Seven

Geometry and experience

From Cairo in mid-December 1919, Swedish mathematician Gösta Mittag-Leffler wrote to Einstein inviting his contribution to a special issue of the journal Acta Mathematica commemorating Henri Poincaré. The French savant – widely acknowledged to be the leading mathematician of his era, as well as an innovative mathematical physicist and influential philosopher of science – had died at the height of his career unexpectedly after minor surgery in 1912. Publication of the memorial issue languished on account of WW I. Still, the more-than-four-year delay furnished ample time for Mittag-Leffler to garner essays from Europe’s leading scientific luminaries, among them H.A. Lorentz and Max Planck. Now he wished to include one from the suddenly famous Albert Einstein. In the five weeks since the announcement in London of the solar eclipse observations confirming his relativistic theory of gravity, a media-created mania made Einstein’s face globally recognized while creating a myth of unfathomable genius that would burden the rest of his life. Not surprisingly, Mittag-Leffler suggested Einstein contribute an essay on those ideas of Poincaré of particular significance in the development of the general theory of relativity, notably “concerning the relations between space, matter, and time”. For unknown reasons, the letter reached Einstein in Berlin only in early April 1920. Nonetheless Einstein promptly replied, agreeing to contribute if allotted sufficient time to write the essay. Then, in late July, he rescinded his promise, pleading that other responsibilities had prevented him from undertaking a needed review of all Poincaré’s writings relevant to the question of geometry and experience (“zu der Frage Geometrie und Erfahrung”).¹ Even so, a seed was planted for renewed reflection on Poincaré. That seed bore fruit in “Geometry and Experience” (“Geometrie und Erfahrung”), given as a public address on January 27, 1921 at the Prussian Academy of Sciences on its annual founder’s day (Friedrichstag) celebration.²

Einstein versus Poincaré

Einstein’s lecture ostensibly revisits the dispute over the foundations of geometry between geometric empiricism and geometric conventionalism, set now in the new context provided by the general theory of relativity. Published in the proceedings of the Prussian Academy on February 3, an expanded version of “Geometry and Experience” appeared as a separate booklet and was sold out within weeks. Within the year it was translated into French, English, Russian, Italian, and Polish. It would become one of the widely reprinted and influential texts essays in 20th-century philosophy of science, one of three Einstein texts included in the 1953 anthology of papers in philosophy of science edited by Herbert Feigl and May Brodbeck, the first such collection to be published in the USA. By 1929 it had been deemed an essential part of the “Scientific World Conception” (Wissenschaftliche Weltauffassung) in the manifesto of the Vienna Circle, written (mostly) by Otto Neurath, Rudolf Carnap, and Herbert Feigl. In a summary section,³ the logical empiricists epitomized the philosophical significance of “Geometry and Experience” in four points:

1. the need for a clear distinction between pure mathematical (axiomatic) geometry and applied geometry, a branch of physics;
2. the definition of the latter as “practical geometry” characterized by the possible positions of a rigid body in space;
3. an insistence that the definition of rigid body has an “empirical basis”, namely, “the preservation of coincidences (equality of spatial intervals)”;

and finally the conclusion

1. 4. that from the standpoint of “practical geometry”, the question regarding the spatial structure of the universe (concerning both metrical structure as well as finite or infinite extension) has an empirical answer.

Fundamental tenets of logical empiricism are indicated in this brief and, as will be seen, rather tendentious précis. Above all, there is an implied rejection of any metaphysics of space. In declaring the question concerning the geometry of physical space to have an unambiguous empirical answer, Einstein was understood not only to invalidate Poincaré’s geometric conventionalism but also to ride the positivist hobby-horse of renouncing metaphysics, the ideological cornerstone of the “Scientific World Conception”. In particular, Einstein had shown the error of any a priori foundation for geometry, in particular the Kantian doctrine that Euclid’s postulates express the “necessity and universality” of the form of outer intuition and so are synthetic a priori conditions of possible experience. To logical empiricism, the collapse of this bastion of the a priori in the theory of space signaled more generally the triumph of empiricism over idealist and metaphysical philosophies; Einstein’s “clear formulations brought order into a field where confusion often prevailed”.⁴ Furthermore, his sharp distinction between empirical “practical” physical geometry and purely formal axiomatic geometry went in tandem with the logical empiricist account of pure mathematics as grounded ultimately in logic. Inspired by Whitehead and Russell’s Principia Mathematica, logical empiricism considered mathematical statements to be reducible to logical statements and purely mathematical truths to be a species of logical truth. As logical truths are paradigmatically analytic, true in virtue of the meanings of the terms they contain, the account of mathematics as logic supported the core logical empiricist thesis that any meaningful statement is either analytic or is an a posteriori synthetic statement, confirmable or refutable by experience. In this way, the logical empiricists would point to “Geometry and Experience” as an illustrious precursor of their dictum that the synthetic a priori statements of metaphysics were literally meaningless. Two decades later, the sharp dichotomy between analytic and synthetic statements was attacked as one of the “dogmas” of empiricism in Quine’s famous 1951 critique of Carnap.⁵

Less than apparent in the above clipped summary is just how the empiricist conception of the geometry of physical space (-time) ostensibly presented by Einstein actually responds to, and defeats, Poincaré’s geometric conventionalism. Seeing this requires unpacking what was stated with considerable compression in 3). “Practical geometry” explicitly rests on Einstein’s concept of the “practically rigid body”. Now the rigid body is a problematic concept (a “child of sorrow” – Schmerzenskind, according to Einstein) already in special relativistic physics. Imagine a very forceful tug on one end of a long completely rigid measuring rod. Since a rigid body has only rigid motions, the fixed spacing between the molecules of the rod will transmit the impulsive force with an indefinitely large (and so possibly superluminal) velocity to the distal end. Nonetheless 3) states that the concept has an “empirical basis” in that observed coincidences between the endpoints of two rigid bodies are preserved when the bodies are translated in space. That statement entails that whenever the endpoints of two “practically rigid” measuring rods are found to coincide in one region of space, they always will be found to do so when the rods are brought together in any other region. Is this really an empirical statement?

A brief reflection should convince that it is not, at least not in any straightforward sense. And so logical empiricism emphasized the “empirical basis” of Einstein’s definition of a rigid body in 3) required a stipulation of “preservation of coincidence”. This states that if two bodies (e.g. rods), whose endpoints are in coincidence at one time at location A, are then separated and translated to a distant location B, their endpoints again will be found in coincidence when compared at B at a later time. The definitional nature of this statement is readily appreciated if one considers that the bodies may travel from A to B at different velocities along distinct, possibly circuitous paths. That the application of pure mathematics in natural science requires stipulations of this sort, here investing a physical object (measuring rod) with the meaning of “practically rigid body” (and of “equality of spatial intervals”) would be enshrined as a central facet of logical empiricism’s account of scientific methodology. The necessary first step in the application of any formal mathematical theory (e.g., pure axiomatic geometry) to empirical phenomena required similar stipulations or “coordinative definitions” associating certain concepts or relations of mathematics with observable objects or processes. That empirical determination of the geometry of physical space rested upon postulation of definitional linkages between formal geometric concepts (“distance”, “straight line”) and physical objects (“measuring rod”, “light ray”) became the logical empiricists’ paradigmatic example of how formal expressions of a mathematized theory acquire empirical meaning in science. Rudolf Carnap in 1927 provided an early, and certainly most graphic, illustration of the significance of the methodology of coordinative definitions. Only through implementing definitions coordinating formal concepts with concrete empirical objects does “the blood of empirical reality” enter through these touch points to flow upward into the most diffuse veins of the hitherto empty theory-schema.⁶

Einstein’s lecture does indeed suggest this – in a way. Yet “Geometry and Experience” is not really concerned with the methodological issues emphasized by logical empiricism. Its message is considerably more tempered, endorsing geometric empiricism after a fashion but merely in the guise of a pro tem strategy. Practically rigid measuring rods and ideal clocks play, at least provisionally, an epistemologically privileged but in principle logically objectionable role in the general theory of relativity. It is no coincidence that in January 1921, the privileged role of rods and clocks in Einstein’s general theory of relativity had become a live issue of contention between Einstein and mathematician Hermann Weyl, a friend and colleague who sought, though in a mathematically speculative way, to reconstruct general relativity without it. What the logical empiricists did not, nor, for the most part, would not mention is the existence of this controversy, the threat it posed to their methodology of coordinative definitions and to the ensuing conception of empiricism in physical science more generally. Considerable stage setting, requiring a detour into the late 19th century debate about the foundations of geometry, will give a clear understanding of the issues at stake. Einstein’s paper is only superficially a pointed intervention on behalf of geometric empiricism in its storied confrontation with geometric conventionalism à la Poincaré. Instead “Geometry and Experience” appropriates that earlier debate, a conflict in any case now outmoded in the different context opened up by the variably curved space-times of general relativity, effectively carrying over the no-longer-suitable terms of the earlier discussion into a new, and considerably more intricate, setting. Interestingly, rather than insisting upon what is novel about the geometry of general relativistic space-times, territory firmly occupied in 1921 by Weyl, Einstein chose to largely mute the controversy, emphasizing the pro tem benefits of a pragmatic justification of the “practical geometry” of rods and clocks while admitting that Weyl (though in the persona of Poincaré) is correct, in principle (“sub specie aeterni”). In the end, what is really at issue in the new situation – unless one knows the backstory – is only dimly perceptible. Like the masked actor in a Nōh play, Einstein relates an illusive and largely symbolic drama, seasoned with stylized elegance (and memorable quotes), directed at an audience that may or may not be able to read between and behind the lines of ritualized presentation.

19th-century geometric empiricism and conventionalism

Einstein’s title echoes the fifth chapter (“Experiénce et géométrie”) of Poincaré’s widely read La Science et l’hypothèsis (1902), an essay famously concluding, “Whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning”.⁷ Geometric empiricism is a 19th-century doctrine, prompted by the mathematical discovery of non-Euclidean geometries earlier in the century and the resulting challenge they presented to the traditional assumption of Euclidean geometry of physical space. Perhaps its most notable proponent was C.F. Gauss (1777–1855), “prince of mathematicians”, whose geometric discoveries, as well as those in many other areas of mathematics, were far in advance of his era. As early as 1816, Gauss formulated a non-Euclidean geometry of constant negative curvature, subsequently named after Bolyai and Lobachevsky, its independent discoverers in the late 1820s; fearing a scandal (“the clamor of the Boetians”, according to Gauss), he did not publish. On the other hand, in his capacity as director of the Göttingen observatory, Gauss devised a precise cartographic survey of the German principality of Hannover. Employing an instrument of his own design to focus and reflect rays of sunlight between three distant mountain peaks (Inselberg, Brocken, Hohenhagen), Gauss used the method of triangulation to pinpoint locations within this great triangle with unprecedented precision. As a matter of fact, historians disagree whether Gauss actually intended to test the Euclidean assumption that the interior angles between the triangular mountain peaks must sum to 180 degrees. Nonetheless, in a paper of 1827 he reported that this was indeed the case while intimating that any empirical test of Euclidean geometry would require triangulations over much greater, indeed stellar, distances. Critical of the Kantian attribution of necessity and universality to the Euclidean axioms as the form of outer intuition, Gauss classified geometry not with arithmetic (which Gauss assumed a priori) but as an empirical science akin to mechanics.

At the time of Poincaré’s essay, around the turn of the century, geometric empiricism was primarily associated (not entirely without qualification, as will be seen) with another of 19th-century Germany’s most notable scientists, physicist and physiologist Hermann von Helmholtz (1821–1894). Helmholtz’s 1866 essay “On the Factual Foundations of Geometry” was his direct response to the posthumous publication of Bernhard Riemann’s 1854 Habilitationsschrift “On the Hypotheses that Lie at the Foundations of Geometry”.⁸ One of the landmarks of modern mathematics, Riemann’s lecture was delivered in Göttingen before the aged Gauss himself who reportedly praised it highly. Riemann outlined a vastly broad conception of geometry, generalizing the concept of space to what he termed “multiplicities” or “manifolds” (Mannigfaltigkeiten) of n dimensions. In such manifolds, afterwards termed Riemannian, distances and angles between points are defined by extending an 1828 theorem of Gauss that determined the intrinsic curvature of a two-dimensional surface (i.e., without regard to an ambient space in which the surface is embedded). Riemann generalized to manifolds Gauss’s differential expression showing, e.g., how distances between finitely separated points on a surface can be measured along a path connecting them by summing up all the small differences in coordinates between adjacent points (e.g., between x and x + dx) on that path. Riemann’s principal assumption required the validity of the Pythagorean theorem only within the infinitesimal region of any point of the manifold, so that the squared distance ds² between two nearby points is equal to the sum of their squared coordinate differences, i.e., $d{s}^2={\displaystyle \sum }_{i}d{x}_i^2$. Riemannian manifolds are accordingly said to be “locally Euclidean”, or in Riemann’s terms, possess “flatness in their smallest parts”. However, Riemann suggested that the flat Euclidean infinitesimal regions might be connected up over finite regions in a manner that may give rise to non-Pythagorean (and so, non-Euclidean) distance relations. He even allowed the possibility that any deviation from flatness (i.e., curvature) in such manifolds might vary with position. Still, he recognized that if solid bodies could freely move around “without distension” (altering their size or shape), this could only be the case in manifolds of uniform (constant or zero) curvature – as on the surface of a sphere or, in the latter case, Euclidean space.

In response to Riemann, Helmholtz sought to show that Riemann’s fundamental hypothesis, namely, that the metrical relations of a manifold are characterized by the above generalized (differential) form of the Pythagorean Theorem, is no mere postulate but might be derived from facts summarizing our experience of measurements with rigid bodies and paths of light rays. In particular, Helmholtz argued that Riemann’s hypothesis could be derived from observations of the arbitrary continuous motions of rigid bodies throughout space. The physical regularities manifested by observed relations of congruence between translated rigid bodies reveal that space is homogeneous, satisfying a condition of “free mobility”. Observed satisfaction of free mobility, i.e., “(t)he independence of the congruence of rigid point-systems from place, location, and the system’s relative rotation” is then “the fact on which geometry is grounded”.⁹ What Riemann stated as a possibility, Helmholtz recognized as fact. On the other hand, Helmholtz acknowledged the notion of a perfectly rigid extended body to be an idealization, exceeding the bounds of experience and not an actual existent. This admission enabled retention of something akin to a transcendental Kantian theory of space in that the free motions of these ideal bodies are expressions of the necessary form of spatial experience, i.e., the space of external intuition.

In general, free mobility should permit construction in space of all figures licensed by the axioms of the geometry. But which axioms are these? Initially Helmholtz concluded they must be those of Euclidean geometry. However Italian geometer Eugenio Beltrami (1835–1899) quickly pointed out that the condition of free mobility does not uniquely single out Euclidean geometry since it obtains also – as Riemann had foreseen – in spaces of constant but (unlike Euclidean space) non-zero curvature. In a footnote added prior to publication, Helmholtz conceded Beltrami’s objection. In effect this meant that the familiar association (“coordination”) of paths of non-refracted light rays to Euclidean straight lines is not obligatory. Rather, in spaces of constant negative curvature (hyperbolic, or Lobachevskian geometry) or those of constant positive curvature (elliptic, or Riemannian spherical geometry), although rays of light may be deemed straight lines, they are not Euclidean straight lines – in the former, the number of parallels to a given straight line through a given point is infinite; in the latter, zero. Yet in virtue of satisfying the condition of free mobility, all these homogeneous spaces of constant curvature are imaginable, i.e., in conformity with spatial intuition, and so permit the familiar practices of measurement and geometric construction. Helmholtz summarized his position in a telling aphorism: “Space can be transcendental without the axioms being so”. Accordingly, he concluded that free mobility remains a necessary presupposition of measurement and so an a priori condition of spatiality or spatial intuition, whereas it remains an empirical question which geometry of constant curvature accurately characterizes the motions of extended rigid figures in space.

Poincaré’s geometric conventionalism targets Helmholtz’s contention that the geometrical axioms of space rest upon presumed facts regarding the free mobility of rigid bodies. Poincaré flatly insisted that the extended rigid body does not exist in reality and in geometry (though not in mechanics) is an impermissible and unphysical idealization. To the concept of distance, for example, there corresponds no perfectly rigid solid body, impervious to deformation through stress, temperature gradients, or indeed, gravitational force. Even so, granting the existence of rigid bodies for the sake of argument, he denied the premise that experiments with such bodies are capable of univocally picking out the geometric axioms characterizing physical space. Rather such experiments can provide information only regarding the mutual relation of these bodies to one another. Finally, following Norwegian mathematician Sophus Lie (1842–1899), Poincaré emphasized that the totality of free continuous motions of presumed rigid bodies has the structure of a mathematical group, now called a Lie group. To say that these motions form a group means that 1) to every translation of a rigid body in one direction there corresponds an inverse operation (a displacement of the same magnitude in the exact opposite direction), 2) that the translations of the body associatively compose (add) together, and that 3) there is an identity, a “motion” that leaves the body in situ. The rigid motions themselves can then be given an explicitly mathematical characterization in terms of the transformations of a particular Lie group acting on a space. In turn, geometrical notions measured in space (e.g., distances and angles) are magnitudes invariant under the action of the group on the space. Geometry accordingly becomes the study of a continuous (Lie) group of motions, the general notion of which Poincaré believed, essentially for reasons of natural selection, to be unconscious and pre-existing innately within us, associated with and informing the motions of our bodies in space.

In the 1880s Lie proved (the so-called Helmholtz-Lie theorem) what Riemann and Beltrami conjectured earlier, that the condition of free mobility is consistent with three distinct groups, corresponding to the three geometries of constant curvature: Euclidean (zero curvature), hyperbolic (Lobachevskian or negative curvature) and elliptical (Riemannian spherical, or positive curvature). Poincaré interpreted Lie’s result to mean that experience alone was unable to distinguish between these; hence, there should be no compulsion to choose one as solely correct. Although experiments made with bodies supposed rigid informs the choice of geometrical axioms for physical space, they do not and cannot determine that choice. Rather, if choice need be made, it is made on grounds of convenience.

Our choice is therefore not imposed by experience. It is simply guided by experience. But it remains free; we choose this geometry rather than that geometry, not because it is more true, but because it is more convenient.¹⁰

As there can be no firm empirical basis for the claim that a single geometry is true or false of physical space, geometry is not to be regarded as an experimental science. On the other hand, Poincaré thought that Euclidean geometry would always be selected on the ground of simplicity; mathematically its simplicity resides in the fact that the group of Euclidean rigid motions alone contains a proper normal subgroup comprising both free translations and rotations about a fixed point. But as one cannot say in general that a particular geometry (say, Euclidean) is true of physical space while another (say, that of Lobachevsky) is false, geometric empiricism can have no clear meaning.

How then does a conventional geometry relate to physics? Poincaré argued that if, in astronomy, the path of a light ray is taken to realize the geometrical notion of “straight line”, observation by itself cannot determine a unique geometry of space. He illustrated this claim with an example pertaining to measurement of the parallax of distant stars, i.e., the angular difference in the apparent position of a star as seen from opposite ends of the Earth’s orbit of the sun, six months apart. On account of the vastness of stellar distances in comparison to the size of the Earth’s orbit, stellar parallax are exceedingly small and difficult to measure; only in 1838 did Friedrich Bessel succeed in measuring the first stellar parallax (the star 61 Cygni). Assuming a Euclidean geometry of physical space, these values are in general quite small, below a certain threshold. Conceivably, however a parallax might be observed to be above this limit (if the geometry of space is hyperbolic or Lobachevskian) or to be negative (if it is the elliptical geometry of Riemann). Supposing observation reveals either of these non-Euclidean values, Poincaré insisted it is still possible to retain Euclidean geometry as the geometry of space by modifying the laws of physics (geometrical optics), namely, that light does not exactly propagate in a Euclidean straight line. More generally, the ideal basic concepts of geometry (e.g., “straight line”) lack any physical meaning in isolation but can be variably interpreted, depending on the content of the assumed physical laws. For this reason, he concluded, “Euclidean geometry has nothing to fear from new experiments”.¹¹

Somewhat surprisingly, neither Poincaré nor Helmholtz took into account the full generality of geometries permitted by Riemann’s theory of manifolds as viable candidates for the geometry of physical space; they restricted attention to spaces of constant curvature wherein alone free mobility is possible. But generic Riemannian geometries extend far beyond the geometries of constant curvature; in particular, they may also possess variable curvature – curvature varying from region to region, and even (smoothly) from point to point. These geometries describe spaces where rigid bodies in principle cannot exist, let alone possess free mobility. The far-sighted Riemann even anticipated the dynamical character of space-time in general relativity, conjecturing that should space be found to have variable curvature, matter might well be the cause. Apparently such ideas were too radical for both Helmholtz and Poincaré; Poincaré in particular took note of them only to deny that the Riemannian geometries of variable curvature could ever be anything more than mathematical curiosities since they cannot be characterized by deductive (synthetic) axiom systems akin to Euclid’s, that is, they “ne pourraient donc jamais être que purement analytiques” (“could never be other than purely analytic”).¹² By 1921, such conservatism regarding the possible geometries of physical space had been swept away by the advent of general relativity, a theory of variably curved space-times, and so the parameters of the dispute between geometric empiricism and conventionalism had been fundamentally transformed. This was the new situation into which Einstein sought to intervene with “Geometry and Experience”.

“Geometry and Experience” 1921

Whereas Poincaré deemed geometry to be the study of the formal properties of a particular continuous group, not an experimental science concerned with imperfect physical realizations of geometrical notions, Einstein began his essay by drawing an apparently related distinction between “axiomatic” and “practical” – i.e., applied – geometry, summarizing the difference in a pithy, oft-quoted comparison:

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.¹³

The declaration expresses the modern axiomatic conception of pure mathematics, exemplified in Hilbert’s 1899 Foundations of Geometry (Grundlagen der Geometrie), a work continually augmented through its seventh (1930) edition and translated into many languages.¹⁴ Hilbert famously considered Euclidean geometry as pertaining to “three systems of things, called points, … lines, and … planes”, primitive terms whose meanings are established purely formally by five kinds of axioms that determine what can be said about them. The axiom groups (incidence, order, congruence, parallels, and continuity) of geometry, as Einstein put it, are “free creations of the human mind”, constrained only by the meta-theoretical requirements of their completeness, mutual consistency and independence. From the outset, the primitive terms of geometry have no reference to the physical world or to the contents of spatial intuition but possess only a contextual meaning bestowed by their occurrence in the axioms and subsequent theorems. This indirect manner of specifying the meaning of primitive terms of a theory had been advanced nearly a century earlier by the French geometer Gergonne, who coined the term “implicit definition” in a paper of 1818. Hilbert gave Gergonne’s method its first completely rigorous implementation. Remarkably, in “Geometry and Experience” Einstein does not at all mention Hilbert but approvingly cites Schlick’s 1918 treatise on general epistemology for its “highly apt” characterization of axioms as “implicit definitions” of the meanings of primitive concepts (§7 of Schlick’s book pointedly refers to Hilbert’s Grundlagen).¹⁵

“Practical geometry” on the other hand is the geometry of the practicing physicist, the application of formal axiomatic geometry to the physical world. Most familiar as the Euclidean geometry of surveying and measurement, it arises in an obvious way from an assumption that solid bodies, e.g., measuring rods, relate to one another as do the ideal rigid bodies of Euclidean geometry. Statements of axiomatic geometry are then interpretable as assertions regarding the relations of physical bodies and Euclidean practical geometry is accordingly a physical science, indeed, “the most ancient branch of physics”, its assertions at best empirical truths, having approximate validity. To Poincaré’s objection that no actual physical bodies are rigid, Einstein countered that experimental practice is able to determine the physical state of a measuring bodies to sufficient accuracy to render their fiduciary metrical behavior with respect to one another free of ambiguity. The practice of spatial measurement is rooted is in this pragmatic assumption and so “all linear measurement in physics (including ‘geodetic and astronomical measurement’) is practical geometry in this sense”.

Having set out the two opposing conceptions of geometry, Einstein made a case for geometric empiricism in the general theory of relativity. The metrical relations of space-time are not a matter of convention but can be empirically established as Euclidean or non-Euclidean using the instruments of practical geometry. In particular, he observed that the (conceptual) availability of practical geometry, i.e., “the possibilities of relative situation”(Lagerungsmöglichkeiten) of practical rigid bodies, had been decisive in taking the momentous step to his theory of gravitation. Briefly alluding to the “rotating disk” thought experiment of 1912 (discussed in Chapter 6), Einstein noted that without its heuristic message, concerning the Lorentz contraction (and so, non-Euclidean behavior) of rigid rods laid along the circumference of a system uniformly rotating with respect to an inertial system, he never should have hit upon the idea that led to the general theory of relativity. As he explained, consideration of the uniformly rotating disk enabled him to see that the geometry of accelerated systems, and so gravitational fields, must be non-Euclidean.¹⁶ The crucial link was provided by the principle of equivalence: the geometry of a gravitational field (a non-inertial system) could not be Euclidean. And in fact the geometry of generic space-times (for arbitrary distributions of matter-energy) in the general theory of relativity is a non-Euclidean Riemannian geometry of variable curvature.

Nonetheless, the general theory of relativity mandated a generalization of Euclidean practical geometry. In the Riemannian context, measuring rods can be considered “practically rigid” only in “infinitesimal” (not extended) regions. The concept of an “infinitesimal rigid rod” arises by considering the notion of a “line segment”, exemplified in the extension bounded by two nearby marks on solid body. Line segments permit comparison in the obvious way by bringing together two solid bodies, on each of which is a bounded line segment. Segments are said to be equal just in case the boundaries of one segment exactly coincide with the boundaries of the other. On the plainly self-evident nature of this claim, another is piggy-backed that the unwary reader may regard as equally self-evident:

If two line segments are found to be equal at one time and at some place, they are equal always and everywhere.¹⁷

Practically rigid rods (of “infinitesimal” extension) physically implement this supposition about equality of line segments. But any assumption of permanent congruence is contentious, though Einstein pointed out that Riemannian practical geometry follows Euclidean practical geometry in adopting it.

A practical geometry suitable for measurements in the theory of relativity must include not only “practically rigid” infinitesimal rods but also “perfect” or ideal clocks. Abstractly considered, in general relativity a clock is simply a periodic physical process by which numbers are assigned to events on the world line of the clock in such a manner that the number of “ticks” of the clock between two events on its world line is directly proportional to the extension of world line between the events. To say that such clocks are “perfect” is to make an assumption analogous to that above for equality of line segments. Namely, if two clocks are found to tick at the same rate at a common initial location but then are separated, they will be found to tick at the same rate when they are brought together again. Einstein asserted “a convincing experimental proof” of this assumption (let us call it constant synchrony) is found in the characteristic frequencies of “natural clocks”, i.e., the discrete pattern of spectral lines uniquely characterizing a chemical element when heated to incandescence, no matter where the element is. The spectrum of hydrogen is, for example, the same in the laboratory as when observed in the light from distant stars (with adjustment for redshifts occasioned by the expansion of the universe, confirmed only some years later in the 1920s).

As underscored by the logical empiricists, with the supposition of “practically rigid bodies” (and “perfect clocks”), the metrical relations of physical space (space-time) are not a matter of conventional stipulation but can be empirically determined from measurements of distances, angles, and durations; in this way, a clear decision can be reached regarding the particular character of the geometry of space-time. The known empirical tests of general relativity rest upon the assumptions of Riemannian practical geometry, in particular the association of the line element ds with measurements made using “practically rigid bodies” and “perfect clocks”. These implements serve as a bridge coordinating the phenomena of gravitational mechanics to a purely mathematical non-Euclidean Riemannian geometry. Einstein summarized the general viewpoint in this way:

To be able to make such assertions (concerning the behavior of real objects), geometry must be stripped of its merely formal-logical character by assigning to the empty conceptual schemata of axiomatic geometry objects of reality that are capable of being experienced.¹⁸

As noted above, logical empiricists regarded this conclusion a canonical expression of their own methodology concerning how the mathematical structures of a physical theory acquire empirical meaning.

Sub specie aeterni

Having argued for an empiricist conception of the geometry of space-time resting upon the “practical geometry” of rods and clocks he implicitly (but elsewhere explicitly) associated with Helmholtz, Einstein proceeded to consider the alternative resting on an objection that actually rigid bodies (even infinitesimal ones) and perfect clocks do not exist while pure, or axiomatic, geometry by itself affirms nothing about the behavior of actual bodies. According to this “more general conception, which characterizes Poincaré’s standpoint”, only the combined system (G+P) of geometry (G) plus physical laws (P) may admit of empirical test. Then G, a particular geometry of space-time, may be conventionally chosen as well as parts of P; however, the remainder of physical laws P (above all, the laws of optics) must be such to ensure that the total system (G+P) is brought into agreement with experience. Somewhat surprisingly in view of the just presented case for empiricism, Einstein conceded that only choice of a total system (G+P) could be considered correct sub specie aeterni.¹⁹ The logical empiricists chose to simply ignore this apparent concession to conventionalism. What is going on? How can Einstein give back with his left hand what his right hand has taken away? Is admission of the correctness sub specie aeterni of the French mathematician’s standpoint on the conventionalist relation between geometry and experience another manifestation of Einstein’s unscrupulous philosophical opportunism?

So many have thought. The puzzle is amplified by philosophers who maintain that the (G+P) formula is better identified with the conventionalism of Poincaré’s contemporary, Pierre Duhem (1861–1916) than that of Poincaré himself.²⁰ Duhem famously argued that a physical theory comprises an entire group of hypotheses in which no single hypothesis considered by itself has observational consequences of its own. A direct implication of this holist “non-separability thesis” is that empirical confirmation or disconfirmation (falsification) pertains only to the theory – an entire collection of hypotheses – as a whole. More precisely, the empirical test of a physical theory implicates the class of hypotheses with an unsettling consequence that, should observation fail to agree with predictions derived from the theory, the only immediate conclusion to be drawn is that at least one hypothesis in the collection requires modification, though said experiment does not tell which. As is the case with other theoretical physicists of the period, there is textual evidence that Einstein held such a conformational holism regarding the test of a physical theory. In fact, there was broad recognition of this thesis, which is almost obvious once physical theories are considered similar to formal axiomatic systems. Moreover, Einstein mentioned Poincaré not Duhem.²¹ Of greater relevance is the fact that Einstein evinced a critically negative attitude, most forcefully in his April 1918 tribute to Planck, toward Duhem’s concomitant thesis of underdetermination of theory by empirical evidence (see Chapter 9). But perhaps above all, it is significant that in “Geometry and Experience” Einstein does not state that the (G+P) holism is correct (as presumably would be the case if he had in mind a Duhemian non-separability thesis pertaining to empirical and semantic content), but only that it is correct sub specie aeterni. That crucial modifying phrase suggests that Einstein’s invocation of Poincaré’s name (not Duhem’s), is both intended and fully appropriate. For the salient issue with Poincaré concerns the standing of the assumptions of practical geometry that, Einstein admitted, do not survive scrutiny sub specie aeterni. Einstein’s choice of the Latin expression is revealing, almost certainly an allusion to Spinoza, with whom the phrases sub specie aeterni (“under the aspect of the eternal”) or sub specie aeternitatis (“under the aspect of eternity”) are familiarly associated. As he had been also for Schopenhauer, Nietzsche, and Wittgenstein, all of whom similarly borrowed the Latin phrase for their own purposes, Spinoza was an intellectual hero of Einstein.²²

Now Einstein’s concession that only the sum G+P admits of empirical test sub specie aeterni is surely not an appeal to the infallible intuition of rational insight. Nor is it an endorsement of Poincaré’s own position regarding a conventional choice of geometry. But it is an explicit admission that the physical assumptions (P) of “practical geometry”, posits concerning the behavior of measuring instruments considered fiduciary (permanent congruence, constant synchrony), are impermissible sub specie aeterni. The surrounding text shows two reasons for this. First, Poincaré’s staunch unwillingness to consider any physical object as “geometrical”, i.e., as ideal, is in principle correct. Such an admission might be thought to be the voice of logical conscience but it has, as will be seen, far-reaching implications for the foundations of physics. Secondly, Einstein has another target in his sights, contrasting the existing standpoint of physical theory with a distantly future, presumably far more complete, point of view. The imperfect state of physical knowledge at the time (P) did not (and does not today) furnish a sufficiently detailed microphysical understanding of bodies to permit derivation of such macroscopic properties as solidity and rigidity. More precisely, the current state of physical knowledge did not permit derivation of the above two assumptions of practical geometry. This is the voice of a pragmatic thinker, expressing a readiness to tolerate (as Weyl, see below, did not) a conceptual blemish in an otherwise aesthetically pleasing and (most importantly) empirically successful theory. The two reasons are clearly distinguished in this passage from “Geometry and Experience”:

The concept of the measuring body as well as in the theory of relativity also the coordinate concept of a measure clock find no exact corresponding object in the real world. It is also clear that the rigid body and the clock do not play the role of irreducible elements in the conceptual edifice of physics, but that of composite structures, which must not play any independent part in the construction of theoretical physics. However, it is my conviction (Überzeugung) that in the present state of development of theoretical physics they must still be employed as independent concepts, for we are still quite far from such secure knowledge of atomic physics (Atomistik) to be able to provide their exact theoretical construction.²³

The first sentence simply concedes Poincaré’s point that geometrical notions have no exact counterparts or physical realizations. As Einstein well knew, there are reasons of principle why there can be no extended rigid bodies already in the special theory of relativity. The invariance of the speed of light, restricting the velocity of possible physical influences, in effect meant that no actual physical body could be perfectly rigid. The second sentence affirms a consideration of more relevance to the general theory of relativity: to accord rods and clocks the status of “irreducible elements” is to place them outside, and so independent of, the laws of gravitation pertaining to all other physical objects, as well as of dynamical laws of matter that might explain the behavior of such complicated structures. Yet the last sentence of the passage affirms a “conviction” that nonetheless in the existing imperfect state of theoretical knowledge, it is still reasonable to consider rods and clocks as de facto independent of these laws. A conviction is typically a matter of faith or belief; later on it even becomes a “sin”. What in “the present state of development of theoretical physics” elicits statement of such a conviction? The answer appears several paragraphs further along, where Einstein offers a justification for the concept of an ideal clock in the theory of relativity:

If two ideal clocks are going at the same rate at any time and at any place (being then in immediate proximity to each other), they will always go at the same rate, no matter where and when they are compared with each other at one place. If this law were not valid for natural clocks, the characteristic frequencies of individual atoms of the same chemical element would not be in such exact agreement as experience demonstrates. The existence of sharp spectral lines is a convincing (überzeugenden) experimental proof of the above-mentioned basic principle of practical geometry. On this ultimately rests the fact that we can meaningfully speak of a metric of the four-dimensional space-time continuum in the sense of Riemann.²⁴

The “conviction” allowing de facto independence of ideal clocks from fundamental theory is then licensed by the “convincing experimental proof” concerning the sharp spectral lines of the chemical elements. The target of this particular argument is not Poincaré but the mathematician Hermann Weyl with whom, in 1921, Einstein was engaged in an ongoing disagreement.

Weyl’s theory of gravitation and electromagnetism

In the spring of 1918, mathematician Hermann Weyl raised an objection of principle to the pragmatic assumptions of Einstein’s practical geometry. Weyl’s geometrical theory of gravitation and electromagnetism essentially rests upon a generalization of the Riemannian geometry of the theory of general relativity. It is highly mathematically sophisticated, termed “the most beautiful of all theories” by the physicist Paul Dirac many years later. But its fundamental idea is both simple and relevant to the argument of “Geometry and Experience”. Recall that vectors have two properties, magnitude and direction. Consider two points p and q at finite separation in a space or manifold. The metric of a Riemannian geometry (and so of general relativity) permits direct comparison of magnitudes of two vectors A at p and B at q but not with respect to directions. In 1917 Italian geometer Tullio Levi-Civita (1873–1941) analytically showed how directional comparisons can be made between vectors at finitely separated points in a Riemannian space. This can be visualized as the “parallel transport” of a fiduciary vector C parallel to A at p along a continuous path from p to q so that at each successive point along the path, C remains parallel to itself at the previous point. At q the angle difference, if any, between B and the transported standard C can be determined. In a curved manifold the result of such direction comparisons will depend on the path taken from p to q. But the fact is that in Riemannian geometry, comparisons of magnitude and direction are treated quite differently. To eliminate this last vestige of “action at a distance” from field physics, Weyl constructed a non-Riemannian geometry to remove the asymmetry: both magnitude and direction comparisons require a “transported” standard. The result, to Weyl’s surprise, was a metric for space-time geometry with degrees of freedom that incorporated not only gravitation (Einstein’s field equations of general relativity) but also electromagnetism. As these forces were the only fundamental interactions known in 1918, Weyl could claim that according to his theory all physics is at base a manifestation of his space-time geometry.

In a bit more detail, Weyl constructed his theory to expressly deny the Einstein-Riemann assumption about the permissibility of distant comparisons of line segments and durations, equivalent to the existence of “practically rigid” infinitesimal rods and “ideal clocks”. In Weyl’s geometry, congruence – the primary metrical concept – is “purely infinitesimal” in that magnitudes (lengths) can be immediately compared only at neighboring points but not “at a distance”. Length comparisons between p and any point q at finite separation are not assumed as in Einstein-Riemann but must be constructed. A comparison of lengths at finitely separated points begins with a comparison at a given point p and a neighboring one p′ “infinitesimally adjacent” to p (more precisely, in the linear tangent space surrounding p). Weyl extended Levi-Civita’s idea of “parallel transport” pertaining to directional comparison to a length comparison whereby a standard of length (normalized unit vector) is “transported” from p to p′. (“Parallel transport” only figuratively involves moving a vector; the operation is defined purely analytically.) A length comparison between a vector at p and one at any distant point q is established by transporting the fiduciary standard point by point along a path p′, p″, p′″, …, q. Since there are (infinitely) many distinct paths between p and q, this procedure is clearly path-dependent, that is, the “distance” between p and q is said to be “non-integrable”. In Weyl’s “purely infinitesimal” standpoint, the twin assumptions of Einstein’s Riemannian practical geometry, pertaining to equality of lengths and of durations independently of place, are no longer permissible postulates. If, as in general relativity they are assumed, they must be understood as empirical facts to be explained by solving the combined field equations for gravitation and matter. In more generic space-times according to Weyl’s geometry, two infinitesimal measuring rods congruent at one space-time point p subsequently transported by different paths to another point q at a finite distance from the first are no longer congruent. This is just the denial of Einstein’s assumption of “practically rigid” measuring rods.

Einstein versus Weyl

Although Einstein immediately accorded Weyl’s theory the accolade of “an achievement of genius of the first rank”, from its inception in early 1918, he inveighed against it, both in public and in private correspondence. Arranging for the initial publication of Weyl’s theory in the Proceedings of the Prussian Academy of Sciences, Einstein appended a short comment stating his belief that the theory could not be in agreement with observation. For according to Weyl’s theory, measuring rods and clocks (or the radii of atoms and their spectral frequencies) are not independent of their position in space and time but rather depend, Einstein argued, on their “prehistory”, more precisely, on the strength of the electromagnetic fields through which their world lines had passed. So if two chemical atoms together at one location are then separated, travel different paths through space-time and are subsequently reunited, they should display different line spectra if one but not the other passed through a strong electromagnetic field. But astronomical observation does not detect this difference; the spectra of all the chemical elements are sharp and do not vary in light sources distributed throughout space (though, as discovered by Hubble in the 1920s, there are Doppler-like shifts in frequency of light from stars moving away from or towards the Earth due to the expanding universe). This then is the objection, not to Poicaré, but to Weyl, reiterated in “Geometry and Experience”:

The above assumption for line elements must also hold good for intervals of clock time in the theory of relativity. Consequently this assumption may be formulated as follows: If two ideal clocks are going at the same rate at any time and at any place (being then in immediate proximity to each other), they will always go at the same rate, no matter where and when they are compared with each other at one place. If this law were not valid for natural clocks, the characteristic frequencies of individual atoms of the same chemical element would not be in such exact agreement as experience demonstrates. The existence of sharp spectral lines is a convincing experimental proof of the above-mentioned basic principle of practical geometry.²⁵

Weyl and others did not find this objection convincing; Weyl held on to his theory until in the early 1920s he became convinced that a continuum-based theory, such as his, could not account for atomic phenomena.²⁶

In September 1920, just a few months before “Geometry and Experience”, at the 86th annual meeting of German scientists at Bad Nauheim a resort spa near Munich, Einstein confronted Weyl’s theory in public, but with a distinctly different argument. The context may provide a reason for this. The meeting became infamous for the intransigent opposition to the theory of relativity expressed by certain well-placed German physicists, above all, Nobel prize winner Philip Lenard. Lenard and Nobel prize winner Johannes Stark, both experimentalists, objected in particular to general relativity, a highly abstract theory employing advanced mathematics unfamiliar to most physicists.²⁷ With this witches’ broth brewing (it came to full boil only after the Nazis seized power in 1933), Einstein in 1920 was understandably sensitive to the fact that there were then but three posed empirical tests of general relativity, only two of which had been satisfactorily met. Despite the dramatic confirming solar eclipse observations of 1919, Einstein was understandably concerned to show that the general theory of relativity had clear ties to observation (through the coordination of the metric tensor to “practically rigid bodies” and “perfect clocks”). Accordingly, he had very plausible expedient reasons for the halfway house pro tem defense of the use of “practically rigid rods”.

Responding to Weyl’s presentation at Bad Nauheim, Einstein reiterated that general relativity is empirically based on measuring-rod geometry. Weyl’s demand that the reliability of rods and clocks must be explained through derivation as solutions to a combined theory of gravitation and matter (electromagnetism) would rob general relativity “of its most solid empirical support and possibilities of confirmation”:

Temporal-spatial intervals are physically defined with the help of measuring rods and clocks. If I consider two (such) structures, then their equality is empirically independent of their prehistory. Upon this rests the possibility of coordinating a number ds to two neighboring world points. Insofar as the Weyl theory renounces this empirically-founded coordination, it robs the theory (general relativity) of its most solid empirical support and possibilities of confirmation.²⁸

This is a different kind of objection to Weyl’s theory, placing more emphasis on the empirical character of general relativity than on confronting Weyl’s theory. Reverting to an injunction Einstein will later make in Oxford, (see Chapter 10) that to know anything about the method of theoretical physics one should examine the theoretician’s deeds and not listen to his words, a more ambivalent position emerges. Despite concerns to defend the use of rods and clocks as necessary to the empirical ties of the general theory of relativity, Einstein would not be constrained by concerns of empiricist methodology in attempting to advance physical theory beyond general relativity. Already in 1923, in one of the first of his many proposals for a unified field theory, the ties of theory to experience are not even considered. In a report on this theory, which starts from a non-metrical (affine) basis, he essentially affirms that observational concerns are completely subordinated to the overriding goal of attaining a theory of greatest mathematical simplicity, a theme that will become increasingly prominent in his later years.

The search for mathematical laws that shall correspond to the laws of nature … resolves itself into the solution of the question: What are the most natural formal conditions that can be imposed upon an affine relation?²⁹

Moreover, in his “Nobel lecture” that same year, he mentioned the “deficiencies of method” in tying relativity theory to observation through the posits of rigid rods and clocks, contrasting it with the “logically purer method” of Levi-Civita, Weyl, and Eddington (who had been spurred by Weyl’s theory to present his own geometrical unification of electromagnetism and gravitation in 1921):

Certainly it would be logically more correct to begin with the whole of the laws and to apply the “stipulation of meaning” to this whole first, that is, to put the unambiguous relation to the world of experience last instead of already fulfilling it in an imperfect form for an artificially isolated part, namely, the space-time metric. We are not, however, sufficiently advanced in our knowledge of nature’s elementary laws to adopt this more perfect method without going out of our depth…. we shall see that in the most recent studies there is an attempt, based on ideas by Levi-Civita, Weyl, and Eddington, to implement that logically purer method.³⁰

Weyl was not merely being unreasonable in sticking to his guns. Einstein clearly recognized that a “logically purer” field physics denying the last vestiges of “action at a distance” must derive, not postulate, the regularities of “practically rigid” measuring rods and “ideal clocks” from underlying field dynamics. Despite his objections, Einstein had to concede that Weyl had touched upon an explanatory sore point and a logical inconsistency in the story Einstein felt compelled to tell about the connection of geometry to experience. Sub specie aeterni something similar to Weyl’s dynamical explanation is a more principled story.

Reprise: Constructive vs. principle theories

Einstein’s long trek to general relativity was spurred by the denial that an irreducibly non-dynamical entity might exist, an “absolute space” that acted in producing inertia-gravitational effects but was not in turn acted upon. Yet rigid rods and perfect clocks, independent of the dynamical laws of physics, are similarly absolute. The similarity becomes palpable in a moment’s reflection: Ideal measuring rods and perfect clocks satisfying the permanent congruence and constant synchrony assumptions yield absolute (though inertial frame-dependent) measures of lengths and durations. In the special theory of relativity, rods and clocks are absolute: Lorentz contraction of practically rigid rods and time dilation of ideal clocks in a moving frame (as determined from practically rigid rods and ideal clocks in a rest frame) are regarded as merely kinematical effects, pertaining not to dynamical changes in these bodies but to the relative motion of reference frames of observers. The permanent congruence and constant synchrony assumptions are obviously satisfied in the inertial frame in which both sets of instruments are initially together at rest, and the one in which they are subsequently rejoined. So the special theory of relativity accordingly treats rods and clocks as ideal bodies independent of the dynamical laws of physics.

General relativity, to the contrary, must also consider rods and clocks in accelerated frames; this is essentially the reason motivating Einstein’s generalization of Euclidean practical geometry of rods and clocks to a Riemannian practical geometry of infinitesimal rigid rods and clocks ticking (generating time) only along their own world line. But in the variably curved space-times of general relativity, the above assumptions guaranteeing the ideal behavior of instruments measuring space and time are considerably more problematic. One can think about it this way. Imagine the microstructure of a solid (albeit “infinitesimal”) rod to be essentially a lattice of atoms, each at constant spacing from its neighbors. Suppose the body to freely fall horizontally through a strong gravitational field. As discussed in Chapter 6, the tidal forces of that field will act differentially along the length of the rod, with the result that particles along its length will traverse non-parallel freefall trajectories. The rod will deform as the rigid connections between the atoms give way. Similarly, in a gravitational field, clocks run more slowly, the stronger the field strengths, the slower the rate of the clock. But how can infinitesimal measuring rods and perfect clocks be immune from the tidal forces of gravitation? And yet they must be if they are stipulated to satisfy the two permanence assumptions.

Recall Einstein’s 1919 distinction between theories of principle and constructive theories (see Introduction) and the superior explanatory force of the latter. The voice of logical conscience in “Geometry and Experience” allowing that “the solid body and the clock do not in the conceptual edifice of physics play the role of irreducible elements, but that of composite structures, which must not play any independent part in theoretical physics” affirms not only the physical inconsistency of exempting certain bodies (rods and clocks) from the dynamical laws of microphysics. It also accords recognition to the superior understanding and explanatory depth that a constructive account of the behavior of these bodies in the theories of relativity could provide, especially within the general theory since the permanence assumptions flagrantly conflict with fact that gravitation acts on all matter. General relativity, a principle theory, does not have the resources to account for the structure of matter; it lacks an explicit theory of matter, an acknowledged gap Einstein unsuccessfully sought to fill over a three-decades-long pursuit of a unified field theory. Correspondingly, there is occasional admission of the unsatisfactory treatment in general relativity of rods and clocks as “theoretically self-sufficient entities”:

Strictly speaking measuring rods and clocks would have to be represented as solutions of the basic equations (objects consisting of moving atomic configurations), not, as it were, as theoretically self-sufficient entities.³¹

Yet immediately following this voice of conscience there is another, that of the pragmatic physicist who has chosen “practical geometry”. It appears both in 1921:

But it is my conviction that in the present stage of development of theoretical physics these concepts (i.e., rods and clocks) must still be invoked as independent concepts, for we are still far from possessing such certain knowledge of theoretical principles as to be able to give exact theoretical constructions of such constructs (solid bodies and clocks).³²

and in 1946:

However, the procedure justifies itself because it was clear from the very beginning that the postulates of the theory are not strong enough to deduce from them sufficiently complete equations … in order to base upon such a foundation a theory of measuring rods and clocks … But one must not legalize the mentioned sin so far as to imagine that intervals are physical entities of a special type, intrinsically different from other variables (“reducing physics to geometry”, etc.).³³

Recently it has been argued that these passages, as well as the admission that Poincaré is correct sub specie aeterni, are to be understood as expressions of Einstein’s “unease” with both theories of relativity as mere “theories of principle”.³⁴ The above remarks then are interpreted as expressing Einstein’s preference for the explanatory superiority of an as yet non-existent constructive account of the macroscopic practical rigidity of rods and practical efficacy of clocks in terms of their underlying microphysical dynamics of the constituent atoms and molecules. The claim has an initial plausibility, particularly in view of Einstein’s several statements about the logical inadmissibility of explicitly assuming, as does practical geometry, rods and clocks to be objects independent of the rest of physics.

But Poincaré may be deemed correct sub specie aeterni for any entirely different reason. A clue lies in the statement quoted above when Einstein agreed in principle with Poincaré’s insistence that no actual body is capable of physically realizing the ideal notions of geometry. Inasmuch as geometry is a formal mathematical discipline, there is no pre-axiomatic understanding of geometrical primitives. Geometry in itself makes no physical assertions; in this conception, metrical concepts are postulated and geometrical primitives (straight line, etc.) are implicitly defined within abstract axiomatic structures. But of course physics (both kinematics and dynamics) presupposes such geometrical notions. So geometry is logically prior to physics. This logical hierarchy of disciplines is of particular relevance here since empirical evidence for any dynamical constructive account of the observed behavior of rods and clocks will make use of geometrical notions. More precisely, since evidence for the dynamical theory in question will be stated in terms of measurable quantities having the dimensions of a length and/or a time, a dynamical or constructive account of rods and clocks must distinguish degrees of freedom for metrical concepts such as length and duration. Poincaré may be ultimately correct for a fundamentally logical reason (What presupposes what?), and not simply as a concession to the potential explanatory superiority of constructive/dynamical treatments of the behavior of rods and clocks, an understanding that in any case does not exist even today.

A fundamental length?

At this juncture an old issue reappears; already entertained by Riemann in 1854, it was subsequently completely ignored by Helmholtz and Poincaré. It concerns the possibility that space at the smallest scales may be discrete rather than continuous. Riemann raised the matter as he considered the question of the applicability of geometry “in the infinitely small”:

Now it seems that the empirical notions on which the metric determinations of space are based, the concept of solid body and that of a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena.³⁵

Riemann’s point concerns the validity of the usual assumption that space is a continuum even down to the scale of “the immeasurably small”, or whether the continuous character of space, assumed by the differential equations of physics, may be only a course-grained approximation, a kind of averaging over an ultimately primitive discrete structure. Yet on the assumption that space is a continuum rest the hypotheses on which the geometry of Riemannian manifolds (including the group-theoretical characterization of rigid body continuous motions) as well as all of classical physics. But if space is fundamentally comprised of discrete parts, then the concept of a spatial continuum, as Mach averred, can be only an economical idea. Certainly if the nature of space (or space-time) were found to be discrete at the smallest scale, it would be an empirical discovery with momentous implications for the relation of geometry to physics. Riemann noted the problem in this way, terming “something else” just the hypotheses or postulates that he placed at the foundation of geometry of manifolds:

In a discrete manifold the principle of metric relations is already contained in the concept of the manifold, but in a continuous one it must come from something else.³⁶

To say “the principle of metric relations is already contained in the concept of a discrete manifold” means that metrical notions (such as distance, length, area, volume) ultimately rest on counting discrete units of space. And if space (and time) is fundamentally comprised of “atoms”, then it is possible to counter the Poincaré objection that no actual physical bodies or objects could serve as physical realizations of the chrono-geometrical concepts of length or duration. In a discrete space-time such notions “already contained in the concept of a discrete manifold”, in the sense that they physically correspond to the primitive elements of space and time themselves.

Should this be the case, the criterion of sub specie aeterni correctness requires modification. Recall that for Poincaré the dual core of that elevated perspective affirms both the logical priority of geometry to dynamics and the “logically purer method” of “put(ting) the unambiguous relation (of the theory) to the world of experience last instead of already fulfilling it in an imperfect form for an artificially isolated part, namely, the space-time metric”. In turn, the logical fact that any dynamics presupposes geometrical notions gives rise to a necessary consistency requirement on any constructive account of rods and clocks. In particular, chrono-geometrical notions – length, area, volume, duration, period – presupposed by the empirical measures that could verify or confirm the derived predictions of the constructive theory cannot be taken from “something else” but must be “already contained in the concept of the manifold”. But this can be the case only if space and time are fundamentally discrete.

In sum, any dynamical construction of the universal behavior of rods and clocks must begin in the “immeasurably small” by positing a fundamentally universal unit length and, corresponding to this, a fundamentally universal duration, say, the time for light to travel this length. Ultimately a constructive theory of microphysics, to explain the practical rigidity of rods and the practical reliability of ideal clocks, cannot presuppose any particular geometry (as Poincaré presupposed Euclidean geometry on grounds of simplicity) but must rest upon notions of a fundamental length and fundamental time from which the geometry of space-time is built up. In the context of a fundamental length, the quantity (G+P) that alone renders space-time geometry empirically testable has been transformed. The quantity is no longer a sum of two distinct terms, one of which may be chosen conventionally adjusting the other as needed, but a fundamental, and non-conventional, unity that is both.

In 1942, some twenty years after “Geometry and Experience”, Einstein explicitly underscored this consistency requirement on constructive theories in a letter to W.F.G. Swann, a British physicist working in the USA who had proposed a quantum dynamical account of rods and clocks in the special theory of relativity:

If one does NOT introduce rods and clocks as independent objects into the theory, then one has to have a structural theory in which a length is fundamental, which then leads to the existence of solutions in which that length plays a determinant (constitutive) role, so that a continuous sequence of similar solutions no longer exists.³⁷

Einstein was well aware that the quantum theory had only a problematic relation to the space-time continuum, and for that reason a “purely algebraic physics” might be adopted.³⁸ However, should there be a fundamental microphysical length (as was independently proposed around that time by Heisenberg in an attempt to circumvent the infinities appearing in solutions of the equations of quantum electrodynamics), the continuum-based physics of the general theory of relativity, and all of Einstein’s attempts to build a unified theory of fields by generalizing upon it, cannot be considered fundamental. Einstein explicitly recognized this possibility in the conclusion of the last of his many letters to his old friend Michele Besso on August 10, 1954:

I certainly admit it is perfectly possible that physics cannot be based upon the notion of the field, that is to say, on continuous elements. But then practically nothing would remain of all my scaffolding – including the theory of gravitation – and also of existing physics.³⁹

In fact physics cannot be based upon Einstein’s notion of a field. This judgment is now orthodoxy in contemporary physics as the result of the dialectical sequence of physical theories from Einstein to the present. Relativity theory (including all classical physics) presupposes a spatial-temporal continuum; classical physics led to quantum physics; quantum physics probed into nether regions of higher and higher energies and smaller and smaller scales, led to theoretical postulate of the fundamental discreteness of space and time to which the geometry of continuous space-time may be a low energy approximation. Today both string theories as well as theories of quantum gravity view space-time at the smallest physically possible scales – of Planck length (10^-33 m) and Planck time (10^-43 sec.), (the time taken for light to travel a Planck length) – as something discrete, perhaps a “quantum foam”. What remains of Einstein’s vision of a unified fundamental theory is perhaps only his tempered belief in the method of mathematical speculation (see Chapter 10).

Summary

Einstein’s admission in 1921 of the sub specie aeterni correctness of Poincaré’s point of view is not a concession to conventionalism, that any geometry may be adopted as a matter of convenience only. It is rather a recognition of the inevitable holist character of any theory that, as a matter of principle, is capable of explaining its own measuring appliances, and so its ties to observation. Poincaré’s position is valued for an unwillingness to consider actual physical objects as “geometrical”, that is, as ideal bodies, and thus as independent of the dynamical laws to which all matter is subject. This is the perspective of a “consistent field theory”. Without a completed theory of matter, however, the adoption of “practical geometry” is a pragmatic way forward to such a theory.

Notes

1Letter of Gösta Mittag-Leffler to Einstein, dated December 16, 1919 (CPAE v. 7, Doc. 218, pp. 308–9; Einstein’s reply to Mittag-Leffler in Stockholm, a one page typed letter copy, noted in “Calendar 1920”, entry for April 12; CPAE, vol. 7, p. 611.

2“Geometrie und Erfahrung”, CPAE 7 (2002), Doc. 52, pp. 383–402; English translation supplement, pp. 208–22, also in Pesic, Peter, Beyond Geometry. New York: Dover Publications, 2007, pp. 147–57.

3As reprinted in Stadler, Friedrich, and Thomas Uebel (eds.) (2012): Wissenschaftliche Weltauffassung. Der Wiener Kreis. Reprint of the first edition on behalf of the Institute Vienna Circle on the occasion of its 20th anniversary. Wien, New York: Springer. Originally published by Arthur Wolf Verlag, Wien, 1929, p. 108.

4Frank, Phillip, Einstein: His Life and Times. New York: Alfred Knopf, 1947; fourth printing January 1953, p. 177.

5Quine, Willard Van Orman, “Two Dogmas of Empiricism”, The Philosophical Review v. 60, no. 1 (January 1951), pp. 20–43.

6Carnap, Rudolf, “Eigentliche und uneigentliche Begriffe”, Symposion Bd. I, Heft 4 (1927), 355–74, p. 373.

7Poincaré, Henri, La Science et L’Hypothèse. Paris: Flammarion, 1968, p. 101; translation by G.B. Halsted, The Foundations of Science, p. 86.

8Pesic, Beyond Geometry, contains translations of Helmholtz (1866) pp. 47–52, and Riemann (1854), pp. 23–40.

9 Ibid., 2007, p. 50.

10Poincaré, “On the Foundations of Geometry”, The Monist (1898), as reprinted in Pesic, Beyond Geometry, pp. 117–46; p. 145.

11Poincaré, La Science et L’Hypothèse, p. 96; translation by G.B. Halsted, The Foundations of Science, p. 81.

12Ibid., p. 63.

13CPAE 7 (2002), Doc. 52, pp. 385–6; Pesic, Beyond Geometry, p. 147.

14Hilbert, David, Grundlagen der Geometrie: Siebente Auflage. Leipzig und Berlin: B.G. Teubner, 1930.

15Schlick, Moritz, Allgemeine Erkenntnistheorie. Berlin: J. Springer, 1918.

16CPAE 7 (2002), Doc. 52, p. 388; Pesic op. cit. (2007), pp. 148–9.

17Ibid., p. 391; Pesic, Beyond Geometry, p. 150.

18Ibid., pp. 387–88; Pesic, Beyond Geometry, p. 148.

19Ibid., p 390; Pesic, Beyond Geometry, p. 149.

20Most recently, Howard, Don, “Einstein and Duhem”, Synthese v. 83 (1990), pp. 363–84.

21There is only indirect evidence that Einstein actually read Duhem’s well-known discussion of the non-separability thesis in Aim and Structure of Physical Theory; see Howard, “Einstein and Duhem”, pp. 368–9.

22During a stay at Leiden less than three months earlier, on the occasion of delivering his inaugural lecture “Ether and the Theory of Relativity”, Einstein used the opportunity to visit the house in which Spinoza lived from 1660–1663 in nearby Rijnsburg, signing the visitors’ book on November 2, 1920. Sometime in 1920, quite possibly triggered by the emotion of that visit, he composed a poem Zu Spinozas Ethik whose first line is “How much do I love that noble man”. (See van Delft, Dirk, “Albert Einstein in Leiden”, Physics Today, April 2006, pp. 57–62).
 A glance at Spinoza’s Ethics, completed in Latin by 1675 but only posthumously published (first in Dutch) after Spinoza’s death in 1677, may illuminate the significance or purpose the qualifying Latin phrase may have had for Einstein. Spinoza sharply distinguished between knowledge based on sense experience (things as they appear to a person from a particular perspective at a particular time) and rational knowledge that reveals the essences of things, according to the nature of reason to regard things as necessary, not as contingent. Sensory knowledge arises “from fortuitous experience” (“experientia vaga”). As sensations correspond both to the nature of external bodies and also to the subjective conditions of the perceiver, sensory experience alone can lead to falsehood and error. Discursive inferential knowledge furnished by reason stands in sharp contrast to sensory knowledge. This rational knowledge is capable of providing adequate ideas of things; the model for such knowledge appears to have been knowledge of necessary truths of mathematics. As applied to nature, rational knowledge must aspire to encompass the causal nexus that necessarily makes a thing what it is. For to have an adequate idea of a thing is to perceive the necessity inherent in nature: not just to know that a thing is, but how it is and why it could not be other than what it is. In later years, Einstein will characterize the intellectual hubris of the theoretical physicist in just this way. Nonetheless, adequate knowledge acquired in this manner requires both concepts and inference and, in this regard, is subordinate to the third and highest level of knowledge, direct intuitive knowledge (“scientia intuitiva”). This type of knowledge yields the adequate idea of a thing conveyed by an immediate, intuitive grasp of how the thing follows necessarily from the attributes of God. All other knowledge is judged an inferior approximation to this superior form exemplifying “the nature of Reason to perceive things under a certain aspect of eternity” (“sub quondam specie aeternitatis”), and so without relation to time. It is the mind’s greatest striving to attain this manner of conception of things, aspiring to an ideal of understanding that is also the mind’s greatest virtue (Ethics 5 p. 25). For to the extent the mind has knowledge of itself and of body under a species of eternity, it has knowledge of God (Ethics 5 p. 30), and so knowledge of the eternal necessity of God’s essence. At this point, however, we must recall that in the original Latin text (but not the Dutch translation) of the Ethics, Spinoza heretically stated (Ethics 4 Preface): “That eternal and infinite being we call God, or Nature (Deus, sive Natura), acts from the same necessity from which he exists”. And, in a string of propositions towards the end of the Ethics, passages that surely appealed to Einstein, Spinoza contrasted the man who is led by opinion or by passions (a “slave”) with the “free man … who lives according to the dictate of reason alone” (Ethics 4 p. 67). The free man is one who becomes progressively detached from the transient interests of the individual in a particular environment and, in proportion to the increase of his knowledge, seeks to view all things sub specie aeternitatis (“under the aspect of eternity”). Through contemplation of the necessary system of nature made possible by this third kind of knowledge, such a man can attain the greatest happiness (beatitudo) (Ethics 5, p. 27).

23“Geometry and Experience”, CPAE 7 (2002), English translation supplement, Doc. 52, pp. 208–22; p. 213; Pesic, Beyond Geometry, pp. 147–57; pp. 149–50.

24Ibid., pp. 391–2; Pesic, Beyond Geometry, p. 150.

25Ibid., pp. 391–2; Pesic, Beyond Geometry, p. 150.

26Weyl, Hermann, “Gravitation und Elektrizität”, Preußischen Akademie der Wissenschaften (Berlin) Sitzungsberichte. Physikalisch-Mathatische Klasse, pp. 465–78; 478–80; translation in L. O’Raifeartaigh, The Dawning of Gauge Theory. Princeton Series in Physics. Princeton, NJ: Princeton University Press, 1997, pp. 24–37. Einstein, “Nachtrag”, Ibid., p. 478, reprinted in CPAE 7 (2002), Doc. 8, p. 61. On the episode and its consequences, see Thomas Ryckman, The Reign of Relativity: Philosophy in Physics 1915–1925. New York: Oxford University Press, 2005.

27Following Lenard and Stark (who would become members of the Nazi party), anti-Semites regarded relativity theory a violation of Aryan “healthy common sense” and a prime example of an abstract formal contamination inflicted on physics by the non-Aryan mind.

28“Discussions of Lectures in Bad Nauheim”, Physikalische Zeitschrift v. 21 (1920), p. 651, as reprinted in CPAE 7 (2002), Doc. 46, p. 352.

29Einstein, “The Theory of the Affine Field”, Nature v. 112 (1923), pp. 448–9; p. 448.

30 Einstein, “Fundamental Ideas and Problems of the Theory of Relativity”, in Nobel Lectures in Physics, 1901–1921. Amsterdam, London, and New York: Elsevier Publishing Co., 1967, pp. 482–90; pp. 483–4.

31“Autobiographical Remarks”, p. 59; p. 61.

32CPAE 7 (2002), Doc. 52, p. 390; Pesic, Beyond Geometry, pp. 149–50.

33“Autobiographical Notes”, p. 58; p. 59.

34Brown, Harvey R., Physical Relativity: Space-Time Structure from a Dynamical Perspective. Oxford, UK: Clarendon Press, 2005.

35Riemann, Bernard, “On the Hypotheses That Lie at the Foundation of Geometry”, as translated in Pesic, Beyond Geometry, pp. 23–40; p. 32.

36Pesic, Beyond Geometry, p. 33.

37Einstein to W.F.G. Swann, January 24, 1942 (EA 20–624); the original German and translation in Amit Hagar, Discrete or Continuous? The Quest for Fundamental Length in Modern Physics. New York: Cambridge University Press, 2014, pp. 153–5.

38See Stachel, John, “The Other Einstein: Einstein Contra Field Theory”, Science in Context v. 6 (1993), pp. 275–90.

39Speziali, Pierre (ed.), Albert Einstein Michele Besso Correspondance 1903–1955. Paris: Hermann, 1979, Doc. 210, pp. 305–7; p. 307.

Further reading

Brown, Harvey R., Physical Relativity: Space-Time Structure from a Dynamical Point of View. New York: Oxford University Press, 2005.

Hagar, Amit, Discrete or Continuous? The Quest for Fundamental Length in Modern Physics. New York: Cambridge University Press, 2014.

Ryckman, Thomas, The Reign of Relativity: Philosophy in Physics 1915–1925. New York: Oxford University Press, 2005.

Weyl, Hermann, The Philosophy of Mathematics and Natural Science. Reprint of the 1949 edition. Princeton, NJ: Princeton University Press, 2009.