The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic ( Hellman 1981, p . 451 468 ).
32.
Frege in particular sought to demonstrate ( see Gottlob Frege, The Foundations of Arithmetic, 1884, and Logicism in Philosophy of mathematics ) that arithmetical truths can be derived from purely logical axioms and therefore are, in the end, logical truths.
33.
This started a period of expansion of logicism, with Dedekind and Frege as its main exponents, which however was brought to a deep crisis with the discovery of the classical paradoxes of set theory ( Cantor 1896, Zermelo and Russell 1900 1901 ).
34.
In the " Principia Mathematica ", Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic.
35.
The importance to logicism of the construction of the natural numbers derives from Russell's contention that " That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected " ( 1919 : 4 ).
36.
Russell continued to defend logicism, the view that mathematics is in some important sense reducible to logic, and along with his former teacher, Alfred North Whitehead, wrote the monumental " Principia Mathematica ", an axiomatic system on which all of mathematics can be built.
37.
Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism : unlike geometry, arithmetic was to be shown to have no basis in " intuition ", and no need for non-logical axioms.
38.
Gottlob Frege began the program of reducing mathematics to logic, continued it with Bertrand Russell, but lost interest in this logicism, and Russell continued it with Alfred North Whitehead in their monumental " Principia Mathematica ", inspiring some of the more mathematical logical posivists, such as Hans Hahn and Rudolf Carnap.
39.
This has not been the definition taken in this article, but the idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to logicism in mathematics ( 19th and 20th centuries ) and the advent of the influence of mathematical logic on philosophy.
40.
About same time as Carnap ( 1929 ), but apparently independently, Fraenkel ( 1928 ) used the word : " Without comment he used the name'logicism'to characterise the Whitehead / Russell position ( in the title of the section on p . 244, explanation on p . 263 ) " ( G-G 2002 : 269 ).
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