Logicism

One of the trends in the foundations of mathematics whose object is to justify mathematics by reducing its initial concepts to those of logic. The idea of reducing mathematics to logic was expressed by G. Leibniz (end of the 17th century). The practical realization of the thesis of logicism was undertaken at the end of the 19th century and the beginning of the 20th century in papers of G. Frege and B. Russell (see [1], [2]). The view of mathematics as part of logic depends on the fact that in an axiomatic system any mathematical theorem can be regarded as an assertion about logical consequences. It just remains to define all the constants that occur in such assertions in terms of logical terms. At the end of the 19th century different forms of numbers, including complex numbers, were defined in mathematics in terms of the natural numbers and operations on them. An attempt to reduce the natural numbers to logical concepts was undertaken by Frege. In Frege's interpretation the natural numbers were the cardinal numbers of certain concepts. However, Frege's system was not free from contradictions. This was clarified when Russell discovered a contradiction in Cantor's set theory (Russell's antinomy) in trying to reduce it to logic. This contradiction stimulated Russell to a reconsideration of the views on logic, which he stated in the form of a ramified theory of types (cf. Types, theory of). However, the construction of mathematics on the basis of the theory of types required the assumption of axioms that would be unnatural to regard as purely logical. One is, for example, the axiom of infinity, which asserts that there are infinitely many individuals, that is, objects of the lowest type. On the whole the attempt to reduce mathematics to logic was not successful. As K. Gödel showed [3], no formalized system of logic can be an adequate basis for mathematics (cf. Gödel incompleteness theorem).

References

[1]	G. Frege, "Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet" , 1–2 , G. Olms, reprint (1962)
[2]	A.N. Whitehead, B. Russell, "Principia Mathematica" , 1–2 , Cambridge Univ. Press (1910–1913)
[3]	K. Gödel, "Ueber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" Monatsh. Math. Phys. , 38 (1931) pp. 173–198
[4]	H.B. Curry, "Foundations of mathematical logic" , McGraw-Hill (1963)
[5]	A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland (1958)