Geometry and Empirical Science C G Hempel

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Geometry and Empirical ScienceAuthor(s): C. G. HempelSource: The American Mathematical Monthly, Vol. 52, No. 1 (Jan., 1945), pp. 7-17Published by: Mathematical Association of America

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GEOMETRY AND EMPIRICAL SCIENCE

C. G. HEMPEL, QueensCollege

1. Introduction. The most distinctive characteristicwhich differentiates

mathematicsfromthe various branches of empirical science, and which ac-countsfor ts fameas thequeen of thesciences, s no doubt thepeculiar certaintyand necessityofits results.No proposition n even the most advanced parts ofempirical cience can ever attain thisstatus; a hypothesis oncerning mattersof empiricalfact" can at best acquire what is looselycalled a high probabilityor a high degreeofconfirmationn thebasis of the relevant evidenceavailable;but howeverwell it may have been confirmed y carefultests, the possibilitycan never be precludedthat it will have to be discarded later in the light ofnew and disconfirmingvidence. Thus, all the theories and hypothesesof em-

pirical sciencesharethisprovisionalcharacterofbeingestablishedand accepted"untilfurther otice,"whereasa mathematicaltheorem, nce proved,is estab-lished once and forall; it holdswith that particularcertaintywhich no subse-quent empiricaldiscoveries,howeverunexpected and extraordinary, an everaffect o the slightest xtent. t is the purposeofthispaper to examinethe na-ture of that proverbial mathematical certainty"withspecial reference o ge-ometry, n an attemptto shed some lighton the question as to the validityofgeometrical theories, nd theirsignificance or our knowledgeof the structureofphysical space.

The natureofmathematicaltruth an be understood hrough n analysis ofthe method by means of which it is established. On this point I can be verybrief: t is the methodof mathematicaldemonstration,which consists in thelogicaldeductionoftheproposition o be provedfrom therpropositions, revi-ously established. Clearly,thisprocedurewould involvean infinite egress nlesssome propositionswere accepted withoutproof; such propositions are indeedfound n everymathematicaldisciplinewhich s rigorously eveloped; they arethe axiomsorpostulateswe shall use theseterms nterchangeably) fthetheory.Geometryprovidesthe historically irst xample of the axiomatic presentationof a mathematicaldiscipline.The classical set ofpostulates,however,on which

Euclid based his system,has proved insufficientor the deductionof the well-knowntheoremsof so-called euclideangeometry; t has therefore een revisedand supplemented n moderntimes, nd at presentvariousadequate systemsofpostulatesforeuclidean geometry re available; the one mostcloselyrelatedtoEuclid's system s probablythat ofHilbert.

2. The inadequacy of Euclid's postulates. The inadequacy of Euclid's ownset ofpostulates llustrates pointwhich s crucialfortheaxiomaticmethod nmodernmathematics:Once the postulates fora theoryhave been laid down,everyfurther ropositionof the theorymust be proved exclusively by logicaldeduction from he postulates; any appeal, explicitor implicit,to a feelingofself-evidence, r to the characteristics f geometricalfigures,r to our experi-ences concerning he behavior of rigidbodies in physical space, or the like, is

7







8 GEOMETRY AND 'EMPIRICAL SCIENCE [January,

strictly rohibited; uch devicesmayhave a heuristic alue inguidingoureffortsto find strictprooffor theorem,but the proof tselfmust containabsolutelyno referenceo such aids. This is particularlymportant n geometry,where our

so-calledintuition fgeometrical elationships,upported by reference o figuresor to previous physical experiences,may induce us tacitlyto make use of as-sumptionswhichareneither ormulatednourpostulatesnorprovable bymeansof them. Consider,for xample,the theorem hatin a triangle he threemediansbisecting hesides intersectn one pointwhichdivides each ofthem n the ratioof 1:2. To prove this theorem,one shows first hat in any triangleABC (seefigure) he inesegmentMN whichconnectsthe centers fAB and A C is parallelto BC and therefore alf as long as the latter side. Then the lines BN and CMare drawn, nd an examinationof thetrianglesMON and BOC leads to the proof

ofthe theorem. n thisprocedure, t is usually taken forgrantedthat BN andCM intersect n a point 0 which lies betweenB and N as well as between C

A

M N

a

and M. This assumption s based on geometrical ntuition, nd indeed, t cannotbe deduced fromEuclid's postulates; to make it strictly emonstrable nd inde-pendent of any reference o intuition, special group of postulates has beenadded to thoseof Euclids, heyare thepostulatesoforder. One of these-to givean example assertsthat if A, B, C are pointson a straight ine 1,and ifB liesbetweenA and C, thenB also lies betweenC and A.-Not even as "trivial" anassumptionas this may be taken forgranted; the systemofpostulates has tobe made so completethat all the requiredpropositionscan be deduced from tby purely ogicalmeans.

Another illustrationof the point under consideration s provided by thepropositionthat triangleswhichagree in two sides and the enclosed angle, arecongruent. n Euclid's Elements, this proposition s presentedas a theorem;the alleged proof,however,makes use ofthe deas ofmotion nd superimpositionoffiguresnd thus involvestacit assumptionswhichare based on ourgeometricintuition nd on experienceswithrigidbodies,but whichare definitely otwar-rantedby-i.e. deduciblefrom-Euclid's postulates. In Hilbert's system, here-fore, hisproposition more precisely:partof t) is explicitlyncludedamongthepostulates.

3. Mathematical certainty. t is this purelydeductive characterof mathe-matical proofwhich forms he basis ofmathematicalcertainty:What therigor-ous proofof a theorem-say the propositionabout the sum of the angles in a







1945] GEOMETRY AND EMPIRICAL SCIENCE 9

triangle-establishes is not the truthof theproposition n questionbut ratherconditional nsight o the effect hat that proposition s certainly rueprovidedthat hepostulatesare true; in otherwords,the proofofa mathematicalproposi-

tionestablishesthe factthat the latter s logically mpliedby thepostulatesofthe theory n question. Thus, each mathematicaltheoremcan be cast into theform

(PI1'2P3 . PN)-> T

where the expressionon the left s the conjunction (joint assertion) of all thepostulates, the symbol on the rightrepresents he theorem n its customaryformulation, nd the arrowexpressesthe relation of logical implicationor en-tailment. Preciselythis characterof mathematicaltheorems s the reason fortheirpeculiarcertainty nd necessity, s I shall now attemptto show.

It is typical of any purely logical deduction that the conclusionto whichit leads simplyre-asserts a properor improper)part of what has alreadybeenstated in the premises.Thus, to illustratethispoint by a veryelementary x-ample, fromthe premise,"This figure s a righttriangle," we can deduce theconclusion, This figures a triangle";but this conclusionclearlyreiterates artof the informationlready containedin the premise.Again, from hepremises,"All primesdifferentrom2 are odd" and "n is a primedifferentrom2," wecan infer ogicallythat n is odd; but thisconsequence merelyrepeatspart (in-deed a relatively mall part) ofthe information ontainedin the premises.The

samesituationprevails nall other ases of ogicaldeduction;and wemay,there-fore, ay that logical deduction-which is the one and onlymethodofmathe-maticalproof-is a techniqueofconceptualanalysis: itdiscloseswhatassertionsare concealed in a given set of premises,and it makes us realize to what wecommittedourselves in acceptingthose premises;but none of the resultsob-tained by thistechniqueevergoes by one iota beyondthe informationlreadycontainedin the initialassumptions.

Since all mathematicalproofs estexclusively n logicaldeductionsfrom er-tainpostulates, tfollows hata mathematical heorem, uchas thePythagorean

theorem n geometry, sserts nothingthat is objectivelyr Mieoreticallyew ascomparedwiththepostulatesfromwhich t is derived, lthough ts contentmaywell be psychologicallyew n the sense that we werenot aware of its beingim-plicitly ontained n thepostulates.

The nature of the peculiarcertainty f mathematics s now clear: A mathe-matical theorem s certainrelativelyo the set of postulates from which it isderived; .e. it is necessarilytrue f those postulates are true; and this is so be-cause the theorem,f rigorously roved,simplyre-asserts artof what has beenstipulatedin the postulates.A truthof this conditionaltype obviouslyimplies

no assertions boutmattersofempiricalfactand can, therefore, everget into

conflictwithany empiricalfindings, ven of the mostunexpectedkind; conse-quently,unlike the hypothesesand theoriesof empiricalscience, it can neversufferhe fate of beingdisconfirmedy new evidence: A mathematical truth s







10 GEOMETRY AND EMPIRICAL SCIENCE [January,

irrefutablyertain ust because it is devoid offactual,or empirical ontent.Anytheoremofgeometry, herefore,when cast into the conditional formdescribedearlier, s analytic in the technical sense of logic,and thus true a priori; i.e. itstruth an be establishedby means ofthe formalmachinery f ogic alone, with-out any reference o empiricaldata.

4. Postulates and truth.Now itmightbe felt hat ouranalysisofgeometricaltruth o fartellsonlyhalf ofthe relevant tory.For whilea geometricalproofnodoubt enables us to assert a propositionconditionally-namely on conditionthat the postulates are accepted-, is it not correctto add that geometry lsounconditionally sserts the truthof its postulates and thus, by virtue of thedeductive relationshipbetweenpostulates and theorems,enables us uncondi-tionallyto assert the truthofits theorems? s it not an unconditional ssertion

ofgeometry hat two pointsdetermine ne and onlyone straight inethat con-nects them,or thatinanytriangle, he sumof theangles equals tworight ngles?That this s definitely ot thecase, is evidencedby twoimportant spects oftheaxiomatic treatment fgeometrywhichwill nowbe brieflyonsidered.

The first f these features s the well-knownfact that in the more recentdevelopment of mathematics,several systems of geometryhave been con-structedwhich are incompatiblewith euclidean geometry,and in which, forexample, the two propositions ust mentioneddo not necessarilyhold. Let usbriefly ecollect ome of the basic facts concerningthese non-euclidean eome-

tries.The postulates on which euclidean geometryrests include the famouspostulateoftheparallels,which, n the case ofplane geometry, sserts n effectthat through verypointP not on a given ine I thereexistsexactlyone parallelto 1, .e., one straight inewhichdoes not meet 1.As thispostulate s considerablyless simplethan the others, nd as itwas also felttobe intuitivelyess plausiblethanthe atter,manyefforts eremade in thehistory fgeometry oprovethatthispropositionneed not be accepted as an axiom,but that it can be deducedasa theoremfrom heremaining ody ofpostulates.All attempts n thisdirectionfailed,however;and finallytwas conclusivelydemonstrated hat a proofoftheparallel principleon the basis of the other postulates of euclidean geometry(even in itsmodern, ompletedform) s impossible.This was shownby provingthat a perfectly elf-consistent eometricaltheory s obtained if the postulateoftheparallels is replaced bytheassumptionthatthrough ny pointP not on agiven straight ine there xist t least twoparallelsto1.This postulate obviouslycontradictsthe euclidean postulate of the parallels, and ifthe latterwere ac-tually a consequence of the other postulates of euclidean geometry, hen thenew set ofpostulateswould clearly nvolvea contradiction,which can be shownnot to be the case. This firstnon-euclideantype ofgeometry,which is calledhyperbolicgeometry,was discovered n theearly20's- f the last century lmost

simultaneously,but independently ythe Russian N. I. Lobatschefskij, nd bythe Hungarian J. Bolyai. Later, Riemann developed an alternativegeometry,knownas ellipticalgeometry,nwhich the axiom oftheparallels is replaced bythepostulate that no line has any parallels. (The acceptance of thispostulate,







1945] - GEOMETRY AND EMPIRICAL SCIENCE 11

however, ncontradistinctiono thatofhyperbolic eometry, equiresthemodi-fication fsome further xioms of euclidean geometry,f consistentnewtheoryis to result.)As is to be expected,manyofthe theoremsof these non-euclideangeometries re at variance with those of euclideantheory;thus, e.g., n thehy-

perbolicgeometry f twodimensions, hereexist,for ach straight ine1,throughany pointP not on 1, nfinitelymany straight ines which do not meet1; also,the sum of the angles in any triangle s less than two right angles. In ellipticgeometry,hisanglesum is always greater han tworight ngles; no twostraightlinesare parallel; and while two differentoints usually determine xactlyonestraight ine connecting hem(as they always do in euclideangeometry), hereare certain pairs of points which are connected by infinitelymany differentstraight ines.An illustration fthislattertypeofgeometry s provided by thegeometricalstructure of that curved two-dimensional pace which is repre-

sented bythesurfaceofa sphere,whentheconceptofstraight ine s interpretedbythatofgreatcircleon thesphere. n this pace, there renoparallellines inceany two greatcircles ntersect; he endpointsofany diameter of thespherearepoints connectedby infinitelymany differentstraight ines," and the sum oftheanglesin a triangle s always in excessoftworight ngles. Also, in thisspace,the ratio between the circumference nd the diameterof a circle (not neces-sarilya great circle) s always less than2ir.

Elliptic and hyperbolicgeometry re not the only types of non-euclideangeometry; ariousothertypeshave beendeveloped; we shall laterhave occasion

to refer o a much moregeneralform fnon-euclideangeometrywhichwas like-wise devisedbyRiemann.The factthat these differentypesofgeometryhave beendevelopedinmod-

ernmathematicsshows clearlythat mathematics cannot be said to assert thetruth fany particular et ofgeometrical ostulates; all thatpuremathematics sinterestedn,and all that it can establish, s the deductiveconsequencesofgivensets ofpostulatesand thus thenecessary ruth f theensuing heorems elativelyto thepostulatesunderconsideration.

A secondobservationwhich ikewise hows thatmathematicsdoes notassertthetruthofany particular et ofpostulatesrefers o the latusoftheconceptsngeometry.here exists, n everyaxiomatized theory, close parallelism betweenthe treatment fthepropositions nd that oftheconceptsof thesystem.As wehave seen, the propositionsfall into two classes: the postulates,forwhich noproof s given,and the theorems, ach of whichhas to be derivedfrom he postu-lates. Analogously, he concepts fall nto two classes: the primitive rbasic con-cepts, for which no definition s given,and the others,each of which has tobe preciselydefined n termsof the primitives. The admission of some unde-finedconcepts is clearly necessary if an infinite egress n definition s to beavoided.) The analogy goes farther:Just as thereexistsan infinityftheoreti-

cally suitable axiom systems forone and the same theory-say, euclidean ge-ometry-, so there lso exists an infinity f theoretically ossiblechoices for heprimitive ermsof that theory; veryoften-but not always-different axioma-tizationsofthesame theory nvolvenot only different ostulates,but also differ-








ent sets of primitives.Hilbert's axiomatizationof plane geometry ontains sixprimitives:point, traight ine, ncidence of a-pointon a line), betweenness asa relationof threepointson a straight ine), congruencefor ine segments, nd

congruence or ngles. (Solid geometry,nHilbert's axiomatization,requires wofurther rimitives, hat of plane and that of incidence of a point on a plane.)All otherconceptsofgeometry, uch as those ofangle,triangle, ircle, etc., redefined n terms of these basic concepts.

But ifthe primitives re not definedwithingeometrical heory,what mean-ing are we to assign to them?The answer is that it is entirelyunnecessarytoconnect any particularmeaningwith them.True, the words "point," "straightline," etc., carry definite onnotationswith themwhich relate to the familiargeometricalfigures,but the validity of the propositionsis completelyinde-

pendentof these connotations. ndeed, suppose that in axiomatized euclideangeometry,we replace the over-suggestive erms"point," "straight ine," "inci-dence," "betweenness," tc.,by theneutral terms object of kind 1," " object ofkind 2," "relationNo. 1," "relationNo. 2," etc., and suppose that we presentthismodifiedwordingof geometryto a competentmathematician or logicianwho, however,knowsnothingof the customary connotationsof the primitiveterms.For this logician, all proofswould clearlyremainvalid, for as we sawbefore, rigorousproof n geometryrestson deductionfrom he axioms alonewithout nyreferenceothecustomary nterpretationf thevariousgeometricalconceptsused. We see therefore hat indeed no specificmeaninghas to be at-tached to theprimitive ermsofan axiomatized theory;and in a precise ogicalpresentationof axiomatized geometrythe primitiveconcepts are accordinglytreatedas so-called logical variables.

As a consequence,geometry annot be said to assertthe truthof its postu-lates, since the latterare formulated n termsof conceptswithoutany specificmeaning; ndeed,for hisveryreason, the postulatesthemselvesdo notmakeanyspecific ssertionwhichcould possiblybe called true or false! In the terminologyofmodern ogic,thepostulatesare not sentences,but sententialfunctionswiththe primitive oncepts as variable arguments.-This point also shows that the

postulates of geometry annot be consideredas "self-evident ruths,"becausewhereno assertion s made, no self-evidence an be claimed.

5. Pure and physicalgeometry.Geometry hus construed s a purelyformaldiscipline;we shall refer o it also as pure geometry. puregeometry, hen,-nomatterwhether t is of the euclidean or of a non-euclideanvariety--dealswithno specific ubject-matter; nparticular, t assertsnothing bout physicalspace.All itstheorems re analyticand thus true withcertainty reciselybecause theyare devoid offactualcontent.Thus, to characterize he mport fpure geometry,

wemightuse the standard form f a movie-disclaimer:No portrayalof the char-acteristicsofgeometricalfigures r of the spatial propertiesor relationships factual physical bodies is intended, and any similaritiesbetween the primitiveconceptsand theircustomarygeometrical onnotations re purelycoincidental.







1945] GEOMETRY AND EMPIRICAL SCIENCE 13

But just as in the case of some motionpictures,so in the case at least ofeuclidean geometry,the disclaimerdoes not sound quite convincing: Histori-cally speaking,at least, euclidean geometryhas its origin n the generalization

and systematization fcertainempiricaldiscoverieswhichweremade inconnec-tion with the measurement f areas and volumes,thepracticeofsurveying, ndthedevelopmentofastronomy.Thus understood,geometryhas factualimport;it is an empirical cience whichmightbe called, invery generalterms, hetheoryofthestructure fphysical space, orbriefly, hysicalgeometry. hat is the rela-tionbetween pureand physicalgeometry?

When the physicistuses the conceptsofpoint, straight ine, incidence, etc.,in statements bout physicalobjects, he obviouslyconnectswitheach ofthem amoreor less definite hysicalmeaning. Thus, the term"point" serves to desig-

nate physical points, i.e., objects of the kind illustratedby pin-points,crosshairs,etc.Similarly, he term"straight ine" refers o straight ines in the senseofphysics,such as illustratedby taut stringsor by the path of light rays in ahomogeneousmedium.Analogously,each ofthe othergeometrical onceptshasa concretephysical meaning n thestatementsofphysical geometry. n view ofthissituation,we can say thatphysical geometry s obtained bywhat is called,in contemporary ogic,a semantical nterpretationfpure geometry,Generallyspeaking, a semantical interpretation f a pure mathematical theory, whoseprimitives re not assigned any specificmeaning,consists n givingeach primi-tive and thus, ndirectly, ach defined erm)a specificmeaningor designatum.In the case of physical geometry,this meaning is physical in the sense justillustrated; t is possible,however,to assign a purelyarithmeticalmeaningtoeach conceptofgeometry; hepossibilityof such an arithmetical nterpretationof geometry s of great importance n the study of the consistency nd otherlogical characteristics fgeometry, ut it falls outside the scope of the presentdiscussion.

By virtueof thephysical nterpretationf theoriginally ninterpreted rimi-tives of a geometrical theory,physical meaning is indirectly ssigned also toeverydefined onceptof thetheory;and ifeverygeometrical erm s now taken

in its physical interpretation,hen every postulate and every theorem of thetheoryunder considerationturns nto a statementof physics.withrespecttowhichthequestionas to truthorfalsitymay meaningfully e raised-a circum-stance which clearlycontradistinguisheshe propositionsofphysicalgeometryfrom hose of the corresponding ninterpreted ure theory.-Consider, forex-ample,thefollowing ostulate ofpure euclideangeometry:For any twoobjectsx, y ofkind 1, thereexistsexactlyone object I ofkind 2 such that bothx and ystand in relationNo. 1 to 1. As long as the threeprimitivesoccurring n thispostulate are uninterpreted,t is obviously meaningless o ask whether hepost-

ulate is true. But by virtueofthe above physical interpretation,he postulateturns nto thefollowingtatement:For any twophysical points x, y thereexistsexactlyone physical straight ine I such that both x and y lie on 1.But this is aphysical hypothesis, nd we may now meaningfully sk whether t is true or








false. Similarly, he theorem bout the sum ofthe angles in a triangle urns ntothe assertion that the sum of the angles (in the physical sense) of a figurebounded by thepaths of three ightrays equals two right ngles.

Thus, the physical interpretation ransforms given puregeometrical he-

ory-euclidean or non-euclidean-into a systemofphysical hypotheseswhich,iftrue,mightbe said to constitute theoryof the structure f physicalspace.But the questionwhether given geometrical heory n physical interpretationis factuallycorrect epresents problemnotofpuremathematicsbut of empiri-cal science; ithas to be settledon the basis ofsuitableexperiments rsystematicobservations.The only assertionthe mathematiciancan make in this contextis this: If all thepostulatesofa givengeometry,n theirphysical nterpretation,are true,then all the theoremsof that geometry,n theirphysical interpreta-tion,are necessarily rue, too, since theyare logicallydeduciblefrom hepostu-lates. It might eem,therefore,hat in orderto decidewhetherphysicalspace iseuclidean or non-euclidean in structure, ll that we have to do is to test therespectivepostulates in theirphysical interpretation.However, this is not di-rectlyfeasible; here,as in the case of any otherphysical theory, he basic hy-pothesesare largely ncapable of a directexperimental est; in geometry, his sparticularly bvious for uch postulatesas theparallel axiom or Cantor's axiomof continuityn Hilbert's system of euclidean geometry,whichmakes an asser-tion about certain nfinite ets ofpointson a straight ine. Thus, the empiricaltest ofa physicalgeometryno less than thatofanyother cientificheoryhas to

proceed ndirectly;namely,bydeducingfrom he basic hypotheses f the theorycertain consequences, or predictions,which are amenable to an experimentaltest. If a test bears out a prediction,then it constitutesconfirmingvidence(though,ofcourse,no conclusiveproof)for he theory;otherwise,t disconfirmsthetheory. f an adequate amountofconfirmingvidencefor theoryhas beenestablished,and if no disconfirmingvidencehas been found,then the theorymay be accepted by the scientist"untilfurther otice."

It is in thecontextof this ndirect rocedure hatpuremathematics nd logicacquire their nestimable mportanceforempiricalscience: While formal ogic

and puremathematicsdo not in themselves stablishany assertions bout mat-tersofempiricalfact, heyprovidean efficientnd entirelyndispensablemachin-ery fordeducing, from abstract theoreticalassumptions,such as the laws ofNewtonianmechanicsor the postulates of euclideangeometrynphysical nter-pretation,consequences concreteand specificenoughto be accessible to directexperimental est. Thus, e.g.,pure euclidean geometry hows that from tspos-tulates theremay be deduced the theorem about the sum of the angles in atriangle, nd thatthis deduction s possibleno matterhowthebasic conceptsofgeometry re interpreted; encealso in the case of thephysical nterpretationf

euclideangeometry.This theorem,n itsphysical nterpretation,s accessibletoexperimental est; and sincethepostulatesofelliptic nd ofhyperbolic eometryimplyvalues differentrom woright nglesfor heangle sumofa triangle, his

particularproposition eems to afford good opportunityfora crucial experi-







19451 GEOMETRY AND EMPIRICAL SCIENCE 15

ment.And no less a mathematicianthan Gauss did indeed perform histest; bymeans ofoptical methods-and thususingthe nterpretationfphysical traightlines as paths of ightrays-he ascertained the angle sum ofa largetrianglede-terminedby three mountaintops. Withinthe limitsof experimental rror,hefound t equal to two right ngles.

6. On Poincare's conventionalism oncerninggeometry.But suppose thatGauss had founda noticeabledeviation from hisvalue; would thathave meanta refutationof euclidean geometry n its physical interpretation, r, in otherwords, of the hypothesisthat physical space is euclidean in structure?Notnecessarily;for hedeviationmighthave been accountedforby a hypothesis othe effects hat thepaths ofthe light rays involvedin the sighting rocesswerebent by some disturbingforceand thus were not actually straight ines. The

same kind ofreference o deforming orces ould also be used if, say, the euclid-ean theorems fcongruenceforplane figuresweretestedin theirphysical nter-pretation by means ofexperimentsnvolvingrigidbodies,and ifany violationsofthe theoremswere found.This point is by no means trivial;Henri Poincare,the great Frenchmathematician and theoreticalphysicist,based on considera-tions of this type his famous conventionalismoncerninigeometry.t was hisopinionthat no empirical est,whatever tsoutcome,can conclusivelynvalidatethe euclidean conceptionofphysicalspace; in otherwords,the validity of eu-clidean geometry nphysicalscience can always be preserved-if necessary,bysuitable changes in the theories of physics,such as the introductionof newhypothesesconcerning eforming r deflecting orces.Thus, the question as towhetherphysical space has a euclidean or a non-euclidean structurewould,becomea matterofconvention, nd the decision to preserveeuclideangeometryat all costswould recommend tself, ccordingto Poincare,by the greater im-plicityof euclidean as comparedwithnon-euclideangeometricaltheory.

It appears, however,that Poincar6's account is an oversimplification. trightly alls attentionto the fact that the testofa physicalgeometryG alwayspresupposes a certainbodyP ofnon-geometrical hysicalhypotheses includingthephysicaltheoryofthe instruments fmeasurement nd observation used in

the test), and that the so-called test of G actually bears on the combinedtheo-retical ystemG*P rather han on G alone. Now, ifpredictions erivedfromG-Parecontradictedbyexperimental indings,hena change n thetheoretical truc-turebecomesnecessary. n classical- hysics,G alwayswas euclidean geometryin its physical interpretation,GE; and when experimental vidence required amodification f the theory, t was P rather than GE which was changed. ButPoincare's assertion that this procedurewould always be distinguishedby itsgreater implicity s not entirely orrect;forwhathas to be takenintoconsidera-tion s thesimplicity fthe totalsystemG P, and not ust that of tsgeometrical

part.And here t is clearlyconceivablethat a simplertotal theory n accordancewith all the relevantempirical evidence is obtainable by going over to a non-euclidean formof geometry atherthan by preserving he euclidean structureofphysicalspace and makingadjustmentsonlyin part P.








And indeed, ust this ituation has arisen n physics nconnectionwith the de-velopmentofthegeneral theory f relativity: f theprimitive erms f geometryare given physical interpretations long the lines indicated before, hen certain

findingsnastronomy epresent ood evidence n favorofa total physical theorywith a non-euclideangeometry s partG. According o this theory, he physicaluniverse t large s a three-dimensional urved space of a very complexgeometri-cal structure; t is finite n volume and yet unbounded nall directions.However,in comparatively mall areas, such as those involved in Gauss' experiment, u-clideangeometry an serveas a good approximative ccount of thegeometricalstructure fspace. The kindofstructure scribed to physicalspace inthistheorymay be illustratedby an analogue in two dimensions;namely, the surface of asphere. The geometrical tructure f the latter,as was pointedout before, an

be described by means ofellipticgeometry,ftheprimitive erm"straight ine"is interpreted s meaning "great circle," and ifthe otherprimitives re givenanalogous interpretations. n thissense,thesurface ofa sphere s a two-dimen-sional curved space of non-euclidean structure,whereas the plane is a two-dimensional space of euclidean structure. While the plane is unbounded in alldirections, nd infiniten size, the sphericalsurface s finiten size and yet un-bounded in all directions: two-dimensional hysicist, ravelling long "straightlines" of that space would never encounter ny boundaries ofhis space; instead,hewould finally eturn o hispointofdeparture,provided thathis life pan andhis techinical acilitieswere sufficientor uch a trip n considerationofthe sizeof his "universe." It is interestingo notethat thephysicists f thatworld,evenif they lacked any intuitionof a three-dimensional pace, could empiricallyascertain the fact that their two-dimensional pace was curved.This might bedone by means ofthemethodoftraveling long straight ines; another, implertest would consist n determining he angle sum in a triangle; again another ndetermining, y means ofmeasuring tapes, the ratio of the circumference f acircle not necessarily great circle)to itsdiameter;this ratio would turnout tobe less than 7r.

The geometrical tructurewhichrelativity hysicsascribestophysical space

is a three-dimensionalnalogue to that ofthe surfaceofa sphere,or,to be moreexact,to that of theclosed and finite urfaceofa potato,whosecurvaturevariesfrompoint to point. In ourphysical universe,the curvatureofspace at a givenpoint s determined y the distribution f masses in itsneighborhood;near largemasses such as the sun, space is strongly urved,while in regionsof low mass-density, he structure fthe universe s approximatelyeuclidean. The hypothe-sis statingthe connection betweenthe mass distribution nd the curvature ofspace at a pointhas beenapproximately onfirmedyastronomical bservationsconcerning hepaths of ightraysin thegravitationalfieldof thesun.

The geometrical heorywhich sused to describethe structure f thephysicaluniverse s of a type that may be characterizedas a generalizationof ellipticgeometry. t was originally onstructedby Riemann as a purelymathematicaltheory,withoutany concretepossibility fpractical applicationat hand. When







1945] FUNCTIONS OF SEVERAL COMPLEX VARIABLES 17

Einstein, n developinghisgeneral theory f relativity,ooked for n appropriatemathematicaltheoryto deal with the structure f physical space, he found nRiemann's abstract systemthe conceptual tool he needed. This fact throws n

interesting idelighton the importanceforscientificprogressof that type ofinvestigationwhich the "practical-minded"man in the streettends to dismissasuseless,abstract mathematicalspeculation.

Of, ourse,a geometrical heory n physical interpretationan neverbe vali-dated with mathematicalcertainty,no matterhow extensivethe experimentaltests to whichit is subjected; like any other theoryofempiricalscience,it canacquire only a moreor less high degreeof confirmation. ndeed, the considera-tions presented n this articleshow that the demand formathematical certaintyin empiricalmatters s misguided nd unreasonable; for, s we saw, mathemati-cal certainty fknowledgecan be attained only at the priceof analyticity ndthus ofcomplete ack offactual content.Let me summarizethis insight n Ein-stein'swords:

"As faras the laws of mathematicsrefer o reality, heyarenotcertain;andas far as they are certain,they do not refer o reality."

FUNCTIONS OF SEVERAL COMPLEX VARIABLES*

W. T. MARTIN, SyracuseUniversity1. Definition f an analyticfunction.Considera domain D in the 2n-dimen-

sional euclideanspace ofncomplexvariablesz1, ,z,. A functionf(zi, , z,.)is said to be analytic in D if n some neighborhood f every point (z1, * * z, )of D it can be represented s the sum of an (absolutely) convergentmultiplepower-series

al,...,j.(z- z)* (Z. -Xni11* * ** n=?

An alternative definitions that f is analytic in D if it has derivatives of all

orders,mixedand iterated, t everypoint of D. These twodefinitionsre veryeasilyseen to be equivalent.

An importantresult due to Osgood J[1] n 1899 states that iff(zi, , z,is boundedn D and ifthen partialderivativesf/Ozi, =1, * * n all exist tevery oint fD, then is analyticnD.

In 1899 and again in 1900 Osgood [1, 21 raised the question as to whethertheboundednessrestriction ould be removed,and in 1906 Hartogs [1 showedthat it actually could be removed.This means that ifa function s analytic ineach variable separately, t is analytic as a function fall n variables. This is a

keyresult in the theory.*This paper s an amplificationf n invitedddressdeliveredt the nnual meetingfthe

Mathematical ssociationfAmericann Wellesley,Massachusetts,nAugust 2,1944.

Documents

Geometry and Empirical Science C G Hempel