In the years 1948 1952 Tarski, along with his students Chin and Thompson, created cylindric algebras as an algebraic logic companion to first-order logic, and in 1956 Paul Halmos (1916 2006) introduced polyadic algebras for the same purpose.
An example of such an algebraic encoding is the Lindenbaum-Tarski algebra, a Boolean algebra corresponding to classical propositional logic.
He invented dozens of different systems of logical syntax, including a syntax based on a generalization of de Morgan's relative product operation, an algebraic syntax that mirrored Boolean algebra to some extent, a quantifier-and-variable syntax that (except for its specific symbols) is identical to the much later Russell-Whitehead syntax.
Another general algebraic notion which applies to Boolean algebras is the notion of a free algebra.
If on the other hand we can show how to construct any algebra satisfying the axioms from the algebra of integers, limiting ourselves only to certain algebraic constructions, then by a theorem of Birkhoff applicable to those constructions we can infer that the axiomatization is complete.
Schröder aimed at a general algebraic theory with applications in many mathematical fields, where the algebra of logic was at the core.
Negative numbers, or even the number zero, had not yet been invented, and the notation was cumbersome (try doing multiplication with Roman numerals).Medieval Islamic scholars such as Muhammad al-Khwarizmi and Omar Khayyam also worked on algebraic problems (and gave us words such as "algorithm" and indeed "algebra" itself).
Thus, we will consider here adherents to the algebraic approach who do hold that the propositions yielded by their algebras are complex and have "parts".
When, in the nineteenth century, algebra and analysis took precedence over geometry, the rectilinear coordinate system of algebraic geometry came to be called "Cartesian coordinates" in honor of Descartes' discovery.
Algebraic QFT has its roots in the work of von Neumann on operator algebras, which was developed by him in an attempt to generalize the Hilbert space framework.
EDWARD FRENKEL is a Russian mathematician working in representation theory, algebraic geometry and mathematical physics.
