Talk:History of type theory

I created this page from the "type theory" page and moved some of the talk items from its talk page to here.

History section

This has only scratched the surface; it could eventually be spun off into its own article, I suppose. I've hidden the following text, because I don't have the (re)sources (e.g. Quine, Tarski) to go further:

Quine reports that Ramsey 1931 urged the "dropping of the ramification and the axiom of reducibility"; Quine opines that "Russell's failure [was] due to this failure to distinguish between propositional functions as notations and propositional functions as attributes and relations.[Quine's commentary before Russell 1908 in van Heijenoort 1967:151-152. Ramsey 1931 The foundations of mathematics and other logical essays, edited by Richard Bevan Braithwaite (Paul, Trench, Trubner, London; Harcourt, Breace, New York); reprinted 1950 (Humanities Press, New York)]."

This is from Hans Riechenbach (1947, reprinted 1980) Elements of Symbolic Logic, Dover Publications, Inc, New York, ISBN:0-486-24004-5

"Although Russell on another occassion,³ suggested the extension of his theory [type theory + axiom of reducibility - ramified types] to a theory of levels of language, this extension ws actually carried through by Ransey⁴, to whom we own the distinction of logical and semantical antinomies, and by Tarski⁵ and Carnap⁶. The ramified theory of types was then dropped¹. An interesting attempt to replace the theory of types by weaker formtion rules was constructed by Quine², whose system employs certain ideas of von Neumann.

³ In his introduction to L. Wittgenstein, Tractatus Logico-Philosophicus, Harcourt, Brace, New York, 1922, p. 23

⁴ F.P. Ramsey, The Foundations of Mathematics, Harcourt, Brace, New York, 1931, p. 20

⁵ A. Tarski, 'Der Wahrheitsbegriff in den Formalisierten Sprachen,' Studia Philosophica, Leopoli, 1935

⁶ R. Carnap, Logical Syntax of Language, Harcourt, Brace, New York, 1937 (German edition, Vienna, 1934)

¹ Cf. B. Russell in his Introduction to the second edition of the Principles of Mathematics, Norton, New York, 1938 [ sic! ]

² W. Quine, Mathematical Logic, New York, 1940, Norton, § 24 and § 29.

Observe Riechenbach's bizarre out-of-sequence history -- Russell had published his 2nd edition to Principia Mathematica in 1927, not 1938. And he has dropped the ramified theory by 1910-1913 with his axiom of reducibility augmented with the "matrix" notion, which he further augments in the 1927 edition (after being exposed to Wittgenstein, cf his references on page xlv-xlvi). He drops the axiom of reducibility but at the loss of "infinite well ordering" as he states at the very end of his 1927 introduction (page xlv). Bill Wvbailey (talk) 16:36, 21 September 2009 (UTC) — Preceding unsigned comment added by Mdnahas (talk • contribs)