Frege, Friedrich Ludwig Gottlob (1848–1925)
German mathematician and philosopher who, inspired by the earlier work of
Leibniz, virtually founded the modern discipline
of mathematical logic. In Die Grundlagen
der Arithmetik (1884), he used set theory
to define the cardinal number of
a given class as the class of all classes that are similar (i.e. can be
placed in a one-to-one correspondence)
to the given class. In Grundgesetze der Arithmetik (2 vols., 1893
and 1903), Frege began attempting to build up mathematics from arithmetic
and symbolic logic on a rigorous and contradiction-free basis. When the
second volume was in the process of being printed, Bernard Russell
pointed out a paradox in Frege's work. The paradox, which became known as
Russell's paradox, stems from the
question: "Is the class of all classes that are not members of itself a
member of itself or not?" The question leads to a contradiction and cannot
be resolved. Frege was thus forced to admit that the foundation of his reasoning
was worthless. As he stated at the end of his work, "A scientist can hardly
encounter anything more undesirable than to have the foundation collapse
just as the work is finished. I was put in this position by a letter from
Mr. Bertrand Russell when the work was almost through the press."
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