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- 1. The Search for Philosophical Novelty
- 2. Machian Positivism
- 3. Kantian and Neo-Kantian Interpretations
- 4. Logical Empiricism
- 5. "Geometrization of Physics": Realism and Transcendental Idealism
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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.

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.

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.

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.

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).

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).

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)).

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.

G_{}=kT_{}, whereG_{}=R_{}1/2g_{}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)).

(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)).

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.

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.

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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- The History of Philosophy Working Group (U. Missouri/Kansas City)

*First published: November 27, 2001*

*Content last modified: November 27, 2001*