What was introduced in von Neumann algebras?
Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
Following the Enlightenment's ideas, the reformers looked to the Scientific Revolution and industrial progress to solve the social problems which arose with the Industrial Revolution. Newton's natural philosophy combined a mathematics of axiomatic proof with the mechanics of physical observation, yielding a coherent system of verifiable predictions and replacing a previous reliance on revelation and inspired truth. Applied to public life, this approach yielded several successful campaigns for changes in social policy.