Einstein on Geometry and Experience
Einstein on Geometry and Experience
Einstein's view on geometry, expressed in his "Geometry and Experience" (1921), seems to have a peculiar ambivalence. On the one hand, he seems to be firmly convinced of the importance of physical geometry, as distinguished from formal or mathematical geometry. On the other hand, he expresses a sympathy with what he takes as Poincare's view, the interdependence of geometry and physics. And it may be noted that a similar ambivalence can be found in Poincare's view on geometry (read carefully Poincare 1952).
Einstein first endorses the view which is often asserted by a number of logical empiricists:
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clearness as to this state of things first became common property through that new departure in mathematics which is known by the name of mathematical logic or "Axiomatics." The progress achieved by axiomatics consists in its having neatly separated the logical-formal form its objective or intuitive content; ... (Einstein, Sidelights on Relativity, Dover, 28-9)
It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the relations of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the co-ordination of real objects of experinece with the emptyu conceptual frame-work of axiomatic geometry. To accomplish this, we need only add the proposition:--solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.
Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. (op. cit., 31-2)
That is to say, we have to draw a sharp distinction between (1) axiomatic geometry and (2) physical geometry; (1) has nothing to do with physical reality, and (2) is related to physical reality via such notions as rigid body or practically-rigid body. This view seems quite close to Reichenbach's view, for instance. Moreover, Einstein emphasizes the importance of this view as regards his own scientific achievements, the general relativity in particular.
I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity. ... In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry. ... If we deny the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H. Poincare:--Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity. Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable ... to retain Euclidean geometry. For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest. (33-4)
As is well known, Einstein's consideration on a rotating system convinced him of the need for a non-Euclidean geometry, as the following figure illustrates (notice that the measuring rod contracts along the circumference):
And notice that Einstein is saying that, because he knew the way for connecting geometry to physical reality by means of the practically-rigid body, he could avoid the error of conventionalism. And he further elaborates this point.
Why is the equivalence of the practically-rigid body and the body of geometry ... denied by Poincare and other investigators? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behaviour, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appears destroyed, and we feel impelled toward the following more general view, which characterizes Poincare's standpoint. Geometry (G) predicates nothing about the relations of real things, but only geometry together with the purport (P) of physical laws can do so. Using symbols, we may say that only the sum of (G)+(P) is subject to the control of experience. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions. All that is necessary to avoid contradictions is to choose the remainder of (P) so that (G) and the whole of (P) are together in accord with experience. (35)
Thus Einstein formulates what he thinks to be Poincare's view, and he is saying that if rigidity is contested, such a view may seem natural and reasonable. Further, Einstein shows a stronger sympathy.
Sub specie aeterni Poincare, in my opinion, is right. The idea of the measuring-rod and the idea of the clock coordinated with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which may not play any independent part in theoretical physics. But it is my conviction that in the present stage of development of theoretifcal physics these ideas must still be employed as independent ideas; ... (35-6)
What does this admission of the "provisional" character of "rods and clocks" imply? It is not entirely clear, but Einstein is at least making a concession to Poincare's view. And, later, Einstein is to argue against Reichenbach's view in Einstein (1949). But for this, see Gruenbaum on Duhem and Einstein.
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References
Einstein, A. (1983) Sidelights on Relativity, Dover (this includes "Ether and the Theory of Relativity", 1920, and "Geometry and Experience", 1921).
Einstein, A. (1949, 1951) "Reply to Criticisms" in Albert Einstein: Philosopher-Scientist (ed. by P.A. Schilpp).
Gruenbaum, A. (1963) Philosophical Problems of Space and Time, Alfred A. Knopf.
Gruenbaum, A. (1974) Philosophical Problems of Space and Time, 2nd ed. (1st edition is reprinted, with extensive supplementary materials), Reidel.
Poincare, H. (1905) Science and Hypothesis, Dover, 1952 (reprint of 1905 translation).
Last modified August 3, 2001. (c) Soshichi Uchii
