In the Tractatus, it is stated that questions about logical formatting cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of logic, (...) that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)

In his book Gabriele Lolli discusses the notion of proof, which is, according to him, the most important and at the same time the least studied aspect of mathematics. According to Lolli, a theorem is a conditional sentence of the form ‘if T then A’ such that A is a logical consequence of T, where A is a sentence and T is a sentence or a conjunction or set of sentences. Verifying that A is a consequence of T generally involves (...) considering infinitely many interpretations; so it is something which is impossible to do in finite terms. Proofs may serve as ‘shortcuts’ in this respect. A proof is defined by Lolli as any finite argument certifying that A is a consequence of T. A proof is a shortcut in the sense that it spares us considering infinitely many interpretations.The reason for such a very general definition of proof is Lolli's strong belief that mathematics is not a rigid system of explicit rules, but rather a set of tools; as a consequence, there is no prescription as to what a proof should or should not be. Actually, mathematics is historically situated and not timeless, and the history of mathematics is the …. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)