Lógos and Máthma 2. Roman Murawski

the theory and the interpretations of the nonlogical vocabulary. Hilbert, as a mathematician, was not interested in establishing precisely the ontological status of mathematical objects. Moreover, one can say that his program was calling on people to turn their mathematical and philosophical attention away from the problem of the object of mathematical theories and turn it toward a critical examination of the methods and assertions of theories. On the other hand, he was aware that once a formal theory has been constructed, it can admit various interpretations. Recall here his famous sentence from a letter to G. Frege of 29th December 1899 (cf. Frege 1976, p. 67):

Yes, it is evident that one can treat any such theory only as a network or schema of concepts besides their necessary interrelations, and to think of basic elements as being any objects. If I think of my points as being any system of objects, for example the system: love, law, chimney-sweep [...], and I treatmy axioms as [expressing] interconnections between those objects, then my theorems, e.g. the theorem of Pythagoras, hold also for those things. In other words: any such theory can always be applied to infinitely many systems of basic elements.15

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      The essence of the axiomatic study of mathematical truths consisted for him in the clarification of the position of a given theorem (truth) within the given axiomatic system and of the logical interconnections between propositions.

      Hilbert sought to secure the validity of mathematical knowledge by syntactical considerations without appeal to semantic ones. The basis of his approach was the distinction between the unproblematic “finitistic” part of mathematics and the “infinitistic” part that needed justification. As is well known, Hilbert proposed to base mathematics on finitistic mathematics via proof theory (Beweistheorie). The latter was planned as a new mathematical discipline in which mathematical proofs are studied by mathematical methods. Its main goal was to show that proofs which use ideal elements (in particular actual infinity) in order to prove results in the real part of mathematics always yield correct results. One can distinguish here two aspects: consistency problem and conservation problem. The consistency problem consists in showing (by finitistic methods, of course) that the infinitistic mathematics is consistent; the conservation problem consists in showing by finitistic methods that any real sentence which can be proved in the infinitistic part of mathematics can be proved also in the finitistic part. One should stress here the emphasis on consistency (instead of correctness).

      To realize this program, one should formalize mathematical theories (even the whole of mathematics) and then study them as systems of symbols governed by specified and fixed combinatorial rules.

      The formal, axiomatic system should satisfy three conditions: it should be complete, consistent and based on independent axioms. The consistency of a given system was the criterion for mathematical truth and for the very existence of mathematical objects. It was also presumed that any consistent theory would be categorical, i.e., would (up to isomorphism) characterize a unique domain of objects. This demand was connected with the completeness.

      The meaning and understanding of completeness by Hilbert plays a crucial role from the point of view of our subject. Note at the beginning that in the Grundlagen der Geometrie completeness was postulated as one of the axioms (the axiom was not present in the first edition, but was included first in the French translation and then in the second edition of 1903). In fact the axiom V(2) stated that it is not possible to extend the system of points, lines and planes by adding new entities so that the other axioms are still satisfied. In Hilbert’s lecture at the Congress at Heidelberg in 1904 (cf. 1905b), one finds such an axiom system for the real numbers. Later, there appears completeness as a property of a system. In lectures “Logische Principien des mathematischen Denkens” (1905a, p. 13) Hilbert explains the demand ←25 | 26→of the completeness as the demand that the axioms suffice to prove all “facts” of the theory in question. He says: “We will have to demand that all other facts (Thatsachen) of the given field are consequences of the axioms”. On the other hand, one can say that Hilbert’s early conviction as to the solvability of every mathematical problem – expressed, e.g. in his 1900 Paris lecture (cf.Hilbert 1901) and repeated in his opening address “Naturerkennen und Logik” (cf. Hilbert 1930b) before the Society of German Scientists and Physicians in Königsberg in September 1930 – can be treated as informal reflection of a belief in completeness.

      In his 1900 Paris lecture, Hilbert spoke about completeness in the following words (see the second problem): “When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science”.

      One can take the “exact and complete description” to be complete enough to decide the truth or falsity of every statement. Semantically such completeness follows from categoricity, i.e., from the fact that any two models of a given axiomatic system are isomorphic; syntactically it means that every sentence or its negation is derivable from the given axioms. Hilbert’s own axiomatizations were complete in the sense of being categorical. But notice that they were not first-order, indeed his axiomatization of geometry from Grundlagen as well as his axiomatization of arithmetic published in 1900 were second-order.

      The demand discussed here would imply that a complete (in this sense) system of axioms is possible only for sufficiently advanced theories. On the other hand, Hilbert called for complete systems of axioms also for theories being developed. One should also add here that Hilbert admitted the possibility that a mathematical problem may have a negative solution, i.e., that one can show the impossibility of a positive solution on the basis of a considered axiom system (cf. Hilbert 1901).

      In Hilbert’s lectures from 1917–1918 (cf. Hilbert 1917–1918), one finds completeness in the sense of maximal consistency, i.e., a system is complete if and only if for any non-derivable sentence, if it is added to the system then the system becomes inconsistent. In his lecture at the International Congress of Mathematicians in Bologna in 1928, Hilbert stated two problems of completeness: one for the first-order predicate calculus (completeness with respect to validity in all interpretations, hence the semantic completeness) and the second for a system of elementary number theory (formal completeness, in the sense of maximal consistency, i.e., Post-completeness, hence the syntactical completeness) (cf. Hilbert 1930a).

Hilbert’s emphasis on the finitary and syntactical methods together with the demand of (and belief in) the completeness of formal systems seem to be the source and reason of the fact that, as Gödel put it (cf. Wang 1974, p. 9), “[...] formalists←26 | 27→considered formal demonstrability to be an analysis of the concept of mathematical truth and, therefore were of course not in a position to distinguish the two”. Indeed, the informal concept of truth was not commonly accepted as a definite mathematical notion at that time. As Gödel wrote in a crossed-out passage of a draft of his reply to a letter of the student Yossef Balas: “[...] a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless” (cf.Wang 1987, pp. 84–85). Therefore, Hilbert preferred to deal in his metamathematics solely with the forms of the formulas, using only finitary reasonings which were considered to be safe – contrary to semantical reasonings which were non-finitary and consequently not safe. Non-finitary reasonings in mathematics were considered to be meaningful only to the extent to which they could be interpreted or justified in terms of finitary metamathematics.16

      On the other hand, there was no clear distinction between syntax and semantics at that time. Recall, e.g., that as indicated earlier, the axiom systems came by Hilbert often with a built-in interpretation. Add also that the very notions necessary to formulate properly the difference syntax-semantics were not available to Hilbert.

      The problem of the completeness of the first-order logic, i.e., the fourth problem of Hilbert’s Bologna lecture, was also posed as a question in the book by Hilbert and Ackermann Gnmdzüge der theoretischen Logik (1928). It was solved by Kurt Gödel in his doctoral dissertation (1929, cf. also 1930)where he showed that the first-order logic is complete, i.e., every true statement can be derived from the axioms. Moreover he proved that,