Mathematical logic and the ideal of apodictic science (Q2753616)

This article attempts to answer the question: ``What are the best, if not the most reliable ways of presenting a scientific theory?'' The historical perspective is given first as follows: The prevailant scheme is that of the deductive method, such as exemplified in Newton's \textit{Principia}, Euclides' \textit{Principles} through Galileo's slight experimental modifications to Hilbert who declared this method to be the ultimate one for mathematics. The work of Gödel and three-valued logic of Łukasiewicz had certainly put this method into a perspective. Lorenzen also distrusts this ideal quoting Pascal who supported science founded upon common language only. It was realized that there are at least two traditions in logic, that of ``mathematics of logic'' and ``the logic of mathematics.'' D'Alambert stressed that it was hopeless to complete a theory by improving a deductive theory, since such a scheme ``always leaves some holes.'' Lewis and Longford noted that ``... in the case of logic, it would appear that the deductive method involves a puzzle: unless logic is taken for granted, we can make no deductions; but it is logic which is to be deduced.'' Intuitionistic logic with Brouwer as its representative is an example of a non-apodictic theory. Brouwer's notion of a logical theory is empirical in nature, namely it is opposite to the Aristotelian notion of a theory, because ``logic is applied mathematics.'' A clear cut interpretation of intuitionistic logic was suggested by Kolmogorov, in 1932, as an activity of solving problems. The logical statement is related to a problem, i.e. to finding the means for solving a given problem. Lorenzen sees any logical statement as causing a conflict between the two parties ... Secondly, the author gives a characterization of an alternative organization and a program for refounding of mathematical logic. In the alternative organization, the principles are heuristic, or better, methodological. This agrees with a Leibniz' suggestion of two basic principles of our reason, i.e. the principle of non-contradiction -- the typical principle of a deductive scheme -- and the principle of sufficient reason -- the typical principle of a heuristic inquiry. Classical logic and intuitionistic logic do not represent together two different theories of informal logic, rather two alternative kinds of organization of mathematical logic as a system. In particular, the polemics between Hilbert and Brouwer, rather than the extremes of two absolutist theories can be seen as two modalities of the presentation of the same subject. The foundations of a scientific theory include more than the options of the kinds of organization; they include one more option, namely that of potential or actual infinity -- the option that concerns the way to formalize predicate calculus. A new foundation of mathematical logic, as a full alternative to formalist mathematical logic may be a theory with group theory as its basic mathematics.