hugo dingler's philosophy of geometry

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HUGO DINGLERS

PHILOSOPHY OF GEOMETRY

ROBERTO TORRETTI

1. The Connection between Geometry and Reality

Hugo Dingler (1881-1954) studied mathematics in Gt-tingen at a time when Felix Klein, David Hilbert and Her-mann Minkowski taught in that university (Dingler 1907).He also studied physics with Wilhelm Rntgen in Munichand, while at Gttingen, he attended Edmund Husserlscourses on philosophy. He obtained a doctors degree in

Munich with a dissertation on differential geometry.From 1920 to 1932 he dealt with physics and its phi-

losophy as Extraordinary Professor at the University. Buthe took no part in the extraordinary physical discoveriesof the time. His numerous books and articles dealt almostexclusively with what we now call foundational problems.

This paper was originally written as a chapter of my Modern Phi-losophy of Geometry, Vol. II. When that book was in effect dismantledby D. Reidels decision to go forward with the publication of Vol. Ialone as a self-contained book (which appeared in 1978 as Philosophyof Geometry from Riemann to Poincar), the materials I had been col-lecting for Volume II were left, so to speak, in the air. Most of themeventually found their way into my Relativity and Geometry(Oxford:Pergamon, 1983); but the chapter on Dingler would not fit in that

context, so I published it separately in Dilogos 32: 85-128 (1978).Bracketed boldface numbers refer to the pages of this publication.In the present version I have corrected a factual mistake regard-ing Dinglers dismissal from his university chair in 1934, which waspointed out to me by Professor Gereon Wolters. I have also correctednumerous misprints and improved the mathematical notation.


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Hugo Dinglers Philosophy of Geometry

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However, they did not discuss the foundations of the newtheories of Relativity and Quanta, but sought to consoli-

date the old physics, based on Euclidean geometry andNewtonian dynamics, into an impregnable and everlastingsystem.1

In 1932, Dingler was appointed to the chair of Philoso-phy at the TH Darmstadt. Two years later he was forcedinto early retirement, allegedly on grounds of economy;though some authors suspect that the decision may have

had something to do with [86] his authorship of a book onJewish culture.2 He subsequently sought to make amendsfor his un-German behavior by writing pro-Nazi trash, butas it is often the case with intellectuals who fawn onparty leaders in a one-party state he labored in vain,for he never quite managed to in ingratiate himself withthe new rulers of his country. Dinglers political antics are

probably no less to blame than his negative stance towardsmodern physics for the almost total oblivion into whichhis work fell outside Germany and Italy after the SecondWorld War.

1Abbreviated references are decoded in the list at the end of thispaper. Abbreviations consisting of an acronym or publication yearfollowed by page number(s) refer to works by Dingler mentioned

at the head of that list. For a full list of Dinglers publications see J.Willer, Relativitt und Eindeutigkeit, pp. 205-209. Willers book is themost complete monograph about Dingler in existence, but I havefound it not altogether reliable on some points. In my opinion thebest critical study of Dinglers epistemology is the introduction byK. Lorenz and J. Mittelstra to EW2. Wagner 1955/56 is a usefulsketch, but Krampf 1955, 1956 and 1971 are too partisan. On thespecific subject of geometry, A. Kamlah 1976 is excellent, thoughits main emphasis lies rather on P. Lorenzens reformulation of

Dinglers theory (see below, pp. 116 ff.). Literature in English is veryscarce. I am acquainted only with a somewhat misleading article byH. Sandborn 1952 and the encyclopedia articles in the Dictionaryof Scientific Biography and in Edwards Encyclopedia of Philosophy.

2 Dingler, Die Kultur der Juden: Eine Vershnung zwischen Religionund Wissenschaft (Leipzig 1919).


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The persistent aim of Dinglers philosophical endeavorswas to vindicate and secure our exact knowledge of nature.

He came to think of it as an ever growing system, continuallybringing new patches and aspects of reality under its sway,but resting on the indestructible, inalterable foundationsprovided by the four ideal sciences of logic, arithmetic,geometry and dynamics. Both historically and systematically,geometry played a key role in this construction. Euclids

Elementswere for a long time the paradigm of exact science,

until the 19th century discovery of alternative systems ofgeometry upset the traditional notions of rationality andbrought the foundations of knowledge into confusion. Thisdiscovery, more than any other historical event, motivatedDinglers own critical enterprise. In his system of science,geometry, as a discipline at once logical and manual,bridges the chasm between thought and reality, paves theway for physics and provides a cue for understanding the

true nature and scope of this science.Dingler discusses geometry and its foundations in many

of his books and in several articles. Two books are spe-cifically devoted to the subject: The Foundations of AppliedGeometry(GAG, 1911) and The Foundations of Geometry(GG,1932).3The former deals with it in the setting of Dinglersearlier epistemology and provides a suitable introduction

to his work. We shall examine it forthwith. The latter willbe studied in Section 4.The discovery of non-Euclidean geometries put an end

to the illusion that geometrical truth can be establishedby sheer deductive reasoning. Infinitely many internallyconsistent yet mutually incompatible axiomatic systemscan be devised that bind what are traditionally regardedas geometrical words into seemingly geometrical [87]

propositions. But each of these systems is by itself an emptylogical edifice (logisches GebudeGAG, p. 5),4which does

3 See also the articles 1907b, 1920b, 1925, 1931, 1935, 1955.4 In GAG, p. 44, the expression is explained as being synonymous

with axiomatic theory.


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not deserve the name of geometry unless it is actuallyconnected with reality.

The connection would be ensured for a particularlogical edifice if its axioms could be said to provide anaccurate description of actual facts. The logical edificewould then express the true geometry of our world. Thispresupposes, however, that the primitive or undefined termsthat occur in those axioms be assigned definite physicalreferents, which is not such a simple matter as nave em-

piricists are wont to believe. Take, for example, Hilbertsaxiom system for Euclidean geometry. Its axioms speakabout three kinds of entities called points, straight linesand planes. What on earth are these entities? One mightfeel tempted to say that a (Euclidean) straight line is anyobject satisfying all the conditions that Hilberts axiomsprescribe for straight lines. But surely the said conditionsdo not even make sense unless one knows already what

are points and planes, and what is to be understood byincidence, betweenness and congruence. We see, by theway, how silly it is to claim that an axiom system such asHilberts implicitly defines its primitive terms, when asa matter of fact it does not even supply fixed criteria forpicking out their referents.

Though Dingler, as we shall see, eventually succumbs in

GAG to a temptation akin to the one we have just rejected,when discussing the referents of Hilberts primitives hecomes up with a different and very important idea. Herecalls that straight lines and planes are being daily con-structed and used by architects and masons, astronomersand engineers. These everyday straights and planes are notusually tested for their conformity with Hilberts axioms butare immediately acknowledged as such if they agree, within

the accepted range of tolerance, with other previously givenstraights and planes, embodied in diverse tools and instru-ments (e.g. rulers, drawing-boards, the carpentry tool calledplane). They all derive ultimately from prototypes madeand kept in the factories of precision instruments.


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Prototype planes are originally produced according tothe following procedure: Take three hard bodies A, B, C

and rub them by turns, pairwise, against one anotherthatis, rub A against B, B against C, C against A, A against Bagain, etc.until three polished surfaces are obtained,each of which fits exactly the other two. The surfaces areviewed as more or less perfect planes, according to [88]the degree of exactness with which they fit each other.A prototype straight line can be obtained by making two

such plane surfaces with a common edge. The proceduredescribed constitutes what Dingler calls an empirical defini-tion or realization of the planeand, indirectly, of thestraight line (GAG, p. 19). The name is justified insofaras any surface will be regarded as plane if a prototypeplane can be applied on it and made to slide over it,within its boundaries, in any direction, while every part of

remains in full coincidence with some part of . Also,if fits any part of a plane surface. (Similar statementsconcerning straight lines will be easily made by the reader.)Dingler emphasizes the peculiar value of this method ofdefinition.

I ask someone: What is a plane. He does not answer aword, he does not give a logical definition of the plane,

expressed through concepts; neither does he produce anobject that has a plane surface. while pointing at the lat-ter []; but he takes three rigid bodies and starts theprocess of making a plane that was described above. Ifhe had just pointed at a prototype plane [Normalebene] Icould not have known that its color or the minor irregu-larities of its texture. etc. do not belong to it as such. Butthrough the method chosen by him I know this at once,

I have at once a notion of what a plane is, or rather ofwhat it ought to be.(GAG, pp. 22f.)

It is clear that we can only get a notion of the plane fromwatching our man making one if we understand what he


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is doing, i.e., if we grasp the rule that guides his behavior.We must be able to see. e.g., that wearing a green jacket is

not prescribed by that rule. But Dingler believes that rulesof human behavior are more or less immediately intel-ligible to us (while natural kinds and laws are not). Thisis a point he sets great store by in his mature philosophy:generality can never be attained by merely opening oureyes and unbiasedly contemplating the events of nature.Universals enter our world and shape our surroundings

only through our understanding of rules and our actingby them. Dingler does not pursue the matter any furtherin this book.

Another question that he fails to discuss, though it issuggested immediately by his exposition, concerns theconsistency of the empirical definition of the plane. Wedo not doubt that a surface that is plane when measured

up against a prototype manufactured by[89]

the aboveprocedure from three bodies A, B and C will also turnout to be plane when compared with a prototype obtainedfrom three other bodies A, B and C. Does our certainty restonly on an empirical regularity. inductively ascertained? Ordoes it stem from some deeper necessity? It is obvious atany rate that sheer conceptual analysis cannot guaranteeour expectations (cf. Bopp 1958, p. 64). We shall return

to the matter later on. when dealing with GG, where itlooms larger than in GAG.

In the book we are now examining. empirical defini-tions notwithstanding their great philosophical interestand their unquestionable practical usefulness are ulti-mately excluded from the foundations of applied geometry.Dingler notes that by picking out the referents of plane

and straight line we do not yet succeed in setting up aconnection between reality and a full system of geometryin which congruence and distance are defined (GAG, p.36). For this we need, according to him, a realization ofthe rigid body. Dingler does not even consider the pos-sibility of a physical geometry in which rigid bodies are


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lacking. (He does in fact mention Riemanns condition ofindependence of lines from the way they lie in space but

does not dwell on its significance.) But he is well awarethat their existence is compatible with any geometry ofconstant curvature. Depending on the actual deportmentof rigid bodies, the geometry of our world may be Eu-clidean or hyperbolic, spherical or elliptic. Evidently, onewill be able to know how rigid bodies behave only if onecan tell which real bodies are rigid. Dingler is never tired

of emphasizing that this is something that one cannotlearn from experience. In order to establish that a givenbody is rigid we must measure the distances between itspoints and verify that they do not change as the bodymoves. But in order to measure distances we must havean acknowledgedly rigid body at our disposal. A prototyperigid body appears thus to be necessary for establishing a

connection between a metric geometry and reality. Dinglerobserves that if the plane and the straight line have beenalready introduced as above, the construction of the rigidbody must take their empirical definition into account,for planes and straights must be preserved in every rigidmotion. This dependence of the full conception of therigid body on that of the plane and the straight line isresponsible, according to Dingler, for the complicated,

seemingly unintuitive form of Euclids fifth postulate (GAG,p. 35). On the other hand he says there [90] exists thealternative of introducing the rigid body directly first, andthen defining the plane and the straight line by means ofit (GAG, p. 33).5Such is the method preferred by him inGAG. If it could be successfully carried out in the way heproposes, it would automatically yield a positive solution

to the consistency problem we raised above regarding the

5 19th century mathematicians successfully characterized Euclideangeometry and the classical non-Euclidean geometries by their respec-tive groups of motions. Cf. Klein 1871, Poincar 1887, Lie 1890.


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empirical definition of the plane. (This might be thereason why Dingler ignores this problem in GAGit is

not worth mentioning, if his theory is correct.)

2. The rigid body

The Foundations of Applied Geometry (GAG) of 1911 reston the theory of science developed a year earlier in a small

book on the Limits and Aims of Science (GZW). Accordingto Dingler science has a double task, namely (a) to ex-

plain, to make understandable whatever exists [] (b)to dominate reality effectively, that is, to actually producedefinite things which did not exist previously (GAG, p.48).6Fulfillment of this task in any specific domain of realitymust begin by pointing out the elementary phenomenon

(Elementarvorgang) into which all phenomena of the domaincan be analyzed in thought and out of which they can bebuilt in fact. In stark opposition to 19th century empiri-cism, Dingler contends that such elementary phenomenacannot be discovered inductively but must be fixed foreach field of research by a rational choice. This choice isguided by a principle that Dingler, after Mach, calls thePrinciple of Economy: the elementary phenomenon is thesimplest conceivable phenomenon in the field.7 Scientifictheory teaches how complex phenomena are constructedfrom the elementary phenomenon. If actual practice de-viates from our theoretical predictions, the anomalies areattributed to unforeseen perturbations following a methodDingler calls Exhaustion.8 Little by little science succeeds

6 The theory of science sketched in GZW, is presented with greaterdetail in the second chapter of GAG.

7 Thus, for example, the elementary phenomenon of optics isillumination from a point-source (GAG, pp. 61 ff.)

8 Exhaustion is the mainstay of Dinglers methodology. Without ita strict systematic and orderly procedure in the formation of physi-


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mains or fields of study, but are roughly parceled out intothem by men. If, following Dingler, we correlate each field

of study with an elementary phenomenon it makes moresense to characterize the former as the set of all phenomenathat can be analyzed into the latter, rather than to definethe latter as the ultimate building block of phenomenain the former. This approach implies, of course, that a par-ticular phenomenon may belong to two or more fields andbe given a different explanation in each in terms of their

respective elementary phenomena. Such inconsistenciescan only be avoided if fields of study are linearly orderedand science is built stepwise so that the theory of the priorand more fundamental fields can never be contradicted bythat of the posterior and less basic ones. Some such ideaof science is found already in Dinglers first book (GKTW,4) but it is not fully [92] developed until much later, in

The Foundations of Physics (GP, first edition 1919; second,much revised edition, 1923) and in Physics and Hypothesis(PH, 1921). Exact science is described in these books asproceeding from a prescientific situation, in which no validgeneral statements can be made, toward a practically unat-tainable but indefinitely approachable goal, in which eachparticular statement of fact would be understood as aninstance of some universal law, inferable from a few simple

axioms. This edifice of exact science, whose foundationswere laid by Euclid and Newton, is called by Dingler thepure synthesis (later, the System). Dingler contrasts thisslowly growing but rock-firm core of human knowledge withthe peripheral, more volatile forms of scientific enterprise.Here fast advances are made by proposing hypothesesthat are experimentally tested in the manner explained

in standard textbooks of scientific methodology. But nomeaningful tests can be carried out, no reliable data canbe extricated from the wavering, drifting store of humanperceptions, unless we can resort to a firm, unambiguoustheoretical framework that regulates the construction of


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scientific instruments and the interpretation of their read-ings. According to Dingler, such a framework can only be

supplied by the pure synthesis, whose results can nevertherefore be undermined by experiment or contradictedby a viable scientific hypothesis. Consequently, the severalcomponents of the Systemlogic, arithmetic, geometry,mechanicscan be built by postulation and exhaustion,in the way described in GAG and cursorily sketchedabove, provided only that each discipline is careful not to

contradict the preceding ones. Geometry, ever faithful tologic and arithmetic, need not fear a clash with other sci-entific theories, for all the rest are subordinate to it. Likea majority of his contemporaries, Dingler believed thatevery experimental measurement involves a measurementof distance and consequently depends on the acceptedgeometry and specifically on the standard of rigidity. The

difficulty mentioned at the beginning of this paragraphdoes not therefore exist for geometry.In fact, the method of exhaustion, by which experi-

mental results that appear to deviate from the theory areinterpreted as the effect of uncontrolled perturbationsand classed as errors, is beautifully illustrated in the fieldof applied geometry. Take the case of surveyors. Dinglerquotes the following passage from Jordans Handbuch der

Vermessungskunde:

If the three angles of a triangle are measured with equalprecision, there will arise, because of errors of measure-ment, a difference between the [93] angle sum and 180.This difference is distributed by equal parts between thethree measured angles.11

11Jordan, op. cit., vol. II, 8th ed. (Stuttgart, 1914), p. 22, quotedin PH, p. 41n. Dingler also refers to vol. I of Jordans work (6thed., 1910), pp. 32 and 208f.


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Clearly, surveyors do not look upon Euclidean geometryas an empirical science, for they correct their experience

by this geometry. In the same book (PH, pp. 26 ff.) andin a separate article on The rigid body (1920a), Dinglertakes up the question of the manufacture of rigid bodiesin the factories of precision instruments. Rigid bodies soldaround the world to scientists and industrialists are madeand judged according to some ultimate standards of rigiditythat Dingler calls autogenous rigid bodies. The latter are

required to comply with Euclidean geometry. Suppose nowthat the exactest available autogenous rigid body appears, onexamination, to violate the laws of that geometry. Supposeall the more obvious sources of error have been eliminated.What can one do under such circumstances? Dingler saysthat one can either (1) regard the fact as the experimentalproof that real space is non-Euclidean or (2) ascribe the

deviation from Euclidean geometry to an unknown causeor to an unknown combination of known causes. Dinglerproposes with tongue in cheek that a questionnaire be sentto the better known manufacturers of precision instrumentsasking them what their reaction would be if the abovecame to pass. He is quite certain that they will all in prac-tice follow alternative (2) even though, if they have beenreading too much philosophy of science, they might not

confess it openly (1920a, p. 490). Moreover, he is certainthat the manufacturers will not hesitate to use Euclideangeometry in all the calculations required for checking theproperties of the autogenous rigid body. Dingler believesthat, living as we do in a vast and broadly unknown world,we are under no obligation to determine the true causeof the anomalous (i.e. counter-Euclidean) behavior of a

body and to correct the anomaly within a fixed length oftime. On the other hand, it is clear that, if such anoma-lies arise, there will never be any dearth of hypotheses forexplaining them. Dingler concludes that the rigid body ismanufactured in the factories that supply all scientific labo-ratories and observatories with their research equipment


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not empirically but by exhaustion of Euclidean geometry(1920a, p. 492).12Hans [94] Reichenbach objected to the

proposed geometrical definition of the rigid body, assert-ing that the latter can also be defined dynamically, namelyas a closed system, in which case it is an open, empiricalquestion whether rigid bodies are actually Euclidean ornot (1922, p. 52). For Reichenbach a closed system is onethat is not subject to the influence of external differentialforces13 and he apparently thought that such systems can

be detected by mutual comparison, without having to resortto a previously established geometry. But, since no bodyis ever totally immune to external differential forces fora significant length of time, if Reichenbachs criterion ofrigidity is adopted the question that must be answered inactual practice is not which bodies are rigid, but which aremore nearly so, i.e., which bodies are subject to a smaller

deformation due to the presence of such forces. Accordingto Dingler the magnitude and sense of the deformationsuffered under given circumstances by different bodiescan only be determined with the aid of the geometricalstandard of rigidity. This standard provides the zero-point

12 See also GP2, p. 140. In GP2, p. 136, Dingler bids us take a lookat a worker making a fine precision instrument. He continually resorts

to geometrical controls drawn from Euclidean geometry.He demands thatthe end surfaces of a screw remain parallel to itself, that all circlesand rectangles satisfy certain well-known geometrical relations, etc.If he notices in his device any deviations from these rules he willnot claim that he has found the empirical proof of non-Euclideangeometry but he will say that there is something wrong and hewill manipulate his device until it is right, i. e. until the greatestpossible agreement with Euclidean geometry is achieved. But thisis precisely the method of exhaustion.

13According to Reichenbach, a differentialor physicalforce is onewhich acts differently on different bodies and can be neutralized byshielding; he contrasts such forces with universal or metric forces,which act equally on all bodies and are not stopped by insulators.Reichenbach maintains that the latter can always be equated to zeroby suitably adjusting geometry. (Reichenbach 1958, pp. 10ff.)


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to which all comparisons of shape must be referred (GP2,pp. 133, 141f).

One can hardly conceive a stronger defense of theconventionality of geometry. But if Dingler is right anygeometry or at any rate any three-dimensional geometry ofconstant curvature can be likewise upheld by exhaustion.It is merely a matter of choosing its motions as charac-teristic of rigidity. Why then Dinglers unmitigated prefer-ence for the Euclidean system? We read in GAG that it is

generally agreed that Euclidean geometry is the simplest(GAG, p. 132). Dingler, however, does not just accept itfrom hearsay, but he seeks to prove it. In several occasionshe gives different arguments for the maximal simplicityof zero-curvature geometry which, if valid, would of [95]course, by the Principle of Economy, uniquely prescribethe choice of it.

In GAG and again twenty years later (1934) he claimedthat Euclidean n-space is simpler than every other n-dimen-sional Riemannian manifold because it alone possesses achart uwith coordinate functions u1,, un, that meets thefollowing requirement: for every integer i (1 in) andevery pair of real numbers aand b in the range of ui, themapping that assigns to each point P on the hypersurfaceui = a the point P on the hypersurface ui = b which has,

except for the i-th coordinate, the same coordinates as P,is an isometry of the former hypersurface onto the latter.I fail to see why a Riemannian manifold that meets thiscondition should in any sense be simpler than one thatdoes not. Moreover, Dinglers proof of his alleged charac-terization of Euclidean space is defective. He argues moreor less as follows: Let the components of the metric tensor

relative to chart ube designated by gjk (1 j,kn). u willmeet the stated condition if and only if the line elementgjk

dujdukis identical for every hypersurface ui= const. Thisimplies that the gjkdo not depend on u

i, so that


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(1)

=g

u

jk

i 0

(1 i,j,kn)

Since this must hold for every value of the index ithe com-ponents gjkof the metric tensor (relative to the chart u) mustbe constant. This can hold only if the manifold is Euclid-ean. The flaw in Dinglers argument is obvious. Take twohypersurfaces ui= aand ui= b, and let their line elements be

represented respectively by gijduiduj and hijduiduj. Since dui= 0 in either case, differences between gijand hij(1 jn)will make no difference in the line elements. Consequentlythe condition (1) need hold only forjik and we are leftentirely in the dark concerning the behavior of the derivativesgij/u

i(1 i,jn). It follows that the gjkneed not be con-stant and the manifold might not be Euclidean. The flaw in

Dinglers proof was pointed out to him by Kurt Reidemeister.Dingler responded by imposing one further requirement onthe chart u: the parametric curves (i.e., the integral pathsof the /ui) must meet orthogonally. This implies that gjk=0 forj k(1 j,kn) on the entire domain of u(1935, p.674). I am not sure that this will save the foregoing argumentbut it certainly suffices to characterize Euclidean n-space.For a chart umeeting boththe stated requirements is what

is usually called a [96] Cartesian coordinate system and itis a well-known fact that such systems can only be definedon Euclidean spaces. But who would venture to assert thata spherical surface is less simple than a cylinder becauseCartesian coordinates can be introduced on the latter, whileon the former one must make do, say, with latitude andlongitude?

In his article On the concept of maximal simplicity(1920b, now in GP2, pp. 101-113), Dingler proposed anexact criterion of maximal simplicity by which, he claimed,the Euclidean metric turns out to be simpler than anyother Riemannian metric. Dinglers criterion of maximalsimplicity is of course essential for all applications of the


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Principle of Economy on which his pure synthesis wasmade to rest at least until the early twenties. It may be

profitably compared with other such criteria proposed bysimplicity-loving contemporary epistemologists. Dinglerscriterion is intended to single out a logical formulationwithin a series or group of such formulations. Logicalformulations (logische Formulierungen) are identified by theirfeatures (Bestimmungen). Features shared by all formulationsin the group are called group-features. All other predicates

of a formulation are said to be individual features. Two ormore features are said to be independent if the negationof any of them is compatible with the conjunction of theothers.14A feature is said to be negative if it consists in theabsence of a feature; otherwise, it is said to be positive.15A feature is said to be elementary (relative to a group offormulations) if it is not a conjunction of independent

features or, if it is such a conjunction, all its componentsjointly determine every formulation of the group whichshares any of them. The simplest logical formulation ofa group is defined to be the one that possesses the leastnumber of positive, independent and elementary individualfeaturesrelative to the group (1920b, p. 428). Dinglersproof that Euclidean geometry is, according to this defini-tion, the simplest Riemannian geometry, is less clear and

straightforward than one might wish, but, if I understandit well, it ultimately rests on the following consideration:the metric tensor of Euclidean n-space is fully [97] deter-mined by the n(n+1)/2 constant values of its componentsrelative to a suitable chart, whereas no other metric ten-

14 Dingler defines independence only for a pair of determinations:they are independent if the negation of one is not incompatiblewith the other. But he subsequently uses this concept as if it werean n-ary relation between determinations (for any n > 1). I presumethat he would have accepted the definition I gave above.

15 Eine negative Bestimmung ist die Aussage, da die Formulier-ung eine gewisse Bestimmung nicht hat (1920b, p. 427). I confessthat I am unable to make any sense of this definition.


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sor of a Riemannian n-space is fully determined by sucha small set of numbers.16The argument is conclusive only

if the constant value of a component of a metric tensorrelative to a chart of the respective n-manifold constitutesan elementary feature of the tensor, in Dinglers sense.Now such a component, even if it happens to be constant,is not just a number, but a scalar field. The statementthat it is everywhere (on the charts domain) equal to afixed real number k can be analyzed into the statement

that its value at an arbitrary point of its domain is k andn statements to the effect that its partial derivatives withrespect to the coordinate functions are all, at that point,equal to zero. These statements are not jointly true ofevery metric tensor of which any of them is true. Conse-quently the statement that was analyzed into them doesnot express an elementary feature (relative to the group

of Riemannian metric tensors). The metric of Euclideann-space does not therefore exhibit only n(n+1)/2 positive,independent, elementary individual features, but at leastn(n+1)2/2provided that the features brought to light inthe preceding analysis are indeed elementary. But in thatcase the Euclidean metric is not the simplest Riemannianmetric but only one of the simplest, according to Dinglerscriterion. For the metric tensor of any maximally symmetric

n-space can be specified in terms of a suitable chart bygiving the value of its components relative to that chartat an arbitrary point of the charts domain and the valueof its partial derivatives at that point with respect to thecoordinate functions. And this makes n(n+1)2/2 features,exactly as in the Euclidean case. We need not probe into

16

The complications in Dinglers actual argument are partly dueto the fact that he tries to prove a stronger proposition, namely,that Euclidean space is the simplest metric space (or, as he puts it,that the Euclidean distance function is the simplest such function).However, he makes some additional assumptions that restrict themetric spaces under comparison to the class of Riemannian n-spaces.See 1920b, pp. 431-33.


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this matter any further. I have dwelt on it so long chieflybecause it serves to illustrate the pitfalls one is bound to

run into when discussing the comparative simplicity of com-plex theories. In Das Experiment (1928) and after, Dinglerno longer looks on simplicity as the decisive criterion thatguides the construction of scientific theories. He expectsinstead that the scientists will to refer all the glitteringand ever changing appearance of nature to a few definite,stable, unambiguously reproducible forms shall suffice to

determine the system of exact fundamental science. Thus,he writes, [98] when we strive to find absolutely unambiguousforms(and such is the unavoidable foundation of all exactscience) we are led perforce to Euclidean geometry (EW2,p. 134).17However, as one of his critics did not fail to pointout, Dingler often uses his new favorite term unambigu-ousness (Eindeutigkeit) much in the same sense in which

he earlier used simplicity (Einfachheit). For instance, hecharacterizes uniformly accelerated motion as the mostunambiguous (eindeutigste) motion that starts from rest,because it contains the least number of defining features(die wenigsten BestimmungenE, p. 117).18

The deadliest objection against the definition of the rigid

17 Further on, Dingler adds that a unique geometry, i.e. so-called

Euclidean [geometry], results einfach aus der Absicht heraus, dieberhaupt erst zur Grndung einer Geometrie hintreibt, aus derAbsicht, in dem vielgestaltigen und flieenden Wirklichen feste,eindeutig bestimmte Formen aufzustellen (EW2, p. 138).

18 See Th. Vogels remarks in von Aster and Vogel (1931), p. 11.Subsequently Dingler regarded the quest for maximal simplicity asa consequence of the supreme scientific principle of unambiguous-ness (Eindeutigkeit). He proves it as follows: Der Geist ist fhig,

ungezhlte Arten von zu realisierenden Planen (Ideen) zu fassen(Grundfhigkeit). Um unter diesen eine eindeutige Auswahl zutreffen, bedarf es eines formalen Prinzips. Da sich aber die Defi-nition einer Idee stets durch Angabe einer (endlichen) Zahl vonBestimmungen vollzieht, die beliebig gro werden kann, so ist daseinzige Mittel einer eindeutigen Festlegung formaler Art ein Mini-mumprinzip fr diese Anzahl (MP, p. 197).


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body by the laws of Euclidean geometry was raised byDingler himself. He called it Wellsteins objection, because

he drew it from Joseph Wellsteins article on the founda-tions of geometry in the Weber-Wellstein Encyclopedia ofMathematics. (The article was published several years be-fore GAG). Wellsteins objection stems from the essentiallyabstract nature of deductive theories. Any consistent theoryadmits several models, i.e. interpretations under which allits theorems come true, and if the theory is not too simple

it can be modeled in many incompatible ways in one andthe same domain of objects. Moreover, any of the modelscan provide a framework for setting up others. Considerthe case in point of Euclidean geometry, that is, the set ofall logical consequences of Hilberts axioms. Let Edenotea model of Euclidean geometry. In other words, E is a setof objects, called points, structured as a Euclidean space.

Consider the collection of all spheres ofE

that pass througha fixed point P. Excise P. There remains a collection ofpunctured spheres. Call each punctured sphere a planeand the intersection of two such planes a straight. Ifwe understand point and incidence as before and con-trive a suitable interpretation of congruence the axioms ofEuclidean geometry will all be satisfied on E\{P}, on the[99] proposed interpretation (Wellstein, GG, pp. 34ff.). LetE denote the set E\{P}, structured as a Euclidean spaceaccording to this interpretation. The Euclidean groupof motions acts of course on E in a definite way. But aparallel translation or a rotation about a point in Ehave next to nothing in common with their namesakes inE. Our example shows clearly that Euclidean geometry isunable to define the rigid body unambiguously (GP2, p.

147). In order to characterize the rigid body as a thing[that] changes in accordance with the laws of Euclideangeometry (of Euclidean motion)19 we must first indicate

19 GAG, p. 132; the context was quoted above on p. 9.


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how the Euclidean system is to be understood in its ap-plications to physical things. Giving up the opinion he

had voiced in GAG, Dingler now acknowledges that theconnection between geometry and reality must be securedbefore the rigid body is defined and cannot be broughtabout by its definition alone. The step-by-step constructionof physical geometry, broached in the second edition ofThe Foundations of Physics(GP2, 1923) and much furtheredin The Experiment (E, 1928), is carried out in full in The

Foundations of Geometry (GG 1933), and again, in a dif-ferent, seemingly more rigorous manner. in The Constructionof the Exact Fundamental Science (AEF; completed in 1944,published posthumously in 1964).20

3 The System and the Untouched

Before presenting the final version of Dinglers philoso-phy of geometry it is advisable to say a few words abouthis mature conception of the system of exact science towhich geometry belongs. This is developed in E and in TheSystem (S 1930), and again in The Method of Physics (MP,1938) and in his posthumous works The Grasping of Reality(EW 1955) and The Construction of the Exact FundamentalScience (AEF, 1964). I take it that Dinglers philosophyof science remained essentially the same throughout thisperiod, although in each new book he shifted perspectivesand changed the terminology, apparently with the aim ofmaking his teachings more perspicuous and persuasive.

Dingler never tired of proclaiming that security is the

essence of science (MP, p. 39; GKTW, pp. 4, 76; GP2, p.32; S, p. 60, etc.). At the frontier posts of knowledge, one

20 See GP2, pp. 147-164: E, pp. 56-109; GG, pp. 5-31; AEF, pp.153-187; see also MP, pp. 95-113, 164 ff.; EW2, pp. 131-142.


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may indeed [100] venture hypotheses, in the interest of aquick advance. But such hypotheses could not be measured

up against facts and would even lack a definite meaning ifscience did not possess an indestructible core of absolutelycertain knowledge. Dingler never reconciled himself withthe modern vision of the boat of science being unbuiltand rebuilt while out at sea.21

He motivates the scientific quest for certainty by anappeal to the supreme ends of mankind. No matter what

these ends are, man must seek to fulfill them in reality,and this implies, according to Dingler, that he must wishto learn more and more ways of unambiguously exertingon this reality unambiguously preconceived effects. Theability to exert such effects he aptly calls the dominationof nature (Die Beherrschung der Wirklichkeit oder der Natur)which is to him the universal goal of all science. For the

stated reason, the pursuit of this goal is a prerequisiteof mans ethical activity (MP, p. 42; cf. S, p. 31).22 Thedomination of nature in the above sense must rest onstable rules and laws stating what has to be done in orderto obtain the desired effects. To establish such laws, withthe widest possible scope, is according to Dingler the chieftask of science (MP, p. 44). How can one achieve a secureknowledge of general laws? Dingler rejects the classical

rationalist conception of self evident universal truths thatare intuitively grasped (PH, p. 118; S, pp. 50f.).

All universal statements are questionable for they referto an unlimited number of cases, which cannot all be per-ceived at once. There are only two kinds of statements that

21 I have not found evidence that Dingler was familiar with OttoNeuraths metaphor, but he does mention with obvious dislike

Poppers comparison of science with a building erected above aswamp on piles of which one knows that they must founder againand again (MP, p. 14; cf. Popper 1959, p. 111).

22 The highest goal of mankind is the subject-matter of ethics.Dingler devoted to it his book Das Handeln im Sinne des hchstenZieles (1935).


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Dingler holds to be indubitable: (a) Statements describingwhatever is directly experienced or lived through, with-

out any explanation or interpretation (Erlebnisaussagen);and (b) statements concerning feasible actions that canbecome and can have become actual or real at any time(HandlungsaussagenMP, p. 34).23 Consequently, if sci-ence is to be thoroughly [101] and securely grounded, allscientific statements must be shown to rest on either ofthese two kinds of uncontroversial carefree statements

(MP, p. 60). However, since induction is a hoax and nouniversal laws can be based on particular statements offact, Erlebnisaussagen cannot provide alone the sought forfoundation of science.24Therefore science must ultimatelyrest on Handlungsaussagen, that is, on proposalsandprogramsfor action. Each such proposal, if at all feasible,can be carried out innumerable times, semper et ubique

(in intention at least), and thus possesses a kind of inbuiltuniversality, stemming from our decision to apply [it]wherever it is appropriate (S, pp. 41,51). Or, as Dinglerputs it in AEF:

In the sphere of that which is subject to our will we canin fact [] bestow universal validity on some statementssimply by voluntarily carrying them out.

(AEF, p. 22)

The fundamental action statement or first propositionof the will (erster Willenssatz) from which all science de-pends is carefully articulated into a system of 22 principlesin AEF (pp. 38-48), under the general title of The Plan.For our present purposes it will suffice to quote a much

23 InDas Systemthese two kinds of statements are called, respectively,(a) Hic-et-nunc-Aussagenor Intellektuationsstzeand (b) Willensaussagenor Realisationsstze (S, pp. 33, 34, 40, 41).

24 On induction, see for example, PH, pp. 102, 132, 137ff.; GP2,pp. 39ff.; S, p. 50; MP, pp. 340ff.; EW2, pp. 141f.; also 1920c, p.130; 1923, pp. 23, 27.


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shorter collection of principles, listed in an earlier workas the demands of an ideal methodology:

(i) Science must provide an absolute and gapless founda-tion for each of its statements (Principle of Founda-tionPrinzip der Begrndung).

(ii) Its several stages and steps must be so ordered thatnone is presupposed by its predecessors (Principle ofOrderPrinzip der Ordnung, also called Principle ofPragmatic OrderPrinzip der pragmatischen Ordnungbe-

cause it forbids pragmatic vicious circles).(iii) It must be possible to reconstruct science in its entiretyat any time, so that, by following the same method againfrom the beginning one arrives at exactly the same re-sults (Principle of Arbitrary ReconstructionPrinzip desbeliebigen Neuaufbaues).

(MP, p. 71)25

[102] The principles of ideal methodology, especiallythe last two, clearly imply that science must have a definitestarting point. This must be found in a prescientific stageof human consciousness, in which no scientific laws, and,generally speaking, no universal statements can be held tobe valid. This stage or sphere of life Dingler called the

25 The first and the last principle are common to all rationalistepistemologies, but the principle of pragmatic order is characteristicof Dingler. A typical violation of this principle is the circle in theempirical foundation of geometry, which arises when geometricalpropositions are empirically tested by means of instruments built inaccordance with a definite geometry (see Dingler 1925). Becauseall exact scientific experiments require instruments embodying thechoice of a geometry, Euclidean geometry is absolutely to be pre-ferred over other geometries of constant curvature (which Dinglerregards, like Helmholtz, as the sole conceivable physical geometries,

being the only ones compatible with the existence of rigid bodies).Since every geometry of non-zero constant curvature involves aparameter namely, the curvature which, in practice, must befixed by measurement, any attempt to construct the fundamentalinstruments of measurement according to such a geometry wouldbe pragmatically circular for it would necessitate the use of thosevery instruments for measuring the parameter (EW2, p.164).


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pre-universal standpoint and also lifes standpoint, be-cause it is the standpoint that we ordinarily take in everyday

life.26Within this sphere we distinguish naturally betweenthat which depends on our own initiatives, our own will,and the Given, i.e., that part of my world in which Ihave not consciously undertaken any intellectual changes(bewute geistige nderungenAEF, p. 30).

As a technical term for the given, Dingler (1942) coinedthe expression das Unberhrte, the Untouched.27 This is

the rock bottom ground beyond which no foundationalinquiry can go, the zero-point from which the methodicalconstruction and reconstruction of science must begin. Inagreement with the third methodological principle, the Un-touched must therefore be recoverable at any time. Dinglerbelieves that its recovery can be effected by abstaining fromall general assumptions and theoretical interpretations, from

every deliberate intellectual addition to what is purelyand simply perceived (EW2, p. 81). We need not dwellhere on the familiar difficulties that beset philosophicalconceptions of the given. We must mention however thanin Dinglers mature view, the Untouched almost coincideswith the world of unsophisticated common sense. When alltheories are put aside, he says, things about me are simplyseen to be out there, quite apart from any considerations

regarding their causal action on my senses. [103] Otherpersons I meet as such, i.e. as living and thinking people,and it were madness to describe them as perceived bodieswhose behavior intimates that they are probably governed byrational souls. The given comprises all the basic faculties(Grundfhigkeiten)including the faculty of speechthat are

26 Vorallgemeinstandpunkt in GP2, p. 8, Lebensstandpunkt inPH, pp. 114f., 122f.; AEF. p. 34; Voreindeutigkeitsstandpunkt inS, p. 54; MWL, pp. 34-38. The earliest formulation, still stronglyinfluenced by 19th century mentalism, in GKTW, pp. 24ff.

27 See also Dingler, AEF, pp. 30 ff.; EW2, p. 80. In E, S and MPDingler had often spoken of untouched nature (die unberhrteNatur).


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necessary for the construction of science. Dingler repeat-edly emphasizes that the givenness of such basic faculties

is not a logical but a practical presupposition of science;they are not a premise to its deductions but a source of itsactual development. Because of this, it is neither necessarynor advisable to analyze them before proceeding to therigorous reconstruction of science. It is enough to exercisethem as required as we go forward.28

Foremost among the basic faculties needed for sci-

ence is the capacity to conceive what Dingler calls ideas(IdeenAEF, p. 54; EW2, p. 62). He describes them asmental forms or structures (geistige Gebilde) that cannot behad as direct perceptions of reality nor as memory imagesof such perceptions (EW2, pp. 109f.; cf. p. 61 and AEF, p.41). Such is, for example, the geometric idea of a widthlessline. Ideas are schematic and can be totally grasped and

analyzed for they contain only what we have deliberatelyput into them. They contrast with perceived realities, withtheir unlimited wealth of detail. Indeed, the infinite full-ness of the latter, as against the essential meagreness ofthe former, is singled out by Dingler as the main criterionfor distinguishing between the outer world of reality andthe inner world of ideas (EW2, p. 75). The very povertyof ideas is what makes them into the mainstay of exact

science. In order to secure its goals one must be able toextract unambiguously defined, reproducible forms fromever flowing nature. But in nature itself there are nosuch univocally definite elements, whose identity can alwaysbe verified with the highest available precision (AEF, p.54). That is why the scientific domination of nature neces-sarily depends on the conception and the deliberate, ever

imperfect but ever improving, realization of ideas. However,

28 That all the natural abilities demanded for the construction ofscience must be given at the outset was pointed out by Dingler inhis earliest book (GKTW, p. 33). He subsequently returned severaltimes to the subject; see MWL, pp. 64ff.; MP, p. 31; AEF, pp. 33ff.


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not any arbitrary idea, not any free [104] creation of thehuman mind will do for this purpose. The ideas of sci-

ence must be built following a strict order, in accordancewith the relevant principles of the Plan, beginning withthe fourfold primordial idea we shall discuss below. Ideasbuilt synthetically and methodically from this one are calledby Dingler ideas of s-science or s-forms (where s stands forsecure). Every attempt at determining something in naturein a truly unambiguous and reliably reproducible way can

only be carried out by means of the realization the ideasof s-science [] Indeed, every exact measuring instru-ment (from the simple meter rod to the interferometer,the quartz clock, etc.) is such a realization of s-forms. S-forms are therefore the only available means of achievingunambiguous statements, exact concepts and laws in thenatural sciences (AEF, p. 55).

The methodical synthesis of s-forms is presented byDingler in AEF. One of the principles of the Plan or firstproposition of the will developed at the beginning of thatbook prescribes that science, in its intellectual part, mustwork exclusively with ideas (AEF, p. 40). These ideas mustbe perfectly definite and it must be possible to reproducethem (mentally) without ambiguity and to recognize themwithout hesitation. They must also be realizable with

increasing precision and their realizations ought to berecognizable as such. The latter requirement implies thatscientific ideas must be specified by properties or relationsdrawn from actual experience (das wirkliche Erleben). Con-sider any such property. It is indeed unambiguous as anexperience, while it is experienced. But there is no way offixing it unambiguously in our minds or outside them; for

mere recollection does not certify identity through timeand there is no guarantee for the constancy of empiricalcircumstances. (None, at any rate, before exact science hasbeen erectedAEF, p. 14.)

Scientific ideas can thereforeonly be specified by perceived relations (Relationserlebnisse).But not any such relation will do, either. Only those per-


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ceived relations are relevant which are wholly independentof the peculiar nature of the related terms (AEF, p. 45).

According to Dingler only one perceived relation or per-ception of relation satisfies this requirement, and that is theperception of difference. This is had whenever we perceivean unqualified something (berhaupt Etwas), which isthereby singled out and characterized as different fromthe rest, the [105] background (AEF, p. 45). This rela-tion is perfectly unambiguous, for there is only one sort

of difference. It can be unambiguously fixed in the mind,and is unambiguously reproducible and recognizable at alltimes (AEF, p. 46). Dingler concludes that the primary s-idea, from which all other such ideas must be synthesized,is the idea of an unqualified, distinct something (EtwasUnterschiedenes berhaupt). This can be regarded (I) in itselfand (II) as limited by the background. Both the something

and its boundary can be viewed as (a) constant and (b)variable. We thus obtain a fourfold idea, which Dinglercalls the Schema:

(Ia) Something distinct, considered in itself, constant.(Ib) Something distinct, considered in itself, variable.(IIa) Something distinct, considered with regard to its

boundary, constant.

(IIb) Something distinct, considered with regard to itsboundary, variable.

The specific kind of something that is given by theSchema in each of these four cases becomes the funda-mental element of a science, when the corresponding caseis dealt with according to the principles of the Plan (AEF,

p. 57). The four sciences in question are:

(Ia) The science of number (Arithmetic).(Ib) The science of time and variables (Analysis).(IIa) The science of space (Geometry).(IIb) The science of motion and its causes (Mechanics).


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It is not possible to sketch here Dinglers constructionof Arithmetic and Analysis (AEF, pp. 57-137, 188-196).29

However, before broaching our chosen subject of geometry,it will be useful to paraphrase Dinglers discussion of therealization of the fundamental idea of arithmetic, namely,the idea of an unqualified, distinct, constant something.A natural object must be produced in which the idea isbrought out as clearly as possible. Such an object necessarilypossesses infinitely many irrelevant properties, which must

be maximally inconspicuous, while its [106] property of be-ing different (from the background) is suitably enhanced.This is achieved as follows: we take a maximally uniformbackground, such as a white leaf of paper, and producein it a difference of the sort described, e.g. a small blackspot. Such a spot meets the stated conditions. One paysno attention to its limits, its color, etc. One sees only that

an unqualified something is here singled out (AEF, p. 58).One can subsequently consider another realization of thesame idea, and view them both as belonging together. Byrepeating this procedure one can obtain a realization ofany positive integer.

4. The Foundations of Geometry

We have seen that young Dingler, in the wake of Ueber-weg and Helmholtz, regarded the twin concepts of rigidbody and rigid motion as the central concepts of geometry.In GAG he sought to define these concepts by means ofthe Euclidean axioms, which, he maintained, determinedthe simplest conceivable distance function. A rigid body,

in this view, is any body such that the (Euclidean) distancebetween any two of its points remains constant as the body

29A related, though greatly improved construction, is given inPaul Lorenzens book,Differential und Integral(1965), of which thereis an English translation.


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kinds of motion, rigid or otherwise, lie beyond its pale.31Dinglers construction no longer aims therefore at defining

the Euclidean rigid body, but at justifying the Euclideanaxioms (in Hilberts version) as necessary consequencesof the only viable unambiguous specification (by meansof the relation of difference) of the primordial idea ofan unqualified something, considered with regard to itsconstant boundary. Notwithstanding this important dif-ference of aim, and a few no less important differences

in execution, Dingler follows in AEF the same steps as inGG, successively specifying the topological ideas of surface,line and point, the affine ideas of plane, straight line andparallelism, and the metric idea of congruence; and thereis little doubt that he regarded the theory developed inAEF only as a more perfect version of the one presented inGG. In either book, anticipating the objection that, in the

end, he has merely produced a new-fangled axiom system,he emphatically asserts that his system differs from all itspredecessors insofar as it is naturally endowed, by the verymanner of its generation, with a definite interpretation.

The starting-point of geometry is very clearly describedby Dingler in his latest summary of the matter: We callsomething bounded, a body; the boundary, we call its sur-face. [] The experience (Erlebnis) of a body bounded

on all sides must be obtained from the Untouched, so tospeak; otherwise we cannot begin at all (EW2, p. 131). Itis clear that we do have the said experience. But is it our[108] only experience of something bounded? Cannot weperceive a surface bounded by lines or a line bounded bypoints? Moreover, though such experiences may indeedsuggest our idea of something bounded, does not the lat-

ter go far beyond them? We cannot visualize, perhaps, but31 In the light of this development it is ironic to recall Dinglers

earlier rejection of the Platonic view, shared by Euclid (and Hil-bert), that movement may not occur in geometry. Dingler haddeclared that this view rested on false epistemological assumptions(GG, p. 23).


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we can certainly conceive with the utmost precision an n-dimensional solid for any integer n> 3. To such questions

Dingler would have probably replied that any boundedobject we might make or handle is a (3-dimensional) bodyand its boundary is what we ordinarily call a surface (EW2,p. 131).32Dingler chooses to regard bodies as distinct fromtheir surfaces. (In other words, a Dinglerian body is alwaysidentical with its interior.) Both in GG and in AEF, Dinglerintroduces at the outset the idea of qualitative spaceor Q-space

(analysis situs space or AS-space in GG), as the backgroundfrom which bodies are separated by their surfaces. Therefollow in GG a few seemingly straightforward definitions:Q-space can always be divided by a surface into two sepa-rate parts; such a surface is called a separation surface. Aseparation surface T may in its turn consist of parts thatlie in different regions of space divided by a separation

surface Twhich does not meet T. If such is not the case,T is called a running surface(Laufflche). Dingler notes thatsuch a surface always has two sides, corresponding to thetwo parts into which it divides space. A running surfacecan always be divided by a line into separate parts. Sucha line is called a separation line. A running line can be de-fined analogously. A running line can always be dividedinto two separate parts by a so called separation punctual.

If P is a punctual such that no two parts of P lie on dif-

32 In a curious passage of MP, p. 105, Dingler sought to justify thechoice of a 3-dimensional space by observing that it was the simplestspace in which arbitrarily many bounded regions can simultaneouslystand in mutual contact, each with all. This, he felt, was required inorder to fulfill the desire for establishing arbitrarily many recipro-

cal causal actions. It is indeed well-known that in a 2-dimensionalspace homeomorphic with the Euclidean plane five or more regionscannot be mutually in contact each with all, except through a point.But it is hard to see why contact through points should not begood enough for transmitting causal influence. One can also takeexception with the assumption that the simplest space is the onewith the smallest admissible dimension number.


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ferent sides of a running surface, P is called a point. InAEF this simple development is replaced by twelve dense

pages of definitions, assumptions and theorems, amountingto a clumsy, skimpy, but for Dinglers purposes probablysufficient topological theory of surfaces. Its declared aimis to eliminate the ambiguities in the everyday concept ofa surface (AEF, p. 160). In the years following the publi-cation of GG, [109] Dingler obviously became aware thatthere are unspeakably many ways of exactly defining that

concept. He consistently prefers what he takes to be thesimpler, more natural alternatives, barring such monstersas a surface with isolated points or with protruding hairsor blades, that might be shaved off, so to speak, by aseparation surface which does not meet the body of whichthe surface in question is the boundary. The definitionsof GG are essentially preserved, but further qualifications

and distinctions are now introduced, such as that betweenopen and closed running lines, and between simply con-nected and multiply connected running surfaces. If weare allowed to take a look at the collection of all pointsdiscernible in Q-space, we shall see that Dinglers theoryfurnishes it with a natural topology, in which Dinglerianbodies are the open sets. That it is a Hausdorff topologyfollows at once from the very definition of a point. Does

Dinglers theory succeed in specifying the topology of Q-space any further? It is hard to tell. We may assume thata body whose boundary is a simply connected runningsurface is always homeomorphic to 3, so that Q-space is athree-dimensional topological manifold. But the only hintwe are given with regard to its global topology is not easyto interpret. Statement 1,22 says that space is unbound-

able (unbegrenzbar). This can be taken to mean merely thatno boundary can be set to space as such; that no surfacecan be found at which, so to speak, every running linewill stop. On this interpretation, 1,22 is innocuous andnot very illuminating. Indeed, every topological manifoldis unboundable in this sense. Dingler adds, however, that


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if space is divided by a separation surface into two bodiesK and K, K will be said to be finite if it contains nothing

of the unboundable of space.33In this case Kis said to befully infinite. On the other hand, if neither K nor Karefinite, they both contain something of the unboundable ofspace and are said to be partially infinite. There is onlyone way in which I can make sense of these baffling defini-tions, and that is by interpreting 1,22 to mean that space isnon-compact; in other words, that not every collection of

bodies comprising all of space includes a finite subcollec-tion that also comprises it. Without doing much violence tothe foregoing definitions one can then reformulate themas follows: K is finite if it is contained in a compact regionof space; otherwise K is infinite (fully or partially, asbefore). Evidently, the assumption that space is non-com-pact is not so indifferent as the assumption that [110] it

has no boundary, and one is entitled to ask what justifiesit. In the methodological comments that follow upon thesection on topology, Dingler observes that his assumptionsin that section are only intended to select and fix amongthe unencompassable multitude of possibilities that are hid-den in the concepts of everyday speech, those which areappropriate (AEF, p. 173). He claims moreover that thechoice made in each case is the only one that could be

unambiguously determined at its respective stage. With themeans available when the choice was exercised no otherstipulation could have been formulated. One may indeedwonder whether this can be said of 1,22, when interpretedas I proposed. For evidently the same means that enableone to postulate that space is non-compact can be usedfor pronouncing it compact, and I cannot see that the

former stipulation is more definite, or simpler, or in anysense more appropriate than the latter. These reflections

33 In einem Krper K kann [. . .] vom Unbegrenzbaren desR[aumes] nichts enthalten sein; dann heit K endlich (AEF, p.161).


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will appear to discredit my interpretation of 1,22. On theother hand, only on that interpretation does Dinglers

theory preclude space from having the topology of projec-tive space, a result that is taken for granted in the sequel.Of course, one may add, any postulate implying this resultwould seem no less arbitrary than the assumption that spaceis non-compact, within a theory whose axioms purportedlylack ontological power, and which, as Dingler puts it,do not say something about the constitution of reality,

but only choose some among its countless possibilities, inorder to determine certain concepts unambiguously, stepby step, in every respect (AEF, p. 175).

The theory of Q-space is followed by the all-importantdefinition of the plane. This is very much the same ineither book. In AEF it reads as follows:

4,5. [] A running surface whose two sides are undis-tinguished (1) in their entirety and (2) with regardto each of their points, is called a plane.

Dingler claims to show from his topological assumptions,that every plane is simply connected (4,54) and dividesspace into two connected (4,53) partially infinite regions(4,55). Also that two different planes cannot share a piece

of surface (4,6) and hence, if they meet at all, they meeton a separation line (4,62). Such a line is called a straight(4,63). Dingler maintains that no further assumptions [111]are necessary in order to prove Hilberts axioms of inci-dence and betweenness. However, most of his proofs areobjectionable.34 We must fix our attention on Definition4,5 itself, which is clarified by Dinglers own paraphrase

34Allow me to paraphrase the proof of 4,54: A plane is a simplyconnected running surface. If the plane E is not simply connectedand P is a point on E there is a neighborhood S of P on E whichcan be embedded in a simply connected running surface. A simplerunning line L1can therefore be drawn which cuts S into two partsand hence also cuts E into two parts. On the other hand (by the


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in EW: a plane is a surface whose two sides can be whollysuperposed at all places, and are thus spatially indistin-

guishable.35 Dingler is quick to point out that, in accor-dance with the Plan, only the relation of difference hasany bearing on this specification of the general idea of aboundary. We note however that not just any differenceorlack of differenceis relevant here, not, say, difference incolor or electric charge, but specifically spatial difference.Again, not every difference that we would ordinarily term

spatial can be taken into consideration; for many suchdifferences e.g. differences in size or shape have notyet been defined and will be defined later with the aid ofthe concept of a plane. Strictly speaking, only topologicaldifferences are available to us. But then, of course, if onlysuch differences are considered, every simply connectedopen separation surface in a topologically Euclidean space

satisfies Dinglers definition of a plane.36

That this was notwhat he had in mind, we gather at once from what he saysabout the practical procedure for the manufacture of planesurfaces, which, he maintains, is justified by Definition 4,5.This is none other than the three-plate method we describedin 1. Now, it is essential for this method that the threeplates be stiff. But how are we to define stiffness? Indeed

definition of multiple connectedness), there must be a simple run-ning line L2 on E which does not cut E in two. But then the tworegions of E can be distinguished by means of these two lines L1and L2, and E does not satisfy the definition of a plane (4,5). Theconcluding sentence is hard to understand: What two regions of Eare being spoken of? The two regions into which it is allegedly cutby L1? But 4,5 does not mention any such regions on a plane, butonly the regions outside it into which the plane divides space. How-ever we need not probe further into this matter, for the italicized

clause also cutsE into two parts is plainly a non sequitur.35 Eine [] Flche, bei der ihre beiden Flchenseiten im

ganzen und an allen Stellen aufeinanderlegbar, d.h. rumlich un-unterscheidbar sind (EW2, p. 133).

36 The two regions into which such a surface divides space arehomeomorphic and hence topologically indistinguishable.


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tion of the plane, for it depends essentially on the as yetundetermined group Gand its action on space. We cannot

expect, therefore, that our definition or Dinglers defini-tion 4,5 can of themselves contribute to specify geometryand geometrical predicates beyond the topological level.Dingler grants as much when he observes, at the end ofthe section on planes and straights, that the concept ofundistinguishability, on which everything rests, has not yetbeen defined in any way (AEF, p. 183).38 We can derive

but little comfort from this remark, for Dingler does notdefine this concept in the sequel. In fact, he resorts againto it, as it stands, in his definitions of parallelism, orthogo-nality and congruence.

[113] To these we must now turn. GG defines parallelline as follows: let be a straight on plane E, and let P bea point on outside ; if a line on passes through P

and the strip limited by and shows no difference oneither extreme to the left and to the right of P, and are said to be parallel. The Dingler parallel to throughP is obviously unique. If we grant that two straights do notmeet at more than one point, it is clear that two Dinglerparallels cannot meet at all (their intersection would oc-cur on one side of P). But there is no apparent reasonwhy any two coplanar lines must intersect if they are not

Dingler-parallel. Dingler, of course, is well aware of it.Bolyai-Lobachevsky geometry in which through any pointoutside a given straight line we shall find a single Dinglerparallel to , but also two Bolyai-Lobachevsky parallels andinfinitely many Euclid parallels to is excluded only byDinglers definition of the deformation-free body.39 This

38

Contrast with this his earlier reliance on our everyday perceptionof difference (1925, p. 323, quoted below in note 49).39 Let be a straight line, P a point outside it, a straight through

P on the plane determined by P and . is parallel to accordingto Euclids definition if and only if and do not meet. To explainparallelism in Bolyai and Lobachevkys terms it is convenient todefine first the angle of parallelism for at P: this is the smallest


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can be rendered as follows: We shall say that a body Kglides along a straight line or along a plane if it moves in

such manner that every point of K that lies on the saidline (plane) at any time continues to lie on it throughoutthe movement; we shall likewise say that K rotates abouta line that passes through it if it moves in such mannerthat any point of K that lies on remains on the samelocation throughout the movement; finally, we shall say thattwo straight lines and meet orthogonally if the figures

formed on either side of are indistinguishable. A bodyK is said to be deformation-freeif and only if (i) whenever Kglides along any straight line through K and any plane through , (a) K also glides along every other plane through and along every parallel to that passes through K andbelongs to any of those planes; (b) if H is the set of pointsof K which lie at some time during the motion on a given

straight line through , then, as K glides along and ,H lies at each moment on a parallel to ; (ii) wheneverK rotates about a straight line through it (a) K eventu-ally returns to its initial position; (b) if G is at some timeduring this motion the intersection of K with a half-plane bounded by , G comes to lie once on each half-planebounded by before returning to ; (c) if P is a point ofK outside which lies at some time during the motion on

a line that meets orthogonally at a point Q, then, if P1and P2 are any two locations of P during the motion, thesegments QP1 and QP2 are [114] indistinguishable except

angle at P whose two sides are (i) a straight through P which doesnot meet and lies on the plane determined by P and , and (ii)the perpendicular from P to . is parallel to according to

Bolyai and Lobachevkys definition if and only if the said angle ofparallelism is the angle formed by and . In Euclidean space allangles of parallelism are straight and the two definitions of parallelismare equivalent. Not so in Bolyai-Lobachevsky space: here angles ofparallelism vary with the distance from P to and are always acute,so that there are infinitely many parallels to through P in Euclidssense but exactly two in Bolyai and Lobachevskys sense.


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by their position. Two segments AB and CD are said to becongruent if a given pair of points on a deformation-free

body can be carried into coincidence first with AB andthen with CD. (GG, pp. 23-25).

Dinglers definition of a deformation-free body is plainlyan attempt to characterize Euclidean translations and ro-tations. We shall allow it to be satisfactory and shall notdiscuss the necessity (or redundancy) of its requirements.We would rather wish to learn what justified Dinglers belief

that such a complex and restrictive set of stipulations wasuniquely determined and preferable to every conceivablealternative. From Helmholtz he acquired the persuasionthat only deformation-free motions can induce a physicallymeaningful distance function on Q-space. This restricts thechoice to the four classical geometries of constant curva-ture: Elliptic, Spherical, Hyperbolic and Euclidean. The

first two are excluded if Q-space is non-compact. (Hencethe need for our proposed interpretation of 1,22). Buthow does Dingler justify his preference for Euclidean andagainst Hyperbolic or Bolyai-Lobachevsky geometry? Heoften notes that the latter is essentially ambiguous, for itcan only be determined up to a constant (we may take thisto be the Schweikart constant, i.e. the distance betweenthe vertex of a right angle and a straight line parallel to

both its sides; or the constant negative Riemannian spacecurvature, etc.see p. 23 n. 25). Euclidean geometry isnot affected by a similar indeterminacy but, as Dingler hadlearnt from Wellstein, in the familiar axiomatic formula-tion it leaves room for a wide variety of interpretations. Itwas his declared aim in GG to overcome this difficulty byhis novel, allegedly nonaxiomatic approach. He may claim

success if, but only if, his definition of the deformationfree body is truly unambiguous. This, as I shall now show,is far from being the case. Let R designate Dinglers Q-space with its natural topology and let Gbe the (abstract)group of Euclidean motions. If Dinglers definition of thedeformation-free body is correct there must be a mapping


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:G R R through which G acts transitively and effec-tively on R. must be such that, if B is any deformation-

free body and H and H are two locations filled by B at twomoments of its history, there is a gGsuch that (g,H) =H. Let :R Rbe an arbitrary homeomorphism. Let :GR R be defined by (g,x) = (g,1(x)). It canbe easily shown that Gacts transitively and [115] effectivelyon R through .40 A body any two of whose locations Kand Kmeet the condition

(h,K ) = K for some hG

will then satisfy Dinglers definition of the deformation-free body. In particular, if agrees with the identity on H,while (H ) is a proper subset of H, Dinglers definitioncan be simultaneously exemplified by the above mentionedbody B that successively occupies H and Hand by another

body B that successively fills H and only part of H. Ofcourse, the planes and straights and parallels mentionedin the definition of the deformation-free body will not bethe same in either case. This is only natural since we haveshown that Dinglers characterization of these geometricalentities depends on the choice of a group and its actionon Q-space. The group is here in both cases the same,

namely, the Euclidean group of motions, but its action isdifferent. Since there are no a priori grounds for prefer-ring to or to , there is no reason for holdingthat B is more properly a deformation-free body than B,or vice versa.

In AEF the treatment of parallelism and congruence israther different. Parallelism is defined for planes. Let 1be a plane that divides space into two regions R

1

and R1

and choose a point P in R1. Let 2be a plane through P

40We must show that for any gand hin G and any xin R, (gh,x)= (g,(h,x)). Now (g,(h,x)) = (g,1 (h,1(x))) = (g,(h,1(x))) = (gh,1(x)) = (gh,x).


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that lies entirely in R1and divides space into two parts R2and R2. 1 must them lie entirely in one of these parts,

say, in R2. The intersection of R1 and R2 is a bodyaslablimited by 1 and 2. If all the directions in whichthis body is unlimited are indistinguishable, 1and 2aresaid to be parallel. Two straight lines, 1and 2are parallelif they are the intersections of two parallel planes with athird plane. It is clear that parallel planes in the forego-ing sense do not meet. Dingler postulates that any two

planes meet unless they are parallel (7,2). He claims thatthe usual parallel postulate can be inferred from this one(8,4; 8,5). Let AB and CD be two segments, lying on twoparallel lines. If AC and BD, or if AD and BC also lie onparallel lines, AB and CD are said to be congruent (8,6).Now, if AB and CD are two arbitrary segments there alwaysexists, on Dinglers assumptions, a segment CB which is

congruent to AB by the foregoing definition.41

Congruence,or the lack of it, between [116] two arbitrary segmentswill therefore be defined if we can provide a criterion ofcongruence for segments with a common endpoint. Thecriterion supplied by Dingler is disappointing: two segmentsPO and PQ, with a common endpoint P, are said to becongruent if they are indistinguishable! (9,1).42

It must be clear by now that Dinglers construction of

geometry as a branch of exact fundamental science doesnot fulfill the high hopes encouraged by his ambitiousprogram. Paul Lorenzen has sought to vindicate Dinglersapproach to the subject in a very illuminating article onThe problem of the foundation of geometry as a scienceof spatial order (1961), apparently written while he was

41 Draw the parallel to AB through C. Draw the parallel to ACthrough B. Let Bdenote their intersection. (The latter must exist,for both lines lie on plane ABC and they are parallel, respectively,to two lines through A).

42 However, it is instructive to read Dinglers definition in thelight of Nevanlinnas treatment of congruence in Nevanlinna andKustaanheimo 1976, part I.


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working on the posthumous edition of AEF.43 We owe toLorenzen an exact definition of the key concept of undis-

tinguishability: Two objects are indistinguishable if everysentence that says something about one implies the sentencewhich says the same about the other. Since implication isexactly defined only for sentences belonging to formalizedor canonic languages, Lorenzens concept of undistinguish-ability is relative to the choice of such a language. In hisdiscussion of geometry, he considers a first-order language

with three sorts of individual constants and variables (stand-ing for points, straights and planes, respectively) and fourprimitive predicates: lies on (a binary relation between apoint and a straight, or between a point and a plane, orbetween a straight and a plane), lies between (a ternaryrelation between a plane and two points), is parallel to (abinary relation between two planes) and is perpendicular

to (a binary relation between a line and a plane). Weneed not worry here about the limitations of first-orderlanguage as a vehicle for geometry (cf. Tarski 1959). Leta and b be any two individual constants of Lorenzenslanguage. We may say that the objects denoted by a andb are geometrically indistinguishable in Lorenzens senseif every sentence S of the language, in which b does notoccur, implies the sentence S obtained by substituting b

for ain every occurrence of the latter in S. Guided by thisgeneral criterion Lorenzen achieves a rigorous formulationof Dinglers definition of the plane. Let the constant de-note a plane, while the constants P and Q denote points.Let S(,P) be any sentence in which the only individualconstants are and P and let S(,Q/P) be the sentenceobtained by substituting Q for P in every occurrence of

P [117] in S(,P). Planes are then characterized by thefollowing two principles of homogeneity:

43 Lorenzens edition of Dinglers AEF appeared in 1964.


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1. Inner homogeneity. If P and Q lie on , S(,P) impliesS(,Q/P).

2. Outer homogeneity. If neither P nor Q lie on , S(,P)implies S(,Q/P).44

Evidently the two principles will suitably restrict the ex-tension of the concept of a plane only if the predicates ofthe language are specified further. As usual, incidence andbetweenness are characterized by axioms. Lorenzen does

not list them but quotes the following two examples:If lies between P and Q, then, for any point X not on , lies between X and P or lies between X and Q.If lies between P and Q and both P and Q lie on the straight there is a point X at which meets.45

Parallelism and orthogonality are characterized by axiom

schemata:I. If point P lies on straight and point Q lies on straight

and and lie on plane and is parallel toplane , S(,,P,) implies S(,,Q/P,/).

II. If point P lies on the straights , and , and isperpendicular to plane and and lie on ,S(,,) implies S(,,/).

(Lorenzen 1961, pp. 136, 135)

Lorenzen adds the following existence postulate: If is any plane and P is any point outside there is aunique plane through P which is parallel to and aunique straight through P which is perpendicular to .In Lorenzens geometrical language congruence betweensegments or point-pairs can be easily defined. We say that

44 Lorenzen 1961, p. 134 gives only the first of these two prin-ciples. The second one is added in the more popular exposition inLorenzen and Schwemmer 1975, p. 227.

45 Lorenzen 1961, p. 136. P E in the first axiom is plainly amisprint for P E.


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two point-pairs A,Band C,D are parallel, if points A, B,C and D

lie on a single plane and there are two parallel

planes 1 and 2, distinct from , such that A and B lieon 1, and C and D lie on 2. Two point-pairs A,BandC,D are said to be orthogonal if A,B lies on a straightperpendicular to some plane through C,D. Four distinctpoints A, B, C and D, such that A,B is parallel to C,Dare said to form a parallelogram ABCD if A,C is parallelto B,D or if A,D is parallel to B,C. If say, A,C and

B,Dare parallel, A,D and B,Care called the diagonalsof ABCD. We can now define:

[118] Two point-pairs A,Band C,Dare congruent if andonly if one of the following conditions is fulfilled:

(i) A,B = C,D.(ii) A point of A,B, say A, is identical with a point of C,D,

say C, and there exists a point X such that ABXD is

a parallelogram with orthogonal diagonals A,X andB,D.

(iii) ABCD is a parallelogram and A,B is parallel toC,D.

(iv) There exists a point-pair X,Ywhich is congruent withboth A,B and C,D.

Lorenzens postulates imply that any perpendicular to

a plane is also perpendicular to every plane parallelto . For let be such a plane and the perpendicularto through P . meets at some point Q. If isa straight such that , and determine a plane which intersects along the straight . is thereby distin-guished among the lines through P on . This, however, isincompatible with axiom schema I.46It follows immediately

46 This can be shown more precisely as follows. Let x and y betwo individual constants or variables standing for points, straight

lines or planes. Let xydenote the intersection or meet of xand

y. Let xy denote their join, by which we mean (i) the planedetermined by them, if x and y are two straights or a straight anda point outside it or (ii) the straight determined by them if x and


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that any two points A and B on a plane and the pointsC and D (where the perpendicular to through A and B

meets a plane parallel to ) determine a rectangle. MaxDehn (1900) showed that the existence of rectangles in aspace will not suffice to mark it out as Euclidean unless itis asserted jointly with the Archimedean axiom.47Lorenzenobligingly adds this axiom to his assumptions. Lorenzenswork certainly throws a new light on the fundamentalconcepts of geometry, placing Dinglers true merits in a

proper perspective, and may even procur

hugo dingler's philosophy of geometry

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