The emergence of discordance: mathematical theories of Brownian movement

Perrin was able to provide two additional, concordant values for N via his experimental study of the mean horizontal displacement and mean rotation of Brownian particles, respectively. What did Perrin think was the importance of these additional determinations of N1 Continuing with the historicist approach and looking at the events from a temporal perspective - as opposed to looking only at the final form of Perrin’s argument - shows that Perrin’s aim in conducting this additional experimental research was not to offer another determination of N, but to use the concordance in order to remove the doubts regarding the molecular-kinetic explanation of Brownian movement that had emerged when the first efforts to experimentally verify Albert Einstein's mathematical work on Brownian motion failed to do so.

In 1905, without even knowing that the phenomenon of Brownian movement had been already observed and studied for around eighty years, Einstein produced a mathematical formula which described the average horizontal displacement that, according to the kinetic theory of heat, the (hypothetical) molecular movement ought to be causing on microscopic particles suspended in a liquid:

where, Xx is the mean horizontal displacement of a suspended (Brownian) particle, t is the time interval during which the displacement is measured, T is the absolute temperature, N is Avogadro’s number, к is the coefficient of viscosity of the liquid, and P the radius of the particle.34

The important thing about Einstein's formula was that it defined the horizontal displacement of a suspended particle without involving its real velocity, which could not be calculated because of the extremely complicated path the particle ought to be describing during a specific time interval. This opened the way for an experimental confirmation, given that all the magnitudes could (theoretically, at least) be experimentally determined. After presenting this equation, Einstein concluded his 1905 paper by hoping ‘that some enquirer may succeed shortly in solving the problem suggested here.’55

The publication of Einstein’s theoretical work on Brownian motion was followed by three independent verification attempts. They were by The Sved- berg in Sweden. Max Seddig in Germany, and Victor Henri in France. They all, and independently of one another, failed to verify the formula. Svedberg argued that his experimental results offered a rough verification of the formula, but his claims were rejected by his contemporaries, including Perrin and Einstein.36 Seddig accepted the failure, but blamed his experimental method.37 The verification failure that came from the quantitative, cinematographic study of Brownian movement, undertaken by Victor Henri at the College de France, in the beginning of 1908, was the failure that had the most impact on Perrin and on the community of French physicists at the time. Contrary to Svedberg and Seddig, Henri concluded that Einstein’s displacement formula did not apply to the Brownian movement of the particles he had experimentally studied.58 Henri’s results were interpreted as a failure to establish the kinetic-molecular movement as the unique cause of the phenomenon of Brownian movement. They were consistent with the position, still defended by many French physicists at the time, that the electric actions exerted from the ions of the liquid on electrically charged suspended particles were an additional cause of the phenomenon.3'7

Perrin believed that the height distribution experiments had established beyond any reasonable doubt the kinetic-molecular explanation of Brownian movement. The failure to experimentally verify Einstein’s displacement formula put in front of him the choice between the inexactness of molecular explanation and the inexactness of the formula. Perrin chose the latter option, believing that some unjustified assumption had entered into Einstein's reasoning. Nevertheless, after suggestions made by Aime Cotton and Paul Lan- gevin, he attempted a verification of the displacement formula by using Brownian particles of exactly known radius which he had used in his height distribution experiments.60

Perrin conducted the first measurements with the help of his doctoral students.61 Surprisingly, the initial displacement measurements offered a satisfactory agreement with the value for N calculated in the height distribution measurements of the same particles. In 1909, Perrin announced a mean value for N, calculated by around 3000 displacement recordings, equal to 70,5.1022. This value was identical with the value for N determined in the height distribution experiments and remained relatively invariant to changes of the various experimental parameters. Perrin used the numerical agreement to support the validity of both the experimental procedures employed to determine the magnitudes appearing in Einstein’s formula and the theoretical assumptions underlying Einstein’s mathematical derivation.62

After verifying Einstein's displacement formula, Perrin saw the possibility of an experimental test of Einstein’s equation for the rotational Brownian movement. Einstein had theoretically demonstrated that the molecular impacts, besides a translational movement, imparted on the suspended microscopic particles also a rotational movement. At the basis of Einstein’s equation of mean rotation was the equipartition of energy theorem, which claimed that, at the same temperature, the mean kinetic energy of rotation of a suspended Brownian particle was equal to its mean kinetic energy of translation, and both equal to the mean kinetic energy of an isolated molecule (and all this independently of the size of the granule). Perrin’s stated aim behind this experimental effort was not another confirmation of the molecular theory of Brownian movement, or another determination of A, but the confirmation of the theoretical assumptions underlying Einstein’s rotation equation:62 in particular, the invariance of the equipartition of energy theorem to changes of the various parameters (especially to changes in the size of Brownian particles).64

Conclusion: Towards a two-way, mutually beneficial, integration of History and Philosophy of Science

In this chapter, 1 have argued for the necessity of the historicist-hermeneutic approach for achieving a mutually beneficial integration of History of Science and Philosophy of Science. As outlined above, aspects of the historicist-hermeneutic approach have been supported by various scholars during the last fifty years. I demonstrated how this approach can be applied concretely to solve one of the most problematic case-studies in philosophy of science: the reasoning underlying Jean Perrin’s argument for molecular reality. I have argued that Perrin’s was a case of multiple determination. Perrin put forward a no-coincidence argument for the existence of molecules, which was based on the agreement between multiple, independent determinations of Avoga- dro's number (and consequently, other molecular magnitudes). The blunt rationale of the argument was the following: it would be a highly improbable coincidence for multiple, independent determinations of molecular magnitudes to achieve concordant results, and yet for there not to be any molecules. The careful application of the historicist-hermeneutic approach, however, shows that there were additional structural elements of Perrin’s argument that were responsible for its exceptional strength and, ultimately, for its success. They were the following:

  • 1 The argument was based on a quantitative multiple determination. That is, the independent determinations concerned specific numerical values of the molecular magnitudes.
  • 2 There was a close agreement between the independent determinations. This agreement became even more striking, if one considered the possible values for the molecular magnitudes that could have been the result of each one of the determinations.
  • 3 There was a (relatively) large number of determinations which converged on the same result.
  • 4 The different determinations were theoretically independent: that is, they were based on independent theoretical assumptions.
  • 5 The different determinations were genetically independent and no effort was made to mutually adjust the numerical values calculated by theoretically independent procedures.
  • 6 The different determinations were based on the investigation of unrelated phenomena.
  • 7 The high quality and reliability of some of the determinations.
  • 8 There was not even one discordant result, despite the large number of determinations.
  • 9 When objections and discordant results which challenged Perrin's determination of molecular magnitudes emerged. Perrin conclusively resolved the discordance.

Following the historicist-hermeneutic approach it is possible to develop a conceptual framework for dealing with the structure and epistemic importance of the multiple determination strategy in scientific practice. The historicist-hermeneutic approach, as employed in Perrin’s case, shows the existence of several structural elements upon which the strength of the nocoincidence argument - the defining feature of the multiple determination strategy - depends. These elements are:

  • 1 The number of determinations: the more the determinations that produce the same result, the stronger the no-coincidence argument.
  • 2 The theoretical independence of the determinations: the more theoretically independent the determination procedures that establish the same result are, the stronger the no-coincidence argument.
  • 3 The genetic independence of the determinations: the more genetically independent the determinations procedures that establish the same result are, the stronger the no-coincidence argument.
  • 4 The reliability of the determinations: the more reliable the determination procedures that establish the same result are, the stronger the no-coin- cidence argument.
  • 5 The quality (or clarity) of the result established by independent determinations: the clearer or more precise is the result upon which the independent determinations agree, the stronger the no-coincidence argument.
  • 6 The quality of the convergence: the more the determination procedures are judged to have established the same result, the stronger the no-coin- cidence argument.
  • 7 The complexity of the independently established result: the more complex is the result that is established by independent determinations, the stronger the no-coincidence argument.
  • 8 The existence of discordant results and/or conflict with accepted knowledge: the less the discordant results and/or the conflict with accepted knowledge, the stronger the no-coincidence argument.

Continuing with the historicist-hermeneutic approach, we can use this preliminary conceptual framew'ork to understand and evaluate the epistemic force of other cases of multiple determination, from past or current science. For example, w'e can use it to understand w'hy in some cases of multiple determination the no-coincidence argument succeeds, whereas in other cases it fails. The implementation of this step will demonstrate the relevance of Philosophy of Science to History of Science. This step is different from the traditional use of philosophical pre-conceptions to interpret the historical material. And this is because from the interaction of this initial conceptual framework with the historical material it is possible to further sharpen and elucidate our initial framework. This could be done, for example, by noticing other structural elements that influence the strength of the no-coincidence argument underlying the multiple determination strategy. The implementation of this step will demonstrate the relevance of History of Science to Philosophy of Science. We can use this more developed framework to elucidate and evaluate other (or even the same) cases of multiple determination.65 And so forth. Our efforts to understand science in its historical dimension are themselves open-ended.

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