## Abstract

In order to show how formal analogies between different physical systems play an important conceptual work in physics, this paper analyzes the evolution of Einstein’s thoughts on the structure of radiation from the point of view of the formal analogies he used as “lenses” to “see” through the “black box” of Planck’s blackbody radiation law. A comparison is also made with his 1925 paper on the quantum gas where he used the same formal methods. Changes of formal points of view are most of the time taken for granted or passed over in silence in studies on the mathematization of physics as if they had no special significance. Revisiting Einstein’s classic papers on the nature of light and matter from the angle of the various theoretical tools he used, namely entropy and energy fluctuation calculations, helps explain why he was in a unique position to make visible the particle structure of radiation and the dual (particle and wave) nature of light and matter. Finally, this case study calls attention to the more general question of the surprising creative power of formal analogies and their frequent use in theoretical physics. This aspect of intellectual creation can be useful in the teaching of physics.

This is a preview of subscription content, log in to check access.

## Notes

- 1.
- 2.
Prepared for the special issue of

*Science and Education*, on the relations between physics and mathematics, this paper is a revised English translation of a previous paper published in French in Gingras (2011). - 3.
- 4.
- 5.
For a detailed survey of the use of models in different disciplines see Morgan and Morrison (1999).

- 6.
See for example Hesse (1966, 8–10), where

*models*are discussed in terms of positive and negative*analogies*. See also W. H. Leatherdale (1974, 1). The confusion between models and analogies is often due to the fact that most authors concentrate on the special cases of analogical models. For a recent analysis, see Bailer-Jones (2009). - 7.
- 8.
For convenience we use modern notations.

- 9.
Einstein’s reflections on those foundations led to his 1906 paper “On the Theory of Light Production and Light Absorption” in which he clarified Planck’s derivation, concluding that “Mr Planck introduced into physics a new hypothetical element: the hypothesis of light quanta”, thus confirming that his own views on the quantum of light and those of Planck were not incompatible as he first thought in 1905; see Einstein (1989, 192, 196).

- 10.
- 11.
We are not discussing here the

*logical*necessity of the analogy used by Einstein but the*historical*fact of its particular use in his argumentation for the particle structure of radiation. For a logical analysis see Dorling (1971). - 12.
For details on this question, see Pais (1982, 68–70).

- 13.
Though it is not the place to develop that argument here, I think the acceptance of the idea of “duality” between particle and wave is closely linked to the Copenhagen interpretation of Quantum mechanics and its underlying operationalist philosophy in which the ontology is defined by measuring instrument. Though always marginal, some major physicists always opposed the idea of “wave-particle duality” as absurd; see for example Landé (1965).

- 14.
Dorling (1971) writes that the use of arguments by analogy “seemed quite uncharacteristic of Einstein” and that it does not occur “in any of his other major writings”. As this paper shows, this is far from being the case.

- 15.
For more details on the relation between Einstein and Bose, see Stachel (2002, 519–538).

- 16.
For en entry into this literature see Gentner et al. (2001).

- 17.
- 18.
For an analysis of the analogy between mechanical and electrical systems, see Gaston Bachelard,

*Le rationalisme appliqué*, Paris, PUF, 1949.

## References

Abiko, S. (2000). Einstein’s theories of fluctuation and the thermal radiation: The first quantum theory through statistical thermodynamics.

*Historia Scientiarum,**10*, 130–147.Bailer-Jones, D. M. (2009).

*Scientific models in philosophy of science*. Pittsburgh: University of Pittsburgh Press.Balashov, Y., & Vizgin, V. (Eds.). (2002).

*Einstein studies in Russia*. Boston: Birkäuser.Baracca, A. (1985). Einstein’s statistical mechanics.

*Revista Mexicana de Fisica,**31*, 695–722.Barcelo, C., Liberati, S. & Visser, M. (2005).

*Analogue gravity*. arXiv:gr-qc/0505065v2, I.Bergia, S., & Navaro, L. (1988). Recurrences and continuity in Einstein’s early research on radiation between 1905 and 1916.

*Archives for the History of Exact Sciences,**29*, 79–99.Blau, S. K. (2005). Black-hole physics in an electromagnetic waveguide.

*Physics Today*, August, 19–20.Brush, S. G. (2007). How ideas become knowledge: The light-quantum hypothesis, 1905–1935.

*Historical Studies in the Physical and Biological Sciences,**37*, 205–246.de Berg, K. C. (1992). Mathematics in science: The role of the history of science in communicating the significance of mathematical formalism in science.

*Science & Education,**1*, 77–87.Dorling, J. (1971). Einstein’s introduction of photons: Argument by analogy or deduction from the phenomena?

*British Journal for the Philosophy of Science,**22*, 1–8.Duck, I., & Sudarshan, E. C. G. (1997).

*Pauli and the spin-statistics theorem*. Singapore: World Scientific.Einstein, A. (1989).

*The collected papers of Albert Einstein, vol. 2, the Swiss years: Writings, 1900*–*1909. English translation*. (A. Beck, Trans.). Princeton: Princeton University Press.Einstein, A. (1993).

*The collected papers of Albert Einstein, vol. 3, the Swiss years, Writings, 1909*–*1911. English translation*. (A. Beck, Trans.). Princeton: Princeton University Press.Einstein, A. (1995).

*The collected papers of Albert Einstein, vol. 5, the Swiss years: Correspondence, 1902*–*1914*.*English translation*. (A. Beck, Trans.). Princeton: Princeton University Press.Einstein, A. (1997a). Quantum theory of the monoatomic ideal gas: Part one and two. In Duck, & Sudarshan, pp. 82–99.

Einstein, A. (1997b). On the quantum theory of the ideal gas. In Duck, & Sudarshan, pp. 100–107.

Garay, L. J., Anglin, J. R., Cirac, J. I., & Zoller, P. (2000). Sonic analog of gravitational black holes in Bose–Einstein condensates.

*Physical Review Letter,**85*, 4643–4647.Gentner, D., Holyoak, K. J., & Kokinov, B. N. (Eds.). (2001).

*The analogical mind: Perspectives from cognitive science*. Cambridge: MIT Press.Gingras, Y. (2001). What did mathematics do to physics?

*History of Science,**39*, 383–416.Gingras, Y. (2011).

*«La valeur inductive des analogies: comment Einstein a vu la lumière à travers le prisme des analogies formelles», dans Hugues Chabot, Sophie Roux, La mathématisation comme problème*(pp. 88–108). Paris: Éditions des archives contemporaines.Hendry, J. (1980). The development of attitudes toward the wave-particle duality of light and quantum theory, 1900–1920.

*Annals of Science,**37*, 59–79.Hesse, M. B. (1966).

*Models and analogies in science*. Notre Dame: University of Notre Dame Press.Holyoak, K. J., & Thagard, P. (1995).

*Mental leaps: Analogy in creative leaps*. Cambridge: MIT Press.Klein, M. J. (1967). Thermodynamics in Einstein’s thoughts.

*Science,**157*, 509–516.Klein, M. J. (1980). No firm foundation: Einstein and the early quantum theory. In H. Woolf (Ed.),

*Some strangeness in the proportion: A centennial symposium to celebrate the achievements of Albert Einstein*(pp. 161–185). Reading, MA: Addison-Wiley.Klein, M. J. (1982). Fluctuations and statistical physics in Einstein’s early work. In G. Holton & Y. Elkana (Eds.),

*Albert Einstein: Historical and cultural perspectives*(pp. 39–58). Princeton: Princeton University Press.Kojevnikov, A. (2002). Einstein’s fluctuation formula and the wave-particle duality. In Balashov & Vizgin, pp. 181–228.

Landé, A. (1965).

*New foundations of quantum mechanics*. Cambridge: Cambridge University Press.Leatherdale, W. H. (1974).

*The role of analogy, model and metaphor in science*. Amsterdam: North-Holland.Maxwell, J. C. (2003).

*The scientific papers of James Clerk Maxwell*(Vol. 1). New York: Dover Publications.Mehra, J., & Rechenberg, H. (1982).

*The historical development of quantum theory, vol. 1, part 2, the quantum theory of Planck, Einstein, Bohr and Sommerfeld: The foundation and the rise of its difficulties 1900–1925*. New York: Springer.Morgan, M., & Morrison, M. (Eds.). (1999).

*Models as mediators: Perspectives on natural and social sciences*. Cambridge: Cambridge University Press.Norton, J. D. (2006). Atoms entropy quanta: Einstein’s miraculous argument of 1905.

*Studies in History and Philosophy of Modern Physics,**37*, 71–100.Pais, A. (1982).

*‘Subtle is the Lord…’ the science and the life of Albert Einstein*. New York: Oxford University Press.Pereira de Ataíde, A. R., & Greca, I. M. (2013). Epistemic views of the relationship between physics and mathematics: Its influence on the approach of undergraduate students to problem solving.

*Science & Education,**22*, 1405–1421.Quale, A. (2011). On the role of mathematics in physics.

*Science & Education,**20*, 359–372.Redhead, M. (1980). Models in physics.

*British Journal for the Philosophy of Science,**31*, 145–163.Schliemann, J., Loss, D., & Westervelt, R. M. (2005).

*Zitterbewegung*of electronic wave packets in III–V zinc-blende semiconductor quantum wells.*Physical Review Letters,**94*, 206801-1.Schützhold, R., & Unruh, W. G. (2005). Hawking radiation in an electromagnetic waveguide.

*Physical Review Letters,**95*(2005), 31301-1.Soler, L. (1999). Les quanta de lumière d’Einstein en 1905, comme point focal d’un réseau argumentatif complexe.

*Philosophia Scientiae,**3*, 107–144.Soler, L. (2001). Les origines de la formule E = hv, ou comment l’analogie est vecteur de nouveauté.

*Philosophia Scientiae,**5*, 89–123.Stachel, J. (2002).

*Einstein from ‘B’ to ‘Z’*. Boston: Birkhäuser.Torregrosa, J., López-Gay, R., & Gras-Marti, A. (2006). Mathematics in physics education: Scanning the historical evolution of the differential to find a more appropriate model for teaching differential calculus in physics.

*Science & Education,**15*, 47–462.Unruh, W. G. (1981). Experimental black-hole evaporation?

*Physical Review Letters,**46*, 1351–1353.Wheaton, B. R. (1983).

*The tiger and the shark: Empirical roots of wave-particle duality*. Cambridge: Cambridge University Press.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Gingras, Y. The Creative Power of Formal Analogies in Physics: The Case of Albert Einstein.
*Sci & Educ* **24, **529–541 (2015). https://doi.org/10.1007/s11191-014-9739-1

Published:

Issue Date:

### Keywords

- Formal Analogy
- Dual Nature
- Light Quantum
- Energy Fluctuation
- Black Hole Evaporation