Formalismo e incompletud

Abstract

The purpose of this article is to realize a philosophical evaluation (from an epistemological, as well as from an ontological point of view) of some of the most important logical theorems that have been discovered during the last 40 years. The theorems I want to refer to are those dealing with the properties of incompleteness and undecidability of formal systems. I fix my attention especially upon Gödel's theorem as it was stated in 1931.

The introduction of this article is dedicated to analyse the reasons why some strictly mathematical propositions (such as Gödel's theorem) have produced such a big philosophical perplexity. In the second section of this essay I offer a brief examination of the logical structure of Gödel's proof (assuming, of course, some knowledge of it). I especially emphasize those aspects of the proof which are essentially methodological: the construction of the Gödelian numeration, the parallel between the process of demonstration and the properties of recursive functions and, most especially, the structure of the "crucial" proposition whose meaning is indirectly studied all through the various sections of this article.

The third section of this study is intended to raise some critical questions against some rather hasty interpretations of the theorem that have been advanced from various different philosophical view points.

It is pointed out that the materialist positions, despite the fact of being the best philosophically oriented, in the sense that they take into account, and make use of, scientific results, overestimate the philosophical value of the theorems of the formal sciences, and assign extralogical powers to merely logical propositions. At the same time they give an excesive importance to the philosophical proyections of the schools of epistemology of mathematics.

It is also shown the incoherence and superficiality of various interpretations of the theorem coming from traditional philosophers who, in a rather extrapolative way, make use of the technical results of metalogic in order to deny the efficiency of formal and mathematical methods.

Finally, I analyse the point of view of different psychological schools, especially that of the so called "genetic psychology" and some other structuralist positions. I consider that these positions give a too complicated account of formal "phaenomena" and, very often, they treat certain mathematical constructions as if they were the psychological processes leading to their acquisition.

The last section of the article is dedicated to characterize a set of statements with philosophical (ontological and epistemological) content. These statements can be validly asserted on the basis of achievements such as Gödel's and other logical theorems. These logical achievements, without having the nature of general philosophical thesis, force the logician to make a choice between: a) to recognize that the mechanical codification of arithmetics is naturally restricted, or b) to continue with that codification but changing the classical notions of truth and deduction.

If the choice is made in favor of the total codification, then the principle of the excluded middle must be abbandonned. The theorems we are dealing with can be considered as a serious obstacle only with respect to a particular method of scientific investigation: axiomatization; and even there, the obstacle is relative only to certain kinds of theories. We could say, in general, that it could be avoided through certain "asynthotical" procedures. Axiomatic systems shouldn't be used from now on, as a unique language definitively closed, but as a series of particular systems so that the logicians and the mathematicians can modify their postulates any time they encounter some difficulties in the language they are working with.

Neither Platonism nor Intuitionism can be justified by the logical theorems we have being considering. My point of view is that it could only be justified a relative use of "universal" concepts, that is to say, the kind of use we give to theoretical terms of natural sciences.