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\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}