With the benefit of hindsight, we can see that Couturat was in broad agreement with the logicism of Russell and Whitehead, while Poincar?anticipated Brouwer's intuitionism.
22.
Dedekind's path to logicism had a turning point when he was able to reduce the theory of real numbers to the rational number system by means of set theory.
23.
Kant's often brief remarks about mathematics influenced the mathematical school known as intuitionism, a movement in philosophy of mathematics opposed to formalism, and the logicism of Frege and Bertrand Russell.
24.
Kurt G�del's incompleteness theorem is sometimes alleged to undermine logicism because it shows that no particular axiomatization of mathematics can decide all statements, although such theorem itself is based in logic.
25.
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic.
26.
In an observation pertinent to Russell's brand of logicism, Perry remarks that Russell went through three phases of realism-- extreme, moderate and constructive ( Perry 1997 : xxv ).
27.
Hilbert dismissed logicism as a " false path " : " Some tried to define the numbers purely logically; others simply took the usual number-theoretic modes of inference to be self-evident.
28.
Philosophical rationalism in its most extreme form is the doctrine that knowledge can ultimately be founded on pure reason, while logicism is the doctrine that mathematical concepts, among others are reducible to pure logic.
29.
Pursuing both Bertrand Russell's logical atomism deconstructing language into supposedly elementary parts and Russell's logicism reducing swaths of mathematics to symbolic logic, logical positivists envisioned both Carnap of the most flagrant " pseudostatements ".
30.
However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which mathematical proofs are Russell's logicism, which defines proof simply in terms of " formal " validity.
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