Pierluigi Graziani

Is a Space Interval a Set of Infinite Points? A Very Old Question

In this paper we will address the question whether a space interval is a set of infinite points . It is a very old problem, but despite its age it is still a live issue, and one we have to confront. We will analyze some topics regarding this question using the most influential objections against it, i.e. The Large and the Small paradox (in particular its Small Horn). We will consider classical contemporary reformulations of the argument (Grunbaum in Philosophy of Science 19:280–306, 1952; Grunbaum in Modern science and Zeno’s paradoxes. Allen and Unwin, London, 1968) and the possible ‘solutions’ to it. Finally, we will propose a new formulation of the paradox and analyze its consequences. In particular, we will bring further arguments supporting the standard thesis that it is possible that a segment of space is composed of a non-denumerable set of indivisible 0-length points.

Are Gandy Machines Really Local

This paper discusses the empirical question concerning the physical realization (or implementation) of a computation. We give a precise definition of the realization of a Turing-computable algorithm into a physical situation. This definition is not based, as usual, on an interpretation function of physical states, but on an implementation function from machine states to physical states (as suggested by Piccinini G, Computation in physical systems. The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/fall2012/entries/computation-physicalsystems. Accessed 5 Dec 2013, 2012). We show that our definition avoids difficulties posed by Putnam’s theorem (Putnam H, Representation and reality. MIT Press, Cambridge, 1988) and Kripke’s objections (Stabler EP Jr, Kripke on functionalism and automata. Synthese 70(1):1–22, 1987; Scheutz M, What is not to implement a computation: a critical analysis of Chalmers’ notion of implementation. http://hrilab.tufts.edu/publications/scheutzcogsci12chalmers.pdf. Accessed 5 Dec 2013, 2001). Using our notion of representation, we analyse Gandy machines, intended in a physical sense, as a case study and show an inaccuracy in Gandy’s analysis with respect to the locality notion. This shows the epistemological relevance of our realization concept. We also discuss Gandy machines in quantum context. In fact, it is well known that in quantum mechanics, locality is seriously questioned, therefore it is worthwhile to analyse briefly, whether quantum machines are Gandy machines.

Theory of Knowing Machines: Revisiting Gödel and the Mechanistic Thesis

Church-Turing Thesis, mechanistic project, and Godelian Arguments offer different perspectives of informal intuitions behind the relationship existing between the notion of intuitively provable and the definition of decidability by some Turing machine. One of the most formal lines of research in this setting is represented by the theory of knowing machines, based on an extension of Peano Arithmetic, encompassing an epistemic notion of knowledge formalized through a modal operator denoting intuitive provability. In this framework, variants of the Church-Turing Thesis can be constructed and interpreted to characterize the knowledge that can be acquired by machines. In this paper, we survey such a theory of knowing machines and extend some recent results proving that a machine can know its own code exactly but cannot know its own correctness (despite actually being sound). In particular, we define a machine that, for (at least) a specific case, knows its own code and knows to be sound.

Continuity of Motion in Whitehead’s Geometrical Space

The paper explores a neglected conception in the foundations of spacetime theories, namely the conception of gunk, point-free spaces inaugurated by De Laguna and Whitehead. Despite the epistemological merits of the proposal they argue that this would have rather unwelcome consequences for the description of motion that is provided by most of our physical theories, even simple ones such as classical mechanics. The tension, they claim, is generated by the following facts: (i) classical mechanics crucially adopts the notion of a point-particle in its description of motion; (ii) sets of (constructed) points in these Whiteheadian spaces turn out to be non-connected; (iii) connectedness is a necessary condition for continuity.