Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...) meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]. (shrink)

In this historical essay, we examine the reciprocal influences of philosophical doctrines and logic, their interrelations with language, and the place of mathematics in these developments.

The author's aim in this biography is to shed light on the contrasts and polarity—yet relationship—between the rational and the irrational in Gödel's work and personality. On the one hand there is the genius logician whose technical work can be said practically to have attained the limits of what rational thought can produce; on the other hand, one is struck, claims the author, by the irrationality in Gödel's personality and psychic structure, such as his belief in the existence of spirits, (...) demons, angels, the devil, and other evil-wishing forces that he thought had haunted and deprived him of a peaceful existence.As is well known from other biographies, Gödel was of a frail physical health and psychic balance, having suffered from his student days on from bouts of depression and ‘nervous’ breakdowns—generally attributed to being overworked—that required at times short hospitalization in a sanatorium and culminating toward the end of his life in a marked deterioration in his mental health that resulted in his starving himself to death out of fear of being poisoned. These events are only mentioned here in passing.The author, who teaches philosophy at the University of Lille, is naturally interested in Gödel's philosophical views as well as in his metaphysical reflections, such as his thoughts about life after death and the survival of an immaterial soul. From the 1940s Gödel kept a philosophical diary that contains 670 handwritten notes, written in the now defunct Gabelsberger shorthand. Having feared controversies whenever his metaphysical/philosophical views went against the spirit of the times, Gödel was never led to a comprehensive publication of these views, preferring to put these notes in the private format of notebooks that are now preserved in his Nachlassat Princeton University. 1 The author had several opportunities to study the two of these notebooks …. (shrink)

Among the aims of the author in this wide-ranging article is to draw attention to the numerous formal sciences which so far have received little scrutiny, if at all, on the part of philosophers of mathematics and of science in general. By the formal sciences the author understands such mathematical disciplines as operations research, control theory, signal processing, cluster analysis, game theory, and so on. First, the author presents a long list of such formal sciences with a detailed discussion of (...) their subject matter and with extensive references to the pertinent literature. Turning to the nature of the formal sciences, the author states that “the formal sciences, though they arose in most cases out of engineering requirements, are sciences and can be pursued without reference to applications”. It is argued, through a wealth of examples, that in a great number of cases the formal sciences permit the attainment of provable certainty about actual parts of the world. As Franklin puts it, “knowledge in the formal sciences, with its proofs about network flows, proofs of computer program correctness, and the like, gives every appearance of having achieved the philosophers’ stone; a method of transmuting opinion about the base and contingent beings of this world into the necessary knowledge of pure reason.” Franklin clearly distinguishes between certainty and necessity: “ g¤g¤g what the mathematician in offering is not, in the first instance, absolute certainty in principle, but necessity. This is how his assertion differs from one made by a physicist. A proof offers a necessary connection between premises and conclusion. One may extract practical certainty from this g¤g¤g but this is a separate step.” Though Franklin explicitly states that there is a gap between necessity and certainty as one passes from mathematical reasoning to applications, the main thrust of the article consists in arguing that the gap is considerably smaller than generally claimed.. (shrink)