Logicism

Logicism or the logicist program refers to a specific position in the philosophy of mathematics. Well as in other philosophical subdisciplines in the first half of the 20 yrs was influential. The approach was first formulated by Gottlob Frege end of the 19th century, and says, in essence, that the mathematics can be traced back to the logic.

The counter-position to the theory of logicism is that it is the logic is a branch of mathematics, so the math is the more fundamental. This position was implicitly represented by the pioneers of mathematical logic in the 19th century, Georg Cantor and George Boole.

Age logicism

Broadly, the logicism can split into two positions:

For 1 ) The first requirement Frege wants to satisfy the need for a scientific foundation of mathematics. Up to Frege's time it was assumed that there are certain unprovable mathematical truths, but had little serious attempt to define and represent these as deriving the other truths of these. ( A significant exception while Frege's example is Euclid with his book " The Elements" ). To carry out his plan, Frege must grasp the notion of proof precise but once. As part of this he creates the first fully explicit formal language and predicate logic, today still in use. With this instrument it is possible Frege to define the concept of number and, starting from elementary arithmetic sentences ( such as " 1 1 = 2"), which had hitherto kept for unprovable to prove.

Ad 2 ) Frege 's system had applied a set of axioms, which he could ascribe the status of self - evident truths. In this axiom system, however, a contradiction is discovered ( the so-called Russell's antinomy ) by Bertrand Russell in 1902. Frege then turns away in disappointment from the logicism. In the following years caused a number of so-called " axiomatic set theories " as Russell 's type theory or the Zermelo -Fraenkel set theory. Although this set to the requirement of an axiomatic foundation of mathematics, at the same time, however, always contain axioms that can not be regarded as logically evident. A particularly clear example is the axiom of infinity, which requires that there are infinitely many objects ( numbers). According to Frege's ideas, such a statement would not axiomatic set, but must be proved by logical means. Gödel's incompleteness theorem proved that every consistent, sufficiently powerful mathematical system contains unprovable but true statements and refuted Frege's position permanently.

Although Frege's logicism must therefore be regarded as a failure in particular because of the second requirement mentioned above, but the first requirement has proved to be extremely fruitful. The tools created by Frege to implement the program have given a decisive impetus of modern logic, with the development of lots of theories, a new branch of mathematics was established.

Neo - logicism

The Neo - logicism of Crispin Wright is based on Frege's theorem.