and repulsive forces. The effluvial and vortex theories produced by a Cartesian spirit were now considered to be refuted and outdated. Coulomb, pursuing this path, discovered his force-law which became paradigmatic for electrostatics. His conception of the flow of electricity in conductors became exemplary, too. He explained leakage of electricity through imperfectly isolating conductors analogous to friction of a body moving on a surface, thus by mechanical resistance of displacement. Coulomb’s interest in friction dates back to his experience of twelve years as a civil engineer. In 1781 he found the law of mechanical friction, equally named after him, the law which puts the force of friction R proportional to the normal force N.” How could this idea be transferred into the flow of electricity? If a body on a plane base is pushed or pulled with a constant force K and with the consequent appearance of a friction R, then for the resulting force follows K+ = K - R, in case K> R; otherwise, K+ = 0. If Coulomb’s law of friction is applied to R, we get: K+ = K - p . N
in which lo is a coefficient depending on the condition of the surface. Coulomb also calls the force of friction a ‘coercitive force’ which unlike the active force of gravitation can only be measured by the maximum of its resistance.12 In 1785 Coulomb transfers these conceptions to charged spheres leaking electricity along imperfectly conducting suspenders. The force F by which the “See Th. Kuhn, Die Enfsfehung des Neuen (Frankfurt/Main, 1977), p. 210 and C. C. Gillispie, ‘Science and Technology’, The New Cumbridge Modern History, Vol. 9 (Cambridge, 1965), p. 145: ‘The ensuing development of the electrical industry was in all the long history of science the first truly portentous application of a major piece of basic research (and not just of rational method) to the occasions of industry’. “See C. S. Gillmor, Coulomb and the Evolution of Physics and Engineering (Princeton, 1971). “Ch. A. Coulomb, ‘36me memoire sur I’electricite et magnetisme’, Histoire et Mkmoires de I’Acadkmie Royale (Paris, 1785). p. 636.
Studies in History and PhiIosophy of Science
108
density D of the sphere’s charge affects the suspending wires is set analogous to the mechanical force K (Fig. 1). F’
F
The force B of the resistance by which a unit of length of the thin suspender opposes the flow of electricity, matches the normal mechanical force N. Thus the total resistance of a wire equals its length s times the ‘normal resistance’ B of the material, if its cross-section is negligibly small. The resulting force, i.e. F - B - s, corresponds to the force F+ by which the direct neighbourhood of the conductor’s end acts upon the charge of the end itself: F’
=
F
-
B - s
in case F > B s; otherwise, F’ = 0. In this way we have found an electrical analogue to the law of mechanical friction. In Coulomb’s time one still believed in the possibility of perfect conduction. The formula allows for this: if B = 0 then the resistance of the whole conductor is 0 and the force which exists on the surface of the sphere does not experience any diminution if measured at the end of the suspender. According to his electrostatic force law, Coulomb proceeds by converting F into B/s2 (more exactly: Dd/(s + r)?, where r is the radius of the sphere) and F’ to ddd/&. He believes that the effect of F can be neglected and, after a somewhat obscure derivation, he obtains the formula d’ = 0’
- B * s.
From this Coulomb concludes: The lengths of different silk threads or of any imperfectly conducting ideoelectric suspenders are proportionate to the squares of the densities as soon as these suspenders isolate completely. And this we have found in accordance with experiment.”
The model of friction can also be transferred to closed circuits. Here, Humphrey Davy’s experiments were of special influence.laO For him, the quantity of electricity which affected the closing wire is considered to be “Ibid., p. 635. ““Phil. Trans. R. Sot. 111(1821). German translation: Gilb. Ann. 71 (1822) 225-261.
109
Logical Reconstruction of Revolutionary Change
dependent on the intensity of the pile diminished by the resistance which the current meets in the closing wire. As already mentioned, Ohm replaces this model by a different one: Fourier’s model of heat conduction. The one-dimensional transition q of heat between two body-elements is here proportional to the difference in temperature du: qx = - kw dt
where k is the inner heat conductibility, From this, Ohm derives: dQ =
0” ox
w the cross section and x the distance.
kw (a’-u) s
dt
k is the specific conductivity; U, u ’ the electroscopic force at the beginning and end of a conductor with length s and cross-section w. The flow of electricity is now interpreted in a manner completely different from the mechanical analogue. After a long and complicated derivation Ohm at long last obtains his law:
wherein S is the ‘size of current’, A the ‘sum of all tensions’ and L the ‘sum of all reduced lengths’ in the circuit. To the ‘pre-theoretical’ physicists of Ohm’s time this model did not seem intuitive enough, by contrast with the frictional model through the aid of which they thought they were able to imagine something more concrete.14 6. Classificational level of the sciences: from electrostatics/-dynamics and chemistry to ‘kinematics of the constant movement of electricity’ Before Ohm the phenomena of electric conduction through different
“Only in the forties of the 19th century, that is some two decades after the establishment of Ohm’s law, did Louis Poisseulle find a hydrodynamical law which says that the following quantity flows through capillaries per unit time:
Q=k
HD’ L
where k is a coefficient depending on material and temperature, H height of fall, D diameter and L the length. If this law had been known before Ohm it might very well have served as an anschauliches model, and a better analogue for Ohm’s law than the model of heat conduction.
110
Studies in History and Philosophy of Science
materials were tackled either by electrostatics or electrodynamics or chemistry: Some researchers were of the opinion that electrical conduction should be viewed as leakage of charged conductors and be explained in some way or other, by Coulomb’s law. On the other hand, one tried to explain higher degrees of conductibility by the strength of the current, an electrodynamical category. In both cases, electricity was thought of as composed of smallest particles, endowed with a force which was inversely proportional to the distance, ideally to the square of the distance. That meant, at least for the atomists, that the way of explaining the microstructure of electrical phenomena was mechanical. Apart from the dynamist, there existed a chemical alternative to this mechanical view. The chemists tried to explain conduction of electricity either as a chemical process or at least as the result of such a process. Ohm did not take sides with any of these conceptions. As a contact-theorist he could not follow any chemical explanation. But equally, he explicitly rejected electrical action at a distance in favour of only short-range forces when (analytically) treating the phenomena of the galvanic circuit. He found a theory which is independent of all former theories treating the so-far disparate relations between intensity, electroscopic force and resistance in the electric circuit. A connection with electrostatics could only be established after Kirchhoff in 1849 had interpreted for the first time the electroscopic force as potential difference.15 Equally, the relation to electrodynamics is not unambiguous. The electromotive force E of the pile is a static category. And Ohm’s theory claims that current intensity Z, a dynamical category, can be brought in connection with the terminal voltage U,, a static category. But since Ampere the existence of terminal voltage had been denied for, or seen incompatible with, the dynamical case. In modern formulation this connection would be: U, = E - Ri * I; where Ri is the internal resistance which usually was of a very considerable size in the piles used in Ohm’s time. We can say now that Ohm’s theory, whilst separated from the established theories, is at the same time also a mixture of disparate theoretical elements taken from very different theories. Thus Ohm’s theory did not live up to any established way of classifying the electric phenomena. Instead of doing ‘normal science’, i.e. normal electrostatics, -dynamics, or chemistry, Ohm declares the problems a class of its own. In this he calls for a completely new classification of the relevant sciences. 7. Organization circuit.
of the scope of theories: from
The applications
the Voltaic pile to the electric
of Ohm’s theory are not only demarcated
‘sPogg. Anna/n. 78 (l&19), 506-513
in a new and
Logical Reconstruction
of Revolutionary Change
111
different way, but also internally organized by a completely new principle. Physicists before and after Ohm see their experiments in different ways: applications which had to appear very heterogeneous when compared with each other are now seen as similar or identical. This falls under the Wittgenstein and Hanson category of ‘seeing as’ and of the ‘gestalt-switch’. In a Voltaic pile closed with a wire atomists and dynamists see a device which produces special electrical phenomena in the closing wire by virtue of its chemical composition and various effects inside the pile. According to this view, the conductivity of the wire, for instance, can depend on the type of the current source used (Voltaic pile, thermo-electric source, friction device. . .), the chemical composition as well as the pile’s ‘intensity’. By a law of conduction was meant a law that integrates all the physical properties or the chemical composition of a body together with all the modifications of conductivity, mentioned above. In addition, the phenomena experienced with the open pile and the closed pile were strictly separated. The classification induced thereby can be found in all the textbooks of the time. Since Ohm’s theory, especially in Wheatstone’s version,” the nature of the current source is irrelevant. It is a ‘black box’, only of interest for its current supply. The same arrangement of experiment which before Ohm was explained solely by the qualities of the pile is now seen as an ‘electric circuit’, an abstract conception under which also the discharge of statically charged bodies may be classified. In the same way, any electric source whatsoever, whether closed or not, can be subsumed under the concept of a circuit. The measurement of tension (‘electroscopic force’) on the open and closed pile can be formulated through one and the same formula: in modern terms, for the closed pile Uk = E - Ri *I,for the open pile E = Uk + Ri =I.In the same way the behaviour of a galvanometer, for instance, is similarly explained by using the conception of the circuit. Between all these applications and numerous other phenomena a similarity relation could be stated which was impossible before Ohm. With the construction of this similarity the road was freed for the investigation of general structural properties of circuits. The formulation of Kirchhoff’s laws” even became a land-mark for graph theory. Possibly one may say that in the changing configuration from ‘Voltaic pile’ to ‘electric circuit’, ‘substance’ is replaced’by ‘function’ in the way that Cassirer thought characteristic of the development of modern thought.”
‘ePhil. Trans. R. Sot. 133 (1843). 303 - 327 and in German in Pogg. Annaln 62 (1844). 499-543. “Pogg. Anna/n 72 (1847), 497 - 508. See N. L. Biggs et ui., Graph Theory 1736- 1936 (Oxford, 1976), p. 131ff. “E. Cassirer, Substanzbegriff und Funktionsbegriff. (Berlin, 1910).
Studies in History and Philosophy of Science
112 8. Level
of the underlying
theory of measurement:
new foundations
for
electric measurement
Ohm’s theory also had drastic effects on the methods of measurement in electrical science. After the general acknowledgement of Ohm’s law all newly proposed methods and instruments for measuring intensity, tension and resistance of a circuit presupposed the validity of Ohm’s law in one way or other. The measurement of the (inner and outer) resistance of a galvanic circuit is a direct derivation from Ohm’s law in all the proposed methods such as Ohm’s himself, Fechner’s, Poggendorff’s and W. Weber’s.” For Wheatstone’s method one even has to presuppose the validity of Kirchhoff’s laws for branched circuits. For the measurement of intensity by a galvanometer, one has to take into account the resistance of the instrument itself. This can only be justified by reference to Ohm’s law. At least until Kohlrausch’s experiments in 1849” the measurement of the ‘electroscopic force’ of the open and closed pile was done exclusively by presupposing Ohm’s law. Here also, the methods of Ohm, Fechner, Wheatstone and Poggendorff are basic. There are at least two reasons why the tension phenomena of a closed circuit were not measured directly by a voltage meter for a very long time: no piles were available which could produce a constant current for a considerable time, and the accuracy of the electrometer was not enough to arrive at an independent confirmation. But even if one measured the tension directly, in the end one again had to take into account the resistance of the measuring instrument itself, thus presupposing Ohm’s concepts. It was only in 1856 that, for the first time, Wilhelm Weber and Rudolf Kohlrausch were able to give a satisfying theory for the absolute measurement of the electromotive force and current intensity without assuming the validity of Ohm’s law beforehand. At the same time, they were the first to express these magnitudes in mechanical units.2’ What was the change then which Ohm’s theory brought about in measurement? First of all, it serves as a theory for measuring resistance, thereby differing from all former theories. Thus, whenever measuring ‘resistance’ one has to presuppose that there already exists a successful application of Ohm’s law. Therefore, ‘resistance’ is a theoretical term relative to Ohm’s theory.22 “See G. Wiedemann, Die Lehre von der Elektrizitiit, and 676ff. and D. Rutenberg, ‘The Early History of Measurement’, Ann. Sci. 4 (1939), 212 -243. _ “Pogg. Annaln 78 (1849), 1 - 21. “Pogg. Annaln 99 (1856). 10 -25. 221n the structuralist view ‘T-theoretical’ is defined theoretical in Tiff from the description of measuring n always that Thas at least one successful application. See l), p. 45ff. and A. Kamlah, ‘An Improved Version
Erkenntnis 10 (1976), 349 - 359.
Band 4, (Braunschweig, 1898). p. 4OOff. the Potentiometer Svstem of Electrical
in the following way: a function n is in a given application of T there follows Sneed (note 3), p. 31ff.. Stegmiiller (note of “Theoretical in a given Theory”‘.
Logical Reconstruction of Revolutionary Change Until
Weber
electromotive
and Kohlrausch, force
It was not
until
Coulombian
the same holds for the measurement
and the terminal
force law as the measuring
theory,
Kirchhoff’s
way of measuring
113
voltage:
instead
one presupposed
analysis
in 1847 that
electrostatic
of taking
the validity
density
one could
of the
Coulomb’s
of Ohm’s law. identify
the
of charge and the Ohmian
way of measuring the ‘electroscopic force’ as the same ‘potential difference’. But this is an empirical hypothesis which could be regarded analytically true only with Weber’s and Kohlrausch’s unifying theory of measurement. Thus, until Weber and Kohlrausch, ‘electromotive force’ exclusively served as a theoretical
term relative
to Ohm’s
theory
and not to any preceding
one.
9. Definition
of the basic terms: from ‘quantity’, ‘intensity’, and ‘conductivity’ to ‘size of current ‘, ‘electroscopic force’ and ‘reduced length’ Everything fundamental
has now been prepared for properly terms at the time of Ohm. Previous
understanding the change of to Ohm three basic concepts
were in use: quantity, intensity and conductivity. Intensity is measured with the electrometer, being only available at the open pile. It was then regarded as a mere static phenomenon which disappeared when discharged. When the Voltaic pile is closed such a static discharge is believed to take place. Thus, the intensity only acts at the moment of closure, in order to overcome the resistance of the closing wire. In a similar way a body slides ‘by itself’ on an inclined plane if its static friction is overcome (kinetic [sliding] friction is slightly less than static friction). The quantity of electricity flowing through the closing wire is indicated and measured by the galvanometer. For its part, it is dependent on the conductivity of those parts of the arrangement which do not excite conception pile: a pile resistances
electricity. Here, it is essential to realize that according to this the quantity per time is also dependent on the initial intensity of the having ever so much quantity, but not intensity enough to overcome will not bring about current flow.
Compared very strange. (‘electroscopic
with these concepts,
Ohm’s new definitions
had at best to appear
His new concepts are called elektroskopische Kraft force’), Stromgr@e (‘size of current’) and reduzierte Liinge
(‘reduced length’). By the latter he understood the length of a standard wire which has exactly the same conductivity as the part of the circuit under investigation. By Leitungsgiite (‘conductive quality’ or specific conductivity) of the standard wire Ohm means the quantity of electricity which flows from one place to another in unit time, but not the velocity of flow, as was often done before him. The size of current S is defined as the quantity of electricity flowing through the cross-section w of a conductor per unit time:
Studies in History and Philosophy of Science
114
S is measured The quantity
by the Multiplikator Q is defined
(galvanometer).
as the ‘sum of the electroscopic
manifestations’.
This would mean that the quantity might have become measurable by means of the electroscope which appeared unthinkable before Ohm. By the ‘electroscopic force’ at a point in the circuit Ohm understands the density of electricity at that point - a conception which has no longer anything to do with the old concept of intensity. This term proved to be the most difficult one to accept for Ohm’s contemporaries. In Ohm’s theory, the Spannung (‘tension’) which is present at the endpoints of a current carrying conductor is the difference of the electroscopic forces at the endpoints. As an axiom, Ohm stipulates this difference to be constant: a = i4 - U’ = constant. For Ohm, this supposition was the most important one of his theory. At first, Ohm stated his law for current carrying wires of finite length. Later, by adding the hypotheses of voltaic contact theory he tried to give a general theory of electric circuits.
II Having sketched the essential levels at which the change in electrical science had taken place I will now try and put forward the set-theoretical predicates of the theories in question. First of all the basic predicate
for Coulomb’s
theory:
P 1 : X is a Coulomb force system iff there is D, d, a, F, such that (1) X = < D, d, a, F>. (2) D is a non-empty finite set of electrically charged bodies. (3) d : D - R +(density of charge of a body). (4) a : D x D + R + (distance between two bodies). * (5) F : D x D + W + (force acting between two bodies). (6) for all p, q E D, where p Z q : F@,q)
=
d(P) * d(q) . eP>q)
This is the basic predicate
of Coulomb’s
theory.
By
f :D + R +we mean a
Logical Reconstruction of Revolutionary Change
115
function from D into the positive real numbers and the asterisk (*) indicates the theoretical term of the predicate. Now, any entity to which this predicate can be attributed is a model of the theory. d and a are T-non-theoretical terms, whereas F is T-theoretical. Coulomb expanded this predicate in order to take into account the case of electric conduction. The resistance or retarding force B a of the wire of length a appears in the same way as the loss of force in the case of mechanical friction. In doing this Coulomb makes use of two new T-theoretical terms: the concept of specific resistance and the concept of the molecular force F’ with which a charged point on a charged body is effected by the charge of its immediate neighborhood: P 2 : X is a Coulomb conducting system iff there is D, d, a, F, F’, B, such that (1) X = < D, d, a, F, P, B > . (2) - (6)asinPl. * (7) F+ : D - W + (molecular force). * (8) B : D x D + R +(specific resistance of the material betweenp E D and q E D). (9) p E D, q E D, p # q, d(q) Q d@). Then:
B@,q)*d.nq) = F@,q) - F+(q) or equivalently: (9’) . . .
B07,q) a@,q)
= d*@)-d*(q).
Humphry Davy transferred this idea to the arrangement which is formed by a closed pile. His theory is not very elaborated and so the following predicate has to be taken with some care. As T-non-theoretical concepts Davy uses the quantity of electricity, the cross-section of the wire, and its temperature, in addition to Coulombian concepts: P 3 : X is a Davy conducting system iff there is D, d, a, F, P B, Q, d, I?, such that (1) X =
Studies in History and Philosophy of Science
116
(12)~ E 0, q E D, d @) 2 d (q), p # q. Then:
If F(p,q)-F+(q)=0
then B@,q) a(q) * 19(4)
= Q @)
d(q)
In 1825 and 1826, Ohm put forward two new theories which are intermediate between the Coulomb conception and his eventual theory.” Though very interesting, we have to omit these theories and reserve the treatment to a later date. In the following predicate representing Ohm’s second article of 1826 the concept of the Ohmian current carrying conductor is introduced. ‘Specific conductivity’ and ‘electroscopic .force’ are the T-theoretical terms of the theory. P 4 : Z is an Ohmian conductor iff there is D, I, w, X, k, u, u 0 such that (1) Z = < D, I, w, X, k, u, u’ >. (2) D is a homogeneous current carrying prismatic conductor. (3) 1 : D + R +(length of the conductor). (4) w:D-‘W +(cross-section of conductor). (5) X : D +W + (strength of electric current). * (6) k : D -*R + (specific conductivity). * (7) u : D +R D(electroscopic force at the beginning of conductor). * (8) u’ : D - R o (electroscopic force at the end of conductor). (9)
X(D) =
(u(D)-
u ‘(D))
. k(D)
. w(D)
IV’)
The next predicate corresponds to Ohm’s theory of 1827, put forward in his Die galvanische Kette, mathematisch bearbeitet, his main work: P 5 : X is an Ohmic circuit iff there is D, E, g, E, T, I, q, a, u, such that (1) X = < D, E, g, =, T, I, g, u, u >. (2) D is a nonempty finite set of homogeneous prismatic conductors. (3) E is a set of points, the endpoints of the conductors. (4) g : D +_E x E; and 2 & E x E is an equivalence relation such that < D, E, g, i > induces a single closed circuit with no branches and no loops, and for any e E E, there is an e’ c E which is equivalent to it (i.e. is in direct contact with it). z’Schweigg. J. Chem. Phys. 44 (1825). llO137 - 166.
118 and Schweigg.
J. Chem.
Phys.
46 (1826),
Logical Reconstruction (5)
of Revolutionary
T is an interval
117
Change
of R D(time-interval).
(6) I : D + R + (length). (7) q : D + R (8) Q : D +W * (9) u : D - R * (10)~ : E -R * (1l)for
+ (cross-section). D(quantity of electricity). + (specific resistance). o (electroscopic
all tE
T, for all eE
du(e) = 0 (condition
force). E:
for stationary
current).
dt (12)for
all e, e’ E E, e # e’ and e 3
1 u(e) - u(e’)l>
0
(contact-theoretical (13)for
e’:
hypothesis).
all t E T, for all s, s’ E D:
dQ(s) q(s)dt
-
dQ(s ‘) (condition
of continuity).
q(s’)dt
(14) be s,E D. Then
for all t E T: dQ(so) .
e=e’
I u(e)-@‘)I
= dt
4s) 4s).q(s)
c sED
e% e’ e,e’EE or: In 1841 the Royal
u Society
decided
=I to regard
.R Ohm’s
law as independent
of
any chemical or contact-theoretical hypothesis.24 After this, Ohm’s law quickly turned into the form which can be taken as still valid today. It was Fechner’s and Wheatstone’s work which was mainly responsible for this development: P 6 : X is an Ohm-Fechner- Wheatstone circuit iff there is LL D2, I, 1, w, k, E, R,, Ri, such that (1) X = < D,, Dz, I, I, w, k, E, R,, Ri >. “i’roc. R. Sot. 4 (184lL 336.
Studies in History and Philosophy of Science
118
(2) D, is the outer (closing) part of the circuit. (3) D2 is a source of current. (4) I : D, x D* - R o (strength of current). (5) I : D, -, R + (length of outer part). (6) w : D, ‘R + (cross-section). (7) k : D, +R + (specific resistance). * (8) E : D2 +R D(electromotoric force). *(9) R,:D,+R + (outer resistance). * (10)R): D, + 1R D(inner resistance). (ll)E(Dz) = (R.(D,) + R,(D,) ) * I(D, Dz). (12)R,(D,)
= k(Q) I(D,) -. ~(DI )
In this predicate we have three different T-theoretical terms: specific conductivity, electromotive force and the inner resistance. P 6 as such is only applicable to one circuit at a time and does not state any connections between different circuits. But a very important part of the lawstatements made by Ohm’s law in pruxi concern sets of circuits. We can account for this by introducing three additional constraints which intuitively say the following: The resistance of any finite part of a circuit remains constant, if this part is included as a proper part in any other electric circuit. And equally the e.m.f. of a source of current remains the same, no matter what circuit it is a proper part of . More specifically: Let o, o’ be Ohm-Fechner-Wheatstone circuits. Then let D,(O) be the external part of the circuit o and D*(w) be the internal part thereof. Then for all Ohm-Fechner-Wheatstone circuits the following constraints hold: If D,(W) = D,(o’), (2) If D*(o) = Dz(o’), (3) If D&J) = D&I’), (I)
then R.(Dj(w)) = RJD(w’)) then Ri(D4w)) = Ri(Dz(o ’ 1) then EU&(w)) = E@(Q)‘))
This concludes the formal part of this paper. III
Having formalized the relevant historical theories in informal set-theoretical terms we go back to the initial question: what can formalization, and especially the ‘structuralist’ one, contribute to the logical reconstruction of the Ohm-episode in the history of electricity? The most important advantage of
Logical Reconstruction the structuralist of the theory grasp
of Revolutionary
approach, in question.
the theory
119
Change
it seems to me, is its ability By a set-theoretical
at one glance.
predicate,
This advantage
to produce
an overview
we can, so to speak,
is not merely
an educational
one. Wittgenstein tried to show that if we want to see how language functions we need an overview of the different uses of words:
really
Our grammar is lacking in this sort of perspicuity. A perspicuous representation produces just that understanding which consists in ‘seeing connections’. . ..The concept of a perspicuous representation is of fundamental significance for us. (Investigations, section 122). I think it is also true in the case of scientific theories that we need a perspicuous mode of representing their essential features in order to understand them and their place in history, i.e. the relations of their internal parts to each other and their connections and relations to other theories. In set-theoretical terms we can give a clear definition
of an application
of a
theory and we can display the theory’s structure in a very precise and unmetaphorical way. By an application of a theory there is meant any entity that can be described by the conceptual apparatus of the theory. We can distinguish two different levels in the structure of the application of a theory: the T-non-theoretical level and the T-theoretical level. At the T-nontheoretical level we can describe what the structure of an entity looks like before one applies the theory to it, i.e. before one has available the special the application of a information supplied by the theory. 25 For example, Coulomb theory (P 1) at the T-non-theoretical charged bodies, their distance from each
level is formed by the set of two other and the density of their
electrical charge. Analogously, for an Ohm conductor (P4) an application at the T-non-theoretical level consists of a conductor, its length, cross-section, and the size of current it carries. And for an Ohmic circuit (P 5) an application consists of a set of conductors which forms a closed cross-sections and the current in the circuit. The respective T-theoretical level of the applications adding
the T-theoretical
functions
circuit,
their lengths,
can be obtained
to the list. The applications
by
of Ohm’s and
Coulomb’s theories already differ at their T-non-theoretical level. The gestaltswitch required when changing from one theory to the other also consists in this difference. Thus, a set-theoretical formalization shows us how a theory organizes its scope and how this organization changed from Coulomb to Ohm. In this way we have elucidated the change which we tried to describe in part II as level 7 (organization of the scope of theories). The formalization also shows us how the meaning of a theoretical term is established. When comparing a Coulomb-type-theory with an Ohm-typeY&x also note 22.
120
theory
Studies in History and Philosophy of Science we can
immediately
see a difference
in the meaning
between
the
Coulomb-term of electrostatic force and the Ohm-term of electroscopic force (resp. tension). Although in both cases an electrometer is involved in measuring
the respective
for the measurement
magnitudes,
differ
from
the theoretical each other.
Thus,
presuppositions by formalizing
contribute to the explication of the meaning of the T-theoretical contributing to level 9 (definition of the basic terms).
necessary we can
terms, thereby
So far we have dealt with metatheoretical predicates which are one-place: ‘organization of the scope of a theory’ and ‘definition of the basic terms of a theory’. Now we will turn to metatheoretical predicates with more than one place, representing intertheoretical relations. First, we have the predicate of a ‘theory being the model of another theory’. Let us assume, that the theories of mechanical friction and of heat-conduction had been formalized respectively. Then the change in model-formation (described at level 5) can now be expressed by stating that the structure of the Coulomb-predicate (P 2) is isomorphic to the structure of the predicate describing the theory of mechanical friction, but not the theory of heat-conduction; and that the structure of the Ohm-predicate (P 5) is isomorphic to the predicate expressing the theory of heat-conduction but not the theory of friction. Another intertheoretical relation would be the classification of a range of phenomena by a theory in relation to a set of other theories. A classification induced by a theory can be seen as in agreement with that of another theory if the non-theoretical terms of the theories are identical. And the classification would be a new one in relation to another theory if the theories differ in at least one non-theoretical term. Let us, in addition, call a new classification by a theory T revolutionary in relation to T' if T introduces such a new nontheoretical term which appears in another theory T", not identical with T or
T'. In this sense, Ohm’s classification is new in relation to Coulomb’s theory, since he does not use ‘density of charge’ as a non-theoretical predicate when describing one and the same phenomenon and uses ‘current’ as a new basic predicate instead. If we take the term ‘electroscopic force’ as a non-theoretical term (which can be done after Kirchhoff, Kohlrausch and Weber) Ohm’s classification is even revolutionary both in relation to Ampere’s and Coulomb’s theory. In the first case, Ohm’s non-theoretical vocabulary differs from Ampere’s in the concept of electroscopic force which is a non-theoretical predicate in another theory, namely Coulomb’s. And in the second case, Ohm uses ‘electroscopic force’ in agreement with Coulomb’s ‘density of charge’ but he includes ‘current’ which is a basic predicate in another theory, namely Ampere’s. As we can easily see, such a change in classification (described on level 6) is most clearly represented by the structure, which is provided by the set-theoretical predicate of a theory.
Logical Reconstruction
of Revolutionary
Change
121
A complete set-theoretical description of a theory T would have to include the description of all those theories on which the measurement of the nontheoretical terms of T is founded. Thus, we can introduce the intertheoretical predicate of a ‘theory T resting on one or more theories of measurement’. Here, we cannot reconstruct all the theories of measurement relevant to our historical episode. But I think it is clear that such a specification of theories of measurement and their change over time from Coulomb to Weber could very well be described in set-theoretical terms. Thus, the ‘structuralist view’ can serve as a reconstruction of level 8 (underlying theory of measurement). If there appear different theories of measurement over time for one and the same non-theoretical term, it can also point out how the meaning of that term has changed (level 9: meaning-change of basic terms). So far, we have tried to show that by formalizing theories in set-theoretical terms one can reconstruct levels 5 - 9. But what about levels 1 - 4? It seems obvious to me that the set-theoretical view cannot contribute anything in reconstructing the change in a theory’s world view (level l), in its standards of theory formation (level 2), in its attitude towards experiment (level 3) and in its view of technical utilization (level 4). If giving a definition of the general term ‘scientific revolution’ would necessarily include change on these and/or similar levels, as I think it does, then it is obvious that revolutionary change could only partly be explicated in ‘structuralist’ terms. The result seems to force upon us the suspicion that there is a general difference in category between levels 1 - 4 and 5 - 9. I think it amounts to the following: levels 5 - 9 describe scientific change at the object-level and levels 1 -4 describe scientific change at the meta-level. At the object-level we describe how the special theories have changed, their concepts, their relations to other theories, the structure of their applications, etc. But at the meta-level we describe traits of a completely new category. We try to find an explication of general terms like explanation, theory, confirmation, law, experiment, usefulness, etc. And I think that the idea of revolutionary change in science did not so much prove itself revolutionary for the philosophy of science by pointing out the historical change in scientific concepts, perceptions, and methods, but by showing that there cannot be given a general explication of metascientific terms which could be adequate for history of science in all its stages. This limitation, I think, cannot be overcome by any philosophy of science which claims to be adequate in reconstructing historical scientific theories, whether using formal means or not.