## Abstract

Around 1930 R. Carnap, H. Hahn and M. Schlick,^{1} largely under the influence of L. Wittgenstein, developed a conception of the nature of mathematics^{2} which can be characterized as being a combination of nominalism and conventionalism and which had been foreshadowed in Schlick’s doctrine about implicit definitions.^{3} Its main objective, according to Hahn and Schlick,^{4} was to conciliate strict empiricism^{5} with the a priori certainty of mathematics. According to this conception (which, in the sequel, I shall call the syntactical viewpoint) mathematics can completely be reduced to (or replaced by) syntax of language.^{6} I.e. the validity of mathematical propositions consists solely in their being consequences^{7} of certain syntactical conventions about the use of symbols,^{8} not in their describing states of affairs in some realm of things. Or, as Carnap puts it: *Mathematics is a system of auxiliary propositions without content or object*.^{9}

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## Gödel’s footnotes

- 1a.Cnf: R. Carnap,
*Erk*. 5 (1935), 30;CrossRefzbMATHGoogle Scholar - 1b.R. Carnap,
*Einheitswissenschaft*, Heft 3 (1934). H. Hahn,*ibid*. Heft 2 (1933), French translations of these papers in*Act. Sci. Ind*. 291,226. Moreover: H. Hahn,*Erk*. 1 (1930), p. 96;Google Scholar - 1c.Erk. 2 (1931), p. 135;Google Scholar
- 1d.
*Krise u. Neuaufbau in den exakten Wissenschaften*, Leipzig 1933. M. Schlick,*Gesammelte Aufsätze*, 1938, p. 145, p. 222.Google Scholar - 2.The terms “mathematical”, “mathematics” throughout this paper are used as synonymous with “logico-mathematical”, “logic and mathematics.” Moreover the term “axiom” is always used in the sense of “formal axiom or rule of inference” and is applied to any assertion that can be used in a complete proof without being [[expressed]] proved, in both contensive and formalized mathematics. The term “contensive” was suggested as a translation of “inhaltlich” by H. B. Curry in [?] [see also Curry,
*Foundations of mathematical logic*, 1963, p. 14].zbMATHGoogle Scholar - 3.Cnf: M. Schlick,
*Allgemeine Erkenntnislehre*, 1. Aufl., 1918, p. 30.zbMATHGoogle Scholar - 4.Cnf.: H. Hahn,
*Act. Sci. Ind*. 226, p. 13, 19 and M. Schlick, l. c, p. 147Google Scholar - 8.E.g., according to Hahn,
*Act. Sci. Ind*. 226, p. 26, the laws of contradiction and of excluded middle express certain conventions about the use of the sign of negation.Google Scholar - 9b.Cnf.
*Erk*. 5 (1935), p. 36;*Act. Sci. Ind.*291, p. 37. The whole passage reads as follows: “Wenn zu der Realwissenschaft die Formalwissenschaft hinzugefügt wird, so wird damit kein neues Gegenstandsgebiet eingeführt, wie manche Philosophen glauben, die den ‘realen’ Gegenständen der Realwissenschaft die ‘formalen’ oder ‘geistigen’ oder ‘idealen’ Gegenstände der Formalwissenschaft gegenüberstellen. Die Formalwissenschaft hat überhaupt keine Gegenstande;sie ist ein System gegenstandsfreier, gehaltleerer Hilfssätze.” [“In adjoining the formal sciences to the factual sciences*no new area of subject matter*is introduced, despite the contrary opinion of some philosophers who believe that the “real” objects of the factual sciences must be contrasted with the “formal”, “*geistig*” or “ideal” objects of the formal sciences.*The formal sciences do not have any objects at all*; they are systems of auxiliary statements without objects and without content”. In Feigl/Brodbeck (eds.),*Readings in the philosophy of science*, New York, 1953, p. 128.] I would like to say right here that Carnap today would hardly uphold the formulations I have quoted (cnf. §42). Moreover some of them were given only by Hahn and Schlick, and probably would never have been subscribed to by Carnap. [[That nevertheless I am discussing them in detail has two reasons, namely: 1. While the program itself and its elaboration, as far as it is feasible, have been presented in detail in several publications, the negative results as to its feasibility in its most straightforward and philosophically most interesting sense have nowhere been discussed. 2. The syntactical program in the form I am presenting it here is a priori perfectly sound and poses an interesting problem. Only a thorough mathematical investigation can decide on its feasibility.]] However, I am not concerned in this paper with a detailed evaluation of what Carnap has said about the subject, but rather my purpose is to discuss the relationship between syntax and mathematics from an angle which, I believe, has been neglected in the publications about the subject. For, while the syntactical program itself and its elaboration, as far as it is possible, have been presented in detail, the negative results as to its feasibility in its most straightforward and philosophically most interesting sense have never been discussed sufficiently.Google Scholar - 10.Cnf. Proc. Lond. Math. Soc, 2 s. vol 25 (1926), p. 338. Reprinted in
*The foundations of mathematics and other logical essays*, by F. P. Ramsey, 1931.zbMATHGoogle Scholar - 13.In consequence of the intuitionistic critique it has become customary to regard, not the theorems of classical mathematics, but only their derivability from certain axioms, to be mathematical truth. But thereby mathematics, to a very large extent, loses its applicability (cnf. §§13–16), unless the consistency of the axioms is known, which however cannot be known on the basis of this “implicationistic” standpoint. For the consistency of a system of axioms is not equivalent to a proposition of the form: B follows from the axioms A. Therefore if only implicationistic mathematics could be replaced by syntax, mathematics as to its applications could not. [[Moreover the delimitation of mathematical truth given by Implicationism is very artificial. For it acknowledges only particular mathematical truths (i.e., propositions of the same type as 2 + 2 = 4 and their consequences of the form: [?]), while we perceive with the same certainty and distinctness that
*a + b*=*b + a*holds for all integers a, b. For a more detailed exposition of implicationism cnf. K. Menger,*Blaetter fuer Deutsche Philosophie*, Bd. 4, 1930, p. 324.]] The concept of “discernible truth” [[(*inhaltliche Richtigkeit*)]], as applied to the propositions of classical mathematics themselves, may be rejected as meaningless. However, what is necessary here is only to imagine some mathematician who believes in the truth of classical mathematics (i.e., in the “natural mathematical intuition” mentioned in footn. 12) and to inquire, whether the consequences he can arrive at on the basis of this belief as to propo-skions considered to be meaningful can also be obtained on the basis of some syntactical interpretation without using mathematical intuition. Cnf. the answer to this question in §41. Moreover, if the syntactical viewpoint with regard to mathematics, in contradistinction to the factual sciences, is to make any sense,*some*concept of truth, other than truth by syntactical convention (let us call it “objective truth”’) must be admitted and the question then is whether, due to some syntactical interpretation, the consequences of the mathematical axioms occurring in the applications, can be obtained without knowing or assuming the objective truth (in the sense admitted) of the mathematical axioms, or at least a considerable number of them. Cnf. the answer to this question given in footn. 34.Google Scholar - 19.I believe that what must be understood by “syntax”, if syntax is not to presuppose some Platonistic realm of all possible combinations of symbols or other abstract entities (cnf. footn. 20), is exactly equivalent to Hilbert’s “Finitism”, i.e., it consists of those concepts and reasonings, referring to finite combinations of symbols, which are contained within the limits of “that which is directly given in sensual intuition” (“
*das unmittelbar anschaulich Gegebene*”), (cnf.*Math. Ann*. 95 (1925), p. 171–173). The section of mathematics thus defined is equivalent with recursive number theory (cnf. Hilbert-Bernays,*Grundlagen der Mathematik*, Bd. 1, p. 20–34 and p. 307–346), except that it may rightly be argued (cnf. P. Bernays,*L’enseignement math*., 34 (1935), p. 61) that exorbitantly great integers must not occur in finitary proofs, because they are theoretical constructions which are as far apart from the “immediately given” (and even from anything given in space-time reality) as the infinite, and, therefore, cannot be known to be meaningful or consistent, unless we trust some abstract mathematical intuition. If restrictions of this kind are introduced (namely to the effect that the integers and the number of elements of combinations referred to in a theorem or its proof must not be above some limit), then the negative results as to finitary consistency proofs mentioned in the sequel remain valid (cnf. footn. 22) and stronger ones probably can be obtained (cnf. §27).Google Scholar - 21.On the occasion of G. Gentzen’s consistency proof for number theory (cnf.
*Forsch. Log. Grundleg. exakt. Wiss*., N. F. Heft 4, 1938) it was ascertained up to which ordinal number definitions and proofs by transfinite induction can be expressed in the formalism of classical number theory. (Cnf. D. Hilbert and P. Bernays,*Grundlagen der Mathematik*, Bd. 2 (1939), p. 360–374). Thereby it became evident that those which cannot be so expressed are not finitary, while, on the other hand, all finitary proofs can be represented as inductions with respect to certain ordinal numbers.Google Scholar - 23a.Cnf. P. Bernays,
*Entretiens de Zurich*, 1938, ed. by F. Gonseth, 1941, pp. 144, 147;Google Scholar - 23b.
*L’enseignement math*., 34 (1935, pp. 68, 69, 94; Rev.*Internat. Phil*, No 27–28, 1954, Fasc. 1–2, p. 2;Google Scholar - 23c.also: G. Gentzen, Trav. oln [?] IX.
*Congr. Int. Phil*, VI, p. 203 (published in:*Act. Sci. Ind*. 535, 1937). In the more recent papers of the formalistic school the term “finitary” has been replaced by “constructivistic”, in order to indicate that it is necessary to use certain parts of intu-itionistic mathematics which are not contained within the limits of that which is directly given in sensual intuition.Google Scholar - 29.That this follows from a consistency proof was formulated as a conjecture by Carnap himself in the discussion at the
*Deutsche Naturforschertagung*in Koenigsberg 1930. (Cnf.*Erk*. 2 (1930), p. 143.)Google Scholar