Principia Mathematica

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Title page of a Principia Mathematica edition

Principia Mathematica ("mathematical principles" or "mathematical foundations") is a work in three volumes on the foundations of mathematics by Bertrand Russell and Alfred North Whitehead , first published between 1910 and 1913. The Principia Mathematica represent the attempt to all mathematical To derive truths from a well-defined set of axioms and inference rules ( inference rules of symbolic logic ). On several hundred pages, a repertoire of terms and symbols is presented, which forms the basis for the later derivation of arithmetic. The derivation of mathematics from logic was intended to refute some of the views widespread up to then about the nature of mathematical knowledge, namely that these are neither empirical nor synthetic a priori (the latter had been assumed by Kant), but of a linguistic nature and thus formally logically justifiable, i.e. analytically a priori.

Subject areas covered

The Principia Mathematica only deal with set theory , cardinal numbers , ordinal numbers and real numbers ; In-depth theorems from real analysis are not included, but towards the end of the third volume it becomes clear that all known mathematics can in principle be developed from the formalism presented.

precursor

An important inspiration and basis of the Principia Mathematica is Gottlob Frege's arithmetic from 1893, the basis of which is a set calculus in which Russell discovered Russell's antinomy , which results from the set of all sets that do not contain themselves . He tried to solve this contradiction and other contradictions of naive set theory with his type theory of 1908, which he then made the basis of the Principia Mathematica .

The second important foundation of Principia Mathematica is the formulary (Formulaire) by Giuseppe Peano in the versions of 1897/98 and 1903; from there Russell took over the symbolic notation and many formulas, already in his type theory.

consequences

The Hilbert program tried to decide positively from 1922 onwards the open question of whether this system of axioms and derivation rules is free of contradictions and whether all true propositions can be derived in this way . Logicians who took part generally based on the Principia Mathematica , such as Paul Bernays and Kurt Gödel , who demonstrated consistency and completeness for subsystems. In 1931, however, Gödel proved in his work On formally undecidable theorems of the Principia Mathematica and related systems I. an incompleteness theorem , which showed that this expectation placed in the Principia Mathematica cannot be fulfilled.

Editions and partial translations

  • Russell, Whitehead: Principia Mathematica . Cambridge University Press, 3 volumes, 1910 to 1913, 2nd edition 1925 to 1927, reprint 1962, ISBN 978-0-521-06791-1 , reprint 1997 ISBN 978-0-521-62606-4
    • The first edition had 666, 742 and 491 pages and appeared in 750 (Volume 1) and 500 copies each (Volume 2, 3). The second edition had 674, 772 and 491 pages.
  • Russell, Whitehead: Introduction to Mathematical Logic , Munich, Berlin: Drei Masken 1932 (translation of the introductions)
  • Alfred North Whitehead, Bertrand Russell, Kurt Gödel: Principia Mathematica. Preface and introductions. Suhrkamp 2008. ISBN 978-3-518-28193-2

literature

  • A. Garciadiego: Bertrand Russell and the origin of set-theoretic paradoxes , Birkhäuser 1992
  • Ivor Grattan-Guinness : AN Whitehead and Bertrand Russell, Principia Mathematica, First Edition (1910-1913) , in: I. Grattan-Guinness (Ed.), Landmark writings in western mathematics (1640-1940) , Elsevier, 2005, p 784-794
  • I. Grattan-Guinness: The search for mathematical roots, 1870-1940. Logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel , Princeton University Press 2000
  • Esther Ramharter, Georg Rieckh: Principia Mathematica in a nutshell . öbvhpt 2007. ISBN 978-3-209-05547-7
  • F. Rodriguez-Consuegra: The mathematical philosophy of Bertrand Russell: origin and developments , Birkhäuser 1991

Web links

Individual evidence

  1. ↑ Russell: Mathematical logic as based on the theory of types (PDF; 1.9 MB), in: American Journal of Mathematics 30 (1908), pp. 222-262.