# Living apart: algebraic logic versus mathematical logic, 1850-1910

## Ivor Grattan-Guinness

(Middlesex University)

The algebraic logicians, largely working out from George Boole and Augustus De Morgan, adopted practices from algebras of their time. They stated laws (such as commutativity) and stressed duality; for theories of collections they drew upon the part-whole method. Content was even similar: Boole's algebra of logic was modelled upon different operators, while De Morgan's logic of relations closely followed functional equations. By and large (especially from Boole) they sought consequences from premises, omitting details of deriviation. Both strands were united in their principal successors, Charles S. Peirce and Ernst Schroeder. They often saw logic as applied mathematics.

The mathematical logicians, especially Giuseppe Peano and his school, and then their British followers Bertrand Russell and A.N. Whitehead, were inspired by mathematical analysis; so their theory of collections was Cantorian, with membership and inclusion explicitly distinguished. They emphasised axioms and definitions and laid out derivations in detail. They hoped to embrace "all" or parts of mathematics within their conception, making mathematics a sort of applied logic. Gottlob Frege was similar in some ways, but he did not gain a wide reputation.

No definitive position over logic and foundations of mathematics emerged during the 1900s, although the mathematical traditions rather eclipsed the algebraic. At that time emerged also (and none too coherently) the metamathematics of David Hilbert, and also model theory in the USA with Edward Huntingdon and others. Further, the paradoxes of set theory came to be recognised as serious issues for all approaches.

**Tuesday 19th October 1993, 14:30**

Seminar Room 322

Department of Computer Science