GENESIS AND STRUCTURE

I wish to thank all my teachers at the Department of Sociology, Delhi School of Economics, especially Prof. Jit Singh Uberoi for having taught the truth and method of science. But for him I could not have produced this paper. Sam Schweber provided a much needed encouragement by sending in fruitful comments on the paper, all of which, unfortunately, I was unable to incorporate. Finally, a heartfelt thanks to my students at Hindu College for welcoming me with a deluge of love and affection.

In the history of physics, the black-body problematic played a crucial role in inaugurating a fundamental innovation in scientific thought—the quantum innovation. The word problematic has been used as Deleuze uses it – “the problematic
refers to the ensemble of the problem and its conditions of realization” (1994:177). In order to reconstruct the history of the black-body problematic, I will look at the specific discursive configuration which actualized the conditions
necessary for its emergence and realization. In this process Kuhn’s theory of history of science will be re-examined and contrary to Kuhn’s argument that rival paradigms emerge *successively* via scientific revolutions, I
propose to show that rival paradigms coexist *simultaneously* and it is the consensus around each and the contradictions between them that produce innovations in science. My critique of Kuhn’s general theory will be carried out
on the basis of his own monograph on the black-body radiation problem (Thomas S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1978).

In this article I try to argue that the theoretical derivation of black-body radiation should be located between three rival paradigms of physics—*mechanics* dealing with ponderable matter, Maxwell’s theory of *electromagnetic* field and *thermodynamics* dealing with heat. There was an unreconciled tension between these three and the black-body radiation problem promised a solution to the union of electrodynamics, mechanics and thermodynamics. Attempts were
made by Boltzmann to reconcile *thermodynamics* with *mechanics* through kinetic theory of gases. Planck sought to reconcile *thermodynamics* with *electrodynamics* through the theory of thermal radiation. That still another
element linking *mechanics* and *electrodynamics* was missing was realized first by Einstein who suggested that efforts to reconcile Newtonian mechanics of discrete *particles* and Maxwellian electrodynamics of continuous *waves* might involve the fundamentals of physics. Boltzmann’s efforts led him to create the subject of ‘statistical mechanics or statistical thermodynamics’ which Planck followed in his derivation of the black-body radiation law
to which Einstein added the concept of light quanta. The simultaneous presence of three rival paradigms, not the successive emergence of unique paradigms, outlined the production of the quantum novelty in physics. In other words, it was precisely
because the theoretical derivation of black-body radiation encapsulated a few fundamental contradictions of physics of the time that it was extremely well suited to inaugurate the quantum innovation.

Contrary to Kuhn’s general methodology, his own case study of black-body radiation does not present classical physics as a monolithic body of doctrine within which the radiation law could be established. In fact, at any point in the history of radiation research it can be shown that a diversity of approaches involving different world views marked its treatment. From the beginning when Kirchoff established the form of the radiation distribution function, F(l,T) , attempts to derive the function saw the coming together of various lines of thought which led to new advances in radiation research. While Lommel (1878) based his understanding on a mechanical model; Michelson (1887) used arguments from kinetic gas theory including Maxwellian statistics of velocity distribution; Wien’s (1896) deduction was based on the second law of thermodynamics and molecular assumptions; Planck brought to his new radiation law (1900) many world views, the mechanical, the electromagentic and the thermodynamical, including Boltzmann’s combinatorials; Lorentz (1903) used the electron concept; Jeans (1905) applied the equipartition theorem and Einstein (1905) took up the notion of energy fluctuations to understand black-body radiation. What emerges from these various attempts is the recurrence of the opposition between three theoretical frameworks—mechanics, thermodynamics and electrodynamics; the opposition between two principles, reversibility and irreversibility; the opposition between two types of laws—dynamical and statistical; and the opposition between macroscopic and microscopic models. The oppositions, dualisms, contrasts or inversions between theories, principles, laws and models form the framework of the black- body problem and the subsequent quantum innovation.

For the argument that follows it is important to remember that for Kuhn (1970), a paradigm defines a field of science and brooks no rival. Kuhn highlights the existence of unequivocal paradigms in order to argue that the succession of paradigms via revolutions is the usual developmental pattern of any mature science. His main point is that work within a well-defined paradigm is more productive of revolutionary episodes of scientific innovations, than work in which no similarly convergent standards are involved. However, as the thermal radiation case proves, a single, homogenous paradigm is quite incapable of generating any revolutions in science. Since opposition, dualism, differentiation and reflection is the method of production of scientific innovation, the concept of universe of discourse is useful in uncovering the field of operation and articulation and which renders them meaningful and efficacious.

I have suspended a purely sequential reading of the black-body narrative in favour of a structural account, that would include the notion of transformation through oppositions. It will be within a ‘structural’ conception that I work out
the genesis of this specific problematic in physics. Cassirer will be our guide, to some extent for he has followed the nature of the quantum development in his book *Determinism and Indeterminism in Modern Physics* (1956).

Cassirer argues that in the nineteenth century the particular problem that gave rise to a new development of thought was the second law of thermodynamics and its conflict with the prevailing mechanical world view. The law was first formulated by Sadi Carnot in 1824 who established that heat always passes from a hot body to a cold one when work is done in a cyclic process. The law was revised and reformulated by Claussius (1865) in terms of the concept of entropy and he restated the two fundamental laws of the mechanical theory of heat as :

- The energy of the universe is constant.
- The entropy of the universe tends to a maximum.

Cassirer writes that from a law of thermodynamics, entropy increase was translated as a general principle of nature, the Principle of Dissipation and Disorder, which contradicted the idea that the world is cyclic and may go on forever in the
same way. The second law expressed the *irreversibility* of physical phenomena and thus contradicted the presuppositions of classical physics which are time *reversible*. He argues that “the entropy law thus came to be a kind of
irrational remainder, a foreigner and intruder in the securely articulated system of classical mechanics and electrodynamics” (1956:76). It was this conflict which unsettled the status of the mechanical world view.

The opposition between the two principles, reversibility and irreversibility, came into view through the laws of mechanics versus the laws of thermodynamics. What this shows is that the initial opposition proves difficult to be ‘resolved’,
it gets transformed and dissipated across different frameworks. The *thermodynamic* principle of irreversibility, as an indicator of evolution giving a mathematical expression for the ‘arrow of time’, was in contrast to the laws
of *mechanics* and *electrodynamics*. The differential equations of Newtonian mechanics are invariant under time reversal and the motion of individual mass points can be reversed simply by giving a minus sign to their reversed velocities;
similarly, Maxwell’s equations of *electrodynamics* too are time reversible.

Alternatives to mechanism were proposed and hotly debated during the 1880s and 90s.

At one end were the energeticists (Mach, Ostwald, Helm) who challenged mechanics, atomism and the autonomy of the second law. At the other end were Claussius, Helmholtz, Boltzmann and others who were attempting to provide a mechanical proof of the
second law by appealing to the microscopic structure of matter and formulating laws of probability to bridge the conflict between *macroscopic* (thermodynamic) *irreversibility* and *microscopic* (mechanical) *reversibility* of molecular motions. Since in the communication of heat energy between innumerable molecules, so small individually and so irregular in their distribution, it is impossible to follow individual molecule motion as one would in a strictly *dynamic* theory, it was necessary to introduce *statistics* to explain the second law. Cassirer argues that in “viewing the entropy law as a probability law there had been introduced into the very concept of law a dualism wholly foreign to its
original meaning” and the introduction of statistical laws at par with dynamic laws was the subject of debate and controversy. For the probability laws were not seen as having “the same epistemological quality and ‘dignity’”
as dynamic laws which were regarded as absolute laws of nature excluding every exception. “But precisely this property would have to be surrendered, if one were to go over to mere probability laws. An event no matter how improbable is still
not an impossible event; not only can it occur, but it will in general occur one day, if we but extend our observations over a sufficiently long period of time” (1956:77).

The tension between dynamic and statistical laws is most easily visible in the 1860’s when Boltzmann and Claussius were trying to give a strictly mechanical interpretation of the second law while Maxwell insisted on the statistical character
of the second law. Maxwell’s argument took the form of what is known as ‘*Maxwell’s Demon’*, that while it would require the action of the demon to produce an observable flow of heat from a cold body to a hotter one,
this process is occurring spontaneously all the time on a submicroscopic scale, perfectly consistent with the laws of mechanics. Maxwell concluded that the chief end of the demon was to show that the second law of thermodynamics has only a statistical
regularity and not a dynamical certainty for systems composed of large number of small molecules. His idea caught on quickly in his circle. William Thomson and P.G. Tait were convinced that the truth of the second law is of the nature of a probability
and not an absolute certainty. This led to discussions on the possibilities inherent in the use of statistics to thermodynamical applications and it was Boltzmann who developed precisely how the second law is related to mechanics creating in the
process the subject of ‘statistical mechanics’ or ‘statistical thermodynamics’.

One can see that solutions to these oppositions were no longer possible within any one conceptual framework, so they were displaced across frameworks. My intention is to show that the combination of these oppositions together with the potential of transformation that such a combination involves led to new advances in scientific thought. In the following pages I will take up the debate between Boltzmann, Planck and Einstein as a framework to locate the problematic of black-body radiation. The debate which occurred between these three scientists, presenting three different relations of two terms each, through the rival theoretical frameworks of mechanics, thermodynamics and electromagnetism, reflects not just a concern with disciplinary boundaries; each side of the debate presented alternative solutions to competing problems. The debate also highlights the spirit of competition present in scientific work. Ironically, in the scientific community, one’s ‘enemies’ are more significant than one’s ‘friends’, for the scientist has to prove the validity of his theory more to his enemy than to his friend. The debate between Planck and Boltzmann was particularly acrimonious with first Planck accusing Boltzmann of wasting his energies on the atomic-kinetic gas theory and later Boltzmann attacking Planck’s proof of irreversibility of radiation. Therefore, contrary to Kuhn’s standpoint, absence of a paradigmatic consensus did not imply that there was no communication or debate or progress. Dialogue in science is not always about agreement or how to come to an agreement but also about how to disagree and how to come to a disagreement. The main participants do not seem to be ‘talking through one another’ in the manner characteristic of Kuhnian paradigm shifts – Boltzmann, Jeans, Lorentz and Einstein were as aware of the exact nature of Planck’s proposal as he was aware of the exact nature of their objections. What allowed the different parties to the debate to enter into a mutually intelligible dialogue and holding different points of views was a universe of discourse which regulates, and not constitutes, what can be said and how it is to be said.

The idea of treating energy as a discrete variable rather than a continuous one was first put forward by Boltzmann in 1872 when he set out to provide a mechanical proof of irreversibility through kinetic gas theory. He argued that from any arbitrary initial distribution of molecular velocities, the effect of molecular collisions must always be to bring the gas to an equilibrium distribution function, the Maxwellian distribution function. His approach was concerned not so much with thermal equilibrium itself as with the irreversible processes by which equilibrium is reached. The result (1872) was his H-Theorem: for non-equilibrium states H is proportional (with a negative proportionality) to entropy; H tends to decrease to a minimum as entropy increases to its maximum value. Once these extreme values are attained, corresponding to the state of thermal equilibrium, the system will stabilize and H will remain constant.

It must be emphasized that while his derivation of the irreversible increase of entropy made use of the statistical distribution of molecular velocities and energies, the result he asserted was supposed to be a certainty and not a probability. The H-Theorem said that entropy would always increase and he presented his theorem in a very deterministic phraseology. The essentially statistical premises of his derivation seemed to vanish without a trace from its results which are shown to have mechanical implications. The juxtaposition is what creates the tension in his text and only reiterates the contradictions. However, Boltzmann was convinced that he had supplied a mechanical proof of the second law of thermodynamics and solved the problem of irreversibility.

In 1876, Joseph Loschmidt argued that no such proof could be valid because all the laws of mechanics are reversible in time. For every mechanically possible motion that leads towards equilibrium, there is another, equally possible, that leads away
from equilibrium and is thus incompatible with the second law. This statement presents what has since been known as the ‘*reversibility paradox*’. Loschmidt concluded that the H-Theorem could not be a deterministic theorem because
there were some initial conditions from which H could for a time increase and entropy decrease. Boltzmann conceded that Loschmidt was quite correct in asserting that entropy decreasing processes existed and entropy decrease depended on special
*initial conditions*. He responded with a statistical interpretation that for some unusual initial conditions it is possible entropy might decrease (and H increase) as time progresses. But those cases are, he wrote, extraordinarily improbable
and for practical purposes may be regarded as impossible.

Provoked by Loschmidt’s criticism, Boltzmann worked out a ‘combinatorial’ definition of entropy (1878). He defined a distribution of molecules over finite cells (in the configuration space of one molecule) as the number of molecules
in each cell, and a *complexion* as the specification for each molecule of the cell to which it belongs. The probability of a given distribution was taken to be proportional to the corresponding number of complexions. Since in the distribution
of molecules, all have the same . *priori* probability (the number of particles in a cell, not their identity, is relevant to the definition of a micro-state), entropy is given as the number of distributions compatible with a given macro-state.
The global macro-state appears not only as the most unique state but also as the most likely to occur and, in the sense that it could be achieved in the largest number of ways, as the final state towards which any evolution will lead, starting
from an arbitrary initial state. In this way, he made entropy a measure of disorder— the tendency towards increasing entropy is simply a tendency towards increasing disorder. Moving beyond Maxwell who had argued that the second law had only
a statistical certainty, Boltzmann made the second law a direct expression of the laws of probability: the entropy, *S*, is proportional to the logarithm of the probability of that state,

*S* = . *log(W)*. He recognized “how intimately the second law is connected to the theory of probability and that the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability” (*in* Klein,1973:73).
But the improbabilities involved in the entropy decreasing processes were tantamount to an impossibility. He chose Thomson’s example to illustrate that one should not expect a mixture of nitrogen and oxygen gases separating in a container
after a month with pure oxygen in the lower half and nitrogen in the upper half of the container, even though from the viewpoint of probability theory that outcome is only extremely improbable, not impossible.

Thus, Boltzmann achieves a subtle reconciliation between macro-irreversibility and micro-reversibility by developing a statistical theory of thermodynamic entropy of a system composed of a vast number of molecules even though the individual molecular motions are described by the reversible laws of mechanics. There is a convergence of the two approaches, the statistical and the dynamical, to the theory of thermal equilibrium. However, the spheres of application of statistical and dynamical laws are separated for Boltzmann in a simple fashion. The statistical procedure is applied solely to the formulation of initial conditions whereas the further course of events is regarded as being governed by strict dynamic laws.

Boltzmann’s probabilistic and microscopic interpretation of entropy was not accepted easily when presented in the last quarter of the nineteenth century. Continuum sciences such as electromagnetism, thermodynamics and acoustics were the vogue.
The energeticists were bitterly opposed to atomistic models because they considered energy rather than matter as the final reality. Moreover, difficulties appeared with Boltzmann’s attempt at bridging the gap between thermodynamics and mechanics
through probability laws that could only be laboriously removed by introducing special hypothetical assumptions. One such hypothesis introduced was that of ‘*molecular chaos’* which implied that certain actual configurations of
molecules within individual cells be prohibited, either initially or as the motion proceeds, configurations which the laws of mechanics taken alone would otherwise allow.

Another objection to the H-Theorem came in 1896 from Ernest Zermelo, Planck’s assistant in Berlin, who developed what has since been known as the ‘recurrence paradox’. Applying a mathematical theorem published by Poincare five years
ago, Zermelo argued that any mechanical system confined in a finite region of space would after a sufficiently long time ultimately return to its initial configuration. This contradicted not only the H-Theorem but also any kinetic theory of heat
because if thermodynamics is thought to obey mechanical laws on the microscopic scale, entropy should behave periodically rather than monotonically. “In such a system”, Zermelo wrote, “irreversible processes are impossible”
(
*in* Kuhn,1978:26). Thus, mechanics could never be the basis of physics for it dealt only with cyclic processes. Such cycles, according to the second law, were not *natural* at all. To this Boltzmann replied in the same journal, *Annalen der Physik*,
that entropy was not simply *mechanical* but also *statistical*.

At this point, Planck continued the debate by asking if probability alone can determine the direction in which a system develops. He answered in the negative. He wrote “probability calculus can serve if nothing is known in advance, to determine
the most probable state. But it cannot serve if an improbable initial state is given, to compute the following state. That is determined not by probability but by mechanics”(*in* Kuhn,1978:27). However, Planck disagreed with Zermelo’s
contention that the entropy law as a natural law is really incompatible with every mechanical interpretation of nature. Planck argued that if one passes from discrete point masses, such as molecules, to mechanical continuum such as an electromagnetic
field, where every part is tied to every other part like an organic whole, then “a strict mechanical significance can be found for the second law” (*ibid.*). Initially Planck was opposed to Boltzmann’s molecular proof of
irreversibility. He believed that like energy, entropy had to be determined not only by the *macroscopic* (*thermodynamic*) state of the system but also by the underlined *microscopic* (*mechanical*) state. His problem was
to find a proper micro model, not one based on discrete mass points of molecular gas theory but on continuous matter like electromagnetic ether. In his programme (1897), he claimed that the reversible equations of electromagnetic wave theory did
explain thermodynamic irreversibility.

To Planck, a derivation of the second law meant a derivation of irreversibility which he thought he could derive by combining thermodynamics and electromagnetism. The locus for such work was provided by black-body radiation on which he started work from 1897 onwards. His main aim was to show that reversible equations of Maxwell’s electromagnetic wave theory could be used to explain irreversible thermodynamic processes. Initially he put forward the model of a single Hertzian resonator in a cavity which he later changed to a model with several resonators at one frequency enclosed in a cavity interacting with the electromagnetic field. If an arbitrary initial distribution of energy is injected into the cavity, then the resonators in the cavity would induce a strictly irreversible evolution of radiation from an incident plane wave to an outgoing spherical wave tending towards spatial homogeneity, isotropy and polychromy. The distribution will move towards equilibrium and entropy would increase until equilibrium is achieved. If one had a formula for the entropy of radiation as a function of the field variables than black-body distribution function would be the one that maximized the total entropy of the radiation in the cavity. For him, a byproduct of this study would be the derivation of the law of spectral energy distribution of thermal energy radiation. He thought he could thus retrieve irreversibility in the form of electromagnetic equations of the field in the empty cavity.

Boltzmann replied that such a miracle could not be performed. The equations of electrodynamics like those of mechanics are invariant under time reversal. All processes that satisfy them can run in either direction and are thus reversible. Any irreversibility in the effect of resonators that Planck finds, Boltzmann commented, derives from his choice of uni-directional initial conditions. Planck conceded Boltzmann’s point and recognized the crucial importance of suitable initial conditions in non-statistical proofs of irreversibility.

In 1898, Planck introduced a special hypothesis for radiation theory, as Boltzmann had for gas theory, that would prohibit some initial and boundary conditions for change to proceed irreversibly. As a physical hypothesis about the distribution of
micro states, the role of ‘*natural radiation*’ is to permit a derivation and a definition of the probability of a state: the states must be so chosen that they provide the probability demanded by the selection of special initial
conditions that preserve the absolute validity of the entropy law. The hypothesis, governing the distribution of initial conditions within individual intervals or regions, determines combinatorial probability and thus entropy: in fact prohibiting
those configurations which violate the second law becomes a means of fixing the relative probability of the states that remain.

Some have read these special hypotheses as a stipulation of randomness (e.g., Stephen Brush, 1976). But Kuhn writes that neither Planck nor Boltzmann’s other contemporaries equated molecular disorder with randomness. Planck’s ‘special assumption’ does not demand that certain actual configurations of the molecules within individual cells be improbable, “but rather that they never occur at all, either initially or as the motion proceeds” (1978:67). In this way, Darrigol (1988) argues that Planck’s generic notion of ‘elementary disorder’ covering both molecular chaos and natural radiation, provides the precondition and also the strict guarantee for the validity of the second law of thermodynamics to which he was firmly attached. Thus, Planck reinterpreted Boltzmann’s combinatorial definition of entropy and probability in a way compatible with the notion of absolute entropy increase. This conveys Planck’s struggle in retaining absoluteness of the entropy principle along with its probabilistic definition of entropy. It indicates how closely both Planck and Boltzmann shared the problem of demonstrating irreversibility. Both started out by seeking a deterministic proof of irreversibility; both were forced to settle for a statistical proof eventually.

The key point of Planck’s derivation (in its 1901) form, in which he is said to have introduced energy quantization, was his adoption of Boltzmann’s atomic-statistical model in toto despite his opposition to him. However, the inner difficulties
of Boltzmann’s gas theory came to light when Planck made an attempt to transfer it to radiation theory. In subdividing the energy continuum into cells or elements of size e, Planck was following Boltzmann who had introduced it (first in
1872) as a mathematical device when presenting a probabilistic derivation of the entropy and velocity distribution of a gas. In his derivation, the precise size of e made no difference. But in Planck’s derivation, the cell size had to fixed
in proportion to resonator frequency with *h* as the constant of proportionality. This restriction puzzled him. But it was for him a restriction on cell size, not on resonator energy and it did not, therefore, bring anything like energy quantization
to mind. The energy elements did not express an intrinsic discontinuity of resonator energy; they had no relation to Einstein’s later light-quanta. Yet after introducing this novelty in his derivation, Planck himself does not identify it
as the fundamental novelty of his theory. Rather the innovation that he thought he had introduced was the constant *h* (later known as Planck’s constant) as a catalyst to perform to harmonious unification of statistical thermodynamics
and classical electrodynamics.

Planck’s attempts to resist micro-(atomic) models, the statistical interpretation of irreversibility and the discontinuous structure of radiation, were all unsuccessful and he gradually realized it. In a lecture given in 1914, Planck finally
declared, “Now, no doubt is left the physicist bound by inductive demonstration: matter is made of atoms, heat is agitation of molecules and the conduction of heat, like every irreversible process, obeys not dynamical but statistical laws,
that is to say probability laws�. On an individual process, the second principle can state something with certainty only to the extent that one already knows that the course of this particular process is not substantially different from the average
course of a great number of processes that all start from the initial state”. In a later talk, (1926), Planck was quite explicit on this point, “The second law of thermodynamics loses its status of principle; it survives only as a
statistical law that is valid not for the properties of an individual system but only for the mean values of a great number of identical macroscopic exemplars of the system” (*in* Darrigol, 1990: 279).

Kuhn’s major interest in reconstructing the history of quantum innovation is to show that even after he arrived at his quantum hypothesis, Planck still saw it as a direct extension of classical physics. It was others, mainly Einstein and Ehrenfest,
who recognized the revolutionary nature of Planck’s derivation that it required the restriction of energy to integral multiples of energy element *hv*, and who were convinced that no classical (mechanical or electromagnetic) model of
black-body radiation could succeed. So while Kuhn (1978,1984) claims that Einstein should be credited with the quantum discovery for consciously making the innovation, it is Planck, who was actually unaware of the nature of innovation he had unknowingly
introduced, to whom the innovation is imputed and after whom the constant, quantum of action *h*, and the black-body radiation law are named. More radical theories than Planck’s (Einstein, Ehrenfest) and more conservative (read classical)
theories than Planck’s (Jeans, Lorentz) brought out the full potential of a theory, scarcely enunciated by its author. It was in this cooperative and competitive enterprise that Plank unknowingly produced a fundamental innovation in scientific
thought. Thus, scientific work, like any other work requires cooperation and team-work in the production of new theories and techniques, yet recognition and reward accrues only to an individual and not to the team. This is the fundamental ‘social
fact’ of the social organization of the scientific profession.

Kuhn interprets Planck’s concern for a dynamical proof of irreversibility as not being central to the actual derivation of the new black-body radiation law which he brought out in December 1900 and for which he is remembered. He recognizes that a dynamical versus a statistical proof of irreversibility was significant in Planck’s work but writes that what provided the actual impetus to the radiation law were the new radiation experiments which invalidated other older laws like that of Wien, Jeans and Rayleigh and simultaneously provided the impulse for Planck to work out a new radiation function (1978: 91-92). There is no doubt that experimental confirmation is necessary to any scientific outcome but to see it as the fundamental basis to scientific innovations will not help us to discover and define the logic of the conditions of its existence. Moreover, if one were to read Planck’s innovation of the quantum by placing it in the matrix of three theoretical frameworks, one is convinced immediately of the centrality of the question of thermodynamic irreversibility versus reversibility in the emergence of the quantum.

After working on the relation between macroscopic thermodynamic quantities and molecular structures and providing a foundation to thermodynamics on the joint basis of the equations of mechanics and the theory of probability, the problem of black-body
radiation stuck Einstein as an illustration of the need to unify the foundations, this time between discrete particles of Newtonian mechanics and continuous field of Maxwell’s electromagnetic theory. In his 1905 paper, *“On a heuristic viewpoint concerning the emission and absorption of light*”,
he proposed that Maxwell’s electromagnetic wave theory might not be the last word on the subject and one should explore the idea that light behaves like a collection of independent, localized particles of energy – the light quanta.
He proceeded from Wien’s idea of 1900 that electromagnetic waves of short and long wavelength differ *qualitatively* as well as *quantitatively* from each other. The long wavelength radiation could be described by the known laws
of electrodynamics but short wavelengths required different laws for explanation. Considering only this short wavelength form of the radiation spectrum, Einstein showed purely thermodynamically that the entropy of black-body radiation in a given
wavelength interval depends upon the volume of the enclosure in exactly the same way that the entropy of an ideal gas depends upon its volume. By interpreting entropy statistically, he recognized that it was simply the independence of the motions
of the gas molecules that produced a particular form for its entropy. His next step was to make a logical leap: if the entropy of radiation has the same form as that of a gas and if the entropy of a gas has that form because it consists of independent
particles, then radiation too must consist of independent particles of energy. Einstein, however, argued for more than the necessity of introducing discontinuities into black-body theory. The concepts of light-quanta and of resonator energy restriction
to *hv* had entered together in Einstein’s papers of 1905 and 1906 and they remained for him parts of a single if unfinished theory. Since the first was abhorrent even to those scientists persuaded of the second’s necessity, disentangling
the two or finding a substitute for both was to be a central task in the further development of the quantum programme.

In the above section, I tried to argue that the black-body problematic should be located in a matrix of competing and opposing paradigms. In Kuhn’s own example of the radiation case, its history clearly reveals (a) the simultaneous presence of opposite and competing paradigms and (b) the coexistence of rival conceptual structures, as not just a historical contingency but a logical necessity, for the articulation as well as the ‘resolution’ of the radiation problem. The articulation of the tensions between rival theories is a necessary condition for the production of a critical discourse. In this way one can see that the rationality of science is to be understood not only historically but also logically. This also explains the uniqueness of the radiation problem in the development of quantum physics. For many commentators have argued that the specific heats of solids at low temperature, a long standing problem in physics and chemistry, could also have provided the impetus for a change in the foundations of physics. But the logical and historical validity of the cavity radiation case as a case involving the three basic conceptual foundations of physics, as I have outlined above, makes it the central factor in the emergence of the quantum innovation. Moreover, this is also the reason why one can’t read the quantum story in a linear way where the black-body radiation case could appear as an ‘anomaly’ or a puzzle in the Kuhnian sense of ‘normal science’, with the above controversy signaling a period of ‘revolutionary science’ and the subsequent quantum innovation as a ‘paradigm shift’ in the history of science. To think of something as an anomaly presumes that it is a divergence or a deviation from the norm. But the pattern of scientific appraisal sketched above reveals not the presence of one norm, from which infractions could be possible, but three fundamental structures of thought. What it shows is the effort of the scientific community to place it in a variety of conceptual frameworks within which the black-body distribution function could be obtained. But this does not indicate in Feyerabend’s terminology the proliferation of incommensurably different conceptual structures. In this controversy because all the three theoretical structures were dominant or neither was dominant, the participants focused their appraisal on comparisons between the explanatory scope of rival theories and outcomes. Therefore, the choice is not between a single monolithic conceptual framework (Kuhn) or an anarchistic welter of incommensurable conceptual frameworks or practices (Feyerabend, social constructivism) but to understand the logic of a scientific situation in terms of dualisms, oppositions and contrasts which form the conditions of existence for the generation of any innovation.

Through these controversies I have also attempted to show that if one were to look outside the textbook tradition, then one would rarely find a completely unchallenged paradigm of scientific research. Recourse to the body of journal literature,
the medium through which natural scientists report their original work, assess and evaluate that done by others and often modify their own work, immediately casts doubt upon Kuhn’s implications of the standard textbook schema of scientific
progress in terms of paradigms and revolutions. The black-body radiation case shows that all the scientists like Boltzmann, Planck or Einstein were reporting their progress on the case, their findings on the subject and their dissension with their
colleagues on the matter, by sending their research publications to various journals. The great diversity of conceptual frameworks encountered, and the meaningful dialogue between them, in the various journals of the time forms the basis of my
critique of the Kuhnian argument for the existence of unequivocal paradigms as a hallmark of a mature science. (Kuhn’s main point is that work within a well-defined paradigm is more productive of revolutionary episodes of scientific innovations,
than work in which no similarly convergent standards are involved.) Of course it is possible that the competing paradigm does not enjoy equal prestige; one of them may be projected as the official paradigm, the orthodoxy, which is endorsed by
the textbooks of the time and the other is formulated as a dissent or a challenge to the orthodoxy that does not find any mention in the textbooks, but is reported in the journals. Thus, in spite of limits put by the orthodoxy to forms of thought
that can be discussed freely and legitimately, they nevertheless don’t disappear completely; they may just go underground, only to appear again. This is because the textbooks partake of the ideal of uniformity or homogeneity to which modern
Western science aspires and thus ignore alternative and oppositional tendencies. In contrast to Kuhn’s claim that modern natural sciences (unlike the arts and the social sciences) aspire to an ideal of uniformity , Cassirer writes (1956)
that the unity of natural knowledge does not demand any such uniformity. He argues that “all scientific thought is dominated and guided by two opposing tendencies that are engaged in a continual process of mutual adjustment. The demand of
‘specification’ is the counterpoise to the claim of ‘homogeneity’. The struggle between these two cannot be decided purely objectively from the nature of the object. It is a dissention and competition that belongs not so
much to the nature of things as to scientific region itself. In this sense, homogeneity and specification were introduced into Kant’s *Critique of Reason*, not as constitutive principles, pertaining to the knowledge of objects but as
regulated principles, as maxims of scientific inquiry” (1956: 80).

Within the scientific discourse, competing theories give conceptual expression to diverse and opposing physical phenomena and thus arises the difficult and paradoxical task of permeating each with the others in a dialectical way and so rendering them
complementary to each other. We saw, for instance, how, Boltzmann’s explanation of *macroscopic thermodynamic* *irreversibility* was conceptualized in the model of *microscopic mechanical reversibility* of molecular motions;
Planck used Boltzmann’s *statistical* definition of entropy in gas theory to justify a *dynamical* interpretation of radiation entropy and derive the radiation law, with Boltzmann’s constant *k* forming the link between
the two; and finally, following Wien’s parallel between the *radiation* pressure and the pressure of a Maxwellian *gas*, Einstein treated radiation itself as a gas of independent particles of energy (quantized). All the scientists
recognized that radiation theory and gas theory were different and yet found it necessary to model one on the other, with Einstein finally transforming the difference into a resemblance. As Deleuze writes that “the question ‘What difference
is there?’ may always be transformed into: ‘What resemblance is there?’” (1994 : 12).

The debate, outlined above, exemplifies Kuhn’s point that a scientific theory is declared invalid only when an alternative candidate is available, contrary to Popper’s claim (1974) that scientific developments proceed by “falsification”
of theory by direct comparison with nature and to Lakatos’ “sophisticated methodological falsifications” where modifications to a “research programme” arise by comparing the revised version with a previous stage in
the development of the selfsame programme (1970:118). Kuhn writes, “the decision to reject one paradigm is always simultaneously the decision to accept another and the judgement leading to that decision involves the comparison of both paradigms
with nature *and* with each other” (1970:77). But while for Kuhn this occurs only during relatively infrequent periods of crisis, for Feyerabend (1970) the proliferation of alternatives to the dominant view goes on all the time in science.
However, despite this difference both accounts fail to provide an intelligible principle of motion of scientific change. The major weakness in Kuhn’s theory of scientific revolutions is his failure to explain why and how the presence of
anomalies (which are always there) will sometimes precipitate a crisis for a paradigm and usher in a revolution. Similarly, in Feyerabend’s theory, given the proliferation of alternatives (all the time), he is unable to explain why and what
changes during discontinuous paradigm shifts.

My reading of the black-body radiation case ends here. It was an attempt to understand the significance of the cavity radiation case in the emergence of quantum physics and to outline the conditions of its existence in the light of current theories of history of science. Scientific discourse, like language, is not just referential; it also exists as a kind of lateral message indicating its own process of formation. Paradigm, construction, research programme, discourse are all different metaphors which help us to understand its formative, reflective and critical character. The dynamism of scientific discourse is such that the universe of discourse becomes at the same time a discourse of the universe.

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DARRIGOL, OLIVIER. 1988. Statistics and combinatorics in early quantum theory. *Historical studies in the physical and biological sciences* 19(1):17-80

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DELEUZE, GILLES. 1994. *Difference and repetition.* London : The Athlone Press.

FEYERABEND, P.K. 1970. Consolations for the specialist. In *Criticism and the growth of knowledge*, ed. I. Lakatos and A. Musgrave. Cambridge : Cambridge University Press.

KLEIN, M.J. 1973. The development of Boltzmannn’s statistical ideas. In *The Boltzmann equation,* ed. E.G.D. Cohen and W.Thirring. Wien : Springer-Verlag.

KUHN, THOMAS S. 1970. *The structure of scientific revolutions*. Chicago : The University of Chicago Press.

-------------------------- 1977. *The essential tension*. Chicago : The University of Chicago Press.

-------------------------- 1978. *Black-body theory and the quantum discontinuity* 1894-1912. New York : Oxford University Press.

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LAKATOS, IMRE. 1970. Falsification and the methodology of scientific research programmes. In *Criticism and the growth of knowledge*, ed. I. Lakatos and A.Musgrave. Cambridge : Cambridge University Press.

POPPER, KARL. 1974. *Conjectures and refutations*. London : Routledge and Kegan Paul.

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