Logicism

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The Logicism or logicist program describes a specific position in the philosophy of mathematics , which was also in other philosophical disciplines in the first half of the 20th century. Influential. The approach was first proposed by Frege Gottlob formulated late 19th century and says, in essence, that the mathematics on logic attributed leaves.

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

Old logicism

Roughly, logicism can be split into two sub-items:

1. All mathematical truths must be traceable to a fixed number of axioms by means of definitions with strict evidence .
2. These axioms themselves must be evident, logical truths; In other words, in Frege's words, they must be “neither capable nor needy of proof” .

To 1.) With the first demand Frege wants to satisfy the need for a scientific foundation of mathematics. Up until Frege's day it was assumed that there were certain unprovable mathematical truths, but hardly any attempt was made to state them and how the remaining truths were derived from them. (A significant exception and at the same time Frege's role model is Euclid with his work “ The Elements ”). In order to carry out his project, however, Frege must first define the concept of proof precisely. In the course of this he created the first fully explicit formal language as well as the predicate logic that is still in use today . With this set of instruments, Frege succeeds in defining the concept of number and, on the basis of this, proving elementary arithmetic theorems (such as "1 + 1 = 2"), which until then had been considered unprovable.

To 2.) Frege had based his system on a number of axioms to which he could ascribe the status of self-evident truths. In this system of axioms, however, Bertrand Russell discovered a contradiction (the so-called Russell antinomy ) in 1902 . As a result, Frege turns away from logicism, disappointed. In the following years a number of so-called “axiomatic set theories” emerged, such as Russell's own type theory or the Zermelo-Fraenkel set theory . Although these implement the requirement of an axiomatic foundation of mathematics, at the same time they always contain axioms that cannot be considered logically evident. A particularly clear example is the axiom of infinity , which requires that there are an infinite number of objects (numbers). According to Frege, such a statement should not have been made axiomatically, but should have been proven by logical means. Gödel's Incompleteness Theorem proved that every consistent, sufficiently powerful mathematical system contains unprovable but true theorems, and it finally refuted Frege's position.

Although Frege's logicism must be viewed as a failure, particularly because of the second above-mentioned requirement, the first requirement has proven to be extremely fruitful. The tools created by Frege to carry out the program gave a decisive impetus to modern logic, with the development of set theories a new branch of mathematics was founded.

Neo-Logicism

Crispin Wright's neo-logicism is based on Frege's theorem .

literature

• Franz von Kutschera, 1989: Gottlob Frege: An introduction to his work , Walter de Gruyter, Berlin; New York, ISBN 3-11-012129-8 .
• Paolo Mancosu, 1998: From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s , Oxford University Press, New York, NY, ISBN 0-19-509632-0 .
• Bertrand Russell, 1912: The Problems of Philosophy (with Introduction by John Perry 1997), Oxford University Press, New York, NY, ISBN 0-19-511552-X .
• Howard Eves, 1990: Foundations and Fundamental Concepts of Mathematics Third Edition , Dover Publications, Inc, Mineola, NY, ISBN 0-486-69609-X .
• I. Grattan-Guinness, 2000: The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and The Foundations of Mathematics from Cantor Through Russell to Gödel , Princeton University Press, Princeton NJ, ISBN 0-691-05858-X .