Gabriele Lolli – författare

Why mathematics is not merely formulaic: an argument that to write a mathematical proof is tantamount to inventing a story.In The Meaning of Proofs, mathematician Gabriele Lolli argues that to write a mathematical proof is tantamount to inventing a story. Lolli offers not instructions for how to write mathematical proofs, but a philosophical and poetic reflection on mathematical proofs as narrative. Mathematics, imprisoned within its symbols and images, Lolli writes, says nothing if its meaning is not narrated in a story. The minute mathematicians open their mouths to explain something—the meaning of x, how to find y—they are framing a narrative. Every proof is the story of an adventure, writes Lolli, a journey into an unknown land to open a new, connected route; once the road is open, we correct it, expand it. Just as fairy tales offer a narrative structure in which new characters can be inserted into recurring forms of the genre in original ways, in mathematics, each new abstract concept is the protagonist of a different theory supported by the general techniques of mathematical reasoning. In ancient Greece, there was more than an analogy between literature and mathematics, there was direct influence. Euclid’s proofs have roots in poetry and rhetoric. Mathematics, Lolli asserts, is not the mere manipulation of formulas.

This book brings together young researchers from a variety of fields within mathematics, philosophy and logic. It discusses questions that arise in their work, as well as themes and reactions that appear to be similar in different contexts. The book shows that a fairly intensive activity in the philosophy of mathematics is underway, due on the one hand to the disillusionment with respect to traditional answers, on the other to exciting new features of present day mathematics. The book explains how the problem of applicability once again plays a central role in the development of mathematics. It examines how new languages different from the logical ones (mostly figural), are recognized as valid and experimented with and how unifying concepts (structure, category, set) are in competition for those who look at this form of unification. It further shows that traditional philosophies, such as constructivism, while still lively, are no longer only philosophies, but guidelines for research. Finally, the book demonstrates that the search for and validation of new axioms is analyzed with a blend of mathematical historical, philosophical, psychological considerations.

This book brings together young researchers from a variety of fields within mathematics, philosophy and logic. It discusses questions that arise in their work, as well as themes and reactions that appear to be similar in different contexts. The book shows that a fairly intensive activity in the philosophy of mathematics is underway, due on the one hand to the disillusionment with respect to traditional answers, on the other to exciting new features of present day mathematics. The book explains how the problem of applicability once again plays a central role in the development of mathematics. It examines how new languages different from the logical ones (mostly figural), are recognized as valid and experimented with and how unifying concepts (structure, category, set) are in competition for those who look at this form of unification. It further shows that traditional philosophies, such as constructivism, while still lively, are no longer only philosophies, but guidelines for research. Finally, the book demonstrates that the search for and validation of new axioms is analyzed with a blend of mathematical historical, philosophical, psychological considerations.

Il libro vuole aiutare a studiare la teoria degli insiemi indicando l'articolazione della teoria, a partire dal concetto di infinito per arrivare alla definizione dei numeri, sia finiti sia infiniti, con la diramazione tra ordinali e cardinali; insiste sulle proprietà degli insiemi numerabili, e sul continuo. Non sostituisce un manuale, perché non ci sono tutte le dimostrazioni ma solo alcune, considerate importanti, che danno il gusto dello stile di questa materia.Ricorda come la teoria sia nata dalle esigenze dell'analisi matematica e come sia legata al problema dei fondamenti; discute il riduzionismo e presenta anche la teoria alternativa rivale delle categorie.Distingue la teoria propria dell'infinito dal linguaggio insiemistico che pervade la matematica.Nelle applicazioni si insiste sul principio di induzione e sulle definizioni induttive, e sulla derivazione delle proprietà degli insiemi finiti, con tutte le definizioni equivalenti di finito, e si indica lo studio delle versioni effettive dei risultati teorici, in particolare la definizione esplicita di funzioni ed enumerazioni, fino gettare un ponte con la teoria della calcolabilità, in vista dell'insegnamento.