Chapter 19. Logic and Set Theory
Logic has been an important part of western mathematics since the time of Plato.
It also has a long history in other cultures, such as the Hindu and Buddhist culture
(see Vidyabhusana, 1971). Logic became mathematized in the nineteenth century,
in the work of mostly British mathematicians such as George Peacock (1791-1858),
George Boole (1815-1864), William Stanley Jevons (1835-1882), and Augustus de
Morgan, and a few Americans, notably Charles Sanders Peirce.
Set theory was the creation of nineteenth-century analysts and geometers,
prominent among them Georg Cantor (1845-1918), whose inspiration came from
geometry and analysis, mostly the latter. It resonated with the new abstraction
that was entering mathematics from algebra and geometry, and its use by the
French mathematicians Borel, Lebesgue, and Baire as the framework for their the-
ories of integration and continuity helped to establish it as the foundation of all
mathematics.
1. Logic
The mathematization of logic has a prehistory that goes back to Leibniz (not pub-
lished in his lifetime), but we shall focus on mostly the nineteenth-century work.
After a brief discussion of the preceding period, we examine the period from 1847 to
- This period opens with the treatises of Boole and de Morgan and closes with
Godel's famous incompleteness theorem. Our discussion is not purely about logic in
the earlier parts, since the earlier writers considered both logical and probabilistic
reasoning.
1.1. From algebra to logic. Leibniz was one of the first to conceive the idea of
creating an artificial language in which to express propositions. He compared formal
logic to the lines drawn in geometry as guides to thought. If the language encoded
thought accurately, thought could be analyzed in a purely mechanical manner:Then, in case of a difference of opinion, no discussion between two
philosophers will be any longer necessary, as (it is not) between
two calculators. It will rather be enough for them to take pen
in hand, set themselves to the abacus and (if it so pleases, at the
invitation of a friend) say to one another: Let us calculate! [Quoted
by Bochenski, 1961, p. 275]
In another place he wrote:Ordinary languages, though mostly helpful for the inferences of
thought, are yet subject to countless ambiguities and cannot do the
task of a calculus, which is to expose mistakes in inference... This
remarkable advantage is afforded up to date only by the symbols of535