Cristina Coppola

Approximate distances, pointless geometry and incomplete information

In this paper an abstract notion of approximate metric space is proposed by means of an interval-valued distance between regions. Regions are interpreted as pieces of information in a space. The resulting theory supports some promising applications to some topics of fuzzy set theory, rough set theory and clustering.

Convergence and fixed points by fuzzy orders

A general approach to fixed point theory is proposed which is related to the notion of fuzzy ordering. This approach extends both the fixed point theorems in metric spaces and the ones in ordered sets.

Multi-valued Logic for a Point-Free Foundation of Geometry

The paper starts from the observation that in the inclusion-based approach to point-free geometry there are serious difficulties in defining points. These difficulties disappear once we reformulate this approach in the framework of continuous multivalued logic. So, a theory of ‘graded inclusion’ is proposed as a counterpart of the usual ‘crisp inclusion’ of mereology. Again, a second theory is considered in which the graded predicates ‘to be close’ and ‘to be small’ are assumed as primitive. In both cases a suitable notion of abstractive sequence and of equivalence between abstractive sequences enables us to define the points. In the resulting set of points a distance is defined in a natural way and this enables a metrical approach to point-free geometry and therefore to go beyond mereotopology.The general idea is that it is possible to search for mathematical formalizations of the naive theory of the space an ordinary man needs to have in its everyday life. To do this we have to direct our attention not only to regions and the related relation of inclusion as it is usual in point-free geometry, but also to those (vague) properties which are geometrical in nature.

Point-free Foundation of Geometry and Multivalued Logic

A. N. Whitehead, in two basic books, considers two dierent approaches to point-free geometry: the inclusion-based approach, whose primitive notions are regions and inclusion relation between regions and the connection-based approach, where the connection relation is consid- ered instead of the inclusion. We show that the latter cannot be reduced to the rst one, although this can be done in the framework of multi- valued logics.

MEREOLOGICAL FOUNDATIONS OF POINT-FREE GEOMETRY VIA MULTI-VALUED LOGIC

We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naive and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science.

Special Issue on point-free geometry and topology. An introduction

In the first section we briefly describe methodological assumptions of point-free geometry and topology. We also outline history of geometrical theories based on the notion of \emph{region}. The second section is devoted to concise presentation of the content of the LLP special issue on point-free theories of space.

Il problem solving come strategia per una diversa gestione dell’errore nell’educazione matematica al primo ciclo

Abstract – Errors in the context of school mathematics are often considered as something to be avoided at all costs: there is a sort of identification between difficulties in mathematical learning and errors and between errors and failure of the teaching strategies. Research into mathematics teaching has for some time brought this wide-spread “epistemology” of error into question. Taking the seminal work of Borasi as their starting point, maths ed-ucators have developed a new epistemology of error in line with a growing attention toward productive thinking in mathematical teaching and learning as well as the role of errors in the development of mathematics. Riassunto – L’errore in matematica nel contesto scolastico e considerato come qualcosa da evitare assolutamente. Questo perche c’e una implicita identificazione tra errore e difficolta, e tra errori degli allievi e fallimento dell’insegnamento. Questa diffusa “epistemologia” dell’errore e stata, da tempo, messa fortemente in discussione dalla ricerca in didattica della matematica. A partire dai lavori di Raffaella Borasi, nella ricerca in didattica della matematica si e sviluppata una nuova epistemologia dell’errore coerente con la crescita dell’attenzione allo stimolo del pensiero produttivo nell’insegnamento e apprendimento della matematica e con il ruolo che gli errori hanno avuto nello sviluppo della matematica. Keywords – error in mathematics, assessment, problem solving, productive thinking Parole chiave – errore in matematica, valutazione, problem solving, pensiero produttivo

The Development of Logical Tools Through Socially Constructed and Culturally Based Activities

In this chapter, we want to focus on the relationships between language and developmental processes of logical tools in “linguistic-manipulative” activities. A didactic path in a primary school classroom was designed based both on cooperation and meanings negotiation in interactional situations and on social-constructivist theories. The activities were deduction tasks. Our research hypothesis was that through cooperative activities related to the shared creation and the manipulation of language, socially and culturally constructed, children will be able to choose various strategies for problem solving. The children worked in cooperative groups and were interviewed using a semi-structured interview based on the explication interview method and on the biographic report. The content analysis and the observations of the activities helped us to identify topics related to the children’s processing and reprocessing of the activities.