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Conventionalism in Geometry and the Interpretation of Necessary Statements

Published online by Cambridge University Press:  14 March 2022

Max Black
Max Black*
Affiliation:
University of Illinois
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Extract

The statements 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 be defined, in terms of the current idiom, as one who adheres to some version, however “weak,” of a principle of verifiability. One, that is, who claims that no statement can have cognitive meaning unless its truth depends, however indirectly, upon the truth of experiential statements; unless it can be provisionally confirrned or refuted by empirical tests. Allowing that some necessary statements have cognitive meaning, then, would be to provide a prima facie case against the validity of the principle of verifiability.

Type
Research Article
Information
Philosophy of Science , Volume 9 , Issue 4 , October 1942 , pp. 335 - 349
Copyright
Copyright © Philosophy of Science Association 1942

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References

1 See R. Carnap, Testability and meaning, Phil. Sci., 3, 419, for a discussion of various forms of a principle of verifiability and the reasons for adopting a “weak” form.

2 It is partly for such reasons that Bertrand Russell, for instance, is led to renounce unconditional allegiance to empiricism. (See Limits of empiricism, Proc. Aris. Soc., 36, 131-156).

3 “there is in every step of an arithmetical or algebraical calculation a real induction, a real inference of facts from facts; and what disguises the induction is simply its comprehensive nature, and the subsequent extreme generality of the language” (Logic, I, 287). See Frege's criticisms of this view in Die Grundlagen der Arithmetik, #9, 10.

4 A. J. Ayer, Language, Truth and Logic, 104. That such a view forces us to regard necessary statements as disguised empirical statements is well argued by C. D. Broad, (Aris. Soc. Supp. Vol. 15, 107). A more recent empirical interpretation of necessary statements is that of N. Malcolm (Are necessary propositions really verbal?, Mind, 49, 204).

5 E.g. the difficulty of reconciling the precision of mathematics with its alleged empirical character. “If it [geometry] were experimental, it would be only approximative and provisional. And what rough approximation!” (Poincaré, Foundations of Science, 79).

6 It is only fair to add that 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, to 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, to respect the customary set of linguistic prescriptions. See #9 below.

7 Cf. W. V. Quine: “What is loosely called a logical consequence of definitions is therefore more exactly describable as a logical truth definitionally abbreviated: a statement which becomes a truth of logic when definienda are replaced by definientia” (Truth by convention, in Philosophical Essays for A. N. Whitehead, 92).

8 But see G. J. Bowdery, Conventions and norms, Phil. Sci., S, 493, for some useful distinctions.

9 Thus Royce finds Poincaré's distinctive contribution to the philosophy of science to consist in his attention to “kinds of hypotheses” which are valuable “despite, or even because of the fact that experience can neither confirm nor refute them” (Intro. to Poincaré, Foundations of Science, 15).

10 This is clearly seen in LeRoy, Duhem, and other French writers. See also the pamphlet Der Wiener Kreis (manifesto of the Vienna Circle) for his influence in the development of neo-positivism. In a fuller discussion of the background of conventionalism, attention would need to be paid also to Wittgenstein's stress on the “tautological” basis of logic and arithmetic.

11 Poincaré himself makes no explicit use of the distinction, whose importance has become clearer since his own work. Its use would have clarified many of his points, notably in the discussions on “Space and Geometry” and “Experience and Geometry” (Op. cit., 66-91).

12 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 “axiom-system” and “calculus.”

13 To give an exact definition of “deductive theory” is a problem of semantics. See R. Carnap, Introduction to Semantics, especially p. 12.

14 This condition need not exclude the occurrence of words, provided they are used in abstraction from any meaning which they may have apart from their use in the system in which they occur.

15 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 to change the deductive theory. In other contexts it might be more convenient to define deductive theories in such a way as to avoid this consequence e.g. by regarding only the axioms and transformation rules as significant in determining the deductive system in question.

17 In 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, in inferring the theorems, (c) formulating explicitly the transformation rules (principles of inference) used by Euclid, (d) replacing words by symbols. In prescribing that such a process of formalization should be performed with a minimum of violence to Euclid's choice of axioms, primitive terms, etc., considerable latitude is of course permitted.

18 Certain types of non-Euclidean geometry require the modification of other axioms, in addition to the parallel axiom. See R. Bonola, Non-Euclidean Geometry, or any text on the subject.

19 “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. In 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).

20 See Poincaré, Op. cit., 59-60 for more detail concerning the contents of the dictionary required.

21 Rougier, Op. cit., 114.

22 Poincaré, Op. cit., 64.

23 “… que doit-on penser de cette question: La géométrie euclidienne est-elle vraie? Elle n'a aucun sens. Autant demander si le système métrique est vrai et les anciennes mesures fausses; si les coordonnées cartésiennes sont vraies et les coordonnées polaires fausses. Une géométrie ne peut pas ětre plus vrai qu'une autre; elle peut seulement ětre plus commode“ (Les géométries non-euclidiennes, Rev. gen. des. Sci, 1891, 769-774).

24 Poincaré does at times speak of geometrical sentences as being definitions in disguise (definitions déguisés), but seems to refer thereby only to 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 Poincaré and other conventionalists to discriminate between the use of “convention” for denoting a choice and the character of what is chosen respectively. When Poincaré says that geometrical axioms are conventions he means that we are free to choose them; but they themselves are neither choices, agreements nor prescriptions. To call what is chosen a convention on the sole ground that we may choose it is as confusing as it would be to call Congress an election on the ground that it is elected.

26 It should be noted, however, that current philosophical discussion concerning the admissibility of the notion of “alternative logics” has to do with applied or interpreted logic, while the present section, like those which have preceded it, is concerned with pure or uninterpreted deductive theories.

27 Or see the deductive arithmetical theories given by Tarski, Op. cit., Chs. 7-9.

28 The situation here is somewhat different than in the case of geometry, since there would probably be more reluctance to admit that a system differing from “ordinary” arithmetic might properly be called an “arithmetic.” But even if common linguistic usage were to withhold the name of “arithmetic” from a system which deviated in any respect from a uniquely designated system there would be nothing to prevent us from imitating Poincaré's arguments concerning geometries. Cf. the effect upon Poincaré'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.

29 The details of such a demonstration are omitted here to save space. The following may suggest the type of transformation which is sufficient. Suppose P contains the term ‘s’ (immediate successor of), but not the converse of s. In Q let s’ correspond to 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 ‘~1s'O’ (it is false that one has the relation s’ to zero). But it is very easy to define s’ in P. We take (x)(y)(xs'y = Dfysx) and define every other primitive term of Q as synonymous with the corresponding unprimed symbol of P. It 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 illustrated in the previous footnote has, of course, no interest for the mathematician exploring the inter-relationships of deductive theories. But the thesis of conventionalism does not require that an “interesting” translation be produced.

31 It is well known that Poincaré himself specifically excluded arithmetic from the scope of his conventionalism: “[to] try to found a false arithmetic analogous to non-Euclidean geometry—it can not be done” (Op. cit., 64). His reasons are based on the peculiar role he ascribes to the principle of mathematical induction: “This rule … is the veritable type of the synthetic a priori judgment … we can not think of seeing in it a convention, as in some of the postulates of geometry.” (Op. cit., 39). But the ground for the exception viz. that 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 Poincaré's sense.

32 Op. cit., 339. The entire section entitled “The objective value of science”, and especially the refutation of the exaggerated conventionalism of LeRoy is pertinent to our discussion.

33 Poincaré, Sur les principes de la geometrie, Rev. d. Met. et d. Mor., 1900, 80.