The Cartesian cogito,epistemic logic and neuroscience: Some surprising interrelations

This paper was first read at the 1988 Annual Meeting of the International Institute of Philosophy in Stockholm, Sweden. The theme of the meeting was Descartes and the Contemporary Philosophy of Mind, which helped to explain the angle of my approach and the first few sentences of the paper. Morally, this paper thus should appear with the rest of the proceedings of the Stockholm meeting in a later number of this journal. However, the interrelations of my paper and Lucia Vaina's are so close that it makes best sense to publish them together.     

What is the axiomatic method?

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation.