There is a noticeable general difference between the sciences and mathematics on the one hand, and the humanities and social sciences on the other. In the former, the factors of integrity tend to dominate more over the factors of ideology. It’s not that scientists are more honest people. It’s just that nature is a harsh taskmaster. You can lie or distort the story of the French Revolution as long as you like, and nothing will happen. Propose a false theory in chemistry, and it’ll be refuted tomorrow.
By Noam Chomsky (via indizombie)
In mathematics, an algebraic set is the set of solutions of a system of polynomialequations. Algebraic sets are sometimes also called algebraic varieties, but normally an algebraic variety is an irreducible algebraic set, i.e. one which is not the union of two other algebraic sets. Algebraic sets and algebraic varieties are the central objects of study in algebraic geometry. The word “variety” is employed in the sense which is similar to that of manifold; the difference is that a variety may have singular points, while a manifold may not. In the Romance languages, both varieties and manifolds are named by the same word, a cognate of the word “variety”.
Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex coefficients (an algebraic object) is determined by the set of its roots (a geometric object). Generalizing this result, Hilbert’s Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specifity of algebraic geometry among the other subareas of geometry.
The Twisted Cubic, pictured above, is an algebraic variety.
I’m going to be really naughty here, and just evaluate the integral over its infinite domain without doing any limits. A pure mathematician would feel very strongly about this.
By Lecturer for a course on series and transforms. (via mathprofessorquotes)