11 research outputs found
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
A Semantic Approach to Illative Combinatory Logic
This work introduces the theory of illative combinatory algebras,
which is closely related to systems of illative combinatory logic. We
thus provide a semantic interpretation for a formal framework in which
both logic and computation may be expressed in a unified
manner. Systems of illative combinatory logic consist of combinatory
logic extended with constants and rules of inference intended to
capture logical notions. Our theory does not correspond strictly to
any traditional system, but draws inspiration from many. It differs
from them in that it couples the notion of truth with the notion of
equality between terms, which enables the use of logical formulas in
conditional expressions. We give a consistency proof for first-order
illative combinatory algebras. A complete embedding of classical
predicate logic into our theory is also provided. The translation is
very direct and natural
On Constructive Axiomatic Method
The formal axiomatic method popularized by Hilbert and recently defended by Hintikka is not fully adequate to the recent practice of axiomatizing mathematical theories. The axiomatic architecture of Topos theory and Homotopy type theory do not fit the pattern of the formal axiomatic theory in the standard sense of the word. However these theories fall under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert constructive. I show that the formal axiomatic method always requires a support of some more basic constructive method
An algebra of behavioural types
Special thanks to Gérard Boudol, Ilaria Castellani, Silvano Dal Zilio, and Massimo Merro, for fruitful discussions and careful reading of parts of this document. Several anonymous referees made useful comments.We propose a process algebra, the Algebra of Behavioural Types, as a language for typing concurrent objects. A type is a higher-order labelled transition system that characterises all possible life cycles of a concurrent object. States represent interfaces of objects; state transitions model the dynamic change of object interfaces. Moreover, a type provides an internal view of the objects that inhabits it: a synchronous one, since transitions correspond to message reception. To capture this internal view of objects we define a notion of bisimulation, strong on labels and weak on silent actions. We study several algebraic laws that characterise this equivalence, and obtain completeness results for image-finite types.publishersversionpublishe
Axiomatic Architecture of Scientific Theories
The received concepts of axiomatic theory and axiomatic method, which stem from David Hilbert, need a systematic revision in view of more recent mathematical and scientific axiomatic practices, which do not fully follow in Hilbert’s steps and re-establish some older historical patterns of axiomatic thinking in unexpected new forms. In this work I motivate, formulate and justify such a revised concept of axiomatic theory, which for a variety of reasons I call constructive, and then argue that it can better serve as a formal representational tool in mathematics and science than the received concept