Mathematics as a Numerical Language

Publisher Summary This chapter discusses the role of mathematics as a numerical language. Constructive mathematics describes or predicts the results of certain finitely performable, albeit hypothetical, computations within the set of integers. Brouwer's intuitionism contains elements that are extremely dubious; free choice sequences and allied concepts admit no ready numerical interpretation. The numerical content of intuitionistic mathematics is diluted by over-reliance on negativistic techniques. The role of negation in predictive mathematics is philosophically secure, if the negative statements having numerical content exist. The chapter provides examples from probability theory, from algebra, and from elementary algebraic topology. Elementary algebraic topology should be constructive, but the definition of the singular co-homology groups gives trouble. The most urgent foundational problem of constructive mathematics concerns the numerical meaning of implication. Constructivists have accepted Brouwer's definitions of the mathematical connectives and quantifiers, implication in particular.