The new system of axioms we propose is based on the foundational theory of De Giorgi et al. Scuola Normale Superiore di Pisa, Preprints di Matematica 26: 1 (1996) slightly modified. In that paper (which is dedicated to a new axiomatic framework for mathematics, informatics and logic) the authors use two kinds of primitive notions: relations and qualities. Since their system is based on the distribution paradigm, they start from distinction. We propose to shift the perspective and to start from (...) unity and then from within unity to pass to distinction; to this end we apply ideas of Lesniewski, Nijhoff International Philosophy Series 44 (1992). We introduce only one kind of entity as a primitive notion, namely relations, and treat qualities as articulations of relations. The new concept of "primary relation" permits the introduction of a dynamic, non-standard form of identity, which we hope will find application in various fields where self-referential structures are required. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

We give an account of stable reasoning, a recent and novel approach to problem solving from a formal, logical point of view. We describe the underlying logic of stable reasoning and illustrate how it is used to model different domains and solve practical reasoning problems. We discuss some of the main differences with respect to reasoning in classical logic and we examine an ongoing research programme for the rational reconstruction of human knowledge that may be considered a successor to the (...) logical empiricists’ programme of the mid-twentieth century. (shrink)

The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in (...) a theoretical model suitable for describing the tran-sition from the conjecturing to the proving phase. In the paper such a model is introduced and worked out to test Lakatos' machinery of proofs and refutations from a new point of view. Abduction and its categorical counter-part, adjunction, allow to explain within a unifying framework most of the phenomenology of conjectures and proofs, encompassing also the method of Greek analysis-synthesis. (shrink)

Starting from the considerations on modern school of some important scholars, which highlight the complexity of the school system, an analytical path is outlined to identify the best strategies by means of a mathematical model. The method followed is the analytical hierarchical one of Saaty that starts from the investigation of the various objectives, criteria and strategies, and indicates procedures to assign qualitative judgments and to transform them into numerical scores. In particular the AHP procedure is applied to find the (...) degree of effectiveness of various strategies for teaching English, in relation to possible contexts that may arise. (shrink)