Principia Mathematica(ID:6961/pri008)

R & W symbolic treatment of maths 

Russell and Whitehead's mathematical formalistic language expressed with great clarity. Based largely on the Peano synax, with extensions, with logical system based on Frege. Provides the intellectual formalism for types.

Ramified Type Theory - "whatever involves all of a collection must not be one of a collection"

The orders of types were eliminated by Ramsay.
Related languages
Boole => Principia Mathematica   Extension of
Cantor set theory => Principia Mathematica   Incorporated some features of
Frege => Principia Mathematica   Strong Reaction to
Peano => Principia Mathematica   Extension of
Principia Mathematica => Lincos   Extension of
Principia Mathematica => New Foundations   Incorporated some features of
Principia Mathematica => Simplified Type Theory   Simplification of
References:
  • Russel, Bertrand "Mathematical Logic as based on a Theory of Types" in American Journal of Mathematics, 30 pp222-262, 1908 view details
  • Whitehead, A. N., and Russell, B. Principia Mathematica. Cambridge, 1910, 1912, 1913. view details
  • Fairouz Kamareddine, Twan Laan and Rob Nederpelt "A History of Types in Logic and Mathematics" view details Abstract: In this talk, we outline the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica (1910-1912) and Church's simply typed lambda-calculus of 1940. We first argue that types have always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's Grundgesetze der Arithmetik for which Russell derived his famous paradox that led him to introduce the first theory of types, the Ramified Type Theory rtt. We discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt upon which Church's simply typed lambda calculus is based.  
          in Workshop on History of Logics, Types and Rewriting Heriot-Watt University, Edinburgh Tuesday 5 December 2000 view details
  • Grattan-Guiness, I. "Mathematics and symbolic logics: an uneasy relationship of the 20th century" view details Abstract: Symbolic logics tend to be too mathematical for the philosophers and too philosophical for the mathematicians. I shall consider this unease as it has manifested during the century, with some emphasis upon Principia mathematica (1910-1913) of Whitehead and Russell.
          in Workshop on History of Logics, Types and Rewriting Heriot-Watt University, Edinburgh Tuesday 5 December 2000 view details