Treatise on Universal Algebra
A Treatise on Universal Algebra With Applications ( A Treatise on Universal Algebra with Applications ) is a work of mathematician and philosopher Alfred North Whitehead from 1898. This book took Whitehead, various algebraic systems in accordance with uniform principles of uniform terms and in a to represent uniform notation. The main systems that he investigated for this were the quaternions of Hamilton , the theory of the expansion of Grassmann and the Boolean algebra of George Boole . Benjamin Peirce had already formulated the goal of achieving a uniform method of representation in the contemporary development of the various algebraic structures . Edward Vermilye Huntington commented on this work: "The algebra designed by Boole was further developed by Schröder and perfected by Whitehead."
Structure of the plant
Whitehead had already started work on his first major work in 1891. In the foreword, Whitehead thanks his friend Andrew Russell Forsyth and his student Bertrand Russell for contributions to the discussion and suggestions for corrections. Because of the Treatise, Whitehead was elected to the Royal Society in 1903 . The work is divided into seven books.
- Book I: Principles of algebraic symbolism, 1--32
- Book II: The algebra of symbolic logic, 33-116
- Book III: Positional manifolds, 117-168
- Book IV: Calculus of extension, 169-269
- Book V: Extensive manifolds of three dimensions, 270-346
- Book VI: Theory of metrics, 347-502
- Book VII: Application of the calculus of extension to geometry, 503-575
In the first book, Whitehead set out the general principles and definitions of his work. The algebra of symbolic logic is largely independent of the rest of the work. Whitehead here refers to Boole, Schröder and Venn . Book three deals with the principles of addition and the general concept of space. Book four deals with the theory of expansion. In the fifth book, the theory of expansion is linked with the manifolds in three-dimensional space as a theory of force. Book six deals with the application of the theory of extension to non-Euclidean geometry based on Whitehead's teacher Arthur Cayley . In addition to Cayley, Whitehead refers to the work of Felix Klein , with whom he was personally acquainted. The seventh book finally shows the application of the theory of extension in normal, three-dimensional Euclidean space.
Whitehead's mathematical writings
- Treatise on Universal Algebra with applications , Cambridge University Press, Cambridge 1910 ( online )
- Sets of operations in relation to groups of finite order , Proc. Roy. Soc. London, 64 (1898-1899), 319-320
- Memoir on the algebra of symbolic logic , Am. J. of Math., 23 (1901), 139-165, 297-316
- On cardinal numbers , Am. J. of Math., 24 (1902), 367-394
- The Axioms of Projective Geometry , Cambridge University Press 1906
- The Axioms of Descriptive Geometry , Cambridge University Press 1907
- Introduction to Mathematics . London: Williams & Norgate 1911
- with Bertrand Russell : Principia Mathematica , three volumes, Cambridge University Press, 1910, 1912, 1913; 2nd edition 1925
- The Philosophy of Mathematics . Science Progress, 5: 234-239 (1910)
- Indication, Classes, Numbers, Validation , in: Mind, New Ser., Vol. 43, No. 171 (July 1934), 281-297. Corrigenda, 543
literature
- A. Dawson: Whitehead's Universal Algebra. In: Michel Weber , W. Desmond (eds.): Handbook of Whiteheadian Process Thought, Volume 2, Ontos, Frankfurt 2008, 67–86
Individual evidence
- ↑ Benjamin Peirce: 'Linear associative algebra', American journal of mathematics, 4 (1881), 97-215. Edited as a book by his son Charles Sanders Peirce , New York 1882. [Original as typescript by B. Peirce 1870]
- ^ Edward Vermilye Huntington: originated by Boole, extended by Schröder, and perfected by Whitehead , in: ders. “New Sets of Independent Postulates for the Algebra of Logic, with Special Reference to Whitehead and Russell's Principia Mathematica”, Transactions of the American Mathematical Society 35 (1933), 274-304, here 278