Principia Mathematica

Principia Mathematica ( " Mathematical Principles " and " Fundamentals of Mathematics " ) is a work in three volumes on the Foundations of Mathematics by Bertrand Russell and Alfred North Whitehead, first published between 1910 and 1913. Principia Mathematica represent the attempt, all mathematical derive truths from a well-defined set of axioms and inference rules ( inference rules of symbolic logic ), as proposed by the Hilbert program. On several hundred pages a repertoire of words and symbols starts by noting that forms the foundation for later derivation of arithmetic. The derivation of mathematics from logic refuted some hitherto widespread beliefs about the nature of mathematical knowledge. One can see it as a proof that they are neither empirically nor were synthetic a priori (the latter Kant had assumed ), but linguistic in nature and thus formally logically justifiable, ie analytic a priori.

Area of ​​interest

The Principia treat only the quantity theory, the cardinal numbers, the ordinal numbers and the real numbers; deeper sets of real analysis are not included, but towards the end of the third volume is clear that the entire known mathematics can in principle be developed from the presented formalism.


An important inspiration and basis of Gottlob Frege's Principia Mathematica forms arithmetic of 1893, whose base is a lot of calculus in which Russell discovered Russell's antinomy itself. From the set of all sets that do not contain themselves results This contradiction and other contradictions of naive set theory he solved by its type theory from 1908 that became the basis of Principia Mathematica.

Another important foundation of Prinicipia Mathematica is the formulary ( Formulaire ) by Giuseppe Peano, as amended in 1903; from her Russell took the symbolic notation and many formulas (already in his theory of types ).


However, it was initially unclear whether this system of axioms and rules of inference is consistent and whether all true propositions could be derived in this way. That this is not possible, showed a few years later Kurt Gödel with his incompleteness theorem, which he expounded in his work About formally undecidable propositions of Principia Mathematica and related systems I..