Generality of algebra
Encyclopedia
In the history of mathematics, the generality of algebra is phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler
and Joseph Lagrange
. According to Koetsier, the generality of algebra principle assumed, roughly, that the algebraic rules that hold obviously for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid. As a consequence, 18th century mathematicians believed that they could derive meaningful results by applying the usual rules of algebra and calculus that hold for finite expansions even when manipulating infinite expansions. In works such as Cours d'Analyse, Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis
.
An example is Euler's derivation of the series
for
. He first evaluated the identity
at
to obtain
The infinite series on the right hand side of this expression diverges for all real
. But nevertheless integrating this term-by-term gives , an identity which is known to be true by modern methods.
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
and Joseph Lagrange
Joseph Lagrange
Count Joseph Lagrange was a French soldier who rose through the ranks and gained promotion to the rank of general officer during the French Revolutionary Wars, subsequently pursuing a successful career during the Napoleonic Wars and winning promotion to the top military rank of General of Division....
. According to Koetsier, the generality of algebra principle assumed, roughly, that the algebraic rules that hold obviously for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid. As a consequence, 18th century mathematicians believed that they could derive meaningful results by applying the usual rules of algebra and calculus that hold for finite expansions even when manipulating infinite expansions. In works such as Cours d'Analyse, Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
.
An example is Euler's derivation of the series
for
. He first evaluated the identityat
to obtainThe infinite series on the right hand side of this expression diverges for all real
. But nevertheless integrating this term-by-term gives , an identity which is known to be true by modern methods.