Conventionalism in Geometry and the Interpretation of Necessary Statements

Conventionalism in Geometry and the Interpretation of Necessary Statements Max Black Philosophy of Science, Vol. 9, No. 4. (Oct., 1942), pp. 335-349. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28194210%299%3A4%3C335%3ACIGATI%3E2.0.CO%3B2-L Philosophy of Science is currently published by The University of Chicago Press.

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CONVENTIONALISM I N GEOMETRY AND T H E INTERPRETATIOK

OF NECESSARY STATEMENTS MAX BLACK

1. DESCRIPTION

O F A NECESSARY STATEMEST.

DIFFICULTY O F RECONCILIIGG THE

MEABISGFULFESS O F NECESSdRP STATEMENTS WITH THE

DEFISIKG P X I N C I P L E S O F EMPIRICISM

The statemelits traditionally labelled "necessary," among them the valid theorems of mathematics and logic, are identified as "those whose truth is independent of experience." The "truth" of a necessary statement has to be independent of the truth or falsity of experiential statements; a necessary statement can be neither confirmed nor refuted by empirical tests. The admission of genuinely necessary statements presents the empiricist with a troublesome problem. For an empiricist may bc defined, in terms of the current idiom, as one who adheres to some version, however "weal<," of a principle of vcriiiabi1ity.l One, that is, who claims that no statement can have cognitive meaning unlcss its truth depends, however indirectly, upon the truth of ercperiential statements; unless it can be provisionally confirmed or refuted by empirical tests. Allo~~ring that some necessary statements have cognitive meaning, then, ~vouldbe to provide a prima facie case against the validity of the principle of ~erifiability.~ In attempting to reconcile the alleged occurrence of necessary statements with a principle of verifiability, an empiricist may proceed in two ways. He may choose to say that there are, strictly speaking, no necessary statemeats; that the expressions so designated are empirical statements in misleading disguise, and that all cognitively meaningful statements, whether labelled "necess51-y" or not, can be tested by experience. This is the position taken by Mill \
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MAX BLACK

the powerful objections advanced by Frege and later critic^.^ But the empiricist who admits this has another resource.

2.

TWO

CONTEMPOIZABY EXAMPLES O F C O N V E S T I O S b l I S T INTERPRETdTION

OF

XECESSARY S T A T E M E ~ T SCRITIC'ISED

I t may be urged that while necessary statements are not devoid of "meaning," in some sense of that slippery term, they embody no truth-claim, are empig of cognitive or "declarative" meaning. Examination will show, the argument proceeds, that all necessary statements record no facts, being expressive rather of linguistic agreement, prescription, or convention. Unlike the empirical interpretation, the latter view remains widzly n e ~ e n t i a l today, being felt by many to constitute, in principle at least, a satisfactory means of interpreting necessary statements on empiricist principles. Two examples mill sufficiently illustrate both the trei:d of this line of argument and the superficial level at which it has been conducted. According to C. D. Kardie, "the conventionalist position . . . is that the laws of logic are prescriptions for the use of IT-ords," (Pwoc. h i s . Soc., 38, 48). Soxany plausible account of the nature of necessary statements must do justice to the felt necessity or certainty which attaches to them upon inspection; it is inadmissible for one who admits that there aTe necessary statements to urge that all of them are of a kind which ~voulclentail absence of the characteristic:; normally connoted by their designation. Hut this is what Hardie in fact does. He finds that the "prescriptions" of logic differ from those of grammar only in bcing capable of formulation in a "syn~bolicsystem," (Ibid., 51); but he fails to explain n h y the mere presentation of p~*escriptionsin the form of a system of symbols manipulated by explicit rules should generate what must, if he is right, be the illusion of certainty. There can be no better reason, on his view, for regarding a mathematic:~lprinciple as "certain" than for attaching the same claim to a rule of chess. LIe reconciles the non-empirical character of necessary statements with an empiricist criterion of meaning only by making nonsense of the distinction between validity and invalidity within the formal ~ciences.~ C. I. Lewis and C. H . Langford say: "The source of the necessary truth of every law of logic is in drjinitions arbitrarily assigned . . . the only truth which logic requires or can state is that which is contained in our own conceptual meanings-what our language or symbolism rcpresents" (Symbolic Logic, 211). Like others who put their trust in definitions, h o ~ ~ e v ethey r , quite fail to inalie clear l1om the arbitrary assignment of the equi~alenceof symbolic expressions E.g. the difiiculty of reconciling the precision of mathematics with its alleged empirical character. "If it [geometry] were experimental, i t would be only approximative and provisional. And what rough approximation!" (PoincarB, Foundations of Science, 79). I t is only fair to add t h a t much the same unsatisfactory procedure is followed by other contemporary defenders of a conventionalist view. None of them explain satisfactorily why a convention, essentially arbitrary in character, should come, as in the case of mathematical principles, t o be regarded as objectively determined in respect of validity. Or why anybody who adopts the conventionalist position should, continue, apart from the inertia of habit, t,o respect the customary set of linguistic prescriptions. See # 9 below.

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337

can generate principles, whether in logic or any other field.7 Nor do they seem to be any more aware than Hardie of the paradox of accounting for necessity in terms of the arbitrariness of definitions. The prime wea!iness of thew and other current versions of conventionalism arises from vagueness and lack of definition. Bnd nowhere is the need for definition more apparent than in connection with the crucial term, whether it, happens to be "convention" or some preferred substitute. The terms in question arc vagrants, whose shifts in contextual location induce a cloud of ambiguities which are commonly and all too lightly disregarded.* The conventionalist, to borro7,v a phrase from Poincark, all too often baptizes a difficulty rather than solves it. A fair verdict on the merits of conventionalism awaits a detailed explanation of how the term "convention" is to be understood.

3. THE

AIMS O F THIS

THE FIELD

PAPER:

OF GEOMETXY

RESTRICTED THESIS CAN BE

RE-STAT8 COSVENTIOKALISbt AS IT APPLIES I S (FOLLOWING P O I N C A I LTO ~ ) ;SEE WHETHER THIS GESERALISED TO APPLY T O ALL SYSTEMS O F NECESTO

SARY STATEXENTS

Conventionalist interpretations of necessary statements derive much of their initial plansibility in the minds of their defenders from the insight which has arisen from attention to the conventional aspects of the organization of knowledge. In this iield, and more especially in the interpretation of geometry, conventionalism has won its most resounding successes. Emphasis upon the conventional nature of geometry is associated especially with the name of Henri Poincar6,Vwhose persuasive and illuminating writings upon a wide range of topics in the philosophy of science may be held largely responsible for the present vogue of conventionalist doctrine.1° The remainder of this paper is intended to promote the clarification of the conventionalist thesis by restating in non-technical language the import and limitations of the conventionalist viewpoint in the field of geometry. By making clear in what sense it is correct to claim that geometry consists of "conventions" we shall be able to see whether Poincark's thesis can be extended to all necessary 7 Cf. W. V. Quine: "What is loosely called a logical consequence of definitions is therefore more exactly describable as a logical t r u t h definitionally abbreviated: a statement

which becomes a truth of logic when definienda are replaced by definientia" (Truth by convention, i n Philosophical E s s a y s for A. N . W h i t e h e a d , 92). 8 B u t see G. J. Bowdery, Conventions and norms, P h i l . S c i . , 8, 493, for some useful distinctions. 9 Thus Royce finds PoincarB's distinctive contribution t o the philosophy of science t o consist in his attention t o "kinds of hypotheses" which are valuable "despite, or even because of the fact t h a t experience can neither confirm nor refute them" (Intro. to PoincarB, Foundations of Science, 15). See also the pam10 This is clearly seen in LeRoy, Duhem, and other French writers. phlet Der W i e n e r K r e i s (manifesto of the Vienna Circle) for his influence in the development of neo-positivism. I n a fuller discussion of the background of conventionalism, attention would need t o be paid also t o Wittgenstein's stress on the "tautological" basis of logic and arithmetic.

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MAX BLACK

statements, and whether such generalisation as i t admits is of a sort to relieve the empiricist's puzzlement about necessary statements. The clarification achieved ~villbe largely negative. For it ~vj11appear that in the sense in which it is correct to claim that geometry consists of conventions the same may be said about all necessary statements. But this sense is not one which mill permit the empiricist to reconcile the non-empirical character of necessary statements with the claims of a principle of verifiability. I t is our aim, in other mords, to show that the wider conventionalist thesis in1 respect of necessary statements derives no support from the restricted conventionalism which is an appropriate product of an analysis of scientific procedures. Whoever wishes to maintain the wider conventionalist thesis may then bc: invited t o say whether the analysed sense of "convcntion" suits his purpose; and if not, to explain in exactly what other sense he wishes the term to be uraderstood. I n what follows we shall rely chiefly upon the exposition of Poincar6, though exegesis will be incidental to the main object of exhibiting a defensible conventionalist interpretation of geometry. (Poincark's numerous essays on the phillosophy of science are conveniently accessible, in translation, in the volumc entitled Foz~ndationsof Science (Science Press, New York, 1929). The critical commentary of 14. Rougier, La pkilosoplzie gbomBtrique de H e n r i Poincarb (Paris, 1920) is valuable in understanding PoincarB's standpoint.) We shall hope to answer four questions: I. In what sense is it correct to say that geometry consists of conventions? 2. In the sense of convention thus established, is it permissible to elaim also that arithmetic and logic consist of conventions? 3. Can the same be said of every system of necessary statements? 4. Does an affirmative answer t o the preceding questions reconcile the nonempirical character of a system of necessary statements with the principle of verifiability?

I n criticising the conventionalist view of geometry, serious confusion will arise unless constant attention is paid to the distinction between "pure" geometry (also referred to on occasion as "mathematical," "abstract," or "uninterpreted" geometry) and "physical" geometry (also known as "applied," "empirical," or "interpreted" geometry).ll I t will be convenient to have before us a brief recapitulation of the character of a deductive theory, of which a pure geometry is a special case. It is to be noted that PoincarB's conventionalist conclusions apply both to pure and to physical geometry, though the import of his view is somewhat differ11 Poincar6 himself makes no expIicit use of the distinction, whose importance has become clearer since his own work. I t s use wouId have clarified many of his points, notably in the discussions on "Space and Geometry" and "Experience and Geometry" (Op. cit., 66-91).

CONVENTIONALISM I N GEOMETRY

339

ent in the two cases. In respect of the former his position is derivative from mathematical researches which establish the reciprocal translatibility (in a sense to be explained) of "alternative" or "contrary" geometries. Attention to the character of the mathematical proof involved ~villallow us to extend the result to a wider class of cases than was considered by PoincarB. The bearing of these results upon physical geometry will then need less detailed attention since lye shall be able to show that conventionalism in respect of physical geometry is a consequence of the conventionalist interpretation of pure geometry. We begin then by describing a "deductive theory."12 Speaking very roughly it may be said that a deductive theory is a "game" played with symbols, arranged in given initial positions, and manipulated in accordance with certain rules of play (or rather the totality of positions obtained in some such game). Not every game will qualify as a deductive system, however: the initial positions must bear some resemblance to the sentences of ordinary language, and the rules of play are to approximate more or less closely to "rules of inference."13 &!toreprecisely, any specimen of a deductive theory will consist of "sentences," i.e. strings of symbols,14 designating either logical connectives, relations and predicates, or individual variables.15 Of the sentences belonging to the deductive theory a select group are distinguished as the axioms (the "initial positions" of our analogy), the relational and predicational signs occurring therein being designated as the primitive terms of the theory. Every other sentence of the system is a theorem, and must be capable of derivation from one or more of the axioms by repeated application of explicit rules of transformation (the "rules of play"). All terms, finally, which are not primitive in the system must be introduced by rules of definitionpermitting every sentence in which they occur to be transformed into sentences composed entirely of primitive terms. A non-primitive term is known as a defined term. A deductive theory, then, is constructed by assigning the axioms, rules of transformation and rules of definition.16 But our account will be incomplete without a specification of the circumstances in which two such sj~stemsare to be regarded as manifestations of a single deductive theory. l2 Here and subsequently I follow the terminology of A. Tarski, Introduction to Logic and the Methodology of Deductive Sciences (New York, 1941). Other synonyms in common

use are LLaxiom-system"and "calculus." l3 To give an exact definition of "deductive theory" is a problem of semantics. See R. Carnap, Introduction to Semantics, especially p. 12. l4 This condition need not exclude the occurrence of words, provided they are used i n abstraction from any meaning which they may have apart from their use in the system in which they occur. l5 For explanation of the terms used here see Carnap, Op. cit., #6 (Survey of some symbols and terms of symbolic logic). 16 As stated here, the addition or omission of definitions without alteration of the axioms or transformation rules suffices t o change the deductive theory. I n other contexts i t might be more convenient t o define deductive theories in such a way as t o avoid this consequence e.g. by regarding only the axioms and transformation rules as significant in determining the deductive system in question.

340

MAX BLACK

Two systems of the sort described are said to be identical if and only if there is a one-one correlation, (a) between the respective primitive terms, and (b) between the defined terms of the two systems, of such a character that on replacing every term by its correlate the axioms, theorems and definitions of the one become transformed into the axioms, theorems and definitions, respectively, of the other. (In other words, ideographic differences in the symbols used, and differences in the order of occurrence of axioms, theorems and definitions are regarded as irrelevant to the definition of a deductive theory.) We have now to explain the reasons for calling some deductive theories "pure geometries." The term "geometry" was for a long time reserved for the science of space i.e. for what would now be termed physical (euclidean) geometry. Suppose some system which has, as a matter of historic fact, been called a "geometry," say that embodied in Euclid's Elements, to be converted into a deductive theory.17 We shall then have one example of a pure geometry. Let the deductive theory obtained by formalizing Euclid's geometry be termed 'E'. By making slight modifications in E we can obtain other deductive theories resembling it in varying degree. Thus by changing the symbolised transform of the so-callcd parallel axiom, we can obtain other deductive theories, H (hyperbolic geometry) and R (elliptic geometry), which it is usual to call (pure) noneuclidean geometries.18 I t is hardly possible to state how much deviation from I3 permits a deductive theory still to qualify as a pure "geometry," since the latter term is applied somewhat arbitrarily, as a matter of technical convenience of classification within mathematics.lg The term "geometry," in fact, when applied to a deductive theory indicates little more than an indefinite degree of resemblance of formal structure in relation to certain familiar and historically important deductive theories. The vagueness of the term "geometry" is, however, of little importance for the present discussion. We may suppose ourselves to be concerned only with the three deductive theories E, H and R, imagined as if presented in full specificity of detail. In referring to these deductive theories as pure geometries we use the designation denotatively with no further commitment to any implied connotation. 17 I n the case of Euclid's Elements this would involve (a) omitting certain definitions and postulates of which no use is made in the elaboration of the geometry, (b) supplying a number of axioms which are needed, though not explicitly formulated, i n inferring the theorems, (c) formulating explicitly the transformation rules (principles of inference) used by Euclid, (d) replacing words by symbols. In prescribing t h a t such a process of formalization should be performed with a minimum of violence t o Euclid's choice of axioms, primitive terms, etc., considerable latitude is of course permitted. l 8 Certain types of non-Euclidean geometry require the modification of other axioms, i n addition t o the parallel axiom. See R. Bonola, Non-Euclidean Geometry, or any text on the subject. 1"The resemblance to Euclid required in a suggested set of axioms has gradually grown less, and possible deductive systems have been more and more investigated on their own account. I n this way geometry has become (what it was formerly mistakenly called) a branch of pure mathematics" (Russell, Principles of Mathematics, 373). Russell himself defines pure geometry with great generality, as "the study of series of two or more dimensions" (Ibid).

CONVENTIONALISM

5.

IN GEOMETRY

341

RELATIOSS O F MUTUAL TEANSLATION BETWEEN EUCLIDEAN AND SON-EUCLIDEAN GEOMETRIES

In explaining the remarkable set of relationships which have been found to hold between E, H and R, it will simplify the discussion if we may suppose each of the deductive theories in question to be constituted by means of different though related symbols. Thus suppose the terms belonging to E are ('A', 'B,, . . . , ;1', 'I,', 'M', . .. , 'P)),

, 'K', are the primitive and 'L', 'M', . . - , 'P', the defined where 'A', 'B', terms. We may then suppose that the corresponding terms of H are ('Al', 'Ijl', . . . , (I
--

-

342

MAX BLACK

every sentence of H becomes transformed into a sentence of E.20 Or expressed in another way, the introduction within E of sufficient new symbols, by means of appropriate definitions, permits us to deduce every axiom of H (and so, mediately, of every theorem of H) as a theorem of E. There are theorems of E, in other words, which on abbreviation by means of nominal definitions differ from the set of axioms of H only in the ideographic character of the symbols of which they are composed. We may add that the converse is true, each of the gcometries E, IP and R being convertible, in the manner described, into each of the others. Poincar& prefers to summarise these remarkable relations by saying that hyperbolic geometry (EI) has been translated into euclidean geometry (E), with the implication that the a,pparently diverse pure geometries are merely alternative for~nulatiocsof the very same system of abstract relationships: "Les th6oremes de Lobatschefski [i. e. of hyperbolic geometry] sur les droits, les triangles, les circles hyperboliques sont des thboremes de g6om6trie euclidienne s'appliquant A des figures autres e t plus compliqu6es que les droits, les circles ordinaires, e t reciproquement ."%l

6.

POINCAR&'S

CONVENTIOXALISM

IS BASED SOLELY OK THE RESULTS

OF THE

PRECEDING SECTIOK

Careful scrutiny of Poincar6's writings will show that his sole positive reason for regarding geometrical statements as conventional in character is the fact of mutual convertibility discussed in the preceding section. Negative grounds for rejecting the view that they are synthetic a priori or empirical are briefly presented;22 but the deciding consideration is the possibility of translation. He returns repeatedly to the analogy of the translation of a text from one language into another, or the expression of a mathematical formula with the aid of alternain making clear the light in which he wishes geomtive systems of co-ordii~ates,~~ etry to be rcgarded. But no explicit characterisation of conventions is supplied. Rougier has a more detailed examination (La notion de convention, Op.cif., 120-123). Having stressed that a convention differs from a proposition, subjectively, in being freely accepted ("un decret, issu d'une libre dbcision de notre esprit, qui se propose B l'acquiescement de notre volont6, toujours maftresse de l'accepter ou de le rejeter"), he distinguishes the particular type of convention relevant to Poincar6's standpoint ("les conventions facultatives") in virtue solely of their capacity for translation into a contrary convention: 20 See Poincar6, O p . cit., 59-60 for more detail concerning the contents of the dictionary required. Rougier, O p . cit., 114. 22 Poincar6, O p . cit., 64. 23 . . que doit-on penser de cette question: La g6om6trie euclidienne est-elle vraie? Elle n'a aucun sens. Autant demander si le systeme m6trique est vrai e t les anciennes mesures fnusses; si les coordonn6es cartksiennes sont vraies e t les coordonn6es polaires fausses. Une g6om6trie ne peut pas &tre plus vrai qu'une autre; elle peut seulement &tre plus cornr~ode" (Les geombtries non-euclidiennes, Rev. gem. dcs. Sci, 1891, 769-774). ([.

CONVEKTIOKSLIS>f IIN GEOMETRY

3 43

"On les reconnAit infailliblement A ce signe: on peut toujours remplacer une convention de cette sorte par unc convention contraire, sans que cela entratne autre chose qu'une simple modification du langage scientifique" (122).

It appears then that in having geometry pronounced to consist of conventions we are to understand only the consequences of the possibility of translation into a "contrary" geometry.24

We have now disentangled Poincark's reply to the first of the questions we have undertaken to answer, via., in what sense pure geometry may be correctly said to consist of conventions. For we are told that euclidean and other types of pure geometry are "conventional" in the sense of being capable of derivation, with the aid of suitable definitions, from the axioms of a "contrary" geometry. Kow if this is all that is meant by designating geometry as a system of conventions, it is not surprising that PoincarB's doctrine, removed from his own context of detailed illustration, should have proved so misleading as a guide to other writers. For in the first place the term "convention" suggests a more definite connotation; the unwary might suppose the conventionalist to be claiming that questions of validity in geometry are decided by the explicit or implicit agreement of mathematical pundits, by appeal to authority rather than reason. Kow this is certainly remote from Poincark's intention; upon the criteria of validity within deductive theories his analysis has no bearing. And as we shall see (in section # 10 below) his view presupposes the unconditional and so non-conventional nature of the principles of inference expressed in the rules of transformation used within deductive theories.25 The value of PoincerB7saccount of geometry is, accordingly, not so much any positive insight into the character of axioms or pure geometries 11-hichhe supplies, as the emphasis and force with which he stresses one consequence of the definition of a pure geometry, the impossibility of assigning preferential validity to any one geometry. This stress upon the arbitrary nature of any preference for a particular pure geometry is of crucial importance in an attempt to extend conventionalism to necessary statements in general. For it is clear that PoincarB is stressing the 24 Poincari: does a t times speak of geometrical sentences as being definitions in disguise (dejinitions dbguisbs), but seems t o refer thereby only t o the part played by the choice of a pure geometry in deciding which empirical definitions shall be employed in physical geometry. 25 The ambiguity arising here is connected with the failure of Poincari: and other conventionalists to discriminate between the use of '(convention" for denoting a choice and the character of what is chosen respectively. When Poincarb says t h a t geometrical axioms are conventions he means t h a t mre are free to choose them; but they themselves are neither choices, agreements nor prescriptions. To call what i s chosen a convention on the sole ground t h a t we may choose it is as confusing as i t would be to call Congress a n election on the ground that i t is elected.

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MAX BLACK

arbitrarg aspects of geometry; he is not maintaining, per absurdurn, that euclidean geometry is "necessarily true" as a prelude to accounting for its unique status in terms of its conventional character. On the contrary he asserts that the question of the unconditional validity of euclidean geometry makes no sense just bccuuse the decision in favour of euclidean geometry is arbitrary. In so far as geonietry is conventional it is not necessary; in so far as it is necessary it cannot, on his view, consist of conventions. Here is foreshadowed the central paradox of unrestricted conventionalism, arising from the attempt to account for what is initially regarded as non-arbitrary and objectively determinate (the unconditional validity of "necessary" statements) in terms of the unreservedly arbitrary (('decisions" about symbolism, embodied in "conventions"). 8.

EXTENSION O F T H E CONVENTIONALIST THESIS TO T H E CASES OF ARITH;\IETIC, LOGIC AND OTHER DEDUCTIVE THEORY

The preceding discussion of the sense in which it is appropriate to regard pure geometry as conventional sho~vsus what must be clone in order to extend the conventionalist thesis to the cases of arithmetic and logic. We should need (a) to describe "alternative" arithmetics and "logics," and (b) to demonstrate the reciprocal translation of arithmetics and logicsz6in a fashion analogous to that of geometry. Let us consider first the case of arithmetic. The initial step, presenting a deductive theory, derived by "abdication of meaning" from the interpreted system commonly understood as denoted by the term "arithmetic," affords no difficulty. We can reproduce the axiom set of PeanolZ7supplemented by explicit statement of the definitions and rules of inference we wish to include. The deductive theory so obtained may be distinguished as 'P'. In order to obtain "alternative" arithmetics we need only to make such relatively minor modifications in the axioms of P as xvill permit us to continue to describe the resulting system as an '(arithmetic," without doing too much violence to common Thus we may replace a single axiom by its contradictory, and so obtain an "alternative" arithmetic, Q, say. Provided the axiom in 26 It should be noted, however, t h a t current philosophical discussion concerning the admissibility of the notion of "alternative logics" has to'do with applied or interpreted logic, while the present section, lilre those which have preceded it, is concerned with pure or uninterpreted deductive theories. 2 7 Or see the deductive arithmetical theories given by Tarslri, Op. cit., Chs. 7-9. 2 8 The situation here is somewhat different than in the case of geometry, since there would probably be more reluctance t o admit t h a t a system differing from "ordinary" arithmetic might properly he called a n "arithmetic." But even if common linguistic usage were t o withhold the name of "arithmetic" from a system which deviated in any respect from a unicluely designated system there would be nothing to prevent us from imitating PoincarB's arguments concerning geometries. Cf. the effect upon PoincarB's discussion if the term "geometry" were to be synonymous with "euclidean-geometry": we should have to use different words but the possibility of translation into a contrary deductive theory would remain unchanged.

CONVENTIONALISM I N GEOMETRY

345

question mas independent of the rest of the axioms in P (and P were consistent) thc new deductive theory would be consistent. A few trials should convince the reader that if we are satisfied with somewhat trivial transformations there neecl be no difficultp in showing that P and Q can be translated into each other in the sense previously e~plained.~+iind similar results can easily be reached in the case of uninterpreted logics. Indeed, there can be little doubt that any deductive theory is capable of translation into 8, '(contrary" deductive theory,30so that PoincarB's thesis admits of extension to all deductive theories without exception. The possibility of translation into a contrary theory TJ-ouldappear to be a generic property of all deductive theories rather than a rneans of distinguishing between sub-classes of such the0rie5.~~ 9. THE

RESULTS OF THE LAST SECTION DO NOT RESOLVE THE EMPIRICIST'S PUZZLE ABOUT NECESSARY STATEMENTS

We turn, finally, to examine h0.i~far the conventionalist position above set forth can assist the empiricist in reconciling the "necessity" of statements with the claims of the principle of verifiability. I t will be remembered that the principal use made by Poincare of his discovery of the "conventional" charactilr of geometry is to stress the arbitrary nature of all considerations of the validity of an entire geometrical system. "lJne [email protected] peut pas &re plus vraie qu'une autre; elle peut seulement &re plus commode." Kow if this were all that the conventionalist were concerned to establish for 29 The det,ails of such a demonstration are omitted here to save space. The following may suggest the type of transformation which is sufficient. Suppose P contains t,he term 's' (immediate successor of), but not the converse of s. I n Q let st correspond t o the converse of the relation denoted by s in P, and let all other terms in Q denote the same relation as the corresponding unprimed letter in P. Then P and Q will be "contrary"; e.g. P will contain the theorem 'IsO' (one is the immediate successor of zero) while Q will contain the theorem '-ls'O1 (it is false t h a t one has the relation s' t o zero). But i t is very easy t o define s' in P . We take (x)(y)(xsly = D, ysx) and define every other primitive term of Q as synonymous with the corresponding unprimed symbol of P. I t is obvious that we shall, thereby, obtain all the axioms of Q as theorems within P. 30 Difficulty in establishing this would arise only if severe restrictions were imposed upon the type of definition to be used in making the translation. The trivial type of transformation illustratcd in the previous footnote has, of course, no interest for the mathematician exploring the intcr-relationships of deductive theories. B u t the thesis of conventionalism does not require t h a t a n "interesting" translation be produced. 31 I t is well 1rnom.n t h a t Poincarb himself specifically excluded arithmetic from the scope of his conventionalism: "[to] try t o found a false arithmetic analogous t o non-Euclidean geometry-it can not be done" ( O p . cit., 64). His reasons are based on the peculiar role he ascribes t o the principle of mathematical induction: "This rule . . . is the veritable type of the synthetic a priori judgment . . . rre can not think of seeing in i t a convention, as in some of the postulates of geometry." (Op. cit., 39). But the ground for the exception vix. t h a t the principle, being equivalent to an "infinity of syllogisms", cannot be deduced solely from logical axioms, is untenable. It is not necessary for a statement to be analytic (derivable solely from logical principles) in order to be a "convention" in PoincarB7s sense.

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plcre geometry, the same result could be reached more directly, though less dramatically, by considering the defining characteristics of a deductive theory. In presenting a system of sentences in the form of a deductive theory me have already allowed an abdication of meaning which forces us to abstain, in that context of discourse, from any consideration of the tmtlr of the sentences taken in isolation. And any question concerning the truth of an entire deductive theory is as senseless as an enquiry concerning the truth of any component axiom or theorem. But the sentences of a deductive theory are not "necessary" per se; what Poincar6 einphasises in his analysis, by labelling certain features of pure geometry as conventional, is precisely what would be traditionally regarded as non-necessary and arbitrary. And the only necessary feature of a deductive theory, viz, the entailment of the theorems by the axioms, does not appear as conventional in his analysis. On the contrary, the non-arbitrary character of the connection between axioms and theorems appears in his stress upon consistency as a test to n hich deductive systems must strbnlit and, still more clearly, in his insistence upon the invariance of the deductive rules of transformation throughout the translations considered. The limits of conventionalism could hardly be expressed inore concisely than in Poincare's own words: ". . . the possibility of translation implies the existence of an invariant. To translate is precisely to disengage this i i ~ v a r i a n t . " ~ ~ Since Poincark's riens presuppose the invariance of the principles of logic, he would be unable to label the latter as conventional without making nonsense of his whole position. (As me might put it, if the principles of logic: themselves are conventions it is entirely a matter of convention whether we take geometry to consist of conventions.) But the generalising conventionalist who wishes to reconcile neccssary statements with a principle of verifiability is committed t o just this extension. To exempt logic from the scope of conventionalism I\-ould be merely to narrow the field of necessary statements. I conclude, therefore, that while Poincarb's findings, in so far as they apply to pule geometry, arc capable of extension to the case of arithmetic and logic, their import is such as to bring small comfort to the empiricist. Even the enlightened anti-empiricist may thank Poincar6 for the demonstration of the unexpectedly wide range of the conventional and arbitrary in geometry, and even be prepared for similar surprises in logic, the remaining branches of mabhematics and other "a priori disciplines." But he will need more persuasive reasons before he is ready to believe that all considerations of validity in the formal sciences can be settled by arbitrary fiat.

10.

T H E BASIS FOR P O I X C A R ~ ' S CONVENTIOXALIST I S T E R P R E T A T I O N O F PHYSICAL

GEOMETRY; T H E RELATIVITY O F T H E FORMULATION O F PHYSICAL LAWS

I S RELATIOX T O T H E CHOICE O F A PURE GEOlIIETRY

The order in which our discussion has been conducted, involving prior attention to the character of a pure geometry, may have imparted a certain air of The entire section entitled "The objective value of science", and espe32 O p . cit., 339. cially the refutation of the exaggerated conveiitionalism of LeRoy is pertinent t o our discussion.

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triviality to Poincark's central thesis. For me seem to have reduced his eontention, so far as it applies to pure geometry, to the negative assertion of the senselessness of attempting to determine whether any pure geometry as a whole is valid. And this we have seen to follo~q-directly from the defining characteristics of a deductive theory, without appeal to the specific relations of inter-translatability upon which PoincarG himself relied. On turning to Poincark's views concerning physical geometry, the essential simplicity of his position, so far as it is relevant to the issues raised at the outset of this paper, is obscured by the interest of the subsidiary de-tail by v~hichhis analysis is illustrated. We cannot reproduce here his discussion of the relations between sensible space and the idealised "space" of geometry or his detailed analysis of the manner in which the physical interpretation of a pure geolnctry requires the choice (within a limited range of indetermination) of a set of "rigid" bodies and associated "displacements without deformation." His conclusions can, however, be stated quite briefly:-In supplying a physical model of a pure geometry we have a t some stage to choose a set of material bodies, to be regarded as "unconstrained" and so fit for use as "measur;ng-rods" and "axes of reference". Subject to certain limitations to which any such choice must conform (c.g. that the non-distorting displacements shall constitute a mathematical group) we have a number of options. If we choose one system of standard bodies we obtain a physical interpretation of euclidean geometry (E); but if me prefer to make another choice we shall obtain a physical interpretation of hyperbolic geometry (H) or elliptic geometry (R). Thus each of the pure geometries can receive a physical realization in terms of the motions of selected sets of material bodies. I t has to be remembered, further, that the chosen set of "rigid" bodies is used as a system of reference in formulating the laws of mechanics and other branches of physics. The normal procedure is to set up quantitative formulac ("la~vs") describing the deviations of all bodies from the motions of the standard "rigid" bodies. W h i c h motions of bodies will be regarded as "constrained" and so in need of explanation (description by means of differential equations) depends upon the initial choice of the rigid "unconstrained" bodies. In this way a new choice of a pure geometry induces a change in the formulation of the laws of physics. For the very same set of motions will receive quite different symbolic expression according to the choice made of the pure geometry of ~vhichthe preferred set of reference bodies will constitute the physical model. "I1 est evident que si on abandonnait la gkomktrie d7Euclide pour celle de Lobatschefslii, on serait oblig6 de modifier l'enonc6 de toutes les lois de la dynamique. De m&me, 17enonc6de ces lois n'est pas la mkme selon que l'on se sert de la langue anglaise ou de la langue f r a n ~ a i s e . " ~ ~ Poincar6's assertion of the conventionality of physical geometry reduces then to the joint assertion of (a) the possibility of finding physical models of each type of pure geometry; (b) the relativity of the expression of physical laws in relation to the choice of a pure geometry, (c) the inter-translatability of the variant formulations so obtained. 33

PoincarB, Sur les principes de la geometrie, Rev. d. Met. et d. iMor., 1900, 80.

348 11.

MAX BLACK CRITICAL COIilMENTS ON T H E DOCTRINE O F T H E I A S T SECTION; ITS CONNECTION WITH THE COXVENTIONALIST ISTERPRETATION O F PURE GEOMETRY

The critical con~nlentspreviously made in connection with PoincarB's conventionalist standpoint concerning pure geometry can be adapted with little alteration to the present context. Thus we may repeat that while establishing the compatibility of commonsense pre-scientific notions concerning space with the selection of a number of mutually incompatible geometries and the arbitrary character of any preference in favour of one rather than any other of these, Poincar6 does not concern himself with tests of validity within any such chosen system. (In the case of a physical geometry the test is, roughly, that of providing the closest approximation to the motions actually discoverable.) Ile shows that the preference for euclidean (physical) geometry rather than non-euclidean (physical) geometry is arbitrary; but he does not tell us how, 01. in what sense, the axioms and theorems of either geometry are discovered to be "true." There is, indeed, a very close connection bet\veen the thesis of conventionalism as applied to pure and to physical geometry respectively, though it is one which is not con~monlystressed. I t is not hard to show that the relativity of physical geometries is a direct consequence of the inter-translatability of pure geometries. We saw that the axioms of H (a non-euclidean geometry) could be regarded, in virtue of a certain correlation between terms, as theorems of E (euclidean geometry). Thus when E receives a physical interpretation (or "model") so that its axioms become converted into empirical statements concerning what may be called "euclidean-rigid" bodies, the axioms of H can be regarded as certain derivative empirical statements of the physical geometry. In other words, the axioms of I1 can then be interpreted as empirical statements about certain bodies which are not rigid in the euclidean sense. And, more generally, whenever we have shotti1 that a deductive theory L can be translated into another theory PC, any physical interprrtation of I< will automatically provide a physical interpretation of L. We can therefore dispense with the detailed considerations which led PoincarB to strev the relativity of physical geometries. In the sense in which he claims that physical geometry is conventional, the very same result will apply for every interpreted deductive theory. There is no reason why \ye should not have "alternative" interpreted arithmetics or logics analogous in every way to the "alternative" geometries in which PoincarB was primarily interested. Gut the very ease with which this extension of Poincark's thesis is achieved makes it useless to the empiricist attempting to reconcile the necessity of necessary statements with ihe claims of empiricism. For if interpreted geometry, or arithmetic, or logic are conventional only in the sense in which the same is true of evclrv interpreted deductive theory without csception, no differential criterion is provided for explaining the admitted differences between empirical and necessary system.

The upshot of our examination will be disappointing to anybody who may have hoped to find in the work of PoincarB indications of a method for admitting

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the meaningfulness of necessary statements without abandoning empiricism. We have seen that thc use of the term "convention" is an unfortunate way of recording tlie interesting relations discussed by PoincarB, whether in their bearing upon "pure" or interpreted deductive theories. Recognition of the fashion in which "contrary" or "alternative" geometries may be capable of reciprocal translation, and even the extension of the same thesis to all deductive theories, hardly commits us to a very far-reaching conventionalism. Nor do we need to talk about "conventions" in drawing attention to the manner in which the relativity of physical geometries is connected with the use of geometry as a standard of reference in the formulation of the laws of physics. For all their interest as contributions to the methodology of science, these results give no aid to those who propose a conventionalist analysis of necessary statements. I t would, no doubt, be unjustifiable to conclude that a conventionalist analysis of necessary statements is not feasible. I t may still be possible so to define the term "convention" that the differences between necessary and empirical statements can be illuminated by its employment. But if this is to be done it will be by means of considerations which derive no support from the analysis of Poincark.

University of Illinois