L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...) fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e un’analisi di alcune delle questioni filosofiche connesse al Problema del Continuo. (shrink)

The problems that exist in relating quantum mechanical phenomena to classical concepts like properties, causes, or entities like particles or waves are well-known and still open to question, so that there is not yet an agreement on what kind of metaphysics lies at the foundations of quantum mechanics. However, physicists constantly use the formal resources of quantum mechanics in order to explain quantum phenomena. The structural account of explanation, therefore, tries to account for this kind of mathematical explanation in physics, (...) and hinges on the following claims: i) scientific models are central in scientific explanation; ii) in some cases the relevant information for the explanation/understanding of a phenomenon P consists in the sole structural properties of the (models displayed by the) theory; iii) in these cases, the interpretation of the formalism in terms of a categorial framework is unessential for the explanation of P and a mathematical model can be at the base of an objective and effective scientific explanation. The present paper will carry a reflection about some issues arising from R.I.G. Hughes and Robert Clifton’s works in the attempt to outline some details of structural explanation. (shrink)

This article examines the possibility of philosophizing about mathematics with children. It aims at outlining the nature of the practice of philosophy of mathematics with children in a mainly theoretical and exploratory way. First, an attempt at a definition is proposed. Second, I suggest some reasons that might motivate such a practice. My thesis is that one can identify an intrinsic as well as two extrinsic goals of philosophizing about mathematics with children. The intrinsic goal is related to a presumed (...) inherent importance of presenting children with some philosophical questions about mathematics. The extrinsic goals consist of first the positive effects such a practice can have on mathematical learning and abilities and second the fostering of children's understanding of philosophical method of inquiry and thinking and therefore of their philosophical thinking competences. Third, some examples found in the literature of previously developed ways of practising philosophy of mathematics with children are presented. This article aims at giving a general outlining picture of the issues surrounding the practice of philosophy of mathematics with children and should therefore be read as an encouragement to further development and studies. (shrink)