Proofs were informal and intuitive, they were rather demonstrations; and the very concept of a proof was of a psychological (and not of a logical) nature. Note that almost no attention was paid to the precization and specification of the language of theories – in fact the language of theories was simply the unprecise colloquial language. One should also note here that in fact till the end of the 19th century, mathematicians were convinced that axioms and postulates should be true statements. It seems to be connected with Aristotle’s view that a proposition is demonstrated (proved to be true) by showing that it is a logical consequence of propositions already known to be true. Demonstration was conceived here of as a deduction whose premises are known to be true, and a deduction was conceived of as a chaining of immediate inferences.
Basic concepts underlying the Euclidean paradigm have been clarified on the turn of the 19th century. In particular, the intuitive (and rather psychological in nature) concept of an informal proof (demonstration) was replaced by a precise notion of a formal proof and of a consequence. This was the result of the development of mathematical logic and of a crisis of the foundations of mathematics on the turn of the 19th century which stimulated foundational investigations.
One of the directions of those foundational investigations was the program of David Hilbert and his Beweistheorie. Note at the very beginning that “this program was never intended as a comprehensive philosophy of mathematics; its purpose was instead to legitimate the entire corpus of mathematical knowledge” (cf. Rowe 1989, p. 200).Note also that Hilbert’s views were changing over the years, but always took a formalist direction.
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Hilbert sought to justify mathematical theories by means of formal systems, i.e., using the axiomatic method. He viewed the latter as holding the key to a systematic organization of any sufficiently developed subject. In “Axiomatisches Denken” (1918, p. 405) Hilbert wrote:
When we put together the facts of a given more or less comprehensive field of our knowledge, then we notice soon that those facts can be ordered. This ordering is always introduced with the help of a certain network of concepts (Fachwerk von Begriffen) in such a way that to every object of the given field corresponds a concept of this network and to every fact within this field corresponds a logical relation between concepts. The network of concepts is nothing else than the theory of the field of knowledge.14
By Hilbert the formal frames were contentually motivated. First-order theories were viewed by him together with suitable non-empty domains, Bereiche, which indicated the range of the individual variables of 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
