# Logic and Modern Philosophy of Mathematics

## Set Theory And The Rise Of Mathematical Logic

Especially from the 1820s with the Frenchman Augustin Louis Cauchy, mathematicians had become more sensitive to the need for rigor in proofs, carefully stating assumptions and definitions and formulating theorems in conditional form. Cauchy's approach was refined from the 1860s by the lectures at Berlin University of Karl Weierstrass. The main context was the calculus and its extension into mathematical analysis. Two consequences are of special import.

Axiomatized logics are far removed from the normal use of logic. In teaching and practice "natural deduction" has recently gained much popularity: state the "local" premises, and deduce conclusions from them using rules for introducing and eliminating connectives. The name was introduced in the 1930s by Hilbert's follower Gerhard Gentzen.

First, in the early 1870s the German Georg Cantor began to develop set theory, initially treating points and numbers but later as a general treatment of collections of things; he even claimed that sets enabled him to define the natural numbers, and by implication to reduce mathematics to sets. His theory differed from the traditional part-whole theory that dates back to the Greeks and that was used by, for example, the algebraic logicians; for Cantor distinguished membership of objects in a set from the inclusion of subsets of objects within it. More controversially, he also incorporated a mathematical theory of the "actual infinite," showing that infinities came in different sizes.

Second, from the 1880s the Italian mathematician Giuseppe Peano began to symbolize as much as possible not only the notions of mathematical analysis (including set theory) but also the logical connectives and predicates with quantification. Thereby he launched "mathematical logic" (his name, in the sense usually adopted today). With an impressive school of followers he came to handle a wide range of mathematical theories this way, and enchanted the young Englishman Bertrand Russell, who came across him in 1900. Russell quickly added a logic of relations to mathematical logic, and converted Cantor's claim that set theory could ground mathematics into the "logicist" thesis (a name provided later by Rudolf Carnap) that mathematical logic could ground sets and thereby (much) mathematics. Russell also found that the German Gottlob Frege had already asserted this thesis for arithmetic and some parts of mathematical analysis, though in a Platonist spirit very different from his own empiricism.

So much for the good news. The bad arrived soon afterwards, in the form of a paradox now named after Russell: a certain set was a member of itself if and only if it was not. Paradoxes are at least a nuisance in mathematical theories: when the theory in question encompasses logic itself, their presence is a disaster. Eventually Russell produced an articulation of logicism in Principia Mathematica (1910–1913), written with Alfred North Whitehead; but his and other paradoxes were avoided rather than solved, and by an unwieldy and epistemologically questionable "theory of types": the ensemble of objects was divided into individuals, sets of individuals, sets of sets of individuals, …, ordered pairs of individuals, …, and so on, and membership was severely restricted.