Editing Philosophy of mathematics (section)

===Logicism===
{{Main|Logicism}}
[[Logicism]] is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic.<ref name=Carnap>[[Rudolf Carnap|Carnap, Rudolf]] (1931), "Die logizistische Grundlegung der Mathematik", ''Erkenntnis'' 2, 91-121. Republished, "The Logicist Foundations of Mathematics", E. Putnam and G.J. Massey (trans.), in Benacerraf and Putnam (1964). Reprinted, pp.&nbsp;41–52 in Benacerraf and Putnam (1983).</ref>{{rp|41}} Logicists hold that mathematics can be known ''a priori'', but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus [[analytic proposition|analytic]], not requiring any special faculty of mathematical intuition. In this view, [[logic]] is the proper foundation of mathematics, and all mathematical statements are necessary [[logical truth]]s.

[[Rudolf Carnap]] (1931) presents the logicist thesis in two parts:<ref name=Carnap/>
#The ''concepts'' of mathematics can be derived from logical concepts through explicit definitions.
#The ''theorems'' of mathematics can be derived from logical axioms through purely logical deduction.

[[Gottlob Frege]] was the founder of logicism. In his seminal ''Die Grundgesetze der Arithmetik'' (''Basic Laws of Arithmetic'') he built up [[arithmetic]] from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts ''F'' and ''G'', the extension of ''F'' equals the extension of ''G'' if and only if for all objects ''a'', ''Fa'' equals ''Ga''), a principle that he took to be acceptable as part of logic.

Frege's construction was flawed. [[Bertrand Russell]] discovered that Basic Law V is inconsistent (this is [[Russell's paradox]]). Frege abandoned his logicist program soon after this, but it was continued by Russell and [[Alfred North Whitehead|Whitehead]]. They attributed the paradox to "vicious circularity" and built up what they called [[ramified type theory]] to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop much of mathematics, such as the "[[axiom of reducibility]]". Even Russell said that this axiom did not really belong to logic.

Modern logicists (like [[Bob Hale (philosopher)|Bob Hale]], [[Crispin Wright]], and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as [[Hume's principle]] (the number of objects falling under the concept ''F'' equals the number of objects falling under the concept ''G'' if and only if the extension of ''F'' and the extension of ''G'' can be put into [[bijection|one-to-one correspondence]]). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.