See also the SEP entry on the algebra of logic tradition.
I took Algebra 1 four times, because I failed it four times, and I still don’t know what algebra even means.
For information on specific branches of algebra, see elementary algebra, linear algebra, and modern algebra.
Algebra is of philosophical interest for at least two reasons.
Universal algebra is the next level of abstraction after abstract algebra.
Algebra is a branch of mathematics sibling to geometry, analysis (calculus), number theory, combinatorics, etc.
MAL shows more clearly than LT how closely Boole’s algebra of logic is based on the common algebra plus idempotent class symbols.
But just barely—adding any new equation to Boolean algebra (without introducing new operations) axiomatizes the inconsistent algebra.
Peacock believed that in order for symbolical algebra to be a useful subject its laws had to be closely related to those of arithmetical algebra.
Such conceptions gradually evolved into “nonstandard” abstract algebras such as quaternions, vectors, linear algebra, and Boolean algebra itself.
A central result in this area is the theorem that a lattice arises as the lattice of subalgebras of some algebra if and only if it arises as the lattice of congruences on some algebra.
Historically the term modern algebra came from the title of the first three editions of van der Waerden’s classic text of that name, renamed simply “Algebra” for its fourth edition in 1955.
It was natural for Boole to want to solve equations in his algebra of logic since this had been a main goal of ordinary algebra, and had led to many difficult questions (e.g., how to solve a 5th degree equation).
Pursuing this direction of research led Stone to ask about the structure of an arbitrary Boolean algebra, a question that he answered by proving that every Boolean algebra is isomorphic to a Boolean algebra of sets.
According to Víctor Sánchez Valencia, the tradition that originated with Boole came to be known as the algebra of logic since the publication in 1879 of Principles of the Algebra of Logic by Alexander MacFarlane (see Sánchez Valencia 2004: 389).
George Boole, (born November 2, 1815, Lincoln, Lincolnshire, England—died December 8, 1864, Ballintemple, County Cork, Ireland), English mathematician who helped establish modern symbolic logic and whose algebra of logic, now called Boolean algebra, is basic to the design of digital computer circuits.
actualism | algebra of logic tradition | Boolean algebra: the mathematics of | logic: algebraic propositional | logic: classical | logic: deontic | logic: hybrid | logic: intensional | logic: many-valued | logic: modal | logic: relevance | logic: temporal | modality: medieval theories of | possible objects | possible worlds | Prior, Arthur
Recent work showed that alternative-based systems don’t need to abandon the elegant uniform algebraic perspective of the classical analysis: while classical entailment gives rise to a Boolean algebra, inquisitive entailment gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators (Roelofsen 2013).
Given the connection between sample spaces and event algebras as defined above, one sees that the event algebra of a refined sample space is larger than the original event algebra: i.e., it contains all the original projectors and others in addition; conversely, a coarsening leads to a new event algebra which is a subset of the old event algebra.
In his Treatise on Universal Algebra, Whitehead took a generalized algebra—called ‘universal algebra’—to be the most appropriate tool for this study or investigation, but after meeting Giuseppe Peano during the section devoted to logic at the First International Congress of Philosophy in 1900, Whitehead and Russell became aware of the potential of symbolic logic to become the most appropriate tool to rigorously study mathematical patterns.
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In his Treatise on Universal Algebra Whitehead took a generalized algebra—called universal algebra—to be the most appropriate tool for this study or investigation but after meeting Giuseppe Peano during the section devoted to logic at the First International Congress of Philosophy in 1900 Whitehead and Russell became aware of the potential of symbolic logic to become the most appropriate tool to rigorously study mathematical patterns