Early Philosophical Interpretations of General Relativity
Each of the following philosophical interpretations of general
relativity selected certain aspects of that theory for favored
recognition. While followers of Mach lauded Einstein's attempt
to implement a "relativization of inertia" in the general theory,
they were much more comfortable with Einstein's operationalist
treatment of concepts in the special theory. Kantians and
neo-Kantians, if freed from strict fealty to the doctrine of the
Transcendental Aesthetic, pointed to the surpassing importance of
certain synthetic "intellectual forms" in the general theory, such as
the principle of general covariance. For logical empiricism, the
philosophical significance of relativity theory was largely
methodological, that conventions must first be laid down in order to
express the empirical content of a physical theory. Finally, within
a few years of its completion in 1915, attempts were made to extend
general relativity's "geometrization" of gravitation to
non-gravitational fields. The first of these, by Weyl, and shortly
thereafter by Eddington, may be distinguished from others, in
particular the many attempts of Einstein, in that they aimed not at a
unified field theory, in the sense of a completely geometrical field
theory of all fundamental interactions, but at reconstructing general
relativity from the epistemological perspectives of transcendental
idealism.
Extraordinary public clamor greeted an announcement of the joint
meeting of the Royal Society of London and the Royal Astronomical
Society on the 6th of November, 1919. To within acceptable margin of
error, astronomical observations during the solar eclipse the
previous May 29^{th} revealed the displacement of starlight
passing near the surface of the sun predicted by Einstein's
gravitational theory of curved spacetime. By dint of having
"overthrown" such a permanent fixture of the cognitive landscape as
Newtonian gravitational theory, the general theory of relativity at
once became a principal focus of philosophical interest and inquiry.
Although some physicists and philosophers initially opposed it,
mostly on non-physical grounds, surveyed here are the principal
philosophical interpretations of the theory accepting it as a
definite advance in physical knowledge. Even so, these include
positions ill-informed as to the mathematics and physics of the
theory. Further lack of clarity stemmed from the scientific
literati who provided differing, and at times, conflicting
mathematical or physical accounts of the theory's fundamental
principles. These are: the principles of equivalence, of general
relativity, of general covariance, and finally what Einstein termed
"Mach's Principle" of the complete relativization of inertia. In
one or another form, all of these controversies have continued into
the present literature of physics and philosophy of physics. (See
e.g., Ohanian (1977); Norton (1993); Friedman (1983); Barbour and
Pfister (1995).) This is not unusual: physical theories, if
sufficiently robust, are rarely, if ever, without unproblematic
aspects, often taken to say different things at different stages of
development. But the very fluidity of physical and mathematical
meaning lent interpretative latitude for inherently antagonistic
philosophical viewpoints seeking vindication, confirmation or
illumination by the revolutionary new theory. Perhaps only
semi-facetiously, Russell (1926, 331) observed that
There has been a tendency, not uncommon in the case of a new
scientific theory, for every philosopher to interpret the work of
Einstein in accordance with his own metaphysical system, and to
suggest that the outcome is a great accession of strength to the
views which the philosopher in question previously held. This cannot
be true in all cases; and it may be hoped that it is true in none. It
would be disappointing if so fundamental a change as Einstein has
introduced involved no philosophical novelty.
It cannot be denied that general relativity proved a considerable
stimulus to "philosophical novelty". But then the question as to
whether it particularly supported any one line of philosophical
interpretation over another also must take into account the fact that
schools of interpretation in turn "evolved" to accommodate what were
regarded as its philosophically salient features. A classic instance
of this is the assertion, to become a cornerstone of logical
empiricism, that relativity theory had shown the untenability of any
"philosophy of the synthetic a priori", despite the fact
that early works on relativity theory by both Reichenbach and Carnap
were written from within that broad perspective. It will be seen
that, however ideologically useful, this claim by no means "follows"
from relativity theory although, as physicist Max von Laue noted in
his early text on general relativity (1921, 42), "not every sentence
of The Critique of Pure Reason" might still be held
intact. What does "follow" from scrutiny of the various philosophical
appropriations of general relativity is rather a consummate
illustration that, due to the evolution and mutual interplay of
physical, mathematical and philosophical understandings of a
revolutionary physical theory, significant "philosophical
interpretations" often are works in progress, extending over many
years.
2.1 In the Early Einstein
In 1912, Einstein's name, together with those of the
Göttingen mathematicians David Hilbert and Felix Klein, was
prominently displayed (in the Naturwissenschaftliche
Rundschau 27, 336) among those joining Mach's in a call for
the formation of a "Society for Positivist Philosophy". Citing the
pressing need of science "but also of our age in general" for a
"comprehensive world view based on the material facts accumulated in
the individual sciences", the appeal appears above all to have been
an orchestrated attempt to buttress Mach's positivist conception
of science in the face of recent realist criticisms of Mach by Max
Planck, then Germany's leading theoretical physicist. More a
declaration of allegiance than an act of scholarly neutrality, it
provides but further evidence of Einstein's youthful enthusiasm
for Mach's writings. Late in life (1949a, 21), Einstein wrote of
the "profound influence" that Mach's Science of
Mechanics (1883) exercised upon him as a student as
well as of the "very great influence" in his "younger years" of
"Mach's epistemological position". Indeed, in first decade or so
of relativity theory, these influences are highly visible. Already in
the special theory of relativity (1905), Einstein's operational
definition of the "simultaneity" of distantly separated events,
whereby clocks are synchronized by sending and receiving light
signals, is closely modeled on the operational definition of "mass"
in Mach's Mechanics. Moreover, occasional
epistemological and methodological pronouncements indicated a broad
consensus with core parts of Mach's epistemology of science,
e.g., "The concept does not exist for the physicist until he has the
possibility of discovering whether or not it is fulfilled in an
actual case" (1917a/1955, 22). Thus relativity theory was widely
viewed as fully compliant with Mach's characterization of
theoretical concepts as merely economical shorthand for concrete
observations or operations.
2.2 A "Relativization of Inertia"?
Machian influences specific to the general theory of relativity
appeared even more extensive. In papers leading up to the definitive
presentation of the general theory of relativity in 1916, Einstein
made no secret of the fact that Mach had been the inspiration for his
epistemologically mandated generalization of the principle of
relativity. Holding, with Mach, that no observable facts could be
associated with the notions of "absolute acceleration" or "absolute
inertia" (i.e., resistance to acceleration), the generalization
mandated that the laws of nature be completely independent of the
state of motion of any chosen reference system. On Mach's death,
Einstein wrote, in a warm obituary, of how close Mach himself had
been, years before, to demanding a general theory of relativity,
quoting extensively from the famous passages in the latter parts of
the Mechanics critical of Newton's "absolute" concepts
of space, time and motion (1916b, 102-3). With this reference in
mind, the physicist Phillip Frank, later to be associated with the
Vienna Circle, observed (1917/1949, 68) that "it is universally
known today that Einstein's general theory of relativity grew
immediately out of the positivistic doctrine of space and motion". In
fact, there are both genuine and spurious Machian motivations
connected with Einstein's principle of general relativity, a
mixture complicated by Einstein's own puzzling remarks regarding
the principle of general covariance.
2.3 Positivism and the "Hole Argument"
A passage from §3 of Einstein's first complete exposition
of the general theory of relativity (1916a) appeared to provide
further grist for the mill of Machian positivism. There Einstein
grandly declared that his requirement of general covariance for the
gravitational field equations (i.e., that they remain unchanged under
arbitrary, but suitably continuous, transformation of the spacetime
coordinates), "takes away from space and time the last remnant of
physical objectivity". An accompanying heuristic "reflection" on the
reasoning behind this claim seemed nothing less than an endorsement
of Mach's phenomenalism. "All our space-time verifications",
Einstein wrote, "invariably amount to a determination of space-time
coincidences....". This is because, Einstein presumed, all results of
physical measurement ultimately amount to verifications of such
coincidences, such as the observation of the coincidence of the
second hand of a clock with a mark on its dial. Observing that such
(topological) relations alone are preserved under arbitrary
coordinate transformation, Einstein concluded that "all our physical
experience can ultimately be reduced to such coincidences". To
Mach's followers, Einstein's illustrative reflection was
nothing less than an explicit avowal of the centerpiece of
Mach's phenomenalist epistemology, that sensations
(Empfindungen), directly experienced sensory perceptions,
alone are real and knowable. Thus Josef Petzoldt, a Machian
philosopher and editor of the 8^{th} edition of Mach's
Mechanics , the first to appear after the general theory of
relativity, noted that Einstein's remarks meant that the theory
"rests, in the end, on the perception of the coincidence of
sensations" and so "is fully in accord with Mach's world-view,
which is best characterized as relativistic positivism" (1921, 516).
However, contemporary scholarship has shown that Einstein's
remarks here were but elliptical references to an argument (the
so-called "Hole Argument") that has only fully been reconstructed
from his private correspondence. Its conclusion is that, if a theory
is generally covariant, the points of the spacetime manifold can have
no inherent primitive individuality (inherited say, from the
underlying topology), and so no reality independent of physical
fields (Stachel (1980); Norton (1984), (1993)). Thus for a generally
covariant theory, no physical reality accrues to "empty space" (or
"spacetime") in the absence of physical fields. This means that the
spacetime coordinates are nothing more than arbitrary labels for the
identification of physical events, or, with rhetorical embellishment,
that space and time have lost "the last remnant of physical
objectivity". Hence this passage was not an endorsement of positivist
phenomenalism.
2.4 "Mach's Principle"
To be sure, for a number of years Einstein expressed the ambition of
the general theory of relativity to fully implement Mach's
program for the relativization of all inertial effects, even
appending the so-called "cosmological constant" to his field
equations (1917b) for this purpose. This real point of contact of
Mach's influence was clearly identified only in 1918, when
Einstein distinguished what he baptized as "Mach's Principle"
(roughly, that inertial effects stem from an interaction of bodies)
from the principle of general relativity which he now interpreted as
the principle of general covariance. Taken together with the
principle of the equivalence, Einstein asserted that the three
principles, were three "points of view" on which his theory rested,
even if they could not be thought completely independent of one
another. Despite Einstein's intent, there is considerable
disagreement about the extent to which, if at all, general relativity
conforms to "Mach's Principle". In part this is due to vagaries
regarding what the Principle actually asserts and then again, to
difficulties in comprehending what physical mechanism might implement
the Principle, however interpreted. How, for instance, could a
body's inertial mass be accounted due to the influence of all
other bodies in the universe? (See the discussions in Barbour and
Pfister (1995)).
2.5 An Emerging Anti-Positivism
As Einstein's principal research activity turned, after 1919,
to the pursuit of a geometrical "unified theory of fields", his
philosophical pronouncements increasingly took on a realist or at
least anti-positivist coloration. Already in (1922, 28) lecturing at
the Sorbonne, Einstein pronounced Mach "un bon
mécanicien" (no doubt in reference to Mach's views
of the relativity of inertia) but "un déplorable
philosophe". Increasingly, Einstein's retrospective
portrayals of the genesis of general relativity centered almost
entirely on considerations of mathematical aesthetics (see Norton
(2000) and §5). On the other hand, positivists and
operationalists alike adopted the Einstein analysis of simultaneity
as relativity theory's fundamental methodological feature. One,
ruefully noting the difficulty of giving an operationalist analysis
of the general theory, even suggested that the requirement of general
covariance "conceals the possibility of disaster" (Bridgman (1949),
354). Finally there was, for Einstein, an understandable awkwardness
in learning of Mach's surprising disavowal of any role as
forerunner to relativity theory in the "Preface", dated 1913, to his
posthumous book (1921) on physical optics, published by Mach's
son Ludwig. Though Einstein died without knowing differently, a
recent investigation has built a strong case that this statement was
forged after Mach's death by his son Ludwig, under the influence
of a rival guardian of Mach's legacy and opponent of relativity
theory, the philosopher Hugo Dingler (Wolters, 1987).
3.1 Neo-Kantians on Special Relativity
In the universities of Imperial and early Weimar Germany, the
philosophy of Kant, particularly the various neo-Kantian schools,
held pride of place. Of these, the "Marburg School" of Hermann Cohen
and Paul Natorp, later Ernst Cassirer, exhibited a special interest
in the philosophy of the physical sciences and of mathematics. But
prior to the general theory of relativity (1915-1916), Kantian
philosophers accorded relativity theory only cursory attention. This
may be seen in two leading Marburg works appearing in 1910,
Cassirer's Substanzbegriff und Funktionsbegriff. and
Natorp's Die Logischen Grundlagen der Exakten
Wissenschaften. Both conform to the characteristic Marburg
modification of Kant that greatly extended the scope of
"transcendental logic", bringing under "pure thought" or
"intellectual forms" what Kant had sharply distinguished as "pure
intuition" and a conceptual faculty of understanding. Of course, this
revisionist tendency greatly transformed the meaning of Kant's
Transcendental Aesthetic and with it Kant's conviction that
space and time were "forms of sensibility" or "pure intuitions a
priori" and so as well, his accounts of arithmetic and
geometry. As will be seen, it enabled Cassirer, some ten years later,
to view even the general theory of relativity as a striking
confirmation of the fundamental tenets of transcendental idealism. In
1910, however, Cassirer's brief but diffuse discussion of "the
problem of relativity" mentions neither the principle of relativity
nor the light postulate nor the names of Einstein, Lorentz or
Minkowski. Rather it centers on the question of whether space and
time are aggregates of sense impressions or "independent intellectual
(gedankliche) forms". Having decided in favor of the latter,
Cassirer goes on to argue how and why these ideal mathematical
presuppositions are necessarily related to measurable, empirical
notions of space, time, and motion (1910, 228-9; 1923, 172-3).
Natorp's treatment, though scarcely six pages is far more
detailed (1910, 399-404). In Marburg revisionist fashion, the
"Minkowski (sic) principle of relativity" was welcomed as a
more consistent (as avoiding "Newtonian absolutism") carrying through
of the distinction between transcendentally ideal and purely
mathematical concepts of space and time and the relative
physical measures of space and time. The relativization of time
measurements, in particular, showed that Kant, shorn of the
psychologistic error of "pure intuition", had correctly maintained
that time is not an object of perception. Natorp further alleged that
from this relativization it followed that events are ordered, not in
relation to an absolute time, but as lawfully determined phenomena in
mutual temporal relation to one another. This is close to a
Leibnizian relationism about time. Similarly, the light postulate had
a two-fold significance within the Marburg conception of natural
science. On the one hand, the uniformity of the velocity of light, deemed
an empirical presupposition of all space- and
time-measurements, reminded that absolute determinations of these measures,
unattainable in empirical natural science, would require a
correspondingly absolute bound. Then again, as an upper
limiting velocity for physical processes, including gravitational
force, the light postulate eliminated the "mysterious absolutism" of
Newtonian action-at-a-distance. Natorp regarded the requirement of
invariance of laws of nature with respect to the Lorentz
transformations as "perhaps the most important result of
Minkowski's investigation". However, little is said about this,
and in fact there is some confusion regarding these transformations
and the Galilean ones they supercede (the former are seen as a "broadening
(Erweiterung) of the old supposition of the invariance of
Newtonian mechanics for a translatory or circular
(zirkuläre, emphasis added) motion of the world
coordinates"(403)). He concluded with an observation that the appearance of
non-Euclidean and multi-dimensional geometries in physics and
mathematics are to be understood only as "valuable tools in the
treatment of special problems". In themselves, they furnish no new
insight into the (transcendental) logical meaning and ground of the
purely mathematically determined concepts of space and time; still
less do they require the abandonment of these concepts.
3.2 Immunizing Strategies
Following the experimental confirmation of the general theory in
1919, few Kantians attempted to retain, unadulterated, all of the
components of Kant's epistemological views. Several examples
will suffice to indicate characteristic "immunizing" strategies (see
Hentschel (1990). The Habilitationsschrift of E. Sellien
(1919), read by Einstein in view of his criticism expressed in an
October, 1919 letter to Moritz Schlick (Howard (1984),625), declared
that Kant's views on space and time pertained solely to
"intuitive space" and so were not touched by the measurable spaces
and times of Einstein's empirical theory. The work of another
young Kantian philosopher, Ilse Schneider, personally known to
Einstein, affirmed that Kant merely had held that the space of
three-dimensional Euclidean geometry is the space in which
Newton's gravitational law is valid, whereas an analogous
situation obtains in general relativity. Furthermore,
Einstein's cosmology (1917b) of a finite but unbounded universe
could be seen as in complete accord with the "transcendental
solution" to the First Antinomy in the Second Book of the
Transcendental Dialectic. Her verdict was that the apparent
contradictions between relativity theory and Kantian philosophy
disappear on closer examination of both doctrines (1921, 71-75).
3.3 Rejecting or Refurbishing the Transcendental Aesthetic
But most Kantian philosophers did not attempt to immunize Kant from
an apparent empirical refutation by the general theory. Rather,
their concern was to establish how far-reaching the necessary
modifications of Kant must be and whether, on implementation,
anything distinctively Kantian remained. Certainly, most at risk
appeared to be the claim, in the Transcendental Aesthetic, that all
objects of "outer" intuition, and so all physical objects, conform to
the space of Euclidean geometry. Since the general theory of
relativity employed non-Euclidean (Riemannian) geometry for the
characterization of physical phenomena, the conclusion seemed
inevitable that any assertion of the necessarily Euclidean character
of physical space in finite, if not "infinitesimal", regions, is
simply false.
Winternitz (1924) is an example of this tendency that may be
singled out on the grounds that it was deemed significant enough to be
the subject of a rare book review by Einstein (1924) . Winternitz
argued that the Transcendental Aesthetic is inextricably connected to
the claim of the necessarily Euclidean character of physical space and
so stood in direct conflict with Einstein's theory. It must
accordingly be totally jettisoned as a confusing and unnecessary
appendage of the fundamental transcendental project of establishing the
a priori logical presuppositions of physical knowledge.
Indeed, these presuppositions have been confirmed by the general
theory: They are spatiality and temporality as "unintuitive schema of
order" in general (as distinct from any particular chronometrical
relations), the law of causality and presupposition of continuity, the
principle of sufficient reason, and the conservation laws. Remarkably,
the necessity of each of these principles was, rightly or
wrongly, soon to be challenged by the new quantum mechanics. (For a
challenge to the law of conservation of energy, see Bohr, Kramers, and
Slater (1924)). According to Winternitz, the ne plus ultra of
transcendental idealism lay in the claim that the world "is not given
but posed (nicht gegeben, sondern aufgegeben) (as a problem)"
out of the given material of sensation. Interestingly, Einstein, late
in life, returns to this formulation as comprising the fundamental
Kantian insight into the character of physical knowledge (1949b,
680).
However, a number of neo-Kantian positions, of which that of Marburg
was only the best known, did not take the core doctrine of the
Transcendental Aesthetic, that space and time are a priori
intuitions, à la lettre. Rather, resources broadly
within it were sought for preserving an updated "critical
idealism". In this regard, Bollert (1921) merits mention for its
technically adroit presentation of both the special and the general
theory. Bollert argued that relativity theory had "clarified" the
Kantian position in the Transcendental Aesthetic by demonstrating
that not space and time, but spatiality (determinateness in
positional ordering) and temporality (in order of succession) are
a priori conditions of physical knowledge. In so doing,
general relativity theory with its variably curved spacetime, brought
a further advance in the steps or levels of "objectivation" lying at
the basis of physics. In this process, corresponding with the growth
of physical knowledge since Galileo, each higher level is obtained
from the previous through elimination of subjective elements from the
concept of physical object. This ever-augmented and revised advance
of conditions of objectivity is alone central to critical
idealism. For this reason, it is "an error" to believe that "a
contradiction exists between Kantian a priorism and
relativity theory" (1921,64). As will be seen, these conclusions are
quite close to those of the much more widely known monograph of
Cassirer (1921). It is worth noting that Bollert's
interpretation of critical idealism was cited favorably by Gödel
(1946/9-B2, 240, n.24) much later during the course of research which
led to his famous discovery of rotating universe solutions to
Einstein's gravitational field equations (1949). This
investigation had been prompted by Gödel's curiosity
concerning the similar denials, in relativity theory and in Kant, of
an absolute time.
3.4 General Covariance: A Synthetic Principle of "Unity of Determination"
The most influential of all neo-Kantian interpretations of general
relativity was Ernst Cassirer's Zur Einsteinschen
Relativitätstheorie (1921). Cassirer regarded the theory as
a crucial test for Erkenntniskritik, the preferred term for
the epistemology of Marburg's transcendental idealism. The
question, posed right at the beginning, is whether the Transcendental
Aesthetic offered a foundation "broad enough and strong enough" to
bear the general theory of relativity. Recognizing the theory's
principal epistemological significance to lie in the requirement of
general covariance ("that the general laws of nature are not changed
in form by arbitrary changes of the space-time variables"), Cassirer
directed his attention to Einstein's remarks, cited in §2
above, that general covariance "takes away from space and time the
last remnant of physical objectivity". Cassirer correctly construed
the gist of this passage to mean that in the general theory of
relativity, space and time coordinates have no further importance
than to be mere labels of events ("coincidences"), independent
variables of the mathematical (field) functions characterizing
physical state magnitudes. Furthermore,in accord with central tenets
of the Marburg Kant interpretation noted above, Cassirer maintained
that the requirement of generally covariant laws was a vindication of
the transcendental ideality of space and time, not, indeed, as "forms
of intuition" but as "objectifying conditions" that further
"de-anthropomorphized" the concept of object in physics, rendering it
"purely symbolic". In this regard, the requirement of general
covariance had significantly improved upon Kant in bringing out far
more clearly the exclusively methodological role of these conditions
in empirical cognition, a role Kant misleadingly assigned to "pure
intuition". Not only has it has shown that space and time are not
"things", it has also clarified that they are "ideal principles of
order" applying to the objects of the physical world as a necessary
condition of their possible experience. According to Cassirer,
Kant's intention with regard to "pure intuition" was
simply to express the methodological presupposition that certain
"intellectual forms" (Denkformen), among which are the
purely ideal concepts of coexistence and succession, enter
into all physical knowledge. According to the development of physics
since the 17^{th} century chronicled in Substanzbegriff
und Functionsbegrif, these forms have progressively lost their
"fortuitous" (zufälligen) anthropomorphic features, and
more and more take on the character of "systematic forms of unity".
From this vantage point, general covariance is but the most recent
refinement of the methodological principle of "unity of
determination" governing the constitution of objects of physical
knowledge, completing the transposition in physics from concepts of
substance into functional and relational concepts. In its wake, the
fundamental concept of object in physics no longer pertains to
particular entities or processes in space and time but rather to "the
invariance of relations among (physical state) magnitudes". For this
reason, Cassirer concluded, the general theory of relativity exhibits
"the most determinate application and carrying through within
empirical science of the standpoint of critical idealism" (1921/1957,
71; 1923, 412).
4.1 Lessons of Methodology?
Logical empiricism's philosophy of science was conceived
under the guiding star of Einstein's two theories of relativity,
as may be seen from the early writings of its founders, for purposes
here, Moritz Schlick, Rudolf Carnap, and Hans Reichenbach. The small
monograph of Schlick, Space and Time in Contemporary
Physics, appearing in 1917, initially in successive issues of
the scientific weekly Die Naturwissenschaften, served as a
prototype. Among the first of a host of "philosophical
examinations" of the general theory of relativity, it was
distinguished both by the comprehensibility of its largely
non-technical physical exposition and by Einstein's enthusiastic
praise of its philosophical appraisal, favoring
Poincaré's conventionalism over both neo-Kantianism and
Machian positivism. The transformation of the concept of space by the
general theory of relativity was the subject of Rudolf Carnap's
dissertation at Jena in 1921. Appearing as a monograph in 1922, it
also evinced a broadly conventionalist methodology combined with
elements of Husserlian transcendental phenomenology. Distinguishing
clearly between intuitive, physical and purely formal conceptions of
space, Carnap argues that, subject to the necessary constraints of
certain a priori phenomenological conditions of the topology
of intuitive space, the purely formal and the physical aspects of
theories of space, can be adjusted to one another so as to preserve
any conventionally chosen aspect. In turn, Hans Reichenbach was one
of five intrepid attendees of Einstein's first seminar on
general relativity given at Berlin University in the tumultuous
winter of 1918-1919; his detailed notebooks survive. The
general theory of relativity was the particular subject of
Reichenbach's neo-Kantian first book (1920), which is dedicated
to Albert Einstein, as well as of his next two books (1924), (1928),
and of numerous papers in the 1920s.
But Einstein's theories of relativity provided far more than
the subject matter for these philosophical examinations; rather
logical empiricist philosophy of science was itself fashioned by
lessons allegedly drawn from relativity theory in correcting or
rebutting neo-Kantian and Machian perspectives on general
methodological and epistemological questions of science. Several of
the most characteristic doctrines of logical empiricist philosophy of
science — the interpretation of a priori elements in
physical theories as conventions, the treatment of the role of
conventions in linking theory to observation and in theory choice,
the insistence on verificationist definitions of theoretical terms
— were taken to have been conclusively demonstrated by Einstein
in fashioning his two theories of relativity. In particular,
Einstein's 1905 analysis of the conventionality of simultaneity
in the special theory of relativity became something of a
methodological paradigm, prompting Reichenbach's own method of
"logical analysis" of physical theories into "subjective"
(definitional, conventional) and "objective" (empirical)
components. The overriding concern in the logical empiricist
treatment of relativity theory was to draw broad lessons from
relativity theory for scientific methodology and philosophy of
science generally, although issues more specific to the philosophy of
physics were also addressed. Only the former are considered here; for
a discussion of the latter, we may refer to Ryckman (forthcoming b).
4.2 From the "Relativized A priori to the "Relativity of Geometry"
A cornerstone of Reichenbach's "logical analysis" of the
theory of general relativity is the thesis of "the relativity of
geometry", that an arbitrary geometry may be ascribed to spacetime
(holding constant the underlying topology) if the laws of physics are
correspondingly modified through the introduction of "universal
forces". This particular argument for metric conventionalism has
generated substantial controversy on its own, but is better
understood through an account of its genesis in Reichenbach's
early neo-Kantianism. Independently of that genesis, the thesis
becomes the paradigmatic illustration of Reichenabch's broad
methodological claim that conventional or definitional elements, in
the form of "coordinative definitions" associating mathematical
concepts with "elements of physical reality", are a necessary
condition of empirical cognition in science. At the same time,
however, Reichenbach's thesis of metrical conventionalism is
part and parcel of an audacious program of epistemological
reductionism regarding spacetime structures. This was first
attempted in his "constructive axiomatization" (1924) of the theory
of relativity on the basis of "elementary matters of
fact"(Elementartatbestande) regarding the observable
behavior of lights rays, and rods and clocks. Here, and in the more
widely read treatment(1928), metrical properties of spacetime are
deemed less fundamental than topological ones, while the latter are
derived from the concept of time order. But time order in turn is
reduced to that of causal order and so the whole edifice of
structures of spacetime is considered epistemologically derivative,
resting upon ultimately basic empirical facts about causal order and
a prohibition against action-at-a-distance. The end point of
Reichenbach's epistemological analysis of the foundations of
spacetime theory is then "the causal theory of time", a type of
relational theory of time that assumes the validity of the causal
principle of action-by-contact (Nahwirkungsprinzip).
However, Reichenbach's first monograph on relativity (1920) was
written from within a neo-Kantian perspective. As Friedman (1999) and
others have discussed in detail (Ryckman, forthcoming a),
Reichenbach's innovation, a modification of the Kantian
conception of synthetic a priori principles, rejecting the
sense of "valid for all time" while retaining that of "constitutive
of the object (of knowledge)", led to the conception of a
theory-specific "relativised a priori". According to
Reichenbach, any physical theory presupposes the validity of systems
of certain, usually quite general, principles, which may vary from
theory to theory. Such "coordinating principles", as they are then
termed, are indispensable for the ordering of perceptual data; they
define "the objects of knowledge" within the theory. The
epistemological significance of relativity theory, according to the
young Reichenbach, is to have shown, contrary to Kant, that these
systems may contain mutually inconsistent principles, and so require
emendation to remove contradictions. Thus a "relativization" of the
Kantian conception of synthetic a priori principles is the
direct epistemological result of the theory of relativity. But this
finding is also taken to signal a transformation in the method of
epistemological investigation of science. In place of Kant's
"analysis of Reason", "the method of analysis of science"(der
wissenschaftsanalytische Methode) is proposed as "the only way
that affords us an understanding of the contribution of our reason to
knowledge" (1920, 71; 1965, 74). The method's raison
d'être is to sharply distinguish between the
"subjective" role of (coordinating) principles — "the
contribution of Reason" — and the "contribution of objective
reality", represented by theory-specific empirical laws and
regularities ("axioms of connection") which in some sense have been
"constituted" by the former. Relativity theory itself is a shining
exemplar of this method for it has shown that the metric of spacetime
describes an "objective property" of the world, once the subjective
freedom to make coordinate transformations (the coordinating
principle of general covariance) is recognized (1920, 86-7; 1965,
90). The thesis of metric conventionalism had yet to appear.
But soon it did. Still in 1920, Schlick objected, both publicly and
in private correspondence with Reichenbach, that "principles of
coordination" were precisely statements of the kind that
Poincaré had termed "conventions" (see Coffa, 1991,
201ff.). Moreover, Einstein, in lecture of January, 1921, entitled
"Geometry and Experience", appeared to lend support to this
view. Einstein argued that the question concerning the nature of
spacetime geometry becomes an empirical question only on certain
pro tem stipulations regarding the "practically rigid body"
of measurement (pro tem in view of the inadmissibility in
relativity theory of the concept "actually rigid body"). In any case,
by 1922, the essential pieces of Reichenbach's "mature"
conventionalist view had emerged. The argument is canonically
presented in §8 (entitled "The Relativity of Geometry") of
Der Philosophie der Raum-Zeit-Lehre (completed in 1926,
published in 1928). In a move superficially similar to the argument
of Einstein's "Geometry and Experience", Reichenbach maintained
that questions concerning the empirical determination of the metric
of spacetime must first confront the fact that only the whole
theoretical edifice comprising geometry and physics admits of
observational test. Einstein's gravitational theory is such a
totality. However, unlike Einstein, Reichenbach's "method of
analysis of science", later re-named "logical analysis of science",
is directed to the epistemological problem of factoring this totality
into its conventional or definitional and its empirical components.
This is done as follows. The empirical determination of the
spacetime metric by measurement requires choice of some "metrical
indicators": this can only be done by laying down a "coordinative
definition" stipulating, e.g., that the metrical notion of a "length"
is coordinated to some physical object or process. A standard choice
coordinates "lengths" with "infinitesimal measuring rods" supposed
rigid (e.g., Einstein's "practically rigid body"). This however
is only a convention, and other physical objects or processes might
be chosen. (In Schlick's fanciful example, the Dali Lama's
heartbeat could be chosen as the physical process establishing units
of time.) Of course, the chosen metrical indicators must be corrected
for certain distorting effects (temperature, magnetism, etc.) due to
the presence of physical forces. Such forces are termed "differential
forces" to indicate that they affect various materials differently.
However, Reichenbach argued, the choice of a rigid rod as standard of
length is tantamount to the claim that there are no non-differential
— "universal" — distorting forces that affect all bodies in
the same way and cannot be screened off. In the absence of "universal
forces" the coordinative definition regarding rigid rods can be
implemented and the nature of the spacetime metric empirically
determined, for example, finding that paths of light rays through
solar gravitational field are not Euclidean straight lines. Thus, the
theory of general relativity, on adoption of the coordinative
definition of rigid rods ("universal forces = 0"), affirms that the
geometry of spacetime in this region is of a non-euclidean kind. The
point, however, is that this conclusion rests on the convention
governing measuring rods. One could, alternately, maintain that the
geometry of spacetime was Euclidean by adopting a different
coordinative definition, for example, holding that measuring rods
expanded or contracted depending on their position in spacetime, a
choice tantamount to the supposition of "universal forces". Then,
consistent with all empirical phenomena, it could be maintained that
Euclidean geometry was compatible with Einstein's theory if only
one allowed the existence of such forces. Thus whether general
relativity affirms a Euclidean or a non-euclidean metric in the solar
gravitational field rests upon a conventional choice regarding the
existence of "universal forces". Either hypothesis may be adopted
since they are empirically equivalent descriptions; their joint
possibility is referred to as "the relativity of geometry". Just as
with the choice of "standard synchrony" in Reichenbach's
analysis of the conventionality of simultaneity, also held to be
"logically arbitrary", Reichenbach recommends the "descriptively
simpler" alternative in which "universal forces" do not exist. To be
sure, "descriptive simplicity has nothing to do with truth", i.e.,
has no bearing on the question of whether spacetime has a
non-Euclidean structure (1928, 47; 1958, 35).
4.3 Critique of Reichenbachian Metric Conventionalism
In retrospect, it is rather difficult to understand the
significance that has been accorded this argument. Carnap, for
example, in his "Introductory Remarks" (1958, vii) to the posthumous
English translation of this work, singled it out on account of its
"great interest for the methodology of physics". Reichenbach himself
deemed "the philosophical achievement of the theory of relativity" to
lie in this methodological distinction between conventional and
factual claims regarding spacetime geometry (1928, 24; 1958, 15), and
he boasted of his "philosophical theory of relativity" as an
incontrovertible "philosophical result":
the philosophical theory of relativity, i.e., the
discovery of the definitional character of the metric in all its
details, holds independently of experience….a philosophical
result not subject to the criticism of the individual sciences."
(1928, 223; 1958, 177)
Yet this result is neither incontrovertible nor an untrammeled
consequence of Einstein's theory of gravitation. There is, first
of all, the shadowy status accorded to "universal forces". A
sympathetic reading (e.g., Dieks (1987)) suggests that the notion
serves usefully in mediating between a traditional a priori
commitment to Euclidean geometry and the view of modern
geometrodynamics, where gravitational force is "geometrised away"
(see §5). For, as Reichenbach explicitly acknowledged,
gravitation is itself a "universal force", coupling to all bodies and
affecting them in the same manner (1928, 294-6; 1958, 256-8). Hence
the choice recommended by "descriptive simplicity" is merely a
stipulation that metrical appliances, regarded as "infinitesimal", be
considered as "differentially at rest" in an inertial system (1924,
115; 1969, 147). This is a stipulation that spacetime measurements
always take place in regions that are to be considered small
Minkowski spacetimes (arenas of gravitation-free physics). By the
same token, however, consistency required an admission that "the
transition from the special theory to the general one represents
merely a renunciation of metrical characteristics" (1924, 115; 1969,
147), or, even more pointedly, that "all the metrical properties of
the spacetime continuum are destroyed by gravitational fields" where
only topological properties remain (1928, 308; 1958, 268-9). To be
sure, these conclusions are supposed to be rendered more palatable in
connection with the epistemological reduction of spacetime structures
in the causal theory of time.
Despite the influence of this argument on the subsequent generation
of philosophers of science, Reichenbach's analysis of spacetime
measurement treatment is plainly inappropriate, manifesting a
fallacious tendency to view the generically curved spacetimes of
general relativity as stiched together from little bits of flat
Minskowski spacetimes. Besides being mathematically inconsistent,
this procedure offers no way of providing a non-metaphorical physical
meaning for the fundamental metrical tensor
g_{},
the central theoretical concept of general relativity, nor to the
series of curvature tensors derivable from it and its associated
affine connection. Since these sectional curvatures at a point of
spacetime are empirically manifested and the curvature components can
be measured, e.g., as the tidal forces of gravity, they can hardly be
accounted as due to conventionally adopted "universal
forces". Furthermore, the concept of an "infinitesimal rigid rod" in
general relativity cannot really be other than the interim stopgap
Einstein recognized it to be. For it cannot actually be "rigid" due
to these tidal forces; in fact, the concept of a "rigid body" is
already forbidden in special relativity as allowing instantaneous
causal actions. Secondly, such a rod must indeed be "infinitesimal",
i.e., a freely falling body of negligible thickness and of
sufficiently short extension, so as to not be stressed by
gravitational field inhomogeneities; just how short depending on
strength of local curvatures and on measurement error (Torretti
(1983), 239). But then, as Reichenbach appeared to have recognized in
his comments about the "destruction" of the metric by gravitational
fields, it cannot serve as a coordinately defined general standard
for metrical relations. In fact, as Weyl was the first to point out,
precisely which physical objects or structures are most suitable as
measuring instruments should be decided on the basis of gravitational
theory itself. From this enlightened perspective, measuring rods and
clocks are objects that are far too complicated. Rather, the metric
in the region around any observer O can be empirically determined
from freely falling ideally small neutral test masses together with
the paths of light rays. More precisely stated, the spacetime metric
results from the affine-projective structure of the behavior of
neutral test particles of negligible mass and from the conformal
structure of light rays received and issued by the observer. (Weyl,
1921) Any purely conventional stipulation regarding the behavior of
"measuring rods" as physically constitutive of metrical relations in
general relativity is then otiose (Weyl, 1923a; Ehlers, Pirani and
Schild (1973)). Alas, since Reichenbach reckoned the affine structure
of the gravitational-inertial field to be just as conventional as, on
his view, its metrical structure, he was not able to recognize this
method as other than an equivalent, but by no means necessarily
preferable, account of the empirical determination of the metric
through the use of rods and clocks (Coffa, 1979; Ryckman (1994),
(1996)).
5.1 Differing Motivations
In the decade or so following the appearance of the general theory
of relativity, there was much talk of a "geometrization" of physics
(Weyl (1918b), (1919); Haas (1920); Lodge (1921)). While these
discussions were largely, and understandably, confined to scientific
circles, they nonetheless brought distinctly philosophical issues
— of methodology, but also of epistemology and metaphysics
— together with technical matters. General relativity revived a
geometrizing tendency essentially dormant within physics since the
17th century. In so doing, it opened up the prospect of a
"geometrization" of physics, the possibility of finding a unifying
representation of all of known physics within a single geometrical
theory of the spacetime continuum. Einstein himself, however, was not
the first to embark on this audacious quest. Rather he initially
followed in the mathematical footsteps of Hermann Weyl, Arthur
Stanley Eddington, and Theodore Kaluza, only gradually (1925)
devising the first of his own "homegrown" geometrical "unified field
theories". Still, by 1923, Einstein had become the recognized leader
of the unification program. (Vizgin (1994), 265)
The first phase of the geometrical unification program essentially
ended with Einstein's "distant parallelism" theory of 1928-1931
(1929), perhaps Einstein's final public sensation (Fölsing
(1997, 605)). Needless to say, none of these efforts met with
success. In a lecture at the University of Vienna on October 14,
1931, Einstein forlornly referred to these failed attempts, each
conceived on a different differential geometrical basis, as a
"graveyard of dead hopes" (Einstein, 1932). By this time, certainly,
the prospects for the geometrical unification program had
considerably waned. A consensus emerged among nearly all leading
theoretical physicists that while the geometrical unification of the
gravitation and electromagnetic fields might be attained in formally
different ways, the problem of matter, treated with undeniable
empirical success by the new quantum theory, was not to be resolved
within the confines of spacetime geometry. In any event, from the
early 1930s, any unification program appeared greatly premature, in
view of the wealth of data produced by the new physics of the
nucleus.
As many will know, the unsuccessful pursuit of the goal of
geometrical unification absorbed Einstein, and his various research
assistants, for more than three decades, up to Einstein's death
in 1955. In the course of it, Einstein's methodology of research
diametrically changed. In place of physical or heuristic principles
to guide theoretical construction, such as the principle of
equivalence, which put him on the path to general relativity, he
increasingly relied on considerations of mathematical aesthetics,
such as "logical simplicity" and the inevitability of certain
mathematical structures under variously adopted constraints. In a
talk entitled "On the Method of Theoretical Physics" at Oxford in
1933, the transformation was stated dramatically:
Experience remains, of course, the sole criterion of the physical
utility of a mathematical construction. But the creative principle
resides in mathematics. In a certain sense, therefore, I hold it true
that pure thought can grasp reality, as the ancients dreamed. (274)
Moreover, the advent and accumulating empirical successes of the new
quantum theory did not dislodge Einstein's core metaphysical
belief in a physical reality conceived as a continuous "total field"
whose components are functions of the spacetime variables, a
geometrical conception of physical reality implied, to be sure, by
general relativity (e.g., (1950), 348). Yet, whatever may have been
Kaluza's philosophical motivations in putting forward his
proposal for geometrical unification, neither Einstein's
mathematical realism nor his metaphysics guided either Weyl or
Eddington, a fact that has often been obscured or ignored in
historical treatments. The geometrical unifications of Weyl (1918a,b)
and Eddington (1921) were above all explicit attempts to comprehend
the nature of physical theory, in the light of general relativity,
from systematic epistemological standpoints that were neither
positivist nor realist. As such they comprise "early philosophical
interpretations" of that theory, although they intertwine philosophy,
geometry and physics in a manner unprecedented since
Descartes. Before turning attention to their "interpretations", it
will be helpful to see how the geometrizing tendency arises within
general relativity itself and to note a few details of the
geometrical unification program that followed in its wake.
5.2 "Geometrizing" Gravity: the Initial Step
Einstein's so-called "geometrization" of gravitational force
in 1915 gave the geometrization program its first, partial,
realization as well as its subsequent impetus. In Einstein's
theory, the fundamental or "metric" tensor
g_{}
of Riemannian geometry appears in a dual role which thoroughly fuses
its geometrical and its physical meanings. As is apparent from the
expression for the "interval" between neighboring spacetime events,
ds^{2} =
g_{}
dx^{}
dx^{}
(here, and below there is an implicit
summation over repeated upper and lower indices), the metric tensor
is at once the geometrical quantity underlying measurable metrical
relations of lengths and times. In this role it ties a mathematical
theory of events in four-dimensional "curved" spacetime to
observations and measurements in space and time. But it is also the
"potential" of the gravitational (or "metrical") field whose value,
at any point of spacetime, depends, via the Einstein Field Equations
(see below), on the presence of physical quantities of
mass-momentum-stress in the immediate region. In the new view, the
idea of strength of gravitational "force" is replaced by that of
degree of "curvature" of spacetime. Such a curvature is manifested,
for example, by the "tidal force" of the Earth's gravitational
field that occasions two freely falling bodies, released at a certain
height and at fixed separation, to approach one another. A freely
falling body is no longer to be regarded as moving through space
according to the "pull" of an attractive gravitational "force", but
simply as tracing out the "laziest" track along the bumps and hollows
of spacetime itself. The Earth's mass (or equivalently, energy)
determines a certain spacetime curvature and so becomes a source of
gravitational action. At the same time, the gross mechanical
properties of bodies, comprising all gravitational-inertial
phenomena, can be derived as the solution of a single system of
generally covariant partial differential equations, the Einstein
equations of the gravitational field. According to these equations,
spacetime and matter stand in dynamical interaction. One abbreviate
way of characterizing the dual role of the
g_{}
is to say that in the general theory of relativity, gravitation,
which includes mechanics, has become "geometrized", i.e.,
incorporated into the geometry of spacetime.
5.3 Extending "Geometrization"
In making spacetime curvature dependent on distributions of mass
and energy, general relativity is indeed capable of encompassing all
(non-quantum) physical fields. However, in classical general
relativity there remains a fundamental asymmetry between
gravitational and non-gravitational fields, in particular,
electromagnetism, the only other fundamental physical interaction
definitely known at the time. This shows up visibly in one form of
the Einstein field equations in which, on the left-hand side, a
geometrical object
(G_{},
the Einstein tensor) built up from the uniquely compatible linear
symmetric ("Levi-Civita") connection associated with the metric
tensor g_{}, and
representing the curvature of spacetime, is set identical to a
tensorial but non-geometrical phenomenological representation of
matter on the right-hand side.
G_{} =
k T_{},
where G_{}
=
R_{}
1/2
g_{}
R
The expression on the right side, introduced by a coupling constant,
mathematically represents the non-gravitational sources of the
gravitational field in a region of spacetime in the form of a
stress-energy-momentum tensor (an "omnium gatherum" in
Eddington's pithy phrase (1919, 63)). As the geometry of
spacetime principally resides on the left-hand side, this situation
seems unsatisfactory. Late in life, Einstein likened his famous
equation to a building, one wing of which (the left) was built of
"fine marble", the other (the right) of "low grade wood" (1936,
311). In its classical form, general relativity accords only the
gravitational field a direct geometrical significance; the other
physical fields reside in spacetime; they are not
of spacetime.
Einstein's dissatisfaction with this asymmetrical state
of affairs was palpable at an early stage and was expressed with
increasing frequency beginning in the early 1920s. A particularly vivid
declaration of the need for geometrical unification was made in his
"Nobel lecture" of July, 1923:
The mind striving after unification of the theory cannot be
satisfied that two fields should exist which, by their nature, are
quite independent. A mathematically unified field theory is sought in
which the gravitational field and the electromagnetic field are
interpreted as only different components or manifestations of the
same uniform field,… The gravitational theory, considered in
terms of mathematical formalism, i.e. Riemannian geometry, should be
generalized so that it includes the laws of the electromagnetic
field."(489)
It might be noted that the tacit assumption, evident here, that
incorporation of electromagnetism into spacetime geometry requires a
generalization of the Riemannian geometry of general relativity,
though widely held at the time, is not quite correct (Rainich (1925);
Misner and Wheeler (1962); Geroch (1966)).
5.4 A "Pure Infinitesimal Geometry"
Still, it wasn't Einstein, but the mathematician Hermann Weyl
who first addressed the asymmetry in 1918 in the course of
refashioning Einstein's theory on the preferred epistemological
basis of a "pure infinitesimal geometry" (Reine
Infinitesimalgeometrie). Holding that direct —
evident, in the sense of Husserlian phenomenology
--comparisons of length or duration could be made at
neighboring points of spacetime, but not, as the Riemannian geometry
of Einstein's theory permitted, "at a distance", Weyl discovered
additional terms in his geometry that he identified with the
potentials of the electromagnetic field. From these, the
electromagnetic field strengths can be immediately derived and so
electromagnetism as well as gravitation could be expressed solely
within the terms of spacetime geometry. As no other interactions were
definitely known to occur, Weyl proudly declared that the concepts of
geometry and physics were the same. Hence, everything in the physical
world was a manifestation of spacetime geometry.
(The) distinction between geometry and physics is an error, physics
extends not at all beyond geometry: the world is a (3+1) dimensional
metrical manifold, and all physical phenomena transpiring in it are
only modes of expression of the metric field, …. (M)atter itself
is dissolved in "metric" and is not something substantial that in
addition exists "in" metric space (1919, 115-16).
By the winter of 1919-1920, for both physical and philosophical
reasons (the latter having to do with his conversion to
Brouwer's "intuitionist" views about the mathematical continuum,
in particular, the continuum of spacetime), Weyl (1920) surrendered
the belief, expressed here, that matter, with its corpuscular
structure, might be derived within spacetime geometry. Thus he gave
up the Holy Grail of the nascent unified field theory program almost
before it had begun. Nonetheless, he actively defended his theory
well into the 1920s, essentially on the grounds of Husserlian
transcendental phenomenology, that his geometry and its central
principle, "the epistemological principle of relativity of magnitude"
comprised a superior epistemological framework for general
relativity. Weyl's postulate of a "pure infinitesimal"
non-Riemannian metric for spacetime, according to which it must be
possible to independently choose a "gauge" (scale of length or
duration) at each spacetime point, met with intense criticism. No
observation spoke in favor of it; to the contrary, Einstein pointed
out that according to Weyl's theory, the atomic spectra of the
chemical elements should not be constant, as indeed they are observed
to be. Although Weyl responded to this objection forcefully, and with
some subtlety (Weyl, 1923a), he was able to persuade neither
Einstein, nor any other leading relativity physicist, with the
exception of Eddington. However, the idea of requiring "gauge
invariance" of fundamental physical laws was revived and vindicated
by Weyl himself in a different form later on (Weyl (1929);see also
O'Raifeartaigh (1997)).
5.5 Eddington's "World Geometry"
Despite Weyl's failure to win many friends for his theory, his
guiding example of unification launched the geometrical program of
"unified field theory", initiating a variety of efforts, all aimed at
finding a suitable generalization of the Riemannian geometry of
Einstein's theory to encompass as well non-gravitational physics
(Vizgin (1994), ch.4). In December, 1921, the Berlin Academy
published Theodore Kaluza's novel proposal for unification of
gravitation and electromagnetism upon the basis of a five-dimensional
Riemannian geometry. But earlier that year, in February, came Arthur
Stanley Eddington's further generalization of Weyl's
four-dimensional geometry, wherein the sole primitive geometrical
notion is the non-metrical comparison of direction or orientation at
the same or neighboring points. In Weyl's geometry the magnitude
of vectors at the same point, but pointing in different directions,
might be directly compared to one another; in Eddington's,
comparison was immediate only for vectors pointing in the same
direction. His "theory of the affine field" included both Weyl's
geometry and the semi-Riemannian geometry of Einstein's general
relativity as special cases. Little attention was paid however, to
Eddington's claim, prefacing his paper, that his objective had
not been to "seek (the) unknown laws (of matter)" as befits a unified
field theory. Rather it lay "in consolidating the known (field) laws"
wherein "the whole scheme seems simplified, and new light is thrown
on the origin of the fundamental laws of physics" (1921, 105).
Eddington was persuaded that Weyl's "principle of relativity of
length" was "an essential part of the relativistic conception", a
view he retained to the end of his life (e.g., (1939, 28)). But he
was also convinced that the largely antagonistic reception accorded
Weyl's theory was due to its confusing formulation. The flaw lay
in Weyl's failure to make transparently obvious that his locally
scale invariant ("pure infinitesimal") "world geometry" was not the
physical geometry of actual spacetime, but an entirely mathematical
geometry inherently serving to specify the ideal of an
observer-independent external world. To remedy this, Eddington
devised a general method of deductive presentation of field physics
in which "world geometry" is developed mathematically as conceptually
separate from physics. A "world geometry" is a purely mathematical
geometry the derived objects of which possess only the structural
properties requisite to the ideal of a completely impersonal world;
these are objects, as he wrote in Space, Time and
Gravitation (1920), a semi-popular best-seller, represented
"from the point of view of no one in particular". Naturally, this
ideal had changed with the progress of physical theory. In the light
of relativity theory, such a world is indifferent to specification of
reference frame and, after Weyl, of gauge of magnitude (scale). A
"world geometry" is not the physical theory of such a world but a
framework or "graphical representation" in whose terms existing
physical theory might be displayed, essentially by the mathematical
identification of known tensors of the existing physical laws of
gravitation and electromagnetism, with tensors of the world
geometry. Such a geometrical representation of physics cannot really
be said to be "right" or "wrong", for it only implements, if it can,
current ideas governing the conception of objects and properties of
an impersonal objective external world. But when existing physics, in
particular, Einstein's theory of gravitation, is set in the
context of Eddington's world geometry, it yields a surprising
consequence: The Einstein law of gravitation appears as a definition!
In the form
R_{} = 0
it defines what in the "world geometry" appears to the mind as
"vacuum" while in the form of the Einstein field equation noted
above, it defines what is there encountered by the mind as
"matter". This result is what was meant by his stated claim of
throwing "new light on the origin of the fundamental laws of
physics". Eddington's notoriously difficult and opaque later
works (1936), (1946), took their inspiration from this argumentation
in attempting to carry out a similar, but algebraic, program of
deriving fundamental physical laws, and the constants occurring in
them, from epistemological principles.
5.6 Meyerson on "Pangeometrism"
Within physics the idealist currents lying behind the "world
geometries" of Weyl and Eddington were largely ignored, whereas
within philosophy, with the notable exception of Émile
Meyerson's La Déduction Relativiste (1925),most
philosophers lacked the tools to connect these readily discernible
currents with their geometrical theories. Meyerson, who had no doubt
of the basic realist impetus of science, carefully distinguished
Einstein's "rational deduction of the physical world" from the
geometrical unifications of gravitation and electromagnetism of Weyl
and Eddington. These theories, as affirmations of a complete
panmathematicism, or rather of a pangeometrism
(§§ 157-58), were compared to the rational deductions of
Hegel's Logic. That general relativity succeeded in
partly realizing Descartes' program of reducing the physical to
the spatial through geometric deduction, is due to the fact that
Einstein "followed in the footsteps" of Descartes, not Hegel
(§133). But pan-geometrism is also capable of
overreaching itself and this is the transgression committed by Weyl
and Eddington. Weyl in particular is singled out for criticism for
seemingly to have reverted to Hegel's monistic idealism, and so
to be subject to its fatal flaw. In regarding nature as completely
intelligible, Weyl had abolished the thing-in-itself and so promoted
the identity of self and non-self, the great error of the
Naturphilosophien.
Though he had "all due respect to the writings of such distinguished
scientists" as Weyl and Eddington, Meyerson took their overt
affirmations of idealism to be misguided attempts "to associate
themselves with a philosophical point of view that is in fact quite
foreign to the relativistic doctrine" (§150). That "point of
view" is in fact two distinct species of transcendental idealism. It
is above all "foreign" to relativity theory because Meyerson cannot
see how it is possible to "reintegrate the four-dimensional world of
relativity theory into the self". After all, Kant's own argument
for Transcendental Idealism proceeded "in a single step", in
establishing the subjectivity of the space and time of "our
naïve intuition". But this still leaves "the four dimensional
universe of relativity independent of the self". Any attempt to
"reintegrate" four-dimensional spacetime into the self would have to
proceed at a "second stage" where, additionally, there would be no
"solid foundation" such as spatial and temporal intuition furnished
Kant at the first stage. Perhaps, Meyerson allowed, there is indeed
"another intuition, purely mathematical in nature", lying behind
spatial and temporal intuition, and capable of "imagining the
four-dimensional universe, to which, in turn, it makes reality
conform". This would make intuition a "two-stage mechanism". While
all of this is not "inconceivable", it does appear, nonetheless,
"rather complex and difficult if one reflects upon it". In any case,
this is likely to be unnecessary, for considering the matter "with an
open mind",
one would seem to be led to the position of those who believe that
relativity theory tends to destroy the concept of Kantian intuition
(§§ 151-2).
Meyerson had come right up to the threshold of grasping the
Weyl-Eddington geometric unification schemes in something like the
sense in which they were intended. The stumbling block for him, and
for others, is the conviction that transcendental idealism can be
supported only from an argument about the nature of intuition, and
intuitive representation. To be sure, the geometric framework for
Weyl's construction of the objective four-dimensional world of
relativity is based upon the Evidenz available in "essential
insight", which is limited to the simple linear relations and
mappings in what is basically the tangent vector space to a point in
a manifold. Thus in Weyl's differential geometry there is a
fundamental divide between integrable and non-integrable relations of
comparison. The latter are primitive and epistemologically
privileged, but nonetheless not justified until it is shown how the
infinitesimal homogenous spaces, corresponding to the "essence of
space as a form of intution", are compatible with the large-scale
inhomogenous spaces (spacetimes) of general relativity. And this
required not a philosophical argument about the nature of intution,
but one formulated in group-theoretic conceptual
form. (Weyl, 1923a,b). Eddington, on the other hand, without the
cultural context of Husserlian phenomenology or indeed of philosophy
generally, jettisoned the intuitional basis of transcendental
idealism altogether, as if unaware of its prominence. Thus he sought
a superior and completely general conceptual basis for the
objective four-dimensional world of relativity theory by constituting
that world within a geometry (its "world structure" (1923)) based
upon a non-metrical affine (i.e., linear and symmetric)
connection. He was then free to find his own way to the empirically
confirmed integrable metric relations of Einstein's theory
without being hampered by the conflict of a "pure infinitesimal"
metric with the observed facts about rods and clocks.
5.7 "Structural Realism"?
It has been routinely assumed that all the attempts at a
"geometrization of physics" in the early unified field theory program
shared something of Einstein's hubris concerning the ability of
mathematics to "grasp" the fundamental structure of the external
world. The geometrical unified field theory program thus appears to
be inseparably stitched to a form of scientific realism, recently
termed "structural realism", with perhaps even an inspired turn
toward Platonism. According to "structural realism", whatever the
"nature" of the fundamental entities comprising the physical world,
only their "structure" can be known as that structure is represented
in the equations of the theory. The sole ontological continuity
across changes in fundamental physical theory is a continuity of
structure, as the equations of the earlier theory can be derived,say
as limit cases, from those of the later. Geometrical unification
theories seems tailored for this kind of realism. For if a
geometrical theory is taken to give a true or approximately true
representation of the physical world, what is geometrically
represented has the definite structure of the fundamental geometrical
relations. But for Weyl and Eddington, geometrical unification was
not, nor could be, such a representation, for essentially the reasons
articulated two decades before by Poincaré (1906,14):
Does the harmony the human intelligence thinks it discovers in
nature exist outside of this intelligence? No, beyond doubt, a
reality completely independent of the mind which conceives it, sees
or feels it, is an impossibility. A world as exterior as that, even
if it existed, would for us be forever inaccessible. But what we
call objective reality is, in the last analysis, what is common to
many thinking beings, and could be common to all; this common
part,...,can only be the harmony expressed by mathematical laws. It
is this harmony then which is the sole objective reality....
In Weyl and Eddington, geometrical unification was an attempt to cast
the "harmony" of the Einstein theory of gravitation in a new
epistemological and explanatory light, by displaying the great field
laws of gravitation and electromagnetism within the common frame of a
geometrically represented objective reality. Their unorthodox manner
of philosophical argument, cloaked, perhaps necessarily, in the
language of differential geometry, has tended to conceal or obscure
conclusions about the significance of a "geometrized physics" that
push in considerably different directions from either instrumentalism
or scientific realism.
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[Please contact the author with other suggestions.]
a priori justification and knowledge |
Einstein, Albert: philosophy of science |
equivalence of mass and energy |
geometry: in the 19th century |
Kant, Immanuel |
Reichenbach, Hans |
space and time: conventionality of simultaneity |
space and time: the hole argument
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Thomas A. Ryckman
tryckman@hotmail.com
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