\documentstyle[12pt]{article} \oddsidemargin -2mm \evensidemargin 6mm \marginparwidth 0pt \marginparsep 0pt \topmargin 0mm \headheight 0pt \headsep 0pt \topskip 0pt \footheight 0pt %\footskip 0pt \textheight 235mm \textwidth 152mm \newcommand{\m}{mathematics} \newcommand{\ma}{mathematical} \newcommand{\ph}{philosophy} \newcommand{\pha}{philosophical} \title{On the distinction proof--truth in mathematics} \begin{document} \author{Roman Murawski\\ {\small Adam Mickiewicz University}\\ {\small Faculty of Mathematics and Comp. Sci.}\\ {\small ul. Matejki 48/49}\\ {\small 60--769 Pozna\'n, Poland}\\ {\small E-mail: {\tt rmur@math.amu.edu.pl}}} \date{} \maketitle Concepts of proof and truth are (even in mathematics) ambiguous. It is commonly accepted that proof is the ultimate warrent for a mathematical proposition, that proof is a source of truth in mathematics. One can say that a proposition $A$ is true if it holds in a considered structure or if we can prove it. But what is a proof? And what is truth? Since Plato, Aristotle and Euclid the best method to justify and to organize mathematical knowledge was considered the axiomatic method. The first mature and most representative example of its usage in \m\ were {\it Elements} of Euclid. They established a pattern of a scientific theory and in particular a paradigm in \m. Since Euclid till the end of the nineteenth century mathematics was developed as an axiomatic (in fact rather a quasi-axiomatic) theory based on axioms and postulates. Proofs of theorems contained several gaps --- in fact the lists of axioms and postulates were not complete, one freely used in proofs various obvious'' truths or refered to the intuition. In fact 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 the theories was simply the unprecise colloquial language. One should also note here that in fact till the end of the nineteenth century mathematicians were convinced that axioms and postulates should be simply 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 nineteenth 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 nineteenth century which stimulated foundational investigations. One of the directions of those foundational investigations was the program of David Hilbert and his {\it 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 \ma\ knowledge''. Note also that Hilbert's views were changing over the years, but always took a formalist direction. 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. The essence of the axiomatic study of mathematical truths was for him to clarify the position of a given theorem (truth) within the given axiomatic system and the logical interconnections between propositions. 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, that is, 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. It has been changing and developing all the time. At the beginning it was understood as one of the axioms of a system, later on as a property that the axioms suffice to prove all facts'' (Thatsachen) of the theory, as the warrant of the solvability of any problem, as exact and complete description'' or as maximal consistency. At the end he distinguished between semantic and syntactical completeness. Kurt G\"odel showed in 1929 the completeness of the first-order logic and in 1930 the incompleteness of the formal system of arithmetic and all richer systems. He stated also that it was precisely his recognition of the contrast between the formal definability of provability and the formal undefinability of truth that led him to his discovery of incompleteness. One should note that G\"odel was convinced of the objectivity of the concept of mathematical truth. He wrote, however, that in consequence of the philosophical prejudices of our times (\ldots) a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless.'' G\"odel's incompleteness theorems and in particular his recognition (before Tarski) of the undefinability of the concept of truth indicated a certain gap in Hilbert's programme and showed in particular, roughly speaking, that (full) truth cannot be established (achieved) by provability and, generally, by syntactic means. The former can be only approximated by the latter. Hence there arose a problem: how should one extend Hilbert's finitistic point of view? Hilbert in his lecture in Hamburg in December 1930 proposed to admit a new rule of inference to be able to realize his program. This rule is similar to the $\omega$-rule, but it has rather informal character and a system obtain by admitting it would be semi-formal. G\"odel pointed in many places that new axioms are needed to settle both undecidable arithmetical and set-theoretic propositions. To give an rough account of how those suggestions and proposals to extend the finitistic point of view do in fact work we shall consider some (technical) results. We restrict ourselves to the case of the arithmetic of natural numbers, more exactly to Peano arithmetic PA. Generally speaking one can obtain completions of PA by: \begin{itemize} \item admitting the $\omega$-rule, \item adding new axioms (in particular reflection principles), \item adding (partial) notion(s) of truth. \end{itemize} In the lecture will be presented a series of results which indicate interconnections between those approaches. \end{document}