Early philosophical interpretations of the general theory ofrelativity selected distinct aspects of that theory for praise orfavored recognition. Positivist followers of Mach initially laudedEinstein’s attempt to implement a “relativization ofinertia”—roughly, that the inertia of a body (inertialmass, local inertial behavior) is an effect due to presence of allother masses—in the general theory. But Machians were daunted bythe theory’s unprecedented distance between mathematicalconstruct and experimental test, ultimately proving more comfortablewith Einstein’s operationalist treatment of the concept ofdistant simultaneity in the special theory. Kantians and neo-Kantians,if freed from strict fealty to the doctrines of the TranscendentalAesthetic and Transcendental Analytic, turned instead to theTranscendental Dialectic. They pointed (as did Einstein himself inlater years) to the surpassing importance of certain “intellectualforms” in the general theory, above all, the principle of generalcovariance. Accordingly, this principle was viewed not as a mereformal principle of coordinate generality but as a “regulativeidea”, a constraint on any fundamental physical theory (roughlyintending that space and time are relational orderings of events interms of co-existence and succession, and that the laws of physicsmust not depend on any particular coordinate system labeling spacetimeevents).

To an emerging logical empiricism, the philosophical significance ofrelativity theory was above all methodological, to the effect thatconventions must first be stipulated in order to express the empiricalcontent of a physical theory. In a more far reaching development,already by its completion in November 1915, an attempt was made byDavid Hilbert to incorporate Maxwellian electrodyamics (the only otherknown physical interaction at the time) into generalrelativity’s geometrization of gravitational attraction. Otherattempts soon ensued of which that of Weyl, and shortly thereafter ofEddington, are distinguished from others, in particular fromEinstein’s many subsequent proposals made over the next threedecades. Weyl and Eddington sought to unify gravitation andelectromagnetism within the confines of broader spacetime geometriesthan Einstein’s, situating geometric unifications of gravitationand electromagnetism within philosophical frameworks of transcendentalidealism drawing, in Weyl’s case, from Husserlian transcendentalphenomenology and in Eddington’s, from a quasi-Kantianstructuralist viewpoint he would later term “selectivesubjectivism”.

• 1. The Search for Philosophical Novelty
• 2. Machian Positivism
  • 2.1 In the Early Einstein
  • 2.2 A “Relativization of Inertia”?
  • 2.3 Positivism and the “Hole Argument”
  • 2.4 “Mach’s Principle”
  • 2.5 Emerging Anti-Positivism
• 3. Kantian and Neo-Kantian Interpretations
  • 3.1 Neo-Kantians on Special Relativity
  • 3.2 Immunizing Strategies
  • 3.3 Rejecting or Refurbishing the Transcendental Aesthetic
  • 3.4 General Covariance: A Synthetic Principle of “Unity of Determination”
• 4. Logical Empiricism
  • 4.1 Lessons of Methodology?
  • 4.2 From the “Relativized A priori” to the “Relativity of Geometry”
  • 4.3 Critique of Reichenbachian Metric Conventionalism
• 5. “Geometrization of Physics”: Platonism, Transcendental Idealism, Structuralism
  • 5.1 Differing Motivations
  • 5.2 The Initial Step: “Geometrizing” Gravity
  • 5.3 Extending Geometrization
  • 5.4 “Pure Infinitesimal Geometry”
  • 5.5 Eddington’s World Geometry
  • 5.6 Meyerson on “Pangeometrism”
  • 5.7 “Structural Realism”?
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Search for Philosophical Novelty

Extraordinary public clamor greeted an announcement of a joint meetingof the Royal Society of London and the Royal Astronomical Society onthe sixth of November 1919. Astronomical observations made by aBritish team five months earlier (during the solar eclipse on May 29)comprised the first empirical test of Einstein’s general theoryof relativity. Lengthy data analysis over the summer of observationsmade under non-ideal conditions had shown that within acceptablemargin of error, light from distant stars passing near the solarsurface had been displaced by the tiny amount (1.75 arcseconds,corresponding to the angle of a right triangle of height 1 inch andbase nearly 2 miles in length) predicted by Einstein’sgravitational theory of curved spacetime. By dint of having“overthrown” such a permanent fixture of the intellectuallandscape as Newtonian gravitational theory, the general theory ofrelativity immediately became a principal focus of philosophicalinterest and inquiry. Although many traditionally-minded physicistsand philosophers would oppose it, the former mostly on non-physicalgrounds such as its abstract mathematical method, the latter becauseit overturned the familiar metaphysics of Newtonian space, surveyedhere are the interpretations of those who recognized the theory as arevolutionary advance not only in physical knowledge but also, quitepossibly, in philosophy. Among them are assessments of the theoryunderstandingly based on the semi-popular writings of Einstein andothers by those lacking expertise in the theory’s technicalaspects. Further lack of clarity stemmed even from the scientificliterati who provided differing, and at times, conflictingmathematical or physical accounts of the theory’s fundamentalprinciples. These are (see below): the principles of equivalence andof what Einstein misleadingly termed “general relativity”(i.e., general covariance), and a third principle baptized asMach’s Principle, that there can be no inertia withrespect to “Space” but only of masses to one another. Inone or another form, these controversies in significant respects havecontinued into the present literature of physics and philosophy ofphysics (see, e.g., Stachel 1980; Friedman 1983; Norton 1993; Barbour& Pfister 1995; Ohanian 2008; Janssen 2014; Ryckman 2017). This isnot unusual: physical theories, if sufficiently robust, are rarely, ifever, without ambiguous or problematic aspects, and are often taken tosay different things at different stages of development. But the veryfluidity of physical and mathematical meaning lent interpretativelatitude to inherently antagonistic philosophical viewpoints seekingvindication, confirmation or illumination by the revolutionary newtheory. At the conclusion of the theory’s first decade, perhapssemi-facetiously Bertrand Russell observed that

There has been a tendency, not uncommon in the case of a newscientific theory, for every philosopher to interpret the work ofEinstein in accordance with his own metaphysical system, and tosuggest that the outcome is a great accession of strength to the viewswhich the philosopher in question previously held. This cannot be truein all cases; and it may be hoped that it is true in none. It would bedisappointing if so fundamental a change as Einstein has introducedinvolved no philosophical novelty. (1926: 331)

General relativity proved a considerable stimulus to philosophicalnovelty. But the question whether it particularly supported any oneline of philosophical interpretation over another must take intoaccount the fact that schools of interpretation in turn evolved toaccommodate what were regarded as general relativity’sphilosophically salient features. A classic instance is the assertion,to become a cornerstone of logical empiricism, that relativity theoryhad shown the untenability of any “philosophy of the synthetica priori”, despite the fact that early works onrelativity theory by both Hans Reichenbach and Rudolf Carnap werewritten from within a broadly Kantian perspective. It will be seenthat, while proving ideologically useful, the claim by no meansfollows from relativity theory although, as physicist Max von Laueobserved in his early text on general relativity (1921: 42),“not every sentence ofThe Critique of PureReason” might still be held intact. What does follow fromscrutiny of the various philosophical appropriations of generalrelativity is a consummate illustration that, due to the evolution andmutual interplay of physical, mathematical and philosophicalunderstandings of a revolutionary physical theory, philosophicalinterpretations often are works in progress, extending over manyyears.

2. Machian Positivism

2.1 In the Early Einstein

Most of Einstein’s early papers (1902–1911) prior to hisnearly exclusive concentration on a relativitistic theory ofgravitation (1909–1915) are devoted not to the theory ofrelativity but with problems posed to classical physics byPlanck’s discovery of his eponymous energy constant in 1900.These early works reveal Einstein to be a strong supporter of LudwigBoltzmann rather than Ernst Mach in the debate over atomism at theturn of the century (Ryckman 2017: chapter 3). Yet in 1912,Einstein’s name together with those of Göttingenmathematicians David Hilbert and Felix Klein, was prominentlydisplayed (in theNaturwissenschaftliche Rundschau 27: 336)among those joining Mach’s in a call for the formation of a“Society for Positivist Philosophy”. Citing the pressingneed of science “but also of our age in general” for a“comprehensive world view based on the material factsaccumulated in the individual sciences”, the appeal appearsabove all to have been an orchestrated attempt to buttressMach’s positivist conception of science in the face of recentcriticisms by Max Planck, then Germany’s leading theoreticalphysicist. More a declaration of allegiance than an act of scholarlyneutrality, it provides evidence of Einstein’s youthfulenthusiasm for at least certain of Mach’s writings. Late in life(1949a: 21), Einstein wrote of the “profound influence”that Mach’sScience of Mechanics (1883) exercised uponhim as a student as well as of the very great influence in his youngeryears of “Mach’s epistemological position”. Alreadyin the special theory of relativity (1905), Einstein’soperational definition of the “simultaneity” of distantlyseparated events, whereby distant clocks are synchronized by sendingand receiving light signals, is closely modeled on the operationaldefinition of mass in Mach’sMechanics. Moreover,occasional epistemological and methodological pronouncements seem toindicate agreement with core parts of positivist doctrines ofmeaningfulness, e.g., “The concept does not exist for thephysicist until he has the possibility of discovering whether or notit is fulfilled in an actual case” (1917a [1955: 22]). Thus thegeneral theory of relativity might be seen as fully compliant withMach’s characterization of theoretical concepts as merelyeconomical shorthand for concrete observations or operations.

2.2 A “Relativization of Inertia”?

Machian influences specific to the general theory of relativityappeared even more extensive. Mach’sidée fixe,that a body’s inertial mass and behavior result from theinfluence of all other surrounding masses (thus eliminating the“monstrous” Newtonian concept of absolute space), wasperhaps the strongest motivation guiding Einstein’s pursuit of arelativistic theory of gravity. In papers leading up to the definitivepresentation of the general theory of relativity in 1916, Einsteinmade no secret of the fact that Mach was the inspiration for anepistemologically mandated attempt to generalize the principle ofrelativity. Holding, with Mach, that no observable facts could beassociated with the notions ofabsolute acceleration orabsolute inertia (i.e., resistance to acceleration), thegeneralization required that the laws of nature be completelyindependent of the state of motion of any chosen reference system. Infact, in striving to completely relativize inertia, Einstein conflateda valid principle of form invariance of the laws of nature (generalcovariance, see below) with a spurious “principle of generalrelativity”, according to which accelerated motions likerotations would be relative to an observer’s state of motion. Ina warm obituary of Mach written within a few days of completing thedefinitive 1916 presentation of his theory, Einstein, quotingextensively from the famous passages in Mach’sMechanics critical of Newton’s “absolute”concepts of space, time and motion, generously avowed thatMach’s understanding of the principles of mechanics had broughthim very close to demanding a general theory of relativity ahalf-century earlier (1916b: 102–3). No doubt with thisreference in mind, physicist Phillip Frank, later to be associatedwith the Vienna Circle, observed that

it is universally known today that Einstein’s general theory ofrelativity grew immediately out of the positivistic doctrine of spaceand motion. (1917 [1949: 68])

But, as noted above, there are both genuine and specious aspectsconnected with Einstein’s “principle of generalrelativity”, a mixture complicated by Einstein’s ownpuzzling remarks regarding the principle of general covariance.

2.3 Positivism and the “Hole Argument”

A passage from §3 of Einstein’s first complete expositionof the general theory of relativity (1916a) appeared to providefurther grist for the mill of Machian positivism. There Einsteingrandiloquently declared the requirement of general covariance for thegravitational field equations (i.e., that they remain unchanged inform under arbitrary, but invertible and suitably continuoustransformation of the spacetime coordinates), “takes away fromspace and time the last remnant of physical objectness”(translating, with Darrigol (2022, p.354) Einstein’s termGegenständlichkeit). An accompanying heuristic reflection onthe reasoning behind this claim seemed nothing less than anendorsement of Mach’s phenomenalism. “All our space-timeverifications”, Einstein wrote, “invariably amount to adetermination of space-time coincidences…”. This isbecause, Einstein presumed, all results of physical measurementultimately amount to verifications of such coincidences, such as theobservation of the coincidence of the second hand of a clock with amark on its dial, or the intersection of the worldlines of two bodies.Observing that such (topological) relations alone are preserved underarbitrary coordinate transformation, Einstein concluded that“all our physical experience can ultimately be reduced to suchcoincidences”. To Mach’s followers, Einstein’sillustrative reflection was nothing less than an explicit avowal ofthe centerpiece of Mach’s phenomenalist epistemology, thatsensations (Empfindungen), directly experienced sensoryperceptions, alone are real and knowable. Thus Josef Petzoldt, aMachian philosopher and editor of the 8^th edition ofMach’sMechanics , the first to appear after thegeneral theory of relativity, noted that Einstein’s remarksmeant that the theory “rests, in the end, on the perception ofthe coincidence of sensations” and so “is fully in accordwith Mach’s world-view, which is best characterized asrelativistic positivism” (1921: 516).

However, contemporary scholarship has shown that Einstein’sremarks here were but elliptical references to an argument (theso-called “Hole Argument”) that has only fully beenreconstructed from his private correspondence. Its conclusion is that,if a theory is generally covariant, the bare points of the spacetimemanifold can have no inherent primitive identity (inherited say, fromthe manifold’s underlying topology), and so no realityindependent of, in particular, the value of the metrical fieldassociated with each point by a particular solution of the Einsteinfield equations (Stachel 1980; Norton 1984, 1993). Thus for agenerally covariant theory, no physical reality accrues to“empty space” in the absence of any physical fields. InEinstein’s rhetorical embellishment, general covariance“takes away from space and time the last remnant of physicalobjectivity”; what he should have said is that spacetime has nointrinsic metrical structure given independently of the distributionof matter [see entry onthe hole argument]. Hence this passage is not really an endorsement of positivistphenomenalism.

2.4 “Mach’s Principle”

For a number of years Einstein expressed the ambition of the generaltheory of relativity to fully implement Mach’s program for therelativization of all inertial effects, even appending the so-calledcosmological constant in an attempt to obtain a fully Machianglobal solution to his field equations (1917b) for this purpose. Thisgenuine point of contact of Mach’s influence was clearlyidentified only in 1918, when Einstein distinguished what he baptizedasMach’s Principle—(too) strongly stated as therequirement that the metric field (responsible forgravitational-inertial properties of bodies) on the left hand side ofhis field equation, iscompletely determined by theenergy-momentum tensor on the right hand side—from the principleof general relativity which he misleadingly interpreted as theprinciple of general covariance. Taken together with the principle ofthe equivalence, Einstein asserted that the three principles, werethree pillars on which his theory rested, even if they could not bethought completely independent of one another. DespiteEinstein’s intent, there is considerable disagreement about theextent to which, if at all, the general theory of relativity couldconform to anything like Mach’s Principle. The Dutch astronomerWillem De Sitter had immediately shown in 1917 that the Einstein fieldequations, even as augmented with the new cosmological constant,permitted matter-free solutions. Machians still have some room formaneuver due to vagaries regarding just what such a principle actuallyrequires. On the other hand, it remains difficult to comprehend justwhat physical process or mechanism might implement the principle,however interpreted. How, for instance, might a given body’sinertial mass be accounted due to the influence of all other masses inthe universe? (See the discussions in Barbour & Pfister 1995.)

2.5 Emerging Anti-Positivism

As Einstein’s principal research activity turned, after 1919, tothe pursuit of a geometrical theory unifying gravitation andelectromagnetism, his philosophical pronouncements increasingly tookon a more realist or at least anti-positivist coloration. Lecturing atthe Sorbonne in April 1922 (1922: 28), Einstein pronounced Mach“un bon mécanicien” (probably a referenceto Mach’s views of the relativity of inertia) but “undéplorable philosophe”. Increasingly,Einstein’s retrospective portrayals of the genesis of generalrelativity centered almost entirely on the success of a strategyemphasizing mathematical aesthetics (see Norton 2000, Ryckman 2014,and§5). Positivists and operationalists alike continued to point to theEinstein analysis of simultaneity as relativity theory’sfundamental methodological feature. One, ruefully noting thedifficulty 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 awkwardnessin learning of Mach’s surprising disavowal of any role asforerunner to relativity theory in thePreface, dated 1913,to Mach’s posthumous book (1921) on physical optics. ThoughEinstein died without knowing differently, a recent investigation hasbuilt a strong case that this statement was forged after Mach’sdeath by his son Ludwig, under the influence of a rival guardian ofMach’s legacy and opponent of relativity theory, the philosopherHugo Dingler (Wolters 1987).

3. Kantian and Neo-Kantian Interpretations

3.1 Neo-Kantians on Special Relativity

In the universities of Imperial and early Weimar Germany, thephilosophy of Kant, particularly the various neo-Kantian schools, heldpride of place. Of these, the Marburg School of Hermann Cohen and PaulNatorp, including Cohen’s student Ernst Cassirer, exhibited aspecial interest in the philosophy of the physical sciences and ofmathematics. Yet prior to the general theory of relativity(1915–1916), Kantian philosophers accorded relativity theoryonly cursory attention. This may be seen in two leading Marburg worksappearing in 1910, Cassirer’sSubstanzbegriff undFunktionsbegriff and Natorp’sDie Logischen Grundlagender Exakten Wissenschaften. Both books conform to thecharacteristic Marburg modification that greatly extended the scope ofKant’s Transcendental Logic, bringing under “purethought” or “intellectual forms” what Kant hadsharply separated in a distinction between the passive faculty ofsensibility and the active faculties of understanding and reason. Ofcourse, this revisionist tendency greatly transforms the meaning ofKant’s Transcendental Aesthetic and with it Kant’sconviction that space and time were forms of sensibility or pureintuitionsa priori and so as well, his accounts ofarithmetic and geometry. As will be seen, it enabled Cassirer, someten years later, to view even the general theory of relativity as astriking confirmation of the fundamental tenets of transcendentalidealism. In 1910, however, Cassirer’s brief but diffusediscussion of “the problem of relativity” mentions neitherthe principle of relativity nor the light postulate nor the names ofEinstein, Lorentz or Minkowski. Rather it centers on the question ofwhether 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 howand why these ideal mathematical presuppositions are necessarilyrelated to measurable, empirical notions of space, time, and motion(1910: 228–9 [1923: 172–3]).

Natorp’s treatment, though scarcely six pages is more detailed(1910: 399–404). In revisionist fashion, the “Minkowski(sic) principle of relativity” was welcomed as a moreconsistent (as avoiding Newtonian absolutism) carrying through of thedistinction between transcendentally ideal, purely mathematical,concepts of space and time and the relative physical measuresof space and time. The relativization of time measurements, inparticular, showed that Kant, once shorn of the psychologistic errorof pure intuition, correctly maintained that time is not an object ofperception. Natorp further alleged that from this relativization itfollowed that events are ordered, not in relation to an absolute time,but only as lawfully determined phenomena in mutual temporal relationto one another, a version of Leibnizian relationism. Similarly, thelight postulate had a two-fold significance within the Marburgconception of natural science. On the one hand, the uniformity of thevelocity of light, deemed anempirical presupposition of allspace- and time-measurements, reminded that absolute determinations ofthese measures, unattainable in empirical natural science, wouldrequire a correspondingly absolute bound. Furthermore, as an upperlimiting velocity for physical processes, including gravitationalforce, the light postulate eliminated the “mysteriousabsolutism” of Newtonian action-at-a-distance. Natorp regardedthe requirement of invariance of laws of nature with respect to theLorentz transformations as “perhaps the most important result ofMinkowski’s investigation”. However, little more is saidabout this, and there is some confusion regarding thesetransformations and the Galilean ones they supersede; the former areseen as a

broadening (Erweiterung) of the old supposition of theinvariance of Newtonian mechanics for a translatory orcircular (zirkuläre, emphasis added) motion ofthe world coordinates. (1910: 403)

He concluded with an observation that the appearance of non-Euclideanand multi-dimensional geometries in physics and mathematics are to beunderstood only as “valuable tools in the treatment of specialproblems”. In themselves, they furnish no new insight into the(transcendental) logical meaning and ground of the transcendental andpurely mathematically determinedconcepts 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 componentsof Kant’s epistemological views. Several examples will sufficeto indicate characteristic “immunizing strategies”(Hentschel 1990). TheHabilitationsschrift of E. Sellien(1919), read by Einstein in view of his criticism expressed in anOctober 1919 letter to Moritz Schlick (Howard 1984: 625), declaredthat Kant’s views on space and time pertained solely tointuitive space; as such Kant’s views were impervious to themeasurable spaces and times of Einstein’s empirical theory. Thework of another young Kantian philosopher, Ilse Schneider, personallyknown to Einstein, affirmed that Kant merely had held that the spaceof three-dimensional Euclidean geometry is the space in whichNewton’s gravitational law is valid, thus no objections to Kantcould be marshalled from the four-dimensional variably curvedspacetimes of general relativity. Furthermore, Einstein’scosmology (1917b) of a finite but unbounded universe was regarded asin complete accord with the “transcendental solution” tothe First Antinomy in the Second Book of the Transcendental Dialectic.Her verdict was that apparent contradictions between relativity theoryand Kantian philosophy disappear on closer examination of bothdoctrines (Schneider 1921: 71–75).

3.3 Rejecting or Refurbishing the Transcendental Aesthetic

In fact many Kantian philosophers did not attempt to immunize Kantfrom an apparent empirical refutation by the general theory. Rather,their concern was to establish how far-reaching the necessarymodifications of Kant must be and whether, on implementation, anythingdistinctively Kantian remained. Certainly, most at risk appeared to bethe claim, in the Transcendental Aesthetic, that all objects of outerintuition, and so all physical objects, conform to the space ofEuclidean geometry. Since the general theory of relativity employednon-Euclidean (Riemannian) geometry for the characterization ofphysical phenomena, the conclusion seemed inevitable that anyassertion of the necessarily Euclidean character of physical space infinite, if not infinitesimal, regions, is simply false.

Winternitz (1924), an example of this tendency, may be singled out onthe grounds that it was deemed significant enough to be the subject ofa rare book review by Einstein (1924). Winternitz argued that theTranscendental Aesthetic is inextricably connected to the claim of thenecessarily Euclidean character of physical space and so stood indirect conflict with Einstein’s theory. The TranscendentalAesthetic accordingly must be totally jettisoned as a confusing andunnecessary appendage of the more fundamental transcendental projectof establishing thea priori logical presuppositions ofphysical knowledge. Indeed, these presuppositions have been confirmedby the general theory: They are spatiality and temporality as“unintuitive schema of order” in general (as distinct fromany particular chronometrical relations), the law of causality andpresupposition of continuity, the principle of sufficient reason, andthe conservation laws. Remarkably, thenecessity of each ofthese principles was, rightly or wrongly, was already challenged bythe developing quantum theory. (The ill-fated 1924 theory of Bohr,Kramers, and Slater attempting to unify the classical electromagneticfield with discontinuous quantum transitions in atoms is a periodchallenge to the law of conservation of energy.) According toWinternitz, thene plus ultra of transcendental idealism layin the claim that the world “is not given but posed (nichtgegeben, sondern aufgegeben) (as a problem)” out of thegiven material of sensation. Significantly, Einstein, late in life,returns to this formulation as comprising the fundamental Kantianinsight into the character of physical knowledge (1949b: 680; Ryckman2017: chapter 10).

However, a number of neo-Kantian positions, of which that of Marburgwas only the best known, did not take the core doctrine of theTranscendental Aestheticà la lettre. Rather,resources broadly within it were sought for preserving an updated“critical idealism”. In this regard, Bollert (1921) meritsmention for its technically adroit presentation of both the specialand the general theory. Bollert argued that relativity theory hadclarified the Kantian position in the Transcendental Aesthetic bydemonstrating that not space and time, but spatiality (determinatenessin positional ordering) and temporality (in order of succession) area priori conditions of physical knowledge. In so doing,general relativity theory with its variably curved spacetime, broughta further advance in the steps or levels of“objectivation” lying at the basis of physics. In thisprocess, corresponding with the growth of physical knowledge sinceGalileo, each higher level is obtained from the previous throughelimination of subjective elements from the concept of physicalobject. This ever-augmented revision of the conditions of objectivityalone is central to critical idealism. For this reason, Bollertclaimed it is “an error” to believe that “acontradiction exists between Kantiana priorism andrelativity theory” (1921: 64). As will be seen, theseconclusions are quite close to those of the much more widely knownmonograph of Cassirer (1921). It is worth noting, however, thatBollert’s interpretation of critical idealism was citedfavorably later by Gödel (1946/9-B2: 240, n.24) during the courseof research which led to his discovery of rotating universe solutionsto Einstein’s gravitational field equations (1949).Gödel’s investigation had been prompted by curiosityconcerning the similar denials, in relativity theory and in Kant, ofan absolute time.

3.4 General Covariance: A Synthetic Principle of “Unity of Determination”

The most influential of all neo-Kantian interpretations of generalrelativity was Ernst Cassirer’sZur EinsteinschenRelativitätstheorie (1921). Cassirer regarded the theory asa crucial test forErkenntniskritik, the preferred term forthe epistemology of the physical sciences of Marburg’stranscendental idealism. The question, posed right at the beginning,is whether the Transcendental Aesthetic offered a foundation“broad enough and strong enough” to bear the generaltheory of relativity. Recognizing the theory’s principalepistemological significance to lie in the requirement of generalcovariance (“that the general laws of nature are not changed inform by arbitrary changes of the space-time variables”),Cassirer directed his attention to Einstein’s remarks, cited in§2.3 above, that general covariance “takes away from space and timethe last remnant of physical objectness ”. Cassirer correctlyconstrued the gist of this passage to mean that in the general theoryof relativity, space and time coordinates have no further importancethan to be mere labels of events (“coincidences”),independent variables of the field functions characterizing physicalstate magnitudes. Indeed, the requirement of general covariance hadsignificantly improved upon Kant by bringing out far more clearly theexclusively methodological role of the transcendental conditions ofempirical cognition, a role Kant misleadingly assigned to pureintuition. Not only does the requirement show that space and time arenot “things”, it has also clarified that the concepts ofspace and time are but “ideal principles of order”applying to the objects of the physical world as a necessary conditionof their possible experience. According to Cassirer, Kant’sintention with regard to pure intuition was simply to expressthe methodological presupposition that certain “intellectualforms” (Denkformen), among which are the purely idealconcepts of coexistence and succession, enter into allphysical knowledge. According to the development of physics since the17^th century previously chronicled inSubstanzbegriffund Funktionsbegriff, these forms have progressively lost their“fortuitous” (zufälligen) anthropomorphicfeatures while more and more taking on the character of“systematic forms of unity”. From this vantage point,general covariance is but the most recent manifestation of amethodological tendency towards “unity of determination”wherein constitution of objects of physical knowledge reveals agrowing but gradual transition from concepts of substance intoconcepts of function. In accord with central tenets of the MarburgKant interpretation noted above, Cassirer maintained that therequirement of generally covariant laws vindicates the transcendentalideality of space and time, not, indeed, as “forms ofintuition” but as “objectifying conditions”further“de-anthropomorphizing” the concept of object in physics.As Cassirer will later argue, this de-anthropomorphizing tendencyultimately renders the concept of object a “purely symbolicform”. The fundamental concept of object in physics no longerpertains to particular entities or processes propagating in space andtime but rather to “the invariance of relations among (physicalstate) magnitudes” represented by generally covariantmathematical objects (tensors). For this reason, Cassirer concluded,the general theory of relativity exhibits “the most determinateapplication and carrying through within empirical science of thestandpoint of critical idealism” (1921 [1957: 71; 1923:412]).

4. Logical Empiricism

4.1 Lessons of Methodology?

Logical empiricism’s philosophy of science was conceived underthe guiding star of Einstein’s two theories of relativity, asmay be seen from the early writings of its founders, for purposeshere, Moritz Schlick, Rudolf Carnap, and Hans Reichenbach. A smallmonograph of Schlick,Space and Time in Contemporary Physics,appearing in 1917 initially in successive issues of the scientificweeklyDie Naturwissenschaften, served as prototype. One ofthe first of a host of philosophical examinations of the generaltheory of relativity, it was distinguished both by the lucidity of itslargely non-technical physical exposition and by Einstein’senthusiastic praise of its philosophical appraisal, favoringconventionalismà la Poincaré over bothneo-Kantianism and Machian positivism. The transformation of theconcept of space within the general theory of relativity was thesubject of Rudolf Carnap’s dissertation at Jena in 1921.Appearing as a monograph in 1922, it also evinced a broadlyconventionalist methodology combined with elements of Husserliantranscendental phenomenology. Distinguishing clearly betweenintuitive, physical and purely formal conceptions of space, Carnapargued that, subject to the necessary constraints of certainapriori phenomenological conditions of the topology of intuitivespace, the purely formal and the physical aspects of theories ofspace, can be adjusted to one another so as to preserve anyconventionally chosen aspect. In turn, Hans Reichenbach was one offive intrepid attendees of Einstein’s first seminar on generalrelativity given at Berlin University in the tumultuous winter of1918–1919; his detailed notebooks survive. The general theory ofrelativity was the particular subject of Reichenbach’sneo-Kantian first book (1920), which is dedicated to Albert Einstein,as well as of his next two books (1924, 1928), and of numerous papersin the 1920s.

But Einstein’s theories of relativity provided far more than thesubject matter for these philosophical examinations. Logicalempiricist philosophy of scienceitself was fashioned bylessons allegedly drawn from relativity theory in correcting orrebutting both neo-Kantian and Machian perspectives on generalmethodological and epistemological questions of science. Several ofthe most characteristic doctrines of logical empiricist philosophy ofscience—the interpretation ofa priori elements inphysical theories as conventions, the treatment of the necessary roleof conventions in linking theoretical concepts to observation, theinsistence on observational language definition of theoreticalterms—were taken to have been conclusively demonstrated byEinstein in fashioning his two theories of relativity. As mentionedabove, Einstein’s 1905 analysis of the conventionality ofsimultaneity in the special theory of relativity became amethodological paradigm for logical empiricism; it promptedReichenbach’s own method of “logical analysis” ofphysical theories into subjective (definitional, conventional) andobjective (empirical) components. An overriding concern in the logicalempiricist treatment of relativity theory was to draw broad lessonsfor scientific methodology and philosophy of science generally,although issues more specific to the philosophy of physics were alsoaddressed. Only the former are considered here; for a discussion ofthe latter, see Ryckman 2007.

4.2 From the “Relativized A priori” to the “Relativity of Geometry”

A cornerstone of Reichenbach’s logical analysis of the theory ofgeneral relativity is the thesis of “the relativity ofgeometry”, that an arbitrary geometry may be ascribed tospacetime (holding constant the underlying topology) if the laws ofphysics are correspondingly modified through the introduction of“universal forces”. This particular argument for metricconventionalism has generated substantial controversy on its own, butis better understood through an account of its genesis inReichenbach’s early neo-Kantianism. Independently of thatgenesis, the thesis becomes the paradigmatic illustration ofReichenbach’s broad methodological claim that conventional ordefinitional elements—“coordinative definitions”associating mathematical concepts of the physical theory with“elements of physical reality”—are a necessarycondition for empirical cognition in the mathematical sciences ofnature. At the same time, Reichenbach’s thesis of metricalconventionalism is part and parcel of an audacious program ofepistemological reductionism regarding spacetime structures. This wasfirst attempted in his “constructive axiomatization”(1924) of the theory of relativity on the basis of “elementarymatters of fact” (Elementartatbestande) regarding theobservable behavior of lights rays, and rods and clocks. Here, and inthe more widely read treatment (1928), metrical properties ofspacetime are deemed less fundamental than topological ones, while thelatter are derived from the concept of time ordering. But time orderin turn is reduced to that of causal order and so the whole edifice ofstructures of spacetime is considered epistemologically derivative,resting upon ultimately basic empirical facts about causal order and aprohibition against action-at-a-distance. The end point ofReichenbach’s epistemological analysis of the foundations ofspacetime theory is then “the causal theory of time”, atype of relational theory of time that assumes the validity of thecausal principle of action-by-contact(Nahewirkungsprinzip).

However, Reichenbach’s first monograph on relativity (1920) waswritten from within a neo-Kantian perspective. As Friedman (1994) andothers (Ryckman 2005) have discussed in detail Reichenbach’sinnovation, a modification of the Kantian conception ofsynthetica priori principles, rejecting the sense of “valid for alltime” while retaining that of “constitutive of the object(of knowledge)”, led to the conception of a theory-specific“relativizeda priori”. According to Reichenbach,any physical theory presupposes the validity of systems of certain,quite general, principles which however may vary from theory totheory. Thesecoordinating principles, as they are thentermed, are indispensable for the ordering of perceptual data; theydefine the objects of knowledge within the theory. The epistemologicalsignificance of relativity theory, according to the early Reichenbach,is to have shown, contrary to Kant, that these systems may containmutually inconsistent principles, and so require emendation to removecontradictions. Thus a “relativization” of the Kantianconception ofsynthetic a priori principles is the directepistemological result of the theory of relativity. But this findingis also taken to signal a transformation in the method ofepistemological investigation of science. In place of Kant’s“analysis of Reason”, “the method of analysis ofscience” (der wissenschaftsanalytische Methode) isproposed as “the only way that affords us an understanding ofthe contribution of our reason to knowledge” (1920: 71 [1965:74]). The method’sraison d’être is tosharply distinguish between the subjective role of (coordinating)principles—“the contribution of Reason”—andthe contribution of objective reality, represented by theory-specificempirical 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 ithas shown that the metric of spacetime describes an “objectiveproperty” of the world, once the subjective freedom to makecoordinate transformations (the coordinating principle of generalcovariance) is recognized (1920: 86–7 [1965: 90]). The thesis ofmetric conventionalism had yet to appear.

But soon it did. Still in 1920, Schlick objected, both publicly and inprivate correspondence with Reichenbach, that “principles ofcoordination” were precisely statements of the kind thatPoincaré had termed “conventions” (see Coffa 1991:201ff.). Moreover, Einstein, in a lecture of January 1921, entitled“Geometry and Experience”, appeared to lend support tothis view. Einstein argued that the question concerning the nature ofspacetime geometry becomes an empirical question only on certainpro tem stipulations regarding the “practically rigidbody” of measurement (pro tem in view of theinadmissibility in relativity theory of the concept “actuallyrigid body”). In any case, by 1922, the essential pieces ofReichenbach’s mature conventionalist view had emerged. Theargument is canonically presented in §8 (entitled “TheRelativity of Geometry”) ofDer Philosophie derRaum-Zeit-Lehre (completed in 1926, published in 1928). In a movesuperficially similar to the argument of Einstein’s“Geometry and Experience”, Reichenbach maintained thatquestions concerning the empirical determination of the metric ofspacetime must first confront the fact that only the whole theoreticaledifice comprising both geometry and physics admits of observationaltest. Einstein’s gravitational theory is such a totality.However, unlike Einstein, Reichenbach’s “method ofanalysis of science”, later re-named “logical analysis ofscience”, is directed to the epistemological problem offactoring this totality into its conventional or definitional and itsempirical components.

This is done as follows. The empirical determination of the spacetimemetric by measurement requires choice of some “metricalindicators”: this can only be done by laying down acoordinative definition stipulating, e.g., that the metricalnotion of length is coordinated to some physical object or process. Astandard choice coordinates lengths with “infinitesimalmeasuring rods” supposed rigid (e.g., Einstein’s“practically rigid body”). This however is only aconvention, and other physical objects or processes might be chosen.(In Schlick’s fanciful example, the Dali Lama’s heartbeatcould be chosen as the physical process to which units of time arecoordinated.) Of course, the chosen metrical indicators must becorrected 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 variousmaterials differently. However, Reichenbach argued, the choice of arigid rod as standard of length is tantamount to the claim that thereare no non-differential—“universal”—distortingforces that affect all bodies in the same way and cannot be screenedoff. In the absence of “universal forces” the coordinativedefinition regarding rigid rods can be implemented and the nature ofthe spacetime metric empirically determined, for example, finding thatpaths of light rays through solar gravitational field are notEuclidean straight lines. Thus, the theory of general relativity, onadoption of the coordinative definition of rigid rods(“universal forces = 0”), affirms that the geometry ofspacetime in a given region is of a non-euclidean kind. The point,however, is that this conclusion rests on the convention governingmeasuring rods. One could, alternately, maintain that the geometry ofspacetime was Euclidean by adopting a different coordinativedefinition, for example, holding that measuring rods expanded orcontracted depending on their location in spacetime, a choicetantamount to the supposition of “universal forces”. Then,consistent with all empirical phenomena, it could be maintained thatEuclidean geometry was compatible with Einstein’s theory if onlyone allowed the existence of such forces. Thus whether generalrelativity affirms a Euclidean or a non-euclidean metric in the solargravitational field rests upon a conventional choice regarding theexistence of non-zero universal forces. Either hypothesis may beadopted since they are empirically equivalent descriptions; theirjoint possibility is referred to as “the relativity ofgeometry”. Just as with the choice of standard synchrony inReichenbach’s analysis of the conventionality of simultaneity, achoice also held to be “logically arbitrary”, Reichenbachrecommends the “descriptively simpler” alternative inwhich universal forces do not exist. To be sure, “ descriptivesimplicity has nothing to do with truth”, i.e., has no bearingon the question of whether the spacetime metric really has anon-Euclidean structure (1928: 47 [1958: 35]).

4.3 Critique of Reichenbachian Metric Conventionalism

In retrospect, it is rather difficult to understand the significancethat has been accorded this argument. Carnap, for example, in“Introductory Remarks” (Carnap 1956 [Reichenbach 1958:vii]) to the posthumous English translation of this work, singled itout on account of its “great interest for the methodology ofphysics”. Reichenbach himself deemed “the philosophicalachievement of the theory of relativity” to lie in thismethodological distinction between conventional and factual claimsregarding spacetime geometry (1928: 24 [1958: 15]), and he boasted ofhis “philosophical theory of relativity” as anincontrovertible “philosophical result”:

the philosophical theory of relativity, i.e., the discoveryof the definitional character of the metric in all its details, holdsindependently of experience.… a philosophical result notsubject to the criticism of the individual sciences. (1928: 223 [1958:177])

Yet this result is neither incontrovertible nor an untrammeledconsequence of Einstein’s theory of gravitation. There is, firstof all, the shadowy status accorded to universal forces. A sympatheticreading (e.g., Dieks 1987) suggests that the notion serves usefully inmediating between a traditionala priori commitment toEuclidean geometry and the view of modern geometrodynamics, wheregravitational force is “geometrized away” (see§5). After all, as Reichenbach explicitly acknowledged, gravitation isitself a universal force, coupling to all bodies and affecting them inthe same manner (1928: 294–6 [1958: 256–8]). Hence thechoice recommended by descriptive simplicity is merely a stipulationthat infinitesimal metrical appliances be considered as“differentially at rest” in an inertial system (1924: 115[1969: 147]). This is a stipulation that spacetime measurements alwaystake place in regions that are to be considered small Minkowskispacetimes (arenas of gravitation-free physics). By the same token,consistency then required an admission that “the transition fromthe special theory to the general one represents merely a renunciationof metrical characteristics” (1924: 115 [1969: 147]), or, evenmore pointedly, that “all the metrical properties of thespacetime continuum are destroyed by gravitational fields” whereonly topological properties remain (1928: 308 [1958: 268–9]). Tobe sure, these bizarre conclusions are supposed to be rendered morepalatable in connection with the epistemological reduction ofspacetime structures in the causal theory of time.

Despite the influence of this argument on the subsequent generation ofphilosophers of science, Reichenbach’s analysis of spacetimemeasurement is plainly inappropriate, manifesting a fallacioustendency to view the generically curved spacetimes of generalrelativity as stitched together from little bits of flat Minskowskispacetimes. Besides being mathematically inconsistent, this procedureoffers no way of providing a non-metaphorical physical meaning for thefundamental metrical tensor \(g_{\mu\nu}\), the central theoreticalconcept of general relativity, nor to the series of curvature tensorsderivable from it and its associated affine connection. Since thesesectional curvatures at a point of spacetime are empiricallymanifested and the curvature components can be measured, e.g., as thetidal forces of gravity, they can hardly be accounted as due toconventionally adopted “universal forces”. Furthermore,the concept of an infinitesimal rigid rod in general relativity cannotreally 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 relativityas allowing instantaneous causal actions. Moreover, such a rod mustindeed be infinitesimal, i.e., a freely falling body of negligiblethickness and of sufficiently short extension, so as to not bestressed by gravitational field inhomogeneities; just how shortdepending on strength of local curvatures and on measurement error(Torretti 1983: 239). But then, as Reichenbach appeared to haverecognized in his comments about the “destruction” of themetric by gravitational fields, it cannot serve as a coordinatelydefined general standard for metrical relations. In fact, as Weyl wasthe first to point out, precisely which physical objects or structuresare most suitable as measuring instruments should be decided on thebasis of gravitational theory itself. From this enlightenedperspective, measuring rods and clocks are structures that are far toophysically complex. Rather, the metric in the region surrounding anyobserver O can be empirically determined from freely falling ideallysmall neutral test masses together with the paths of light rays. Moreprecisely stated, the spacetime metric results from theaffine-projective structure of the behavior of neutral test particlesof negligible mass and from the conformal structure of light raysreceived and issued by the observer (Weyl 1921). Any purelyconventional stipulation regarding the behavior of measuring rods asphysically constitutive of metrical relations in general relativity isthen otiose (Weyl 1923a; Ehlers, Pirani and Schild 1973; Geroch 1978).Alas, since Reichenbach reckoned the affine structure of thegravitational-inertial field to be just as conventional as itsmetrical structure, he was not able to recognize this method as otherthan an equivalent, but by no means necessarily preferable, account ofthe empirical determination of the metric through the use of rods andclocks (Coffa 1979; Ryckman 2005: chs. 2 & 4; Giovanelli2013b).

5. “Geometrization of Physics”: Platonism, Transcendental Idealism, Structuralism

5.1 Differing Motivations

In the decade or so following the appearance of the general theory ofrelativity, there was much talk of a reduction of physics to geometry(e.g., Hilbert 1917; Weyl 1918b, 1919; Haas 1920; Lodge 1921). Whilethese discussions were largely, and understandably, confined toscientific circles, they nonetheless brought distinctly philosophicalissues—of methodology, but also of epistemology andmetaphysics—together with technical matters. Einstein’smathematical representation of gravitational field potentials by themetric tensor \(g_{\mu\nu}\) of a variably curved spacetime geometrywas quickly termed a “geometrization of gravitationalforce”. Weyl and others acclaimed Einstein as reviving ageometrizing tendency essentially dormant within physics since the17th century. In so doing, Einstein supposedly opened up the prospectof a complete geometrization of physical theory, the possibility offinding a unifying representation of all of known physicalinteractions within the frame of a unique metrical theory of thefour-dimensional spacetime continuum. Einstein himself, as will beseen, turned out to be highly critical of the idea of“geometrizing physics” even as the general theory ofrelativity was its inspiration. This was certainly the case for theGöttingen mathematician David Hilbert (1915, 1917). In the veryweek in late November 1915 when Einstein presented his completedgravitational theory to the Berlin Academy (on Thursday), Hilbert (theprevious Monday) proposed to the Göttingen Academy a schematicgenerally covariant axiomatization coupling Einstein’sgravitational theory with a relativistic electromagnetic theory ofmatter due to German physicist Gustav Mie. Hilbert’s theorycould not possibly succeed (however, that matter is not fundamentallyelectromagnetic in nature was only shown in the 1930s by the quantumphysics of the atomic nucleus). It is remembered today mostly for theso-called “Einstein-Hilbert action” in the usualvariational formulation of Einstein’s theory. But at the timeHilbert viewed his theory as a triumph of his “axiomaticmethod” as well as a demonstration that

physics is a four-dimensional pseudogeometry [i.e., a geometrydistinguishing spatial and temporal dimensions] whose metricdetermination \(g_{\mu\nu}\) is bound, according to the fundamentalequations … of my first [1915] contribution, to theelectromagnetic quantities, that is, to matter. (1917: 63, translationby the author)

However, this implied reduction of physics to geometry was cruciallyobtained within the epistemological frame of what Hilbert termed“the axiomatic method”; its intended significance is thatof a proposed solution to the 6th (“the axiomatization ofphysics”) of the famous 23 mathematical problems posed byHilbert at the 1900 International Congress of Mathematicians in Paris(Brading & Ryckman 2008). Others pursued a different path, seekingthe reduction of physics to geometry by generalizing beyond thepseudo-Riemannian geometry (a Riemannian geometry distinguishingbetween space and time dimensions) of general relativity (HermannWeyl, Arthur Stanley Eddington) or by employing Riemannian geometry infive dimensions (Theodore Kaluza). Whatever may have beenKaluza’s philosophical motivations (van Dongen 2010:132–5), neither mathematical realism nor Platonism played a rolein Weyl’s (1918a,b), and following Weyl, Eddington’s(1921) generalizations of Riemannian geometry. Their proposals wereabove all explicit attempts to comprehend the nature of fundamentalphysical theory in the light of general relativity, from systematicepistemological standpoints neither positivist nor realist. As suchthey comprise early philosophical interpretations of that theory,although they intertwine philosophy, geometry and physics in a mannerunprecedented since Descartes.

After completing general relativity, Einstein long entertained hopesthat matter might be described in geometrical terms by a theoryunifying gravitation and electromagnetism. But he stoutly andrepeatedly resisted proclamations of any reduction of physics togeometry (e.g., 1928: 254; Giovanelli 2013a). Theoretical unificationwas overriding goal; no special significance would accrue to anysuccessful unifying theory whose physical objects, motions, andinteractions are described in geometrical terms (Lehmkuhl 2014).Einstein nonetheless followed both approaches to theoreticalunification, that of generalizing Riemannian geometry and by addingextra dimensions. By 1923, Einstein was the recognized leader of theunification program (Vizgin 1985 [1994: 265]) and by 1925 had devisedhis first “homegrown” geometrical “unified fieldtheories” (Sauer 2014). The first phase of the geometricalunification program essentially ended with Einstein’s“distant parallelism” theory of 1928–1931 (e.g.,1929), an inadvertent public sensation (Fölsing 1993 [1997:605]). Needless to say, none of these efforts met with success. In alecture at the University of Vienna on October 14, 1931, Einsteinforlornly referred to his failed attempts, each conceived on adistinct differential geometrical basis, as a “graveyard of deadhopes” (Einstein 1932). By this time, certainly, the viableprospects for the geometrical unification program had considerablywaned. A consensus emerged among nearly all leading theoreticalphysicists that while unification of the gravitation andelectromagnetic fields might be attained in formally distinct ways,the problem of matter, treated with undeniable empirical success bythe new quantum theory, was not to be resolved solely within theconfines of classical fields and spacetime geometry. In any event,from the early 1930s, prospects of any type of unification programappeared greatly premature in view of the wealth of data produced bythe new quantum mechanics of the nucleus.

Still, unsuccessful pursuit of the goal of geometrical unificationabsorbed Einstein and his various research assistants for more thanthree decades, up to Einstein’s death in 1955. In the course ofit, Einstein’s research methodology underwent a dramatic change(Ryckman 2014; Ryckman 2017: chapters 9 and 10). In place ofphysically warranted principles to guide theoretical construction,such as the equivalence between inertial and gravitational mass thathad set him on the path to his greatest success with generalrelativity, Einstein increasingly relied on considerations ofmathematical aesthetics, of “logical simplicity”, and theinevitability of certain mathematical structures under variousconstraints, adopted essentially for philosophical reasons. In a talkentitled “On the Method of Theoretical Physics” at Oxfordin 1933, the transformation was stated dramatically:

Experience remains, of course, the sole criterion of the physicalutility of a mathematical construction. But the creative principleresides in mathematics. In a certain sense, therefore, I hold it truethat pure thought can grasp reality, as the ancients dreamed. (1933:274)

Even decades of accumulating empirical successes of the new quantumtheory did not dislodge Einstein’s core metaphysical conceptionof physical reality of continuous field functions defined on aspacetime manifold (“total field”) where particles and theconcept of motion are derived notions (e.g., 1950: 348).

5.2 The Initial Step: “Geometrizing” Gravity

The “geometrization of gravitational force” in 1915 gavethe geometrization program its first, partial, realization as well asits subsequent impetus. In Einstein’s theory, the fundamental or“metric” tensor \(g_{\mu\nu}\) of Riemannian geometryappears in a dual role which thoroughly fuses its geometrical and itsphysical meanings. As is apparent from the expression for thedifferential interval between neighboring spacetime events,\(\textit{ds}^{2}= g_{\mu\nu} \textit{dx}^{\mu} \textit{dx}^{\nu}\)(here, and below there is an implicit summation over repeated upperand lower indices), the metric tensor is at once the geometricalquantity underlying measurable metrical relations of lengths andtimes. In this role it ties a mathematical theory of events infour-dimensional curved spacetime to observations and measurements inspace and time. But it is also the potential of the gravitational (or“metrical”) field whose value, at any point of spacetimearises as a solution of the Einstein Field Equations (see below) forgiven physical quantities of mass-momentum-stress in the immediatesurrounding region. In the new view, the idea of strength ofgravitational force is replaced by that of degree of spacetimecurvature. This curvature is manifested, for example, by the tidalforce of the Earth’s gravitational field that occasions twofreely falling bodies, released at a certain height and at fixedseparation, to approach one another. A freely falling body (ideally,an uncharged test particle of negligible mass) in a gravitationalfield follows a geodesic path (that is, a curve of extremal length ina generally curved pseudo-Riemannian spacetime). The body is no longerto be regarded as moving through space according to the pull of anattractive gravitational force, but simply as tracing out the laziest(longest or slowest possible) four-dimensional trajectory between twofinitely separated spacetime events. In consequence, in generalrelativity the equation of motion of a free body is ageodesicequation according to which the body’s spacetime (four-)acceleration vanishes identically and its free fall becomesindistinguishable from inertial motion. According to this equation, afree body moves on a geodesic path in both thepresence andabsence of a gravitational field. This is possible becausethe equation contains a multi-index term (not a tensor) called aconnection allowing either a gravitational field or aninertial field according to a given choice of spacetime coordinates.As a result, in general relativity there is no observer-independent(coordinate-free) way to partition the combined inertial-gravitationalfield into its separate components. The gross mechanical properties ofbodies, comprising all gravitational-inertial phenomena, can bederived as the solution of a single linked system of ten generallycovariant partial differential equations, the Einstein equations.According to these equations, spacetime and matter stand in dynamicalinteraction. One abbreviate way of characterizing the dual role of the\(g_{\mu\nu}\) is to say that in the general theory of relativity,gravitation, which includes mechanics, has becomegeometrized, i.e., incorporated into the geometry ofspacetime. Yet Einstein objected to claims that in general relativitygravitation isreduced to geometry as that statement’snatural interpretation severs the unification of inertia andgravitation that to Einstein comprised the theory’s coreachievement.

5.3 Extending Geometrization

In making spacetime curvature dependent on distributions of mass andenergy, general relativity is indeed capable of encompassing all(non-quantum) physical fields. However, in classical generalrelativity there remains a fundamental asymmetry between gravitationaland non-gravitational fields, in particular, electromagnetism, theonly other fundamental physical interaction definitely known until the1930s. This shows up visibly in one form of the Einstein fieldequations in which, on the left-hand side, a geometrical object(\(G_{\mu\nu}\), the Einstein tensor) built up from the uniquelycompatible linear symmetric (“Levi-Civita”) connectionassociated with the metric tensor \(g_{\mu\nu}\), and representing thecurvature of spacetime, is set identical to a tensorial butnon-geometrical phenomenological representation of matter on theright-hand side.

The expression on the right side, introduced by a coupling term thatcontains Newton’s gravitational constant, mathematicallyrepresents the non-gravitational sources of the gravitational field ina region of spacetime in the form of a stress-energy-momentum tensor(an “omnium gatherum” in Eddington’s (1919:63) pithy phrase). As the geometry of spacetime principally resides onthe 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 (theright) of “low grade wood” (1936: 311). In its classicalform, general relativity accords only the gravitational field a directgeometrical significance; the other physical fields resideinspacetime; they are notof spacetime.

Einstein’s dissatisfaction with this asymmetrical state ofaffairs was palpable at an early stage and was expressed withincreasing frequency beginning in the early 1920s. A particularlyvivid declaration of the need for geometrical unification was made inhis “Nobel lecture” of July 1923:

The mind striving after unification of the theory cannot be satisfiedthat two fields should exist which, by their nature, are quiteindependent. A mathematically unified field theory is sought in whichthe gravitational field and the electromagnetic field are interpretedas only different components or manifestations of the same uniformfield,… The gravitational theory, considered in terms ofmathematical formalism, i.e., Riemannian geometry, should begeneralized so that it includes the laws of the electromagnetic field.(489)

It might be noted that the tacit assumption, evident here, thatincorporation of electromagnetism into spacetime geometry requires ageneralization of the Riemannian geometry of general relativity;though widely held at the time, it is not quite correct (Rainich 1925;Misner & Wheeler 1957; Geroch 1966).

5.4 “Pure Infinitesimal Geometry”

It wasn’t Einstein, but the mathematician Hermann Weyl who firstaddressed the asymmetry in 1918 in the course of reconstructingEinstein’s theory on the preferred epistemological basis of a“pure infinitesimal geometry” (ReineInfinitesimalgeometrie). Holding thatdirect—evident, in the sense of theWesensschau of Husserlian phenomenoloy—comparisons oflength or duration could be made at near-by points of spacetime, butnot, as the Riemannian geometry of Einstein’s theory allowed,“at a distance”, Weyl discovered an expanded geometry withadditional terms that, following Einstein’s example, he simplyformally identified with the potentials of the electromagnetic field.From these, the electromagnetic field strengths can be immediatelyderived. Choosing an action integral to obtain both the homogeneousand the inhomogeneous Maxwell equations as well as Einstein’sgravitational theory, Weyl could express electromagnetism as well asgravitation solely within the confines of a spacetime geometry. As noother interactions were definitely known to occur, Weyl proudlydeclared that the concepts of geometry and physics were the same.Hence, everything in the physical world was a manifestation ofspacetime geometry.

(The) distinction between geometry and physics is an error, physicsextends not at all beyond geometry: the world is a \((3+1)\)dimensional metrical manifold, and all physical phenomena transpiringin it are only modes of expression of the metric field, ….(M)atter itself is dissolved in “metric” and is notsomething substantial that in addition exists “in” metricspace. (1919: 115–116, translation by the author)

By the winter of 1919–1920, for both physical and philosophicalreasons (the latter having to do with his largely positive reaction toBrouwer’s intuitionist views about the mathematical continuum,in particular, the continuum of spacetime—see Mancosu &Ryckman 2005), Weyl (1920) surrendered the belief, expressed here,that matter, with its corpuscular structure, might be derived withinspacetime geometry. Thus he gave up the Holy Grail of the nascentunified field theory program almost before it had begun. Nonetheless,he actively defended his theory well into the 1920s, essentially onthe grounds of Husserlian transcendental phenomenology, that hisgeometry and its central principle, “the epistemologicalprinciple of relativity of magnitude” comprised a superiorepistemological framework for general relativity. Weyl’spostulate of a “pure infinitesimal” non-Riemannian metricfor spacetime allowed the “gauge” (by which Weyl meant ascale of length; the term has a different meaning today) to vary ateach spacetime point. But it met with intense criticism. Noobservation spoke in favor of it; to the contrary, Einstein pointedout that according to Weyl’s theory, the atomic spectra of thechemical elements should not be stable, as indeed they are observed tobe. Although Weyl responded to this objection forcefully, and withsome subtlety (Weyl 1923a), he was able to persuade neither Einstein,nor any other leading relativity physicist, with the exception ofEddington. However, the idea of requiringgauge invariance offundamental physical laws was revived and vindicated by Weyl himselfin a different form later on (Weyl 1929; see Ryckman 2005: chs. 5& 6; O’Raifeartaigh 1997; Scholz 2001, 2004; Afriat 2017,Other Internet Resources). Weyl’s 1918 generalization ofRiemannian geometry witnessed a resurgence in the 1970s and itsframework continues to be a productive resource in theoretical physics(Scholz, 2018).

5.5 Eddington’s World Geometry

Despite failure to win many friends for his theory, Weyl’sguiding example of unification launched the geometrical program ofunified field theory, initiating a variety of efforts, all aimed atfinding a suitable generalization of the Riemannian geometry ofEinstein’s theory to encompass as well non-gravitational physics(Vizgin 1985 [1994: chapter 4]). In December 1921, the Berlin Academypublished Theodore Kaluza’s novel proposal for unification ofgravitation and electromagnetism upon the basis of a five-dimensionalRiemannian geometry. But earlier that year, in February, came ArthurStanley Eddington’s further generalization of Weyl’sfour-dimensional geometry, wherein the sole primitive geometricalnotion is the non-metrical comparison of direction or orientation atthe same or neighboring points. In Weyl’s geometry the magnitudeof 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 samedirection. His “theory of the affine field” encompassedboth Weyl’s geometry and the semi-Riemannian geometry ofEinstein’s general relativity as special cases. Little attentionwas paid however, to Eddington’s claim, prefacing his paper,that his objective had not been to “seek (the) unknown laws (ofmatter)” 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 onthe origin of the fundamental laws of physics” (1921: 105).

Eddington was persuaded that Weyl’s “principle ofrelativity of length” was “an essential part of therelativistic conception”, a view he retained to the end of hislife (e.g., 1939: 28). But he was also convinced that the largelyantagonistic reception accorded Weyl’s theory was due to itsconfusing formulation. The flaw lay in Weyl’s failure to maketransparently obvious that the locally scale invariant (“pureinfinitesimal”) “world geometry” was not thephysical geometry of actual spacetime, but an entirely mathematicalconstruct inherently serving to specify the ideal of anobserver-independent external world. To remedy this, Eddington deviseda general method of deductive presentation of field physics in which“world geometry” is developed mathematically asconceptually separate from physics. A “world geometry” isa purely mathematical construction the derived objects of whichpossess only structural properties requisite to the ideal of acompletely impersonal world; these are objects, as he wrote inSpace, Time and Gravitation (1920), a semi-popularbest-seller, represented “from the point of view of no one inparticular”. Naturally, this ideal had changed with the progressof physical theory. In the light of relativity theory, such a world isindifferent to specification of reference frame and, after Weyl, ofgauge of length (scale). A world geometry is not the physical theoryof such a world but a framework or “graphicalrepresentation” within whose terms existing physical theorymight be displayed, essentially through a purely formal identificationof known tensors of the existing physical laws of gravitation andelectromagnetism with those derived within the world geometry. Such ageometrical representation of physics cannot really be said to beright or wrong, for it only implements, if it can, current ideasgoverning the conception of objects and properties of an impersonalobjective external world. But when existing physics, in particular,Einstein’s theory of gravitation, is set in the context ofEddington’s world geometry, it yields a surprising consequence:The Einstein law of gravitation appears as a definition! In the form\(R_{\mu\nu} = 0\) it defines what in the “world geometry”appears to the mind as “vacuum” while in the form of theEinstein field equation noted above, it defines what is to beencountered by the mind as “matter”. This result is whatis meant by Eddington’s stated claim of throwing “newlight on the origin of the fundamental laws of physics” (seeRyckman 2005: chapters 7 & 8). Eddington’s notoriouslydifficult and opaque later works (1936, 1946), carried his viewpointfurther in attempting to show that “the substratum of everythingis of mental character” (1928, p.281). In the aftermath ofquantum mechanics and Dirac’s relativistic theory of theelectron, Eddington pursued an algebraic, not geometric, programseeking to derive fundamental physical laws, and in particular theconstants occurring in them, froma priori epistemologicalprinciples. This philosophy, termed “selectivesubjectivism” (1939), argued that fundamental physical theory isnot capable of revealing the world as it is in itself but merelyrepresents relations between subjectively selected observables andhence reflect the interpretation the physicist imposes on the data ofobservation.

5.6 Meyerson on “Pangeometrism”

Within physics the idealist currents lying behind the world geometriesof Weyl and Eddington were largely ignored, whereas within philosophy,EÉmile Meyerson’sLa DéductionRelativiste (1925) was a notable exception in considering thephilosophical perspectives of both authors at considerable length.Meyerson, who had no doubt concerning the basic realist impetus ofscience, carefully distinguished Einstein’s “rationaldeduction of the physical world” from the speculativegeometrical unifications of gravitation and electromagnetism of Weyland Eddington. These theories, as affirmations of a completepanmathematicism, or rather of apangeometrism(1925: §§ 157–58), were compared to the rationalistdeductions of Hegel’sLogic. That general relativitysucceeded in partly realizing Descartes’ program of reducing thephysical to the spatial through geometric deduction, is due to thefact that Einstein “followed in the footsteps” ofDescartes, not Hegel (1925: §133). Butpan-geometrism isalso capable of overreaching itself and this is the transgressioncommitted by both Weyl and Eddington. Weyl in particular is singledout for criticism for seemingly to have reverted to Hegel’smonistic idealism, and so to be subject to its fatal flaw. Inregarding nature as completely intelligible, Weyl had abolished thething-in-itself and so promoted the identity of self and non-self, thegreat error of theNaturphilosophien.

Though he had “all due respect to the writings of suchdistinguished scientists” as Weyl and Eddington, Meyerson tooktheir overt affirmations of idealism to be misguided attempts“to associate themselves with a philosophical point of view thatis in fact quite foreign to the relativistic doctrine” (1925:§150). That “point of view” is in fact two distinctspecies of transcendental idealism. It is above all“foreign” to relativity theory because Meyerson cannot seehow it is possible to “reintegrate the four-dimensional world ofrelativity theory into the self”. After all, Kant’s ownargument for Transcendental Idealism proceeded “in a singlestep”, 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 theself”. Any attempt to “reintegrate” four-dimensionalspacetime into the self would have to proceed at a “secondstage” where, additionally, there would be no “solidfoundation” such as spatial and temporal intuition furnishedKant at the first stage. Perhaps, Meyerson allowed, there is indeed“another intuition, purely mathematical in nature”, lyingbehind spatial and temporal intuition, and capable of “imaginingthe four-dimensional universe, to which, in turn, it makes realityconform”. This would make intuition a “two-stagemechanism”. While all of this is not“inconceivable”, it does appear, nonetheless,“rather complex and difficult if one reflects upon it”. Inany 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 thatrelativity theory tends to destroy the concept of Kantian intuition.(1925: §§ 151–2)

Meyerson had come right up to the threshold of grasping theWeyl-Eddington geometric unification schemes in something like thesense in which they were intended. The stumbling block for him, andfor others, is the conviction that transcendental idealism can besupported only from an argument about the nature of intuition, andintuitive representation. To be sure, the geometric framework forWeyl’s construction of the objective four-dimensional world ofrelativity is based upon theEvidenz available in“essential insight”, which is limited to the simple linearrelations and mappings in what is basically the tangent vector space\(T_P\) at a pointP in a manifold. Thus in Weyl’sdifferential geometry there is a fundamental divide between integrableand non-integrable relations of comparison. The latter are primitiveand epistemologically privileged, but nonetheless not justified untilit is shown how the infinitesimal homogeneous spaces, corresponding tothe “essence of space as a form of intuition”, arecompatible with the large-scale inhomogeneous spaces (spacetimes) ofgeneral relativity. And this required not a philosophical argumentabout the nature of intuition, but one formulated in group-theoreticconceptual form (Weyl 1923a,b). Eddington, on the other hand,without the cultural context of Husserlian phenomenology or indeed ofphilosophy generally, jettisoned the intuitional basis oftranscendental idealism altogether, as if unaware of its prominence.Thus he sought a superior and completely generalconceptualbasis for the objective four-dimensional world of relativity theory byconstituting that world within a geometry (its “worldstructure” (1923)) based upon a non-metrical affine (i.e.,linear and symmetric) connection. He was then free to find his own wayto the empirically confirmed integrable metric relations ofEinstein’s theory without being hampered by the conflict of a“pure infinitesimal” metric with the observed facts aboutatomic spectra.

5.7 “Structural Realism”?

It has been routinely assumed that all the attempts at ageometrization of physics in the early unified field theory programshared something of Einstein’s hubris concerning the ability ofmathematics to grasp the fundamental structure of the external world.The geometrical unified field theory program thus appears to beinseparably stitched to a form of scientific realism, termed“structural realism”, with perhaps even an inspired turntoward Platonism. According to one (now termed“epistemic”) form of “structural realism”,whatever the intrinsic character or nature of the physical world, onlyits structure can be known, a structure of causal or other modalrelations between events or other entities governed by the equationsof the theory. The gist of this version of structural realism wasfirst clearly articulated by Russell in his Tarner Lectures at TrinityCollege, Cambridge in 1926. As Russell admits, it was the generaltheory of relativity, particularly in the formulation given withinEddington’s world geometry, that led him to structuralismregarding cognition of physical reality (Russell 1927: 395). Russell,however, rested the epistemic limitation to knowledge of the structureof the physical world on the causal theory of perception. As such,structural features of relations between events not present in thepercepts of these events could only be inferred according to generallaws; hence, posits of unobserved structural features of the world areconstrained by exigencies of inductive inference. Moreover,Russell’s structural realism quickly fell victim to a ratherobvious objection lodged by the mathematical Max Newman (see entry onstructural realism).

In its contemporary form, structural realism has both an epistemic andan “ontic” form, the latter holding essentially thatcurrent physical theories warrant that the structural features of thephysical world alone are ontologically fundamental (Ladyman & Ross2007). Both versions of structural realism subscribe to a view oftheory change whereby the sole ontological continuity across changesin fundamental physical theory is continuity of structure, as ininstances where the equations of an earlier theory can be derived, sayas limiting cases, from those of the later. Geometrical unificationtheories seems tailored for this kind of realism. For if a geometricaltheory is taken to give a true or approximately true representation ofthe physical world, it provides definite structure to relationsposited as fundamental and presumably preserved in any subsequentgeometrical generalization. It is therefore instructive to recall thatfor both Weyl and Eddington geometrical unification was not, nor couldbe, such a representation, for essentially the reasons articulated twodecades before by Poincaré:

Does the harmony the human intelligence thinks it discovers in natureexist outside of this intelligence? No, beyond doubt, a realitycompletely independent of the mind which conceives it, sees or feelsit, is an impossibility. A world as exterior as that, even if itexisted, would for us be forever inaccessible. But what we callobjective reality is, in the last analysis, what is common to manythinking beings, and could be common to all; this commonpart,…, can only be the harmony expressed by mathematical laws.It is this harmony then which is the sole objective reality….(1906: 14)

In Weyl and Eddington, geometrical unification was an attempt to castthe harmony of the Einstein theory of gravitation into a newepistemological light, displaying the field laws of gravitation andelectromagnetism within the common frame of a geometricallyrepresented physical reality. Their unorthodox manner of philosophicalargument, cloaked, perhaps necessarily, in the language ofdifferential geometry, has tended to conceal or obscure conclusionsabout the significance of a geometrized physics that push inconsiderably different directions from either instrumentalism orscientific realism.

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