The Emergence of the Intuition of Truth in Mathematical Thought

In this contribution regarding the emergent character of the intuition of truth in mathematics, S. Galvan examines the feasibility of applying the category of emergence to the hierarchy of formal systems and to the hierarchy of the evidence systems which are their correlates. This hierarchy is a direct consequence of the phenomenon of incompleteness, in its various forms that result from Godel's theorems. Hence the contribution explores the meaning of the most significant form of incompleteness resulting from Godel's theorems, namely the omega-incompleteness of arithmetic theories, and examines some methods able to remedy this form of incompleteness. In particular, two ways are explored to overcome the incompleteness of primitive recursive arithmetic PRA: the first consists in the use of an explicit Tarskian theory of truth and the second is hased on a notion of truth as non-finitary evidentiability. In both cases, the determination of the theory's consistency always requires the emergent character of truth from derivability; in the first case, due to the non-finitary extensional character of the model-theoretical notion of truth (inasmuch as the objectual domain is understood in an actualistic sense) and, in the second, because of the abstract intensional character of the soundness proof within a non-conservative extension of PRA. To operate according to principles hased on forms of evidence that are abstract and open to novelty is very different from operating in accordance with the computational logic of a machine. The Galvan' contribution shows how the mind follows its own paths in discovering the truth and the modalities of its justification, this being due to the fact that this undertaking cannot be restricted to the use of only concrete contents and their finite combinations. The author argues that it follows that a cognitive system endowed with the capability of intending the abstract is characterized by a relationship to the ohject which is not one of reception but of presence. The object is known insofar as it is present to the system and not as a mere source of stimuli elaborated by the system. This relationship shows the mark of the emergent novelty, which is the mark of the emergence of consciousness, which in its turn is a pre-condition af the presence af an object to the cognitive system.