Geometry and Empirical Science
Author(s): C. G. Hempel
Source: The American Mathematical Monthly, Vol. 52, No. 1 (Jan., 1945), pp. 7-17
Published by: Mathematical Association of America
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GEOMETRY AND EMPIRICAL SCIENCE
C. G. HEMPEL, QueensCollege
1. Introduction. The most distinctive characteristic which differentiates
mathematics from the various branches of empirical science, and which accountsforits fameas the queen of the sciences,is no doubt the peculiar certainty
and necessityof its results. No propositionin even the most advanced parts of
empiricalscience can ever attain this status; a hypothesisconcerning"matters
of empirical fact" can at best acquire what is loosely called a high probability
or a high degree of confirmationon the basis of the relevant evidence available;
but however well it may have been confirmedby careful tests, the possibility
can never be precluded that it will have to be discarded later in the light of
evidence. Thus, all the theoriesand hypothesesof emnew and disconfirming
pirical science share this provisionalcharacterof being establishedand accepted
"until furthernotice," whereas a mathematical theorem,once proved, is established once and forall; it holds with that particular certaintywhich no subsequent empirical discoveries,however unexpected and extraordinary,can ever
affectto the slightestextent. It is the purpose of this paper to examine the nature of that proverbial 'mathematical certainty"with special referenceto geometry,in an attempt to shed some light on the question as to the validity of
geometricaltheories,and theirsignificancefor our knowledge of the structure
of physical space.
The nature of mathematicaltruthcan be understoodthroughan analysis of
the method by means of which it is established. On this point I can be very
brief: it is the method of mathematical demonstration,which consists in the
logical deduction of the propositionto be proved fromotherpropositions,previously established. Clearly,thisprocedurewould involve an infiniteregressunless
some propositionswere accepted without proof; such propositionsare indeed
found in every mathematical disciplinewhich is rigorouslydeveloped; they are
the axioms or postulates(we shall use these termsinterchangeably)of the theory.
Geometryprovides the historicallyfirstexample of the axiomatic presentation
of a mathematical discipline.The classical set of postulates, however,on which
forthe deduction of the wellEuclid based his system,has proved insufficient
known theorems of so-called euclidean geometry;it has thereforebeen revised
and supplementedin moderntimes,and at presentvarious adequate systemsof
postulates for euclidean geometryare available; the one most closely related to
Euclid's systemis probably that of Hilbert.
2. The inadequacy of Euclid's postulates. The inadequacy of Euclid's own
set of postulates illustratesa point which is crucial forthe axiomatic method in
modern mathematics: Once the postulates for a theory have been laid down,
every furtherpropositionof the theory must be proved exclusively by logical
deduction fromthe postulates; any appeal, explicit or implicit,to a feelingof
self-evidence,or to the characteristicsof geometrical figures,or to our experiences concerningthe behavior of rigid bodies in physical space, or the like, is
7
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GEOMETRY AND 'EMPIRICAL SCIENCE
[January,
strictlyprohibited;such devices may have a heuristicvalue in guidingour efforts
to finda strictprooffora theorem,but the proofitselfmust contain absolutely
no referenceto such aids. This is particularlyimportantin geometry,where our
so-called intuitionof geometricalrelationships,supportedby referenceto figures
or to previous physical experiences,may induce us tacitly to make use of assumptionswhichare neitherformulatedin our postulates nor provable by means
of them. Consider,forexample, the theoremthat in a trianglethe threemedians
bisectingthe sides intersectin one point which divides each of them in the ratio
of 1:2. To prove this theorem,one shows firstthat in any triangleABC (see
figure)the line segmentMN whichconnectsthe centersofAB and A C is parallel
to BC and thereforehalf as long as the latter side. Then the lines BN and CM
are drawn,and an examinationof the trianglesMON and BOC leads to the proof
of the theorem.In this procedure,it is usually taken forgranted that BN and
CM intersectin a point 0 which lies between B and N as well as between C
A
M
N
a
and M. This assumptionis based on geometricalintuition,and indeed, it cannot
be deduced fromEuclid's postulates; to make it strictlydemonstrableand independent of any referenceto intuition,a special group of postulates has been
added to those of Euclids,theyare the postulates of order. One of these-to give
an example asserts that if A, B, C are points on a straightline 1, and if B lies
between A and C, then B also lies between C and A.-Not even as "trivial" an
assumption as this may be taken forgranted; the system of postulates has to
be made so complete that all the required propositionscan be deduced fromit
by purelylogical means.
Another illustration of the point under consideration is provided by the
propositionthat triangleswhich agree in two sides and the enclosed angle, are
congruent. In Euclid's Elements, this proposition is presented as a theorem;
the alleged proof,however,makes use of the ideas of motionand superimposition
of figuresand thus involves tacit assumptionswhich are based on our geometric
intuitionand on experienceswith rigidbodies, but which are definitelynot warranted by-i.e. deducible from-Euclid's postulates. In Hilbert's system,therefore,this proposition(more precisely:part of it) is explicitlyincludedamong the
postulates.
3. Mathematical certainty.It is this purely deductive character of mathematical proofwhich formsthe basis of mathematicalcertainty:What the rigorous proof of a theorem-say the propositionabout the sum of the angles in a
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1945]
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9
triangle-establishes is not the truthof the propositionin question but rathera
conditional insightto the effectthat that propositionis certainlytrue provided
thatthe postulates are true; in otherwords,the proofof a mathematicalproposition establishesthe fact that the latter is logically implied by the postulates of
the theoryin question. Thus, each mathematical theoremcan be cast into the
form
aa.
PN)-> T
(PI1'2P3
where the expressionon the left is the conjunction (joint assertion) of all the
postulates, the symbol on the right representsthe theorem in its customary
formulation,and the arrow expresses the relation of logical implicationor entailment. Precisely this character of mathematical theoremsis the reason for
theirpeculiar certaintyand necessity,as I shall now attempt to show.
It is typical of any purely logical deduction that the conclusion to which
it leads simplyre-asserts(a proper or improper)part of what has already been
stated in the premises. Thus, to illustratethis point by a very elementaryexample, fromthe premise, "This figureis a right triangle,"we can deduce the
conclusion,"This figureis a triangle"; but this conclusionclearlyreiteratespart
of the informationalready contained in the premise.Again, fromthe premises,
"All primes differentfrom2 are odd" and "n is a prime differentfrom2," we
can inferlogically that n is odd; but this consequence merelyrepeats part (indeed a relativelysmall part) of the informationcontained in the premises.The
same situationprevailsin all othercases of logical deduction; and we may, therefore,say that logical deduction-which is the one and only method of mathematical proof-is a techniqueof conceptual analysis: it discloses what assertions
are concealed in a given set of premises,and it makes us realize to what we
committedourselves in accepting those premises; but none of the results obtained by this technique ever goes by one iota beyond the informationalready
contained in the initial assumptions.
Since all mathematicalproofsrestexclusivelyon logical deductionsfromcertain postulates,it followsthat a mathematicaltheorem,such as the Pythagorean
new as
theoremin geometry,asserts nothing that is objectivelyor Mieoretically
its
content
it
is
may
from
which
although
derived,
with
the
postulates
compared
new in the sense that we were not aware of its being imwell be psychologically
plicitlycontained in the postulates.
The nature of the peculiar certaintyof mathematicsis now clear: A mathematical theoremis certain relativelyto the set of postulates fromwhich it is
derived; i.e. it is necessarilytrue if those postulates are true; and this is so because the theorem,if rigorouslyproved, simplyre-assertspart of what has been
stipulated in the postulates. A truthof this conditional type obviously implies
no assertionsabout mattersof empiricalfact and can, therefore,never get into
conflictwith any empiricalfindings,even of the most unexpected kind; consequently, unlike the hypothesesand theories of empirical science, it can never
sufferthe fate of being disconfirmedby new evidence: A mathematical truthis
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GEOMETRY AND EMPIRICAL SCIENCE
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irrefutablycertainjust because it is devoid of factual,or empiricalcontent.Any
theoremof geometry,therefore,when cast into the conditional formdescribed
earlier,is analytic in the technical sense of logic, and thus true a priori; i.e. its
truthcan be established by means of the formalmachineryof logic alone, without any referenceto empirical data.
4. Postulates and truth.Now it mightbe feltthat our analysis ofgeometrical
truthso fartells only halfof the relevantstory.For while a geometricalproofno
doubt enables us to assert a proposition conditionally-namely on condition
that the postulates are accepted-, is it not correctto add that geometryalso
unconditionallyasserts the truth of its postulates and thus, by virtue of the
deductive relationshipbetween postulates and theorems,enables us unconditionally to assert the truthof its theorems?Is it not an unconditionalassertion
of geometrythat two points determineone and only one straightline that connects them,or that in any triangle,the sum of the angles equals two rightangles?
That this is definitelynot the case, is evidenced by two importantaspects of the
axiomatic treatmentofgeometrywhichwill now be brieflyconsidered.
The firstof these features is the well-knownfact that in the more recent
development of mathematics, several systems of geometry have been constructed which are incompatible with euclidean geometry,and in which, for
example, the two propositionsjust mentioned do not necessarilyhold. Let us
brieflyrecollectsome of the basic facts concerningthese non-euclideangeometries. The postulates on which euclidean geometryrests include the famous
postulate of the parallels, which,in the case of plane geometry,asserts in effect
that througheverypoint P not on a given line I thereexists exactly one parallel
to 1,i.e., one straightline whichdoes not meet 1.As this postulate is considerably
less simple than the others,and as it was also feltto be intuitivelyless plausible
than the latter,many effortswere made in the historyof geometryto prove that
this propositionneed not be accepted as an axiom, but that it can be deduced as
a theoremfromthe remainingbody of postulates. All attempts in this direction
failed,however; and finallyit was conclusivelydemonstratedthat a proofof the
parallel principle on the basis of the other postulates of euclidean geometry
(even in its modern,completed form)is impossible.This was shown by proving
that a perfectlyself-consistentgeometrical theoryis obtained if the postulate
of the parallels is replaced by the assumptionthat throughany point P not on a
givenstraightline I thereexistat least two parallels to 1.This postulate obviously
contradicts the euclidean postulate of the parallels, and if the latter were actually a consequence of the other postulates of euclidean geometry,then the
new set of postulates would clearlyinvolve a contradiction,which can be shown
not to be the case. This firstnon-euclideantype of geometry,which is called
hyperbolicgeometry,was discovered in the early20's-of the last centuryalmost
simultaneously,but independentlyby the Russian N. I. Lobatschefskij,and by
the Hungarian J. Bolyai. Later, Riemann developed an alternative geometry,
known as elliptical geometry,in which the axiom of the parallels is replaced by
the postulate that no line has any parallels. (The acceptance of this postulate,
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GEOMETRY AND EMPIRICAL SCIENCE
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however,in contradistinctionto that of hyperbolicgeometry,requiresthe modificationofsome furtheraxioms of euclidean geometry,ifa consistentnew theory
is to result.) As is to be expected,many of the theoremsof these non-euclidean
geometriesare at variance with those of euclidean theory;thus, e.g., in the hyperbolicgeometryof two dimensions,thereexist,foreach straightline 1,through
any point P not on 1, infinitelymany straightlines which do not meet 1; also,
the sum of the angles in any triangle is less than two rightangles. In elliptic
geometry,this angle sum is always greaterthan two rightangles; no two straight
lines are parallel; and while two differentpoints usually determineexactly one
straightline connectingthem (as they always do in euclidean geometry),there
are certain pairs of points which are connected by infinitelymany different
straightlines. An illustrationof this latter type of geometryis provided by the
geometrical structure of that curved two-dimensionalspace which is represented by the surfaceof a sphere,when the concept of straightline is interpreted
by that ofgreat circleon the sphere. In thisspace, thereare no parallel linessince
any two great circlesintersect;the endpointsof any diameterof the sphere are
points connected by infinitelymany different"straightlines," and the sum of
the angles in a triangleis always in excess of two rightangles. Also, in thisspace,
the ratio between the circumferenceand the diameter of a circle (not necessarilya great circle) is always less than 2ir.
Elliptic and hyperbolicgeometryare not the only types of non-euclidean
geometry;various othertypeshave been developed; we shall later have occasion
to referto a much more general formof non-euclideangeometrywhichwas likewise devised by Riemann.
The fact that these different
types of geometryhave been developed in modern mathematics shows clearly that mathematics cannot be said to assert the
truthofany particularset ofgeometricalpostulates; all that pure mathematicsis
interestedin, and all that it can establish,is the deductive consequences ofgiven
sets of postulates and thus the necessarytruthof the ensuingtheoremsrelatively
to the postulates underconsideration.
A second observationwhich likewiseshows that mathematicsdoes not assert
the truthof any particularset of postulates refersto theslatus oftheconceptsin
There exists,in everyaxiomatized theory,a close parallelismbetween
geometry.
the treatmentof the propositionsand that of the concepts of the system.As we
have seen, the propositionsfall into two classes: the postulates, for which no
proofis given,and the theorems,each ofwhichhas to be derived fromthe postulates. Analogously,the concepts fall into two classes: the primitiveor basic concepts, for which no definitionis given, and the others, each of which has to
be preciselydefinedin terms of the primitives.(The admission of some undefined concepts is clearly necessary if an infiniteregress in definitionis to be
avoided.) The analogy goes farther:Just as there exists an infinityof theoretically suitable axiom systemsforone and the same theory-say, euclidean geometry-, so therealso exists an infinityof theoreticallypossible choices forthe
primitivetermsof that theory;very often-but not always-different axiomatizationsof the same theoryinvolve not only different
postulates, but also differ-
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GEOMETRY AND EMPIRICAL SCIENCE
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ent sets of primitives.Hilbert's axiomatization of plane geometrycontains six
primitives: point,straightline, incidence (of a-point on a line), betweenness(as
a relation of three points on a straightline), congruenceforline segments,and
congruenceforangles. (Solid geometry,in Hilbert's axiomatization,requirestwo
furtherprimitives,that of plane and that of incidence of a point on a plane.)
All other concepts of geometry,such as those of angle, triangle,circle,etc.,are
definedin termsof these basic concepts.
But if the primitivesare not definedwithingeometricaltheory,what meaning are we to assign to them? The answer is that it is entirelyunnecessaryto
connect any particular meaning with them. True, the words "point," "straight
line," etc., carry definiteconnotations with them which relate to the familiar
geometrical figures,but the validity of the propositions is completely independent of these connotations. Indeed, suppose that in axiomatized euclidean
geometry,we replace the over-suggestiveterms "point," "straightline," "incidence," "betweenness,"etc.,by the neutral terms'object of kind 1," " object of
kind 2," "relation No. 1," "relation No. 2," etc.,and suppose that we present
this modifiedwording of geometryto a competent mathematician or logician
who, however, knows nothingof the customary connotations of the primitive
terms. For this logician, all proofs would clearly remain valid, for as we saw
before,a rigorousproofin geometryrests on deduction fromthe axioms alone
withoutany referenceto the customaryinterpretationof the various geometrical
concepts used. We see thereforethat indeed no specificmeaning has to be attached to the primitivetermsof an axiomatized theory;and in a precise logical
presentationof axiomatized geometrythe primitive concepts are accordingly
treated as so-called logical variables.
As a consequence, geometrycannot be said to assert the truthof its postulates, since the latter are formulatedin terms of concepts without any specific
meaning;indeed,forthisveryreason,the postulates themselvesdo not make any
specificassertionwhich could possibly be called trueor false! In the terminology
of modernlogic, the postulates are not sentences,but sententialfunctionswith
the primitiveconcepts as variable arguments.-This point also shows that the
postulates of geometrycannot be considered as "self-evidenttruths,"because
where no assertionis made, no self-evidencecan be claimed.
5. Pure and physical geometry.Geometrythus construedis a purelyformal
A pure geometry,then,-no
discipline; we shall referto it also as pure geometry.
matterwhetherit is of the euclidean or of a non-euclideanvariety--deals with
no specificsubject-matter;in particular,it asserts nothingabout physicalspace.
All its theoremsare analytic and thus truewith certaintypreciselybecause they
are devoid offactual content.Thus, to characterizethe importofpure geometry,
we mightuse the standard formof a movie-disclaimer:No portrayalof the characteristicsof geometricalfiguresor of the spatial propertiesor relationshipsof
actual physical bodies is intended, and any similaritiesbetween the primitive
concepts and theircustomarygeometricalconnotationsare purelycoincidental.
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But just as in the case of some motion pictures, so in the case at least of
euclidean geometry,the disclaimer does not sound quite convincing: Historically speaking, at least, euclidean geometryhas its originin the generalization
and systematizationof certainempiricaldiscoverieswhichwere made in connection with the measurementof areas and volumes, the practice of surveying,and
the developmentof astronomy.Thus understood,geometryhas factual import;
it is an empiricalscience whichmightbe called, in verygeneralterms,the theory
What is the relaof the structureof physical space, or briefly,physicalgeometry.
tion between pure and physical geometry?
When the physicistuses the concepts of point, straightline, incidence,etc.,
in statementsabout physical objects, he obviously connectswith each of thema
more or less definitephysical meaning. Thus, the term "point" serves to designate physical points, i.e., objects of the kind illustrated by pin-points,cross
hairs, etc.Similarly,the term "straightline" refersto straightlines in the sense
of physics,such as illustratedby taut stringsor by the path of light rays in a
homogeneousmedium. Analogously,each of the other geometricalconcepts has
a concretephysical meaningin the statementsof physical geometry.In view of
this situation,we can say that physical geometryis obtained by what is called,
in contemporarylogic, a semantical interpretationof pure geometry,Generally
speaking, a semantical interpretationof a pure mathematical theory, whose
primitivesare not assigned any specificmeaning,consists in giving each primitive (and thus, indirectly,each definedterm)a specificmeaning or designatum.
In the case of physical geometry,this meaning is physical in the sense just
illustrated; it is possible, however, to assign a purely arithmeticalmeaning to
each concept of geometry;the possibilityof such an arithmeticalinterpretation
of geometryis of great importance in the study of the consistencyand other
logical characteristicsof geometry,but it falls outside the scope of the present
discussion.
By virtueof the physicalinterpretationof the originallyuninterpretedprimitives of a geometricaltheory,physical meaning is indirectlyassigned also to
everydefinedconcept of the theory;and if every geometricaltermis now taken
in its physical interpretation,then every postulate and every theorem of the
theoryunder considerationturns into a statement of physics. with respect to
whichthe question as to truthor falsitymay meaningfullybe raised-a circumstance which clearly contradistinguishesthe propositionsof physical geometry
fromthose of the correspondinguninterpretedpure theory.-Consider, forexample, the followingpostulate of pure euclidean geometry:For any two objects
x, y of kind 1, thereexists exactly one object I of kind 2 such that both x and y
stand in relation No. 1 to 1. As long as the three primitivesoccurringin this
postulate are uninterpreted,it is obviously meaninglessto ask whetherthe postulate is true. But by virtue of the above physical interpretation,the postulate
turnsinto the followingstatement: For any two physical points x, y thereexists
exactly one physical straightline I such that both x and y lie on 1. But this is a
physical hypothesis,and we may now meaningfullyask whether it is true or
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GEOMETRY AND EMPIRICAL SCIENCE
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false. Similarly,the theoremabout the sum of the angles in a triangleturnsinto
the assertion that the sum of the angles (in the physical sense) of a figure
bounded by the paths of three lightrays equals two rightangles.
Thus, the physical interpretationtransformsa given pure geometricaltheory-euclidean or non-euclidean-into a system of physical hypotheseswhich,
if true, mightbe said to constitutea theoryof the structureof physical space.
But the question whethera given geometricaltheoryin physical interpretation
is factuallycorrectrepresentsa problemnot of pure mathematicsbut of empirical science; it has to be settledon the basis of suitable experimentsor systematic
observations. The only assertion the mathematician can make in this context
is this: If all the postulates of a given geometry,in theirphysical interpretation,
are true, then all the theoremsof that geometry,in their physical interpretation, are necessarilytrue,too, since they are logicallydeducible fromthe postulates. It mightseem, therefore,that in orderto decide whetherphysicalspace is
euclidean or non-euclidean in structure,all that we have to do is to test the
respective postulates in their physical interpretation.However, this is not directlyfeasible; here, as in the case of any other physical theory,the basic hypotheses are largelyincapable of a direct experimentaltest; in geometry,this is
particularlyobvious forsuch postulates as the parallel axiom or Cantor's axiom
of continuityin Hilbert's system of euclidean geometry,which makes an assertion about certain infinitesets of points on a straightline. Thus, the empirical
test of a physical geometryno less than that of any otherscientifictheoryhas to
proceed indirectly;namely,by deducingfromthe basic hypothesesof the theory
certain consequences, or predictions,which are amenable to an experimental
test. If a test bears out a prediction,then it constitutes confirmingevidence
(though,of course,no conclusiveproof) forthe theory;otherwise,it disconfirms
the theory.If an adequate amount of confirmingevidence fora theoryhas been
established, and if no disconfirmingevidence has been found, then the theory
may be accepted by the scientist"until furthernotice."
It is in the contextof thisindirectprocedurethat pure mathematicsand logic
acquire their inestimable importancefor empirical science: While formallogic
and pure mathematicsdo not in themselvesestablish any assertionsabout matand entirelyindispensablemachintersof empiricalfact,theyprovide an efficient
ery for deducing, from abstract theoretical assumptions, such as the laws of
Newtonian mechanicsor the postulates of euclidean geometryin physical interpretation,consequences concrete and specificenough to be accessible to direct
experimentaltest. Thus, e.g., pure euclidean geometryshows that fromits postulates there may be deduced the theorem about the sum of the angles in a
triangle,and that this deduction is possible no matterhow the basic conceptsof
geometryare interpreted;hence also in the case of the physical interpretationof
euclidean geometry.This theorem,in its physical interpretation,is accessible to
experimentaltest; and since the postulates of ellipticand ofhyperbolicgeometry
fromtwo rightangles forthe angle sum of a triangle,this
implyvalues different
particular propositionseems to afforda good opportunityfor a crucial experi-
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GEOMETRY AND EMPIRICAL SCIENCE
15
ment. And no less a mathematicianthan Gauss did indeed performthistest; by
means ofoptical methods-and thus usingthe interpretationofphysicalstraight
lines as paths of lightrays-he ascertained the angle sum of a large triangledeterminedby three mountain tops. Within the limits of experimentalerror,he
found it equal to two rightangles.
6. On Poincare's conventionalismconcerninggeometry.But suppose that
Gauss had founda noticeable deviation fromthis value; would that have meant
a refutationof euclidean geometryin its physical interpretation,or, in other
words, of the hypothesis that physical space is euclidean in structure? Not
necessarily;forthe deviation mighthave been accounted forby a hypothesisto
the effectsthat the paths of the lightrays involved in the sightingprocess were
bent by some disturbingforce and thus were not actually straight lines. The
same kind of referenceto deformingforcescould also be used if, say, the euclidean theoremsof congruenceforplane figureswere tested in theirphysical interpretationby means of experimentsinvolvingrigid bodies, and if any violations
of the theoremswere found. This point is by no means trivial; Henri Poincare,
the great French mathematician and theoreticalphysicist,based on considerations of this type his famous conventionalismconcerninig
geometry.It was his
opinion that no empiricaltest,whateverits outcome,can conclusivelyinvalidate
the euclidean conceptionof physical space; in other words, the validity of euclidean geometryin physical science can always be preserved-if necessary,by
suitable changes in the theories of physics, such as the introductionof new
hypothesesconcerningdeformingor deflectingforces.Thus, the question as to
whetherphysical space has a euclidean or a non-euclidean structure would,
become a matterof convention,and the decision to preserve euclidean geometry
at all costs would recommenditself,according to Poincare, by the greatersimplicity of euclidean as compared with non-euclideangeometricaltheory.
It appears, however, that Poincar6's account is an oversimplification.It
rightlycalls attentionto the fact that the test of a physical geometryG always
presupposes a certainbody P of non-geometricalphysical hypotheses(including
the physical theoryof the instrumentsof measurementand observationused in
the test), and that the so-called test of G actually bears on the combined theoreticalsystemG*P ratherthan on G alone. Now, ifpredictionsderived fromG-P
are contradictedby experimentalfindings,then a change in the theoreticalstructure becomes necessary. In classical-physics,G always was euclidean geometry
in its physical interpretation,GE; and when experimentalevidence required a
modificationof the theory,it was P rather than GE which was changed. But
Poincare's assertion that this procedurewould always be distinguishedby its
greatersimplicityis not entirelycorrect;forwhat has to be taken into consideration is the simplicityof the total systemG P, and not just that of its geometrical
part. And here it is clearlyconceivable that a simplertotal theoryin accordance
with all the relevant empirical evidence is obtainable by going over to a noneuclidean formof geometryratherthan by preservingthe euclidean structure
of physical space and making adjustments only in part P.
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And indeed,just thissituationhas arisen in physicsin connectionwiththe developmentof the general theoryof relativity:If the primitivetermsofgeometry
are given physical interpretationsalong the lines indicated before,then certain
findingsin astronomyrepresentgood evidence in favorof a total physical theory
with a non-euclideangeometryas part G. Accordingto this theory,the physical
universeat large is a three-dimensionalcurvedspace of a verycomplexgeometrical structure;it is finitein volume and yet unbounded in all directions.However,
in comparativelysmall areas, such as those involved in Gauss' experiment,euclidean geometrycan serve as a good approximativeaccount of the geometrical
structureofspace. The kind ofstructureascribed to physicalspace in this theory
may be illustratedby an analogue in two dimensions; namely, the surface of a
sphere. The geometricalstructureof the latter, as was pointed out before,can
be described by means of ellipticgeometry,if the primitiveterm "straightline"
is interpretedas meaning "great circle," and if the other primitivesare given
analogous interpretations.In this sense, the surfaceof a sphere is a two-dimensional curved space of non-euclidean structure,whereas the plane is a twodimensional space of euclidean structure.While the plane is unbounded in all
directions,and infinitein size, the spherical surface is finitein size and yet unbounded in all directions:a two-dimensionalphysicist,travellingalong "straight
lines" of that space would never encounterany boundaries of his space; instead,
he would finallyreturnto his point of departure,provided that his lifespan and
his techinicalfacilitieswere sufficientforsuch a trip in considerationof the size
of his "universe." It is interestingto note that the physicistsof that world,even
if they lacked any intuition of a three-dimensionalspace, could empirically
ascertain the fact that their two-dimensionalspace was curved. This mightbe
done by means of the method of travelingalong straightlines; another,simpler
test would consist in determiningthe angle sum in a triangle; again another in
determining,by means of measuringtapes, the ratio of the circumferenceof a
circle (not necessarilya great circle) to its diameter; this ratio would turnout to
be less than 7r.
The geometricalstructurewhichrelativityphysicsascribes to physical space
is a three-dimensionalanalogue to that of the surfaceof a sphere,or, to be more
exact, to that of the closed and finitesurfaceof a potato, whose curvaturevaries
frompoint to point. In our physical universe,the curvatureof space at a given
point is determinedby the distributionof masses in its neighborhood;near large
masses such as the sun, space is stronglycurved, while in regionsof low massdensity,the structureof the universeis approximatelyeuclidean. The hypothesis stating the connectionbetween the mass distributionand the curvature of
space at a point has been approximatelyconfirmedby astronomicalobservations
concerningthe paths of lightrays in the gravitationalfieldof the sun.
The geometricaltheorywhichis used to describethe structureof the physical
universe is of a type that may be characterized as a generalization of elliptic
geometry.It was originallyconstructedby Riemann as a purely mathematical
theory,withoutany concretepossibilityof practical application at hand. When
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1945]
FUNCTIONS OF SEVERAL COMPLEX VARIABLES
17
Einstein, in developinghis generaltheoryofrelativity,looked foran appropriate
mathematical theoryto deal with the structureof physical space, he found in
Riemann's abstract system the conceptual tool he needed. This fact throwsan
interestingsidelight on the importance for scientificprogress of that type of
investigationwhich the "practical-minded"man in the streettends to dismissas
useless, abstract mathematicalspeculation.
Of,course, a geometricaltheoryin physical interpretationcan never be validated with mathematical certainty,no matter how extensive the experimental
tests to which it is subjected; like any other theoryof empiricalscience, it can
acquire only a more or less high degree of confirmation.Indeed, the considerations presentedin this article show that the demand formathematicalcertainty
in empiricalmattersis misguidedand unreasonable; for,as we saw, mathematical certaintyof knowledgecan be attained only at the price of analyticityand
thus of complete lack of factual content. Let me summarizethis insightin Einstein's words:
"As far as the laws of mathematicsreferto reality,theyare not certain; and
as far as they are certain,they do not referto reality."
FUNCTIONS OF SEVERAL COMPLEX VARIABLES*
W. T. MARTIN, SyracuseUniversity
1. Definitionof an analyticfunction.Consider a domain D in the 2n-dimen, z,. A functionf(zi, * , z,.)
sional euclidean space ofn complexvariables z1,
is said to be analytic in D if in some neighborhoodof every point (z1, * * z,z)
of D it can be representedas the sum of an (absolutely) convergentmultiple
power-series
i11* * ** n=?
al,...,j.(z-
z)*
(Z.
-Xn
An alternative definitionis that f is analytic in D if it has derivatives of all
orders,mixed and iterated,at every point of D. These two definitionsare very
easily seen to be equivalent.
, z,)
An importantresult due to Osgood J[1]in 1899 states that if f(zi,
j =1, * * n all existat
is boundedin D and if then partialderivatives
Of/Ozi,
every
pointofD, thenf is analyticin D.
In 1899 and again in 1900 Osgood [1, 21 raised the question as to whether
the boundednessrestrictioncould be removed,and in 1906 Hartogs [1 ] showed
that it actually could be removed. This means that if a functionis analytic in
each variable separately,it is analytic as a functionof all n variables. This is a
key result in the theory.
* This paperis an amplification
ofan invitedaddressdeliveredat theannualmeetingofthe
on August12, 1944.
MathematicalAssociationofAmericanin Wellesley,Massachusetts,
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