Logicism

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.^[1] Bertrand Russell and Alfred North Whitehead championed this theory fathered by Gottlob Frege. Frege gave up on the project after Russell recognized a paradox exposing an inconsistency in naive set theory. Russell and Whitehead continued on with the project in their Principia Mathematica.^[2] Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms of Zermelo-Fraenkel set theory (or one of its extensions, such as ZFC), which has no known inconsistencies (although it remains possible that inconsistencies in it may still be discovered).

Kurt Gödel's incompleteness theorem is sometimes alleged to undermine Logicism because it shows that if mathematics is consistent, some of its theorems are not derivable; since first-order logic is both consistent and all of its theorems are derivable, mathematics cannot be a simple extension of first-order logic. However, one can argue that the basic spirit of Logicism remains valid, though in a somewhat less powerful sense than was originally thought.

Logicism was key in the development of Analytic philosophy in the twentieth century.

Neo-logicism

Neo-logicism describes a range of views claiming to be the successor of the original logicist program. ^[3] More narrowly, it is defined as attempts to resurrect Frege's programme through the use of Hume's Principle.^[4] Two of the major proponents of neo-logicism are Crispin Wright and Bob Hale.^[5]

See also

• Nominalism
• Mathematical fictionalism

References

1. ^ Logicism
2. ^ "Principia Mathematica" article in the Stanford Encyclopedia of Philosophy.
3. ^ n.dvi
4. ^ PHIL 30067: Logicism and Neo-Logicism
5. ^ http://www.st-andrews.ac.uk/~mr30/papers/EbertRossbergPurpose.pdf

External links

• Logicism

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