Weyl, Reichenbach and the Epistemology of Geometry

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0039- 3681 94)00040-9

Weyl Reichenbach and the Epistemology of

Geometry

T. A. Ryckman

...physics has in the main contented itself with studying the abridged edition of

the book of nature. A. S. Eddington.

1

The Epistemology of Geometry

denotes a Fragestellung which has an

ancient pedigree going back at least to the Pythagoreans. We pick up the story

in a version deriving from Kant, or rather from the demise of Kantian

orthodoxy concerning the synthetic a priori status of geometric concepts, and

extending through Helmholtz and the conventionalisms of Poincare and

Reichenbach. In so far as geometry continues to be held to be the science of

space, the problematic of the epistemology of geometry concerns the eviden-

tial basis of the endeavor to determine the geometrical properties of space,

considered as the arena in which physical objects and processes are located. As

is well-known, this problematic is transformed by the use, in general relativity,

of a (pseudo-) Riemannian geometry of variable curvature whose metric in a

given region of space-time is dynamically dependent upon the (local) distribu-

tion of mass and energy. But less well-known are the questions of principle

posed to Einsteins solution

to the problem of the epistemology of

geometry, based upon his 1915 theory of general relativity (GTR), by Hermann

Weyls attempt to expand GTRs geometrization of gravity into a unified

theory encompassing all known forces and interactions. Weyls program, in

the period 1918-1924, employed a generalization of Riemannian geometry

expressly designed to challenge the congruence assumptions underlying

Riemannian geometry (and hence GTR). Significantly, Weyls criticisms arose

from an epistemological perspective according to which these assumptions

*Department of Philosophy, Northwestern University, Evanston, IL 60208-1315, U.S.A

Received 26 March 1993; in inal or m 8 March 1994.

A Generalization of Weyls Theory of the Electromagnetic and Gravitational Fields, Proceed-

ings of the Royal Society of London A99 (1921), lOk-127; see p. 108.

Pergamon

Stud. Hist. Phil. Sci., Vol. 25, No. 6, pp. 831-870, 1994

Copyright 0 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved

0039-3681/94 $7.00+00.00

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Studies

i n Hi story and Phil osophy of Science

appeared as an entirely avoidable blemish in Riemanns own naturphiloso-

phische

aspirations; in the context of GTR, Weyls views fundamentally

question the legitimacy of Einsteins tacit assumption of the existence of

practically rigid bodies to which the central theoretical structure of GTR,

the metric interval ds, was normed, i.e. held to correspond.

In brief, Einstein had, as a practical expedient, followed Helmholtz in

postulating the existence of de facto rigid bodies corresponding to metrical

concepts (in particular, congruence).2 To be sure, Einstein was well aware that

acute thinkers, notably Poincare, had demonstrated that the use of physical

objects and processes as correlates for geometrical concepts was not at all

innocent, and that one could thus only arbitrarily attribute geometrical

properties to space itself as distinct from the space implicated by the behavior

of these objects and processes. Poincares assessment, Einstein would note (see

Section 4 below), was unimpeachable

sub specie aetern i .

Indeed, this lofty

vantage point, momentarily occupied already by Weyl, is none other than that

of a systematic theory, a unified theory of fields, to whose discovery Einstein

would unsuccessfully devote the largest portion of the remainder of his life. Yet,

at least for the moment, for Einstein the empirical basis of GTR rested upon the

coordination of the ds to infinitesimal rigid bodies or material processes

(atomic spectra), a supposition which, if not completely theoretically satisfying,

was in perfect accord with the observable facts about transported measurement

bodies.

In any case, this explicit acknowledgement of the seemingly unavoidable role

of physical indicators (rigid bodies, test particles, atomic clocks, light

rays) in providing an evidential basis for belief in a particular geometrical

characterization of space, pointed to an apparent need for a clear distinction

between pure and applied geometries, and for an epistemological account

for the formers relation to the latter. By the first decade or so after the

inception of general relativity in 1915, there had emerged two scientifically

respectable responses to these new developments in the problem of the

epistemology of geometry.

The first, building upon the climate of pluralism created by the 19th century

discovery of new taxa of geometry, notably, projective and non-Euclidean

geometries, developed a purely formal axiomatic conception of geometry.

Associated primarily (and somewhat unfairly) with the name of Hilbert, it

The affiliation with Helmholtz is established by Einstein himself in a little-known essay,

Nichteuklidische Geometrie und Physik. LXe Neue Rundschau 36 (1925), 1620. Id like to thank

Don Howard for calling my attention to this essay. Einsteins (and Reichenbachs, see Section 4

below) association with the geometric views of Helmholtz is surprising, given the fact that

Helmholtzs conception permits only geometries of constant curvature, not the geometries

(Riemannian and the Weyl-Eddington non-Riemannian varieties) of variable curvature of GTR.

?See Hans Freudenthal, The Main Trends in the Foundations of Geometry in the 19th Century,

in E. Nagel, P. Suppes, and A. Tdrski (eds), Logic, Methodology and Philosophy qf Science;

Proceedings of the 1960 Confrrence (Stanford, CA, Stanford University Press, 1962) pp. 613321.


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Weyl , Rei chenbach and the Epistemol ogy of Geomet ry

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becomes part of the epistemological orthodoxy of logical empiricist philosophy

of science due to Einsteins famous maxim in his lecture Geometry and

Experience to the Prussian Academy in January 1921:

. ..as far as the propositions of mathematics refer to reality, they are not certain; and

as far as they are certain, they do not refer to reality.4

Geometry, under this purely formal axiomatic conception, becomes simply a

mathematical science, i.e. a body of theorems deduced from a set of axioms.

As to why the name geometry is given to some mathematical sciences and not

to others, Oswald Veblen, an important proponent of this view, provided the

only reasonable answer:

(B)ecause the name seems good, on emotional and traditional grounds, to a sufficient

number of competent people.5

The governing conception here is a complete rejection of the traditional

viewpoint, according to which geometry singles out, by privilege of intuition,

any particular subject matter, notably, that of space; that this has been true of

geometry in the past is simply an artifact of culture.

Since geometry has thus been eviscerated of all non-formal content, we arrive

rather naturally at the epistemological juncture intended by Reichenbach to

canonically summarize Einsteins views on the relation of mathematical

concepts to the physical world: the purely formal objects of mathematics

(geometry) are coordinated to (zugeordnet) physical objects6 the paradigm

case being the coordination of metric concepts to measuring rods and clocks.

As could be argued at length, in adopting the language of coordination to

characterize the relation of mathematical concepts to physical entities in

theoretical physics, both Reichenbach and Einstein show the influence of the

formalist epistemology of Schlicks Allgemeine Erkenntnislehre of 1918. Here,

while conscientiously rooting out any cognitive role for, or reliance on, notions

of intuition, Schlick had proposed an account of scientific cognition wholly in

4Geometrie and Erfahrung, Sitzungsberichte.

Preussische Akademie der Wissenschaften,

Physikalische-Mathematische Klasse, 1921, pp. 123-130; separately issued in expanded form and

translated by W. Perret and G. B. Jeffrey, as Geometry and Experience, in

Sidelights on Relativity

(New York, E. P. Dutton, 1923) pp. 27-56; see p. 28.

s0swald Veblen and J. H. C. Whitehead, The Foundations of Differential Geometry (Cambridge,

Cambridge University Press, 1932) p. 17.

6Geometrie and Erfahrung (op.cit.,

note

4).

p. 125: It is clear that the system of concepts of

axiomatic geometry alone cannot make any assertions as to the relations of real objects...which we

will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped

of its merely logical-formal character in that empirical (erlebbare) objects of reality must be

coordinated (zugeordnet) to the empty conceptual schema of the axiomatic geometry.

See Don Howard, Einstein, Kant, and the Origins of Logical Empiricism, in W. Salmon and

G. Wolters (eds),

Language, Logic, and the Structure of Scientt c Theories

(Pittsburgh, University

of Pittsburgh; Konstanz, Universitat Konstanz, forthcoming), and, on Schlicks epistemological

employment of this concept, my Conditio Sine Qua Non? Zuordnung in the Early Epistemologies

of Cassirer and Schlick,

Synthese 88

(199 I), 57-95.


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834 St udi es i n Hi story and Phi l osophy of Science

terms of implicitly defined concepts and of their Zuordmmg to real objects or

objects of experience. The Hilbertian resonances of his characterization of

cognition are pointed out by Schlick himself, who also claimed to have thus

provided a complete refutation of Kantian epistemology. We might note in

passing that precisely this sharp distinction between formal science and

empirical content was to become the hallmark of logical empiricism under the

rubric of a sharp analytic/synthetic distinction, a distinction in which, unlike in

Kant, there is no longer any cognitive role for intuition, either pure or

empirical.

The other approach differs perhaps more in ideology or epistemological

orientation than in other respects. Its leading figures are Riemann, Lie, Klein,

Weyl and Cartan; its main mathematical tenet is not the formal axiomatic

approach to geometry but rather its continuous (i.e. Lie) group-theoretic

characterization.* While not committing the original Kantian sin of positing a

necessarily Euclidean structure to the space of intuition as a condition of

experience, neither does it abjure talk of intuition and space altogether. Indeed,

the preference for local (i.e. differential and tangent space) characterizations

governed the approach here to what was still referred to as the problem of

space

(Das Raumproblem),

although, at least in Weyls terms, this is now to

be conceived as the problem of space itself, not merely a doctrine of the

configurations possible in

space.

9 Naturally, such an approach is not math-

ematically incompatible with a purely axiomatic approach, but epistemologi-

tally, it is quite at variance with the axiomatic viewpoint in its insistently local

chauvinism. Here a kind of naturphilosophische demand is made that the world

be understood in terms of its behavior in the infinitely small; only there will be

found the simple elementary laws which genuine understanding requires.O In

Weyls case, the epistemological advantage of such an approach is obvious:

only the spatially-temporally coincident and the immediate surrounding

On the classification, see 8. Cartan, Le R6le de la Thborir des Groupes de Lie dans Lbvolution

de la GkomPtrie Modern, Comptes Rendus de la CongrPs internationale, Oslo 1936 I, pp. 92.-103;

reprinted in Oeuvres Complgtes, 2nd edn, partie III 2 (1984), 1373-1384. It is Cartan who will

employ Lie groups most broadly as a tool of unification in geometry with his theory of generalized

spaces: roughly, a space of tangent spaces (a.k.a. a connection) such that two infinitely near

tangent spaces are related by an infinitesimal transformation of a given Lie group.

Space-Time Matter (op.cit., note 26 below), p. 102.

For example, The Lie theory [of continuous groups] belongs in that great train of thought

(Gedankenzug) which would understand the world from its behavior in the infinitely small and

which has proved itself so fruitful because only the passage back to the infinitely small leads to

simple elementary laws, Its standpoint is intuitively thoroughly natural; for example, instead of

describing the mobility in Euclidean space of a rigid body rotating about a fixed point according

to the requirement that the location of the body at each moment proceed from its initial location

through an operation of the Euclidean rotation group, it is natural to conceive of the bodys

continuous rotation as an integral succession

(Aneinanderreihung)

of infinitesimal operations of this

group, which follow upon one another according to the single elements of time. Hermann Weyl,

Kontinuierliche Gruppen und ihre Darstellungen durch Lineare Transformationen, in Atti de1

Congress0 Internazionale dei Marematici, Bologna 3-10 Settembre 1928 I, pp. 233-246; see p. 243.


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Weyl , Rei chenbach and the Epistemol ogy of Geomet ry

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neighborhood at once have a clearly exhibited meaning in intuition. * Intuition

(Anschauung) here has a linkage to

immediate insight or evidence that

reflects, loosely speaking, rather more of an Husserlianiz than a Kantian

nuance. Indeed for Weyl, since Evidenz is the sole source of all insight, the

formalist epistemology of Schlick will simply remain not understandable.3

Weyls epistemological objections to formalism are further revealed, of course,

in the foundations of mathematics. In the very same year (1918) in which he

publishes the initial papers on his unified theory of gravitation and electromag-

netism (see below), Weyl will also seek to counter the trend toward formalism

associated with his teacher Hilbert, basing even the justification of mathemati-

cal propositions not on proof, but on

immediate insight which furnishes

the experience of truth.14

But no less a striking contrast, elucidated below,

shows up in the foundations of GTR, a distinction that Einstein himself

encapsulated in another context in 1919 in pointing to the difference between

kinds

of theories in physics: thus the general theory of relativity, is, on the one

hand with Pauli, Reichenbach and Einstein (at least in this period of his career),

viewed as a theory ofprinciple whose basis lies in empirical fact, or on the other,

Nur das raumzeitliche Zusammenfallen und die unmittelbare raumzeitliche Nachbarschaft

haben einen in der Anschauung ohnes weiteres Klar aufweisbaren Sinn. Geometrie und Physik

(The Rouse Ball Lecture at Cambridge University, May, 1930) Die Nuturwissenschaften 19 (1931),

49-58; see p. 49.

I cannot here give Weyls tortuously complex epistemological views the attention they deserve,

but two points may be insisted upon: (1) to label Weyl an Husserlian (as is often done) does little

to further understanding; (2) Weyls own research activity in the exact sciences provides the arena

for interpretation of his epistemological reflections. Consider the relationship between these two

passages: The world comes only into consciousness in the general form of the consciousness which

is there (welche da ist); a penetration of Being and essence (Durchdringung des Seins und Wesens),

of the this and thus (Dies und So). (The intimate understanding of this penetration, it may be

remarked, is according to my conviction the key to all philosophy) [Das Verhaltnis der kausalen

zur statischen Betrachtungsweise in der Physik, Schweizerische Medizinsche Wochenschrzft 34

(1920), 131-741; see p.738.1 In eld theory the space-time continuum plays in a certain sense the role

of substance, if we conceive of the opposition of Substance and Form as that of the this and thus

(Dies und So).

. .According to field theory the world description consists, to use the terminology

of Hilbert, in here - so relations - the here represented through the space-time coordinates, the

so through the state magnitudes. Was ist Materie? Zwei Aufsiitze zur Naturphilosophie (Berlin,

Springer, 1924) p. 42.

%ee Weyls review of Schlicks ANgemeine Erkenntnislehre (1918) in Jahrbuch iiber die

Fortschritte der Mathematik 46 (1923), 59-62:

esp. p.

60:

To the reviewer it is not understandable

how anyone who has ever attained an insight (Einsicht) can be satisfied with this point of view [i.e.,

with Schlicks conception of the essence of knowledge as lying in the merely designative or

semiotic conception of concepts-TR].... In as much as [Schlick] ignores intuition, in so far as it

goes beyond the merely perceptual-like sensuous, in such wide measure, he frankly discards

Tverwirf;) evidence (Evi>eni) which is still the sole source of all insight.

14Das Kontinuum (Leiuzig, Veit. 1918). n. 11. footnote: The famous Dedekind essay Was sind und

was sollen die Zahlen begins (Foreword to the 1st edition) with the statement: What is proveable

should not be believed in science without proof. This remark is certainly characteristic of the mode

of thinking of most mathematicians; nevertheless, it is a preposterous principle. As if such an

indirect collection of grounds, though we call it a proof, is capable of awakening any belief

without our assuring ourselves in immediate insight

(unmittelbarer Einsicht)

of every single step.

This (and not the proof) generally remains the last source of justification of knowledge; it is the

experience of truth (Erlebnis der Wahrheit).


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Studies in H istory and Phil osophy of Sci ence

with Weyl, as a constructive theory, built up from a hypothetical and ideally

simple starting point.i5

In this paper, I will seek to elucidate this difference by contrasting the views

of Weyl and Reichenbach on what can thus, only with some ambiguity, be

called the epistemology of geometry. To do so, I will situate the develop-

ment of Reichenbachs geometric conventionalism as occurring against the

backdrop of, and in no small measure in response to, an explanatory strategy

adopted by Weyl to defend his broadening of the general theory of relativity,

incorporating the electromagnetic field into the metric of space-time, against

what appeared at the time to be a decisive objection raised by Einstein. In

particular, my aim is to show that the central mechanism by which Reichenbach

created his metaphilosophical theory of equivalent descriptions (destined to

become part of the methodological edifice of logical empiricist philosophy

of science) is, in its appeal to a little-understood distinction between

differential and universal forces, targeted on Weyls theory. More exactly,

it is directed against Weyls explanatory account of how it is that we

do observe the congruence behavior of transported measuring rods (or the

constancy of frequencies of spectral lines of atoms) that we do, despite the

existence of the central theoretical structure in this unified theory, a space-time

with a non-Riemannian metric, according to which vector transference is not

generally integrable. But beyond establishing such an historical linkage, this

case study aims to demonstrate how an influential and exceedingly general

(global) characterization of theoretical underdetermination arose as a one-

sided, hence questionable, extrapolation from a critical foundational contro-

versy within the nexus of the revolutionary ideas of GTR itself: a story that

should serve to renew our skepticism of this manner of practising philosophy of

science.

In what follows, a summary exposition of Weyls theory stressing its

philosophical motivations precedes (in Section 3) an account of Weyls reply

to the initial criticisms of Einstein and Pauli that his theory conflicts with

observation. We then turn in Section 4 to an examination of the somewhat

tempered position of Einstein in his lecture of January, 1921, Geometry

and Experience,

contrasting this with the response of Reichenbach as

tracked through the emergence of his characteristic form of geometric

conventionalism.

The distinction is drawn by Einstein in an essay What is the Theory of Relativity? at the

request of The Times of London, 28 November, 1919, reprinted in I deas and Opi nions (New York,

Bonanza Books, 1954) pp. 227-232. I have been prompted to apply the distinction here by the

discussions (which point to Einsteins switch, in the case of GTR, from the former to the latter) in

two papers of John Stachel, Special Relativity from Measuring Rods, in R. S. Cohen and

L. Laudan (eds) Physics, Phil osophy and Psychoanaly sis; Essays in Honor of Adol f Gr ii nbaum,

(Dordrecht, D. Reidel, 1983) pp. 255-273; and Einstein and the Quantum: Fifty Years of

Struggle, in R. Colodny (ed.), From Quark s t o Quasars: Phil osophical Probl ems ofModernPhysics,

(Pittsburgh, University of Pittsburgh Press, 1986) pp. 3499385.


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Weyl , Rei chenbach and the Epist emol ogy of Geometr y

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Hermann Weyls expansion of the theory of general relativity in 19 18-l 923

was, following the inception of general relativity by Einstein in 1915-1916, the

first and historically most important attempt to offer on its basis a unified

theory of all known physical forces, at that time including only gravitation

and the forces of the electromagnetic field.

16 In so doing, Weyl inaugurated

efforts which were to occupy Einstein, unsuccessfully, until his death, and

created a tool, local gauge invariance which, in somewhat different form,

continues to be employed by contemporary physics of elementary particle

interactions. What is not widely remembered today is that Weyl, who was from

19 16 one of the most prominently strong supporters of GTR, had a philosophi-

cal understanding of GTR that was not only distinctively different, in crucial

places, from that of Einstein,17 but also completely at variance with the received

philosophical interpretation of GTR by logical empiricism which was largely

the creation of Hans Reichenbach and which, until recently, was passed down

to subsequent generations of philosophers of science as a lasting achievement,

part of the canon of scientific philosophy.18

Weyls unified theory, which Weyl himself actively promulgated until

the mid-1920s and the advent of Quantum Mechanics, is really only

16Hilberts famous notes on the foundations of physics of November 1915 and December 1916

(Kiinigl. Gesellsrhafi d Wissenschaften zu Giittingen. Nachrichten. Math.-Phys. Klasse 1915,

pp. 396407; 1917, pp. 53-76) are not unification attempts in the same sense, since although

Hilbert (using invariant theory and a variational method) derived Mies equations of electrody-

namics from the field equations of gravitation (independently discovered by Hilbert in November,

1915) the Mie theory is valid only in the limiting case of special relativity; for discussion, see

J. Mehra, Einstein, Hilbert, and the Theory of Gravitation (Dordrecht/Boston, D. Reidel, 1974)

pp. 2430.

A very fundamental difference concerns the interpretation of the relativity of motion where

Weyl takes issue with Einsteins relativist response to Lenard at the Bad Nauheim Naturforscher

Versammlung in September 1920: If, with Einstein, one can adapt a coordinate system to all

particles such that simultaneously one can transform them all to a state of rest, it no longer makes

sense to speak of relative motion. . .The principle that matter generates the field can therefore only

be straightforwardly maintained if the concept of motion admits in itself a dynamic moment. The

analysis of the concept of motion revolves not around the opposition absolute or relative but rather

around

kinematic 0; dynamic. Die Relativitiitstheorie auf her Naturforscherversammlung in Bad

Nauheim, Jahresberichte der Deutschen Mathematische- Vereiniaunp (1922). DD. 51-63 and 62-63;

also, As long as one ignores the guiding field (Fiihrungsfeld), one can speak neither of relative nor

of absolute motion; only with regard to the guiding field does the concept of motion gain a content.

Relativity theory, correctly understood, will not eliminate absolute motion in favor of relative, but

it denies (vernichtet) the kinematic conception of motion and replaces it through the dynamic.

Massentragheit und Kosmos, in

Zwei Aufsiitze zur Naturphilosophie (op.cit.,

note 12) p. 66; on the

concept of guiding field, see Section 2 below.

See especially, J. A. Coffa, Elective Affinities: Weyl and Reichenbach, in W. Salmon (ed.),

Hans Reichenbach: Logical Empiricist (Dordrecht, D. Reidel, 1979). pp. 2677304. I would like

to record here my debt for the stimulation provided by this paper, the only one in the literature on

its subject matter. Coffa focuses on the conflict of the respective viewpoints regarding the

interpretation of relativity (in particular, relative motion), expressly leaving aside their disagree-

ment over the empirical determination of the metric,

which is our topic. Coffa modestly said his

paper is no more than an invitation to pursue a topic (p. 267) but it is certainly much more than

that.


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838 Studies in History and Philosophy of Science

understandable as a bold, but ultimately unsuccessful attempt to remove, from

a perspective that could be described as Riemanns own, a

Schoenheitsfehler9

from Riemanns theory of manifolds, the mathematical framework in whose

terms GTR is cast. This framework, it should be noted, uneasily coexists within

the problematic of the epistemology of geometry: for indeed it transforms the

very meaning of geometry as the term applies to the theory of physical space.

And it is this transformed conception of geometry, in the current consensus,

that has won the day in GTR.

Weyls theory, or at least its motivation, really begins with Riemanns famous

1854

Hubilitationsschrif

On the Hypotheses which Lie at the Foundations of

Geometry.20 Here Riemann sketched, in briefest outline, an approach to the

mathematical investigation of the concept of space by conceiving of spatial

objects, which he called multiply extended magnitudes or, what we know

today as n-dimensional topological manifolds. For Riemann, physical space,

the space of our encounters with external objects, must be initially considered

as merely a special case of a triply extended (3-dimensional) manifold. The

concept of extension in itself determines no particular geometry or system of

metric relations. Of course, Riemann did not ignore the question of metric

relations in space. He allowed that various systems of geometric axioms were

possible but that none was logically (or for that matter, cognitively) necessary,

and each was empirically contingent in its description of space. Tantalizingly,

he even questioned the continuity of space, suggesting, in an obscure passage,

that if space is in fact continuous, its metric relations may stem from something

external to space, namely, matter. And he speculated that the viability of

familiar metric concepts, such as the rigidity of measuring rods, may break

down in the domain of the infinitely small. 21 The mathematical significance of

Riemanns address is, of course, of the highest order, containing as it does in

embryo, the foundations of differential topology and geometry. But it is not an

overestimation of Riemanns paper, scarcely 20 pages long with very few

This characterization is Dirk J. Struiks, Schouten, Levi-Civita, and the Emergence of Tensor

Calculus, in David E. Rowe and John McCleary (eds),

The History of Modern Mathematics,

Vol. 2 (Boston and New York, Academic Press, 1989). DU. 98-108: see D. 104.

Largely unknown to English readers, Weyl alsoe&ed an edition of Riemanns essay, B.

Riemann, Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, Neu herausgegeben und

erltiutert van H. WeyI (Berlin, J. Springer, 1919); Dritte Auflage, 1923. Weyls editorial

elucidations of Riemanns text are an important source for his own epistemological and

noturphilosophische views,

But now it seems that the empirical concepts, in which spatial measure-determinations are

grounded, the concepts of rigid body and light ray, appear to lose their validity in the (domain of

the) infinitely small. It is therefore easily imaginable that the measure-relations of space in the

infinitely small are not in accord with the presuppositions of geometry and in fact this must be

supposed as soon as the phenomena might thereby be more simply explained. Concerning this

passage (at pp. 19-20), Weyl observed: The complete understanding of the concluding remarks of

Riemann concerning the underlying basis of the metric relations of space (inner Grund der

Massverhiiltnisse des Raumes) was first disclosed to us by Einsteins general theory of relativity,

ibid., pp. 4641.


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Weyl, Reichenbach and the Epistemology of Geometry 839

formulas, to attribute to it an equal philosophical importance.22 For, by

recognizable intention and with the advantage of hindsight, it also provides the

resources for a complete refutation of Kants doctrine of space. But this can

perhaps best be seen in the work of Weyl on general relativity we are about to

consider.

In his own mathematical development of these ideas, written a few years

later, in 1861, for a prize competition in Paris (he didnt win ), Riemann

more or less conjured up an expression (a quadratic differential form) which

is a generalization of the familiar Pythagorean theorem and which gives

metrical relations in such a manifold.23 It contains, in addition to the

differentials of the coordinates, functions of the coordinates as coefficients to

the coordinate differentials which in the general case may have different

values at different points of space. Such an expression provides a metric not

only for curved spaces, but also for spaces of variable curvature. Riemanns

ideas about variably curved space were indeed revolutionary; no less a

mathematician than Henri Poincare, writing in the early 1890s and again in

his well-known Science and Hypothesis of 1903, expressed the judgement that

these spaces were but mathematical curiosities and, in his words, could

never be other than purely analytic, that is, not axiomatizable, hence

unsuitable candidates for geometry in any conventionally chosen combi-

nation of geometry and physics as a theory of physical space.24 In his 1861

prize essay, Riemann had considered only the case of manifolds in which the

direction, but not the length or magnitude of a vector, was altered under

**The assessment here relies on the case made in Gregory Nowak, Riemanns Habilitarionsvor-

rrag and the Synthetic A priori Status of Geometry, in David E. Rowe and John McCleary (eds),

The History of Modern Mathematics, Vol. 1 (Boston and New York Academic Press. 1989).

pp. 1748.

See R. Farwell and C. Knee, The Missing Link: Riemanns Commentario, Differential

Geometry and Tensor Analysis,

Historia Mathematics 17 (1990), 223-255.

A standard history

writes that the Riemann expression of the distance element d.?=C,,g,,dxdxk was, for Riemann,

an article of faith, essentially; J. A. Coolidge, A History of Geometrical Methods (Oxford,

Clarendon Press, 1940), p. 410. Of course, its validity represents the supposition of the validity

of the Pythagorean theorem in the domain of the infinitely small; Weyl will provide a proof of

the uniqueness

of the Pythagorean measure of the line element in this domain, which

establishes that it is the only measure under which the (infinitesimal) orthoeonal groun is

a volume-preserving invariant,

- -I

Die Einzigaragkeit der Pythagoreischen Massbestimmung,

Mathematische Zeitschrift 12 (1923),

114146; a slightlv more readable oresentation is in

Mathematische Analyse des Raumproblems; Vorlesun~en- gehalten in Barcelona und Madrid

(Berlin, J. Springer, 1923) lectures 7 and 8; see the brief discussion towards the end of Section 2

below.

Z4Les Geometries non-euclidiennes, Revue g&&ale des Sciences pures et appliqukes 2 (1891)

7699774; this essay is reprinted as chapitre III of La Science et LHypothPse (Paris, 1903;

reprinted by Flammarion, Paris, 1968), cited passage at p. 73: Ces geometries de Riemann, si

interessantes a divers titres, ne pourraient done jamais &tre que purement analytiques et ne se

preteraient pas a des demonstrations analogues a celles dEuclide. Incidentally, the only in-print

English translation of this work (by W. J. Greenstreet, published by Dover, New York) garbles

this passage (at p. 48) completely inverting its meaning, as was already pointed out by Bertrand

Russell in his review of this translation, Mind 14 (1905), 412418.


10/40

840

Studies in History and Philosophy of Science

transport in the manifold.25 In the general case, transporting a vector around

a closed curve and returning it to its point of origin alters its direction but

not length; hence, length is integrable, but direction is not. Such manifolds,

possessing as metric the positive-definite quadratic form mentioned above,

are known as Riemannian manifolds.

All very well, but what has this to do with Weyl? As Felix Klein had done

some twenty years before, Weyl recognized in Riemanns differential geometry,

which Weyl was soon to generalize, a fundamental epistemological postulate:

that genuine understanding or explanation of the processes of nature studied in

physics is produced only through conceptions regarding the behavior of nature

in the infinitely sma11.26 For Weyl, Riemanns passage from a Euclidean

distant (Fern) to a Riemannian

local (N&e) geometry is exactly

parallel to the transition in physics from action-at-a-distance theories to the

physics of fields. We can understand, e.g. Coulombs law or Ohms law which,

in Weyls terms, are

Fernwirkunggesetzen

(action-at-a-distance laws), only

because these are derivable from Maxwells equations of the electromagnetic

field. Similarly, Euclidean geometry formulated to meet the requirements of

continuity is Riemannian geometry, in which there is a local Euclidean

structure in the neighborhood of each point. The transition from Euclidean

distant geometry to Riemannian differential geometry (and, as we shall see,

on to a non-Riemannian pure infinitesimal geometry), as well as the passage

from action-at-a-distance physics to a physics of action by immediate contact

(Nuhewirkungsphysik) both illustrate the epistemological gain in understanding

brought by restricting ones considerations to behavior in the infinitely small.

Why is this? Because in the passage to the infinitely small, all problems can be

linearized: this is the approach of the differential calculus, infinitesimal

25That Riemann considered only this restricted case, corresponding to, in Weyls theory, the

absence of an electromagnetic field (F,k=O, see below) was due to the historical development of

Riemanns views in Gausss theory of surfaces; see Weyls letter to Einstein of 19 May 1918 (cited

in N. Straumann, Zunz Ursprung der Eichtheorien bei Hermann WeyL Physikalische Bliitter 43

(1987), 414-421, at p. 416. For an extended discussion of this development, see Weyls edition of

Riemanns essay (op.&. , note 20), pp. 3538.

26The principle, to understand the world from its behavior in the infinitely small, is the driving

epistemological motive of contiguous-action physics

(Nahewirkung physik)

as well as Riemannian

geometry, but it is also the driving motive in the grandiosely directed remaining life-work of

Riemann, above all, the theory of complex functions. Raum-Zeit&Materie (Berlin, J. Springer,

1918) p.82, repeated through the 5th edition (1923) p. 86, of this work; the 4th edition, translated

into English as Space-Time Matter by Henry L. Brose (London, Meuthen and Co. 1923; reprinted

by Dover, 1952) translates the passage somewhat differently on p. 92. In a Vienna lecture in 1894,

Klein had made the same observation, linking the names of Riemann and Faraday: As in physics,

the banishment of actions-at-a-distance has led to the explanation of the phenomena through the

inner forces of a space-filling aether, so in mathematics, the understanding of functions arises from

their behavior in the infinitely-small, especially, therefore from the differential equations which

satisfy them. . .If I might dare to so sharply press the analogy, then I would say

that Riemann in

the domain

of

mathematics and Faraday in the domain

of

physics stand parallel. Riemann und

seine Bedeutung fur die Entwicklung der modernen Mathematik, Jahresbericht der

Deufschen Mathematiker- Vereinigung, Vol. 4 (18945) equivalent to Gesammelte Mathematische

Ahhandlungen, Vol. 3 (Berlin, J. Springer, 1923) pp. 482497, at p. 484.


11/40


geometry and field physics.27 Linearizing promotes mathematical tractability,

but thereby we also gain in intuitive clarity, we come to understand.

But there is also an interesting and unusual connection between the

epistemological requirement to understand nature through the infinitely small

and Weyls version of holism, i.e. to the inseparability of physics and geometry

and to the impossibility of verifying single statements independently of the

nexus of theoretical commitments in which they are embedded. In an example

he repeatedly uses, Weyl argues that it is field physics (or rather the Wechsel-

wirkung

between matter and field) which illustrates the unavoidability of

epistemological holism in fundamental physics, that is the physics of the

interactions of matter and field. The example concerns, on the one hand, the

so-called ponderomotive force exerted upon a test charge by the electro-

magnetic field, and on the other, the generation of field strengths by the test

charge. The laws and interactions here constitute a cycZe: the field acting upon

the test charge, which in turn disturbs the surrounding field. Weyls use of this

example occurs in the context of arguing that physics and geometry are an

inseparable whole.28 Dynamical Wechselwirkung (interaction) is the character-

istic mode of action transmission in field theories. Einstein had of course shown

that the inertial and gravitational structure of the world is not geometrically

rigid but active and dynamic; it is an interactive field like the electromagnetic

field and this analogy always governs Weyls presentation of general relativity.

Just as the

Wechselwirkung

between a charged body and the surrounding

electrical field can only be understood in terms of the contiguous-action

(Nahewirkung)

represented in the partial differential equations of Maxwells

theory, so also the reciprocal influence between what Weyl was to term the

Fiihrungsfeld (guiding field, the inertial-gravitational structure of the world)

and a massive body within it calls out for explanation in the same terms. As we

shall see, from a purely infinitesimal standpoint of differential geometry and

the contiguous-action of field physics, the apparently congruent behavior of

transported measuring rods should not be accepted as brute fact-to do so

shows a complacency toward the remnants of the old Euclideanferngeometrisch

analysis of nature. Weyl will thus demand that the behavior of measuring rods

27Mathematische Analyse des Raumprohlems (op.cit., note 23), pp. 17-18.

*Raum-Zeit-Materie,

9 8

(op.cit., note 26) p. 60 (and repeated verbatim in all subsequent

editions): Space-Time+Matter, p. 67: The distribution of the elementary quanta of matter provided

with charges fixed once and for all (...) determine the field. The field exerts upon charged matter a

ponderomotive force. . .The force determines, in accordance with the fundamental law of

mechanics, the acceleration, and hence the distribution and velocity of the matter at the following

moment. We require this whole network of theoretical considerations to arrive at an experimental test

~ if we assume what we directly observe is the motion of matter. . .We cannot merely test a single

law detached from this theoretical fabric The connection between direct experience and the

objective element behind it, which reason seeks to grasp conceptually in a theory, is not so simple

that every single statement of the theory possesses an immediate, intuitively verifiable meaning. We

will . .see that geometry, mechanics and physics comprise in this manner an inseparable theoretical

unity....


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842

Studies

in History and Philosophy of Science

and clocks be

derived

from a systematic theory, i.e. a fully dynamical theory

of matter-field interactions.

Weyl was among the first strong supporters of the general theory of

relativity,29 and one of the first to lecture on general relativity, which he did

in the summer of 1917 in Zurich where he was professor of mathematics.

Weyl was then 31 years old and had just been released from the German

army into which he had been drafted in 1915. Upon returning to Zurich, he

had immediately immersed himself in Einsteins theory while looking for

something to whet his mathematical curiosity.30 These lectures were written

up and published in the spring of 1918 under the title Raum-Zeit-Materie;

the book was to go through five editions by 1923. The fourth edition was

(rather poorly) translated into English in 1922 and is still in print, while a

German edition (the 7th) is also still in print. This was the first textbook (if

it can be called that ) on general relativity and it won high praise from

Einstein, who wrote in a rare book review that every page reveals the

unerring hand of the master. In the first edition, Weyl had not yet

introduced his expanded theory and he largely adhered to the geometrical

development of general relativity which followed Einsteins own exposition in

proceeding from the basic structure of a four-dimensional Riemannian

manifold with a special case of the Riemann metric, called a Lorentz metric,

which takes account of the light cone (hence, causal) structure of space-

time.32 But no sooner was the book out then Weyl, spurred on by a 1917

paper by the Italian geometer, Tullio Levi-Civita, conceived of a stunning

generalization of Einsteins theory. As great as his respect was for Einsteins

theory, Weyl now believed, from the

naturphilosophische

standpoint of the

29Writing to Weyl on 23 November 1916, Einstein remarked, clearly with Weyl (and Hilbert) in

mind, that the average mental power of its [i.e., GTRs] supporters by far surpasses that of its

opponents. This is a kind of objective evidence for the naturalness and sensibleness of the theory,

translated and quoted in Sigurdsson (op. cit., note 30 below), p. 161.

30Biographical details can be found in Skuli Sigurdsson,

Hermann Weyl, Mathematics and

Physics, 1900&1927, Harvard Ph.D. dissertation, 1991.

3Deutsche Literaturzeitung,

Vol. 25 (21 June, 1918), Columns 372-373. Einsteins review of the

first edition is remarkable for its effusive praise; he commends Weyl in particular for his success in

integrating the field equations of gravitation, a treatment showing how simplifying and clarifying

is the work of the born mathematician there. Interestingly, towards the end, Einstein expresses

some reservation concerning the conception Weyl gives of the relations obtaining between the

expressions of theoretical physics and reality:

ich mit dem Verfasser nicht ganz iibereinstimme

beziigich der Auffassung des... Verhiiltnisse. welches zwischen den Aussagen der theoretischen Physik

und der Wirklichkeit besieht.

32The notable difference from Einsteins treatment is that Weyl, under the influence of

Levi-Civita, has already introduced the notion of a geodetic coordinate system (p. 100 ff.) in

which the components of any (contravariant, covariant) vector are unaltered by an infinitesimal

parallel displacement (infinitesimalen Parullelverschiebung); the transformation law of this parallel

displacement (involving the Christoffel symbols of the second kind) is then used to briefly

characterize a geodetic vector field, which in the 4th and 5th editions of

Raum-Zeit Materie

(op.&., note 26) will be baptized as the affine connection (also, guiding field, Fiihrungsfeld) of

the manifold. Weyl goes on to express the Riemann curvature tensor (p. 108) in terms of the

three-index Christoffel symbol (of the 2nd kind), i.e. the components of the affine connection.


13/40


purely infinitesimal, that Einstein had achieved only a partial victory over

ferngeometrisch

or

fernvorstellung

prejudices and the hoary old metaphysical

concept of substance.

Levi-Civitas paper, though purely mathematical, was a first product of

the new attention Einsteins theory had brought to differential geometry;

Levi-Civita himself had, together with his teacher, Giorgio Ricci, published

the first systematic treatment of the tensor calculus in a paper in 1901.33 In

his 1917 paper,34 Levi-Civita showed how the fundamental notion of

Riemannian curvature could be non-metrically expressed in terms of the

concept of an infinitesimal displacement of a vector parallel to itself.35 Weyl

generalized Levi-Civitas result, removing an unnecessary embedding restric-

tion, and in two papers, published still in early 1918, laid the framework for

his unified theory. In the first of these, he sketched his unified theory which

incorporates electromagnetism into the metric of space-time; in the second he

provided a mathematical exposition of what he termed a pure infinitesimal

geometry which removed, as he put it, the last

ferngeometrisch

inconsistency,

a remnant of the Euclidean past, from Riemanns theory of

manifolds.36 From now on, Weyls position will be that A genuine Local-

Geometry (wahrhafte Nahe-Geometrie) can only be acquainted with a prin-

ciple of transport of length from one point to another infinitely adjacent to

it; in other words, Riemanns geometry goes only half-way towards the ideal

of a pure infinitesimal geometry since it assumes a path-independent, distant

comparison of line elements.37 In Weyls new geometry,

both

direction (like

Riemann)

and

length (unlike Riemann) are not, in general, integrable. In

such a Weyl space, as it is now known, lengths at one and the same point

alone can be compared but not from one point to another lying at a finite

distance. The epistemological attraction for Weyl of a purely infinitesimal

s3MOthodes de calcul differential absolu et leurs applications, in

Mathemati sche Annaien,

Vol. 54, pp. 125-201. This paper was an important resource for Einstein (and Grossmann) in the

mathematical development of general relativity.

Nozioni di parallelism0 inuna varieta qualunquee conseguente specificazione geometrica della

Curvatura Riemanniana, in

Rendicont i de1 Cir colo di Mat emat ica di Palermo 42

(1917).

351f a (contravariant here, but alternately covariant) vector A at the pointsxk is parallelly

displaced to an indefinitely adjacent point xk + dx,, then the resulting vector A 6A is determined

by the expression

6A = - {,k} Xdx,, ({j,k} = 1-;, = I-,)

where the 40 independent magnitudes Y,, (equivalently, Christoffel symbols of the second kind) are

expressed in accordance with Riemannian geometry by the metric tensor and its first derivatives: for

a clear discussion, see L. Silberstein, The Theory of Relat iv it y, 2nd edition (London, Macmillan,

1924), pp. 346ff. In Levi-Civitas paper, a construction is given in which the manifold where

parallel-transport is defined is embedded in a ten-dimensional flat Euclidean space.

36Gravitation und Elektrititlt, in

Si t zungsberi chte. Preussischen A kademi e der W i ssen-

schaft en, Physik ali sche-M athemati sche Kl asse (1918) pp. 465480; Reine Infinitesimalgeometrie,

Malhemati sche Zeit schri ft 2 (1918),

pp. 384411, p. 385.

37Gravitation und Elektrizitat (ibid.), p. 466; Raum-Z eit 6Mat eri e, 3rd Edn (Berlin, J. Springer,

1919), p. 91; Space-Time-M att er (op.& ., note 26) p. 102.


14/40

844


foundation for differential geometry was direct: it was fully

daccord

with the

principle that understanding nature stemmed from formulating its behavior

in the infinitely small.

Since there is no longer an invariant unit of length, in order to have a metric

at all in a Weyl space, it is necessary first to specify an arbitrary unit length at

each point, that is, to gauge (umeichen) the space; mathematically, this is

accomplished by adding to the structure of the Weyl space a pseudo-vector

field which assigns to each point of the space a unit vector. This essentially

means that the choice of a local coordinate system also involves the choice of

a local unit of length. Compatibility of the affine structure developed from the

notion of infinitesimal vector parallel transport with the metric requires

permitting a special choice of gauge (geodetic gauge) for the infinitesimal

neighborhood of a point P such that vector length I is unaltered in parallel

displacement from

P.

But in general, a vectors length changes under transport

from P x) to P(x + dx) where the standard of length at P is carried over.

This is given by a similarity transformation,

dl = -ldy,

(1)

where 9 is a function of position independent of the displaced distance.

However, the arbitrarily chosen gauge at P may be altered according to

I = 2

where the gauge factor A is an always positive continuous function of the

coordinates. In this case, in place of (l), we have

dl = - Pd#

(1)

where

dyl = dq - d;llA.

Weyl shows that if the desired condition that dl vanishes at

P

is met, then

dq = C p,dx

which is to say that dy is a linear diffeiential form.

Characterization of the metric therefore requires two fundamental forms,

the quadratic differential form of Riemannian geometry,

ds = gikdxidXk

(Einstein summation convention)

and the linear one just defined. These are defined up to the gauge transfor-

mations

ds = ids and

d# = dy, - d log il (d log A = dAl2)

Weyl would thus demand that natural laws not only satisfy the Einstein

condition of general covariance, i.e. invariance under arbitrary (but sufficiently

continuous) transformations of the coordinates, but also the more restrictive


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Weyl, Reichenbach and the Epistemology of Geometry

845

condition of gauge invariance.38

By the vector operation curl (rot), Weyl

then defines from v, an antisymmetric tensor, which provides a measure of the

extent of the non-integrability at each point (that is, the departure from

Riemannian geometry there). In components it is written

where q is the function already noted, with the consequence that when Fjk=O

the length of a vector is integrable (path independent), i.e. the special case of

Riemannian geometry is recovered.

Now Weyls discovery was that this tensor is formally identical to the

electromagnetic field tensor (the so-called Faraday tensor) and thus to the first

system of Maxwells equations if q is seen as the potential of the electromagnetic

field. It is therefore only natural to identify, as Weyl immediately did, these two

structures. The startling meaning of this identification is that

all

the phenomena

of nature are represented in the 14 independent coefficients pi and g, of the now

expanded metric; these in turn are to be deduced from a single universal

world-law (i.e. an action law) of the highest mathematical simplicity.39 In

*Space-Time-M utt er (op. cit ., note 26), p. 286; more restrictive since Weyl now demands that

laws of nature be of gauge weight 0. This additional invariance means that the general solution

of the field equations now contain 5 arbitrary functions (instead of 4, as in Einsteins theory), hence

that there be 5 identities (superfluous equations) between the 14 field equations. In accordance

with the theorem of integral invariants (Noether theorem), according to which a conserved quantity

corresponds to each invariant of a Lagrangian system, Weyl demonstrates, in an extremely

complicated discussion, that just as conservation of energy-momentum corresponds to coordinate

invariance, conservation of electricity (both the electric potential and the current satisfy the

boundary condition %++/%i=O) corresponds to gauge invariance, as is reflected in the fact that the

Maxwell equations are of gauge weight 0. See Raum-Zeit -M aterie 5th Aufage (op.cit., note 26),

pp. 3088317. Weyl takes this demonstration to have provided a strong formal argument in favor

of the theory: as origin of the conservation law of electric charge a priori must be expected one new,

single arbitrary function involving an invariance property (Invarianzeigenschuff) of the field laws.

Geomet ri e und Physik op.& . , note 1 l), p. 54.

Reine Infinitesimalgeometrie (op.cit., note 36) p. 385, footnote 4: I am audacious enough to

believe that the totality of physical phenomena permit deduction out of a single universal world-law

Weltgesetz

of highest mathematical simplicity. In a mode of procedure very much in the style of

the Mie-Hilbert theory of matter, Weyl would then seek in the next few years, without success, to

find a single action principle expressing the force law governing the interaction between the g,, and

the pi; this is to have the form of an integral invariant (Lagrangian density), additively combining

a gravitational and an electromagnetic component; see Space-Ti me-Mut t er, Sections 35536 and the

later. somewhat chastened, abbreviated approach in Raum-Zeit -M aterie 5th AuJage (op. cit ., note

26), Section 40. This failure to univocally determine a world-function played no small part in the

critical reception of Weyls theory. Given the widespread use of Lagrangian and Hamiltonian

methods today, it is instructive to observe that Pauli

(ap.cit.,

note 54 below), already in 1921,

expressed skepticism towards what he saw as the too little empirically oriented Gottingen

approach of seeking such a Weltfunktion: it is not at all self-evident from a physical point of view,

that physical laws should be derivable from an action-principle. It would, on the contrary, seem far

more natural to derive the physical laws from purely physical requirements, as was done in

Einsteins theory... (p. 201). Some years later, Eddington expressed a similar opinion in the face of

the difficulties posed by quantum mechanics: Least action seems to fail here because it attempts to

jump by a formal device a territory comprising the problems of electron structure and quantum

theory requiring other conceptions than those of field physics.

Universe: Electromagnetic-

Gravitational Schemes,

Encyclopedia Britannica,

13th Edn (1926), pp. 907-908. For a critical

assessment of Weyls failure to find a suitable Weltfunktion, see P. Bergmann, Int roducti on to the

Theory of Relat iv it y (New York, Dover, 1976; first published in 1943), ch. 16.


16/40

846


setting out to remove what Weyl saw as a glaring Schiinheitsfehler in Riemanns

theory of manifolds, Weyl had produced a geometric theory unifying electricity

and gravitation. Since these were the only physical forces recognized in

1918, Weyl was led to triumphantly proclaim the unity of geometry and

physics:

Everything real

(Wirkliche)

that transpires in the world is a manifestation of the

world-metric: Physical concepts are none other than those of geometry.40

Such declarations elicited no small amount of criticism from Einstein and

others; despite their stridency, what Weyl seems to have meant is simply that it

is the task of geometry to investigate the essence of metric concepts, while

physics is concerned to determine the law (i.e. the integral invariant

formulated as an action principle) according to which the actual world is

singled out from all other possible four-dimensional metric spaces.41

Now for Weyl, the essence of the concept of metric lies in the concept of

congruence, which is a purely infinitesimal concept;42 here the metric is

viewed solely as a structure of the continuous field whereupon the concepts of

vector and tract

(M-e&e)

employed for its characterization have in them-

selves nothing to do with material measuring rods.43 Within the epistemology

of a purely contiguous physics and of purely infinitesimal geometry, Weyl

distinguishes the nature of the metric, which is a priori and the same at each

point of space, from its orientation from one point to another indefinitely

nearby, this being variable and pace Riemann and Einstein, locally dependent

upon the distribution of matter.

As a corollary to these epistemological foundations of exact science, Weyl

will take up the so-called

Raumproblem

whose Helmholtz-Lie solution,

based as it is on the

ferngeometrisch

postulate of free mobility, is neither

adequate to the non-homogeneously curved spaces permitted by Riemanns

theory of manifolds nor survives the scruples of Weyls stricter purely

infinitesimal point of view. In so doing, he locates the essence of the metric,

hence of congruence, in the idea of the infinitesimal orthogonal group, and he

is able to prove, under the assumption that the metric uniquely determines the

affine connection, that the group of infinitesimal rotations at a point (for n

dimensions) that can be characterized as the set of linear transformations

leaving a non-degenerate quadratic differential form invariant is the only group

Reine Infinitesimalgeometrie

op.cit.,

note 36) p. 385.

411bid.:

The sole distinction between geometry and physics is this: that geometry investigates

generally what lies in the essence of metric concepts, while physics determines the law through

which the actual world is singled out from among all possible four-dimensional metric spaces of

geometry and explores its consequences.

@Die Einzigartigkeit der Pythagoreischen Massbestimmung (op.cit., note 23) pp. 114115.

43Ueber die physikalischen Grundlagen der erweiterte Relativitatstheorie (op. cit., note 66

below), p. 473.


17/40


whose mappings of a vector onto itself are volume true, i.e. accord with the

concept of congruence. This demonstrates the uniqueness of the Pythagorean

determination of measure, hence its a prioricity and provides a new solution

to the Raumproblem.44 For our purposes, I wish here only to call attention to

the thoroughgoing character of Weyls,

naturphilosophische

perspective, which

unites physics and geometry; and thus to show how deeply rooted is his

philosophical opposition to what will become the standard philosophical view

of measurement in GTR.

Beginning with the third edition of Raum-Zeit-Materie, published in 1919,

Weyl also presents a development of Einsteins theory from the perspective of

his more general geometry. It is worth spending a few moments here to sketch

this mathematical characterization of general relativity, both because it creates

the mathematical machinery for other attempts (including Einsteins own,

which occupied him to the end of his life in 1955) to expand general relativity

into a unified theory of fields, and because it has become more or less standard

in contemporary accounts of general

relativity.

45 In Einsteins original treat-

ment, general relativity is developed by assuming Riemannian geometry at the

outset. But Weyl now develops the differential geometry on which general

relativity is based in three stages.

At the most fundamental level is the concept of a topological manifold,

corresponding (as in Riemann) to the notion of an empty world.46 To such a

manifold is added an affine connection based, as in Levi-Civita, on the

notion of infinitesimal parallel vector shift; physically, an affine space is

interpreted as a world endowed with a gravitation-inertial field, to which

Weyl gives the suggestive name,

Fiihrungsfeld

(guiding field) and in

which affine geodesics are instantiated as the inertial trajectories of

uncharged test particles, and gravitational

force as occasioning (slight)

departures from these inertial trajectories. Significantly, now the components

of the affine connection (the 40 quantities I;J, and not those of the metric

tensor g,, are taken as the field strengths of the gravitational field; indeed,

this produces the only consistent way in which to view the Principle of

Equivalence (since the I:, is not a tensor, and hence can be locally

44Die Einzigartigkeit der Pythagoreischen Massbestimmung (qcit., note 23); Mathematische

Analyse des Raumproblems (note 23), lectures 7 and 8. Weyl uses a method that combines Lies

theory of continuous groups with infinitely small virtual variations of the mappings onto itself of

an n-dimensional vector body radiating from a point. For a recent discussion of Weyls solution,

see the paper by Erhard Scheibe in W. Deppert et al. (eds), Exact Sciences and their Philosophical

Foundations (op.cit., note 49 below).

45A fundamental text in this regard is E. Schriidinger, Space-Time Structure (Cambridge,

Cambridge University Press, 1950).

?n Reine Infinitesimalgeometrie

(op. cit.,

note 36), p. 286, Weyl refers to a topological manifold

as an empty world, sometimes also referring to it as an amorphous continuum, but such a

manifold is not to be viewed as structureless, since the topological skeleton determines the

connectivity of the manifold in the large. Philosophie der Mathematik and Naturwissenschaft,

pp. 63 and 66 (op.cit., note 66 below).


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848


transformed away). Finally comes the notion of a metric space in which it

is possible to speak of the distance (respectively, time interval) between two

point-events. This is a world endowed with a causal structure and either (as

in Einstein) a path-independent transport of length, or, with Weyl, only a

path-dependent

one.

A crucial consideration is that the metrical structure be compatible with

the affine structure so that affine geodesics are also geodesics of the metrical

space. Weyl proved that the metric univocally determines the affine connec-

tion; this relation of compatibility he calls the basic facts

(Grundtatsache)

of

infinitesimal geometry.47 Hence, even though the notion of metric is logi-

cally the more fundamental conception, what is most significant about this

treatment is that it shows how Einsteins unification of gravitation and

inertia (from the Principle of Equivalence) can be conceived solely as the

structure of the affine connection of the manifold of space-time. This

amounts to, as it were, peeling away the metrical structure of the manifold

to reveal the structure responsible for the effects of inertia and gravitation

(hence Weyls term, Fiihrungsfeld, guiding field).48 This approach was

subsequently adopted by Einstein and has become more or less standard.

What is not standard is that, for Weyl, there are two different kinds of

curvature of space-time, one corresponding to the gravitational-inertial field,

the other corresponding to the electromagnetic field, which also enters into

consideration as part of the metrical structure.49

3.

When, in early March, 1918, Weyl first informed Einstein of his result,

Einstein wrote back that he thought it a a stroke of genius of the first

rank.50

But within a few days, Einstein was to formulate an objection that, as

47Raum-Zeit&A4aterie,

3rd Edn (op.&., note 37) p.111; see the discussion at the end of Section

2 below and the Appendix to this paper.

sCoffa (op.&., note 18) p. 279, uses this unstripping metaphor.

491n addition to providing the basis of the modern mathematical treatment of GTR, Weyl also

proposed a method for solving the problem of measurement in GTR (see below), made

fundamental contributions to formulating the equations of motion, discovered a large class of exact

solutions to the Einstein-Maxwell field equations, and helped to create relativistic cosmology.

These contributions are surveyed in .I. Ehlers, Hermann Weyls Contributions to General

Relativity,

in W. Deppert et

al.

(eds),

Exact Sciences and their Philosophical Foundations,.

Proceedings of the Hermann Weyl Congress, Kiel, 1985

(Frankfurt am Main-Bern. Verlag Peter

Lang, 1988) pp. 833105. In Ehlers assessment,

Apart from Albert Einstein, nobody has

contributed more to the conceptual clarification of the general theory of relativity than Hermann

Weyl. (See pp. 8485.)

5In a postcard to Weyl dated 6 April 1918; quoted in Straumann

(op.&.,

note

25),

p. 415.

Eddington (op.cit., note 52 below), regarded Weyls theory as unquestionably the greatest advance

in relativity after Einsteins work (p.198). Of course, Weyls work was the spur for Eddingtons

own efforts toward unification.


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849

we shall see, proved to be decisive (to which we will come in a moment).

Nonetheless Weyls paper was communicated by Einstein to the Prussian

Academy of Sciences (after some negotiation by Einstein on Weyls behalf with

Walter Nernst, who presumably didnt understand it) and it was published,

with an appended note by Einstein outlining his objection (which Nernst had

insisted on) together with a reply by Weyl, in the proceedings of the Academy

in May 191LL51

Weyls theory, put forward in two explanatory versions, received an

unusual reception from the theoretical physics community, beginning with

Einstein himself. This response still awaits an adequate historical treatment;

and it seems to tell much about authority and how challenges to authority

are met within science. There is, above all, the manner of Einsteins reaction.

The received view is that Einstein simply pointed out that Weyls theory was

not in accord with the empirical facts and that was the end of the matter.

But Einsteins response-over the next six or so years -was considerably

more ambiguous and the actual sequence of events is considerably more

complicated.

It is of course true that Einstein expressed this objection privately in

correspondence with Weyl and then publicly, in the aforementioned note

appended to Weyls Prussian Academy paper, and then again indirectly in the

well-known popular lecture entitled Geometry and Experience given in

January, 1921, to the Prussian Academy. This objection was taken up,

amplified, and broadcast to the wider physics community by Wolfgang Pauli jr.,

in the canonical encyclopedia article on relativity he wrote in 1920 (when he was

20 ). The objection seemed to show that Weyls theory was not in agreement

with observation.

In fact, in its original version, Weyls theory predicted effects that, as

Eddington showed, were below the threshold of observations2 and, as it later

turned out, within the limits of quantum mechanical uncertainties.53 As Pauli

himself had helped to show in one of his first scientific publications, Weyls

theory, though containing field equations of the fourth (and not the second)

order, yielded Einsteins field equations as a special case and agreed on two

Gravitation und Elektrizitat (op. cit., note 36); for Nernsts obstruction, see Sigurdsson (op. cit.,

note 30), pp. 166167.

The Mathematical Theory

of

Relativity (Cambridge, Cambridge University Press, 1923) p. 207.

After making some order of magnitude assumptions which set Fik comparable to the force at the

surface of an electron, Eddington concludes, Thus dN1 [the ratio of the change in length of an

infinitesimally transported vector to its original length at P-TR] would be far below the limits of

experimental detection. Paulis text is cited in note 55 below.

53For example, a standard text reports,

The strength of Einsteins objection seems not

as powerful now as at the time when it was raised, since we know the classical physics does

not describe atomic phenomena without certain quantum-theoretical modifications. R. Adler,

M. Hazin and M. Schiffer. Introduction to General Relativity, 2nd Edn (New York, McGraw-Hill,

1975) p. 506.


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850


predictions derivable from Einsteins theory: the gravitational red shift and the

advance of the perihelion of Mercury.54

What then was Einsteins objection? We do observe that measuring rods

retain their length under transport in electromagnetic fields;

prima facie,

this is

evidence that Riemanns geometry, not Weyls, is the geometry of space-time.

More substantively, Einstein argued as follows: if Weyls theory is correct, then

the spectral lines emitted by atoms would not be very sharp and well-defined as

in fact they are observed to be. For if two atoms, say of hydrogen, are once

together in one space-time region and then transported via different paths to

another region of space-time where they are brought together again, then,

according to Weyls theory, we should generally observe a difference in their

spectral lines corresponding to their past histories, i.e. to the differing values of

the electromagnetic field strengths in the space-time regions they occupied in

the interim. Pauli, indeed, showed that no matter how small the initial

difference in the spectral lines of the two atoms posited by Weyls theory, this

difference would increase indefinitely in the course of time.55 But astronomi-

cal observation tells us that hydrogen atoms everywhere in the heavens exhibit

the same spectral signature. So Weyls theory did not, apparently, correspond

to the facts of observation.

At the time, this objection was largely taken as decisive by the community of

physicists, though, as has been noted, perhaps it should not have been. In no

small measure, this is of course due to the enormous prestige of Einstein as the

%ee Merkurperihelbewegung und Stralenablenkung in Weyls Gravitationstheorie, Verhand-

lungen der Deutschen Physiknlischen Gesellschaf 21122 (1919), 742-750. In another early paper Zur

Theorie der Gravitation und der Elektrizitlt von Hermann Weyl,

Physikalischen Zeitschrift 20

(1919), 457467, Pauli took up the issue of the action function, seeking in particular, a function

yielding static, spherically symmetric singularity-free solutions (corresponding to the atomic

composition of matter). Weyl did succeed in finding solutions containing singularities, i.e.

corresponding to an electrical particle in which the phase quantity a, (the electrostatic potential) has

a singularity at the radial center, which briefly led him to proclaim, Matter is accordingly a true

singularity of the field. (See Space-Time Matter, pp. 298-300.) However, the failure to find

singularity-free solutions, hence to resolve the problem of matter, coupled with the fact that the

differential equations of Weyls theory were so complicated it was not possible to integrate them,

were ultimately the stumbling blocks on which Weyls theory foundered. See the article by

V. Bargmann, Relativity, in M. Fierz and V. Weisskopf (eds),

Theoretical Physics in the Tlventieth

Century; A Memorial Volume to Wovgang Pauli (New York, Interscience Publishers, 1960)

pp. 1877198.

Taking the electrostatic potential p to be a function of time, Pauli derives, from Weyls gauge

transformations, the equation r = roe@,

where r is the proper time and a is a factor of

proportionality. Pauli explains this equation as follows: Let two identical clocks C,, Cz, going at

the same rate, be placed at first at the points P,, at an electrostatic potential (o,. Let the clock C,,

then be taken to point Pz, at potential pa, for t seconds, and then finally returned to P,. The result

will be that the rate of clock C,, compared with that of clock C,, will be increased or decreased,

respectively, by a factor exp[-a((o, - q,)t] (depending on the sign of

a

and of (o, - p,). In

particular, this effect should be noticeable in the spectral lines of a given substance, and spectral

lines of definite frequencies could not exist at all. For, however small a is chosen, the differences

would increase indefinitely in the course of time, according to [this equation]. Theory of Relativity

(New York, Pergamon Press, 1958) p.196; the original German text was published in 1921 as

Relativitltstheorie in the Encykloptidie der mathematischen Wissenschuften V19 (Leipzig, B.C.

Teubner).


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Weyl , Rei chenbach and the Epistemol ogy of Geomet ry 851

discoverer of general relativity. In fact, and as hinted above, in several letters to

Weyl during 1918 and 1919, Einstein reveals that his opinion of Weyls theory

is far less one-sided than his public posture would suggest.56 And there can be

little question that recurring bouts of doubt as to the total invalidity of Weyls

approach, and of the related generalization offered by Eddington are evident.

For even as he repeatedly remarked during the period 1919-1925 that this route

led only to exasperating dead-ends, he continued to explore his own variants of

the Weyl/Eddington unification schemes in the initial steps of what was to be a

three-decade-long futile effort to construct a unified theory of fields.57 It is

almost as if Einstein could finally quiet his own nagging self-doubts only by

exhausting all the possibilities he deemed reasonable within what he called the

Weyl-Eddington complex of ideas.58

Despite his enormous respect for Einstein, Weyl was not persuaded by the

Einstein objection. Instead, he adopted a two-pronged argumentative strategy

to counter it, in effect, producing a second explanatory version of his theory. On

the one hand, he provocatively responded that the behavior of physical objects

such as rods and clocks or, for that matter, atoms, has as such nothing to do

with the ideal metric notions defined by vector transference:

The functioning of these instruments of measurement is however a physical occur-

rence whose course is determined through laws of nature and which has as such

56Excerpts from letters in 1918 and 1919 are quoted in Straumann (op.&., note 24).

Banesh Hoffman also speaks of Einsteins official argument against Weyls theory; Albert

Einstein; Creator and Rebel (New York, Viking, 1972), p. 223. The unofficial side of Einsteins

objections to Weyl emerge clearly in his correspondence with M. Besso, his friend in Zurich

who was also a friend of Weyl and favorably disposed to the logic of Weyls theory; see his

letters to Besso of 4 December and 12 December 1918 and of 26 July 1919 in P. Speziali (ed.)

(note 58 below). F. Herneck has uncovered a recording Einstein made in Vienna on 14 January

1921, in which Einsteins warning, that unless the interval element ds is connected with the

observable facts, a

reality-strange

Wirklichkeitsfremden)

theory is the result, is taken

as an indirect criticism of Weyl; Einstein und seine Welrbild (Berlin, Der Morgen, 1976),

pp. 103-108.

s71n addition to the paper of 1919 (op.&.

,

note 68 below), Einsteins engagement can be

tracked through a large number of papers in the 1920s in the

Sitzungsberichte

of the

Preussichen

Akademie der Wissenschaften, Physikalische-Mathematische Klasse: Ueber eine naheliegende

Ergiinzung des Fundaments der allgemeinen Relativitiitstheorie (1921) pp. 261-264; Zur

all,emeinen RelativitSitstheorie (1923), pp. 32-38 and 7677; Zzrr afien Feldtheorie (1923),

pp- 1377140; Einheitliche Feldtheorie v& Gravitation und Elektriz& (1925), pp. 414419, as

well as in The Theory of the Affine Field, Nature 112 (1923), 448449 and in Eddingtons

Theorie und Hamiltonisches Prinzip, an Anhang to the German translation of Eddingtons

work (op. cit., note 52), Relativitatstheorie in M&hematischer Behandlung (Berlin, J. Springer,

1925), pp. 367-371. The conclusion there,

essentially no different from similar verdicts

expressed since 1919, reveals the continuing attraction of the Weyl-Eddington approach: For

me, the end result of this consideration unfortunately consists of the impression that the

Weyl-Eddington deepening of geometric foundations is capable of bringing us no progress

in physical knowledge; hopefully, future development will show that this pessimistic opinion

has been unjustified. As we shall see, Reichenbach will take to heart the first clause of this

opinion.

Letter to Besso 25 December 1925, P. Speziali (ed.), Albert Einstein Michele Besso Correspon-

dance, 1903S1955 (Paris, Hermann, 1972), p. 215.


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852


nothing to do with the ideal process of congruent transplantation of world intervals

(Verpjlanzung von Weltstrecken).59

For Pauli, this meant that although there was no longer a direct contradic-

tion with experience, there also no longer existed an immediate connection

between electromagnetic phenomena and the behavior of measuring rods and

clocks; hence, the connection between electromagnetism and the world metric

posited in Weyls theory becomes purely formal. Eddington similarly inter-

prets Weyl as thus giving up any claim to characterize the geometry of the real

world, by providing only a graphical representation, i.e. a kind of conven-

tional representation, of world geometry.60 After all, if geometrical relations

are not concerned with measuring rods and clocks, what are they concerned

with?61

Weyls point, however, though Paulis response does not directly address it,

was that such a response ducks an explanatory burden that is essential from the

viewpoint of a consistent theory of fields or a systematic theory of field-matter

interactions.62 For it is just wrong-headed or, as Weyl expressly states,

perverse to use physical bodies such a rods and clocks, which are indicators

of the gravitational field, as at the same time instruments to stipulate metric

relations. To do so is just to treat as a definition (rigid rod, clock) what

should be explained, i.e. should be derivable from a systematic theory.63 In terms

of such a systematic theory, Einsteins

dejinition

of measure determinations in

the metrical field with the help of measuring rods and clocks has validity only

as a preliminary connection to experience just as does the definition of electrical

field strengths as the ponderomotive force on a unit charge. In order, in Weyls

terms to close the circle, it is

necessary, once a suitable action law has been set up, to

prove

that here, the charged

body under the influence of the electric magnetic field, there, the measuring rod under

the influence of the metrical field, exhibit, as consequences of the action laws, that

59Eine Neue Erweiterung der Relativit%tstheorie,

Annalen der Physik 59 (1919), 101-133, at

p. 113.

The Mathematical Theory of Relativity (op.cit.,

note

52),

Section

83

(Natural geometry and

World geometry). The new view entirely alters the status of Weyls theory. Indeed it is no longer

a hypothesis, but a graphical representation of the facts, and its value lies in the insight suggested

by this graphical representation. The antithesis of natural geometry versus graphical represen-

tation will recur in Reichenbachs examination of Weyls theory (see below). Pauli subsequently

also finds this interpretation to characterize the situation very well; see his review of the German

translation of Eddingtons book in Die Naturwissenschaften 13 (1926), 273-274 and also his letter

to Eddington of 20 September 1923, thanking Eddington for sending his book; A. Hermann

et al.

(eds) Wolfgang Pauli Wissenschaftlicher Briefwechsel, Bd. 1 1919-1929 (New York/Heidelberg,

Berlin, Springer, 1979), pp. 115-l 19.

As Coffa (opcit., note 18), pp. 283-284, forcefully puts this objection.

62Weyl changes his own interpretation of his theory along this axis; see Feld und Materie

(op.cit., note 66 below) and the 5th edition of Raum-Zeit-Materie (op.cit., note 26), Section 38.

63Eine neue Erweiterung der Relativitltstheorie (op.cit., note

.59),

p. 113: If then these

instruments [i.e. measuring rods and clocks] also play an unavoidable role as indicators of the

metric field...then it is apparently perverse (verkehrt) to define the metric field through indications

taken directly from them. As we shall see, Reichenbachs discussion concludes just the opposite.


23/40


853

behavior we had originally utilized for the physical definition of the field

magnitudes.G4

General relativity, in the conception of Einstein, is not, of course, systematic

in this sense, relying as it does on the notion of a practically rigid rod that

corresponds to congruence at a distance (i.e. path independent transport of

length). However, from the perspective of Weyls elegant generalization of

Riemannian geometry, the fundamental metric concept, congruence, is only to

be properly conceived as a purely infinitesimal concepV5 to do so renders

Einsteins practically rigid rod an unprincipled and gratuitous assumption.

As we shall see in a moment, this assumption is built into Reichenbachs neo-

conventionalist account of the geometry of space and time, with an argument

that has Weyls alternative squarely in its sights.

Clearly, Weyl had to account for why we do observe the congruence-

preserving behaviors of rods and clocks that we do, as well as for the constancy

of spectral lines of atoms. He does so (and this is the other component of Weyls

explanatory strategy) by invoking a dualism regarding the manner in which

physical quantities are determined, a distinction which he sees as reprieving his

theory from the empirical r

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Weyl, Reichenbach and the Epistemology of Geometry

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