Carlo Cellucci

Giaquinto’s book is a philosophical examination of how the search for certainty was carried out within the philosophy of mathematics from the late nineteenth to roughly the mid-twentieth century. It is also a good introduction to the philosophy of mathematics and the views expressed in the body of the book, in addition to being thorough and stimulating, seem generally undisputable. Some doubts, however, could be raised about the concluding remarks concerning the present situation in the philosophy of mathematics, specifically Zermelo's iterative concept of set as a foundation for set theory, Simpson's reverse mathematics, Feferman’s Predicativist Programme, and the cognitive foundations of mathematics.

La logica matematica di Russell, di K. Gödel.--Che cos'è il problema del continuo di Cantor?, di K. Gödel.--Osservazioni al Convegno su i problemi di matematica per il secondo centenario di Princeton, di K. Gödel.--Matematica e logica, di A. Church.--Osservazioni sulla definizione e sulla natura della matematica, di H.B. Curry.--Sull'infinito, di D. Hilbert.--Il programma di Hilbert, di G. Kreisel.--Fondamenti storici, principi e metodi dell'intuizionismo, di L.E.J. Brouwer.--Disputa, di A. Heyting.--L'intuizionismo in matematica, di A. Heyting.--Verso un nominalismo costruttivo, di N. Goodman e W.V. Quine.--Un mondo di individui, di N. Goodman.

This article examines the current justifications of deductive inferences, and finds them wanting. It argues that this depends on the fact that all such justification take no account of the role deductive inferences play in knowledge. Alternatively, the article argues that a justification of deductive inferences may be given in terms of the fact that they are non-ampliative, in the sense that the content of the conclusion is merely a reformulation of the content of the premises. Some possible objections to this view are discussed and found inadequate.

The aim of this article is to show that intuition plays no role in mathematics. That intuition plays a role in mathematics is mainly associated to the view that the method of mathematics is the axiomatic method. It is assumed that axioms are directly (Gödel) or indirectly (Hilbert) justified by intuition. This article argues that all attempts to justify axioms in terms of intuition fail. As an alternative, the article supports the view that the method of mathematics is the analytic method, a method originally used by the mathematician Hippocrates of Chios and the physician Hippocrates of Cos, and first explicitly formulated by Plato. The article examines the main features of the analytic method, and argues that, while intuition plays an essential role in the axiomatic method, it plays no role in the analytic method, either in the discovery or in the justification of hypotheses. That the method of mathematics is the analytic method involves that mathematical knowledge is not absolutely certain but only plausible, but the article argues that this is the best we can achieve.