Abstract
Henri Poincaré and Albert Einstein are exemplars of the fundamental investigator referred to in the grand manner as philosopher-scientist. Their philosophical and scientific thoughts were linked and their interest in fundamental issues led them to probe the process of thinking itself. The quotations in the epigraphs to this chapter and to the Introduction bespeak their commitment to this connection as a guiding theme in their scientific research. This Ariadne’s thread is best thrown into perspective through the realization that their individual philosophies of science are composed of an epistemology of the origin of knowledge and an epistemology of scientific theories. In fact, the case of Poincaré and Einstein indicates that it is necessary—and, in my opinion, revealing of the structure of scientific theories—to separate the broad discipline of epistemology into two parts: (1) the construction of prescientific knowledge, that is, the origins of knowledge, which may be referred to as theory of knowledge; and (2) the relations of the scientist’s knowledge of the world of perceptions to the structure of a scientific theory and the study of what a scientific theory is, hereinafter to be referred to as epistemology or scientific epistemology.1 Both aspects of epistemology include an analysis of concept formation.
The logical point of view alone appears to interest [Hilbert]. Being given a sequence of propositions, he finds that all follow logically from the first. With the foundation of this first proposition, with its psychological origin, he does not concern himself.
H. Poincaré (1903)
The whole of science is nothing more than a refinement of everyday thinking. It is for this reason that the critical thinking of the physicist cannot possibly be restricted to the examination of concepts of his own specific field. He cannot proceed without considering critically a much more difficult problem, the problem of analyzing the nature of everyday thinking.
A. Einstein (1936)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
To some extent this distinction has been previously made by Toulmin (1972), who discusses an epistemics and a philosophical epistemology, and by McMullin (1970).
Analysis of the connections among von Helmholtz, Marius Sophus Lie, and Poincaré remains to be written. Briefly, in the 1860s von Helmholtz demonstrated that the axioms of Euclidean geometry could be deduced from the laws of motion of rigid bodies, along with the assumption that a space of any number of dimensions is determined by that number of coordinates; similarly, non-Euclidean geometries are the laws of motion of other sorts of bodies. It was essentially the lack of rigor of von Helmholtz’s results that spurred Lie in 1867 to begin developing the subject of continuous groups of transformations. As Lie wrote to Poincaré in a letter of 1882: “(Riemann and) v. Helmholtz proceeded a step further [than Euclid] and assumed that space is a Zahlen-Mannigfaltigkeit [manifold or collection of numbers]. This standpoint is very interesting; however it is not to be considered as definitive” (italics in original). Thus, instead of studying in an approximate manner how geometries are generated by rotations or translations of rigid bodies, Lie developed the means to analyze rigorously how points in space are transformed into one another through infinitesimal transformations—that is, the subject of continuous groups of transformations. Although Poincare quickly realized the fundamental importance of Lie’s work for both mathematics and philosophy, he disagreed with Lie’s own interpretations of the origins of geometry and the notion of space. The reason is that for both von Helmholtz and Lie the matter of the group (i.e., the Zahlen-Mannigfaltigkeit, or spatial coordinates) existed prior to the group: “For me, on the contrary, the form exists prior to matter” (Poincaré, 1898a). Thus, continued Poincaré, using the group-theoretical approach the origin of geometry can be analyzed without assuming beforehand the existence of both space and geometry. Poincaré agreed with von Helmholtz that an analysis of the origins of geometry should take account of these perceptions, but disagreed with von Helmholtz’s and Lie’s beliefs that a Euclidean geometry joined with physical laws is empirically testable (Helmholtz, 1876; Lie, letter of 1882 to Poincaré).
Piaget, for example, believes otherwise: “Poincaré had the great merit of foreseeing that the organization of space was related to the formation of the ’group of displacements,’ but since he was not a psychologist, he regarded this group as a priori instead of seeing it as the product of a gradual formation” (1969).
See Poincaré (1902a) for a dictionary between Euclidean and non-Euclidean geometry, “as we would translate a German text with the aid of a German-French dictionary.” Assuming that Euclidean geometry is self-consistent, then the researches of Beltrami, Cayley, and Klein were interpreted as proving that nonEuclidean geometries were also self-consistent. Poincaré (1902a) asks, “Whence is this certainty derived, and how far is it justified?” He considered this problem “not insoluble.” It was resolved in 1900 by Hilbert who reduced the axioms of geometry to those of an arithmetic system; such a system was assumed to be self-consistent. This assumption was in turn shattered in 1931 by the work of Kurt Gödel.
Poincaré went on to write that “from the law of conservation of energy Leibnitz deduced that of conservation of momentum without doubt because he assumed that the energy, invariable in the absolute motion, should be invariable in relative motion.” This assumption, indigenous for classical mechanics, led Poincaré in (1900c) to invoke the ad hoc hypothesis of a “complementary force” in order to impose the invariability of energy and to save action and reaction in an inertial reference system that contains an emitter of unidirectional radiation. For using Lorentz’s local time coordinate and the Galilean spatial transformations, Poincaré essentially came upon the transformation law for the energy of a pulse of radiation accurate to order (v/c), which means that energy is not an invariable quantity. Before special relativity there was some confusion between quantities that are conserved (such as energy and momentum) and quantities that are invariable or invariant (such as electric charge). See Miller (1980, 1981b).
With the advent of general relativity theory, Calinon’s speculations on the physical possibility of non-Euclidean geometries were revived as having been prophetic (e.g., Robertson, 1949). Calinon’s essay of (1897) is discussed in Capek (1976). Suffice it to say that only Calinon (1885) deserves further study.
Just as he had proceeded with geometry, Poincaré (1902a) asked whether the principles of mechanics are a priori or obtained from experiment. For example, if the principle of inertia were a priori, then the Greeks would not have taken the preferred motion to be circular motion. Nor could this principle have been obtained from experiment because no one has ever actually observed a purely force-free motion (i.e., an inertial reference system).
Poincaré (1908a) comments on Hertz’s mechanics, which is formulated in a non-Euclidean geometry, thus: It quite seems, indeed, that it would be possible to translate our physics into the language of geometry of four dimensions. Attempting such a translation would be giving oneself a great deal of trouble for little profit, and I will content myself with mentioning Hertz’s mechanics, in which something of the kind may be seen. Yet it seems that the translation would be less simple than the text, and that it would never lose the appearance of a translation, for the language of three dimensions seems the best suited to the description of our world, even though that description may be made, in case of necessity, in another idiom.
See also Poincaré (1908a), where he writes that the earth’s “absolute velocity . . . has no sense,” rather one must mean “its velocity in relation to the ether.”
The reason is that in Lorentz’s theory the ether acts on matter with no inverse reaction. Poincaré discussed this problem in print (1900c) and in correspondence with Lorentz (Miller, 1980). It turns out that Poincaré was more concerned with the Lorentz theory’s violation of Newton’s third law than the ad hoc contraction hypothesis because this law was a convention. In order to preserve Newton’s third law Poincaré was willing to introduce into Lorentz’s theory several hypotheses, one of which was as ad hoc as the hypotheses of contraction (see Note 16, above). However, owing to Lorentz’s extension of his theory in 1904 to the electron itself, in such a way as to explain the failure of all extant ether-drift experiments, as well as Kaufmann’s data on high–velocity electrons, Poincaré was willing to discard Newton’s third law in electromagnetism because it impeded progress (1904a). In other words (see Figure 1.1), without being empirically disconfirmed, Newton’s third law lost its status as a convention because it was no longer “fertile.”
Rayleigh (1902) and, in an improved version, Brace (1904) sent light in two orthogonal directions through an isotropic crystal at rest in the laboratory. Owing to the hypothesis of contraction the indices of refraction should differ in these two directions. To one part in 10 decimal places Rayleigh found no difference, and Brace pushed the accuracy to one part in 13 decimal places. Trouton and Noble (1903) tried to measure the turning couple on a charged condenser at rest in the laboratory. A turning couple was assumed to be present owing to the motion of the laboratory through the ether. This motion was expected to increase the electrostatic energy density between the plates because the charges on the plates are also the source of convection currents. A measurement of the turning couple would permit determination of the earth’s velocity relative to the ether. No turning couple was measured.
Thus, contrary to Guidymyn (1977), Poincaré never considered the principle of relativity in the physical sciences to be a convention. Further evidence for this point will be offered in the section, “On Poincaré’s Post–1905 Thoughts on Geometry, Classical Mechanics, and the Physical Sciences.”
Einstein’s (1907a) contains his first published recollections on the state of physics in 1905. Study of this paper, in conjunction with others of his early papers and correspondence, leads to the following description of his assessment of the state of physics in 1905. For a more detailed analysis see Miller (1981b). His previous research on fluctuation phenomena led him by the end of 1904 to conclude that light has both wave and particle properties, and Lorentz’s electromagnetic theory could explain only the wave mode. His research on Brownian motion led him to conclude that mechanics cannot be the basic theory because in volumes the order of the electron’s higher-order derivatives of the acceleration cannot be disregarded. Thus, neither the electromagnetic nor mechanical world-pictures could succeed. But restricting considerations to large volumes and not discussing the constitution of matter, the laws of electromagnetism and mechanics could be used with confidence; this is what he did in the special relativity paper.
See the letter of 29 September 1909 of Einstein to Arnold Sommerfeld that is quoted in Stachel (1980). Einstein, however, almost certainly embarked on this path independently of Poincaré. The properties of a rotating disc was a much-discussed problem in 1909. It concerned the notion of a rigid body as defined in classical mechanics in contrast with the deformations proposed by Lorentz and by Einstein [see Miller (1981b), Chapter 7].
Einstein’s description in “Physics and Reality” of the origin of geometry is analogous to those in his (1921a, b), but more detailed. Here he emphasized the importance of the “concept” of the practically rigid body. In “Physics and Reality” Einstein omitted Poincarés name altogether in these discussions, and in the “Autobiographical Notes” of 1946 he singled out only Hume and Mach as helpful to his thinking toward the special theory of relativity. In the “Replies to Criticisms,” however, he rectified this omission. There Einstein exhibited once again his deep understanding of Poincaré by giving his name to the “nonpositivist” in an imaginary dialogue with the logical empiricist philosopher Hans Reichenbach, who was chastised severely.
This letter was first discussed by Holton (1978), and in more detail in Holton (1979).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer Science+Business Media New York
About this chapter
Cite this chapter
Miller, A.I. (1984). Poincaré and Einstein. In: Imagery in Scientific Thought Creating 20th-Century Physics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-0545-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4684-0545-3_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4684-0547-7
Online ISBN: 978-1-4684-0545-3
eBook Packages: Springer Book Archive