Constructivism (mathematics)

Abstract: 7. The Topology of Metric Spaces. 8. Algebra. 9. Finite Type Arithmetic and Theories of Operators. 10. Proof Theory of Intuitionistic Logic. 11. The Theory of Types and Constructive Set Theory. 12. Choice Sequences. 13. Semantical Completeness. 14. Sheaves, Sites and Higher Order Logic. 15. Applications of Sheaf Models. 16. Epilogue. Bibliography. Index.

Abstract: Sets and orderings: sets functions relations and orderings more about sups and infs sets of sets - filters topologies constructivism and choice nets and convergences. Algebra: elementary algebraic systems concrete categories the real numbers linearity convexity boolean algebras logic and intangibles. Topology and uniformity: toplogical spaces separation and regularity axioms compactness uniform spaces metric and uniform completeness Baire theory positive measure and integration. Topological vector spaces: norms normed operators generalized Riemann integrals Frechet derivatives metrization of groups and vector spaces barrels and other features of TVSs duality and weak compactness vector measures initial value problems.

Abstract: Publisher Summary In this chapter, constructive is meant as finitism, constructive recursive analysis, and intuitionism. The chapter discusses constructivism as the study of a special area in the whole of mathematical experience. The principal aspects of constructivism discussed in the chapter are the role of logic and abstract concepts; reductions to quantifier-free statements; and interpretation of the logical operations; intensional aspects; the validity of Church's thesis; continuity axioms, the possibility of a theory of continuous; usefulness of the subjectivistic interpretation; and the quest for explicit definability. The chapter also discusses Markov's schema; connection between validity for intuitionistic predicate logic and mathematical assumptions. It presents the existence of classical counterparts to problems of constructive mathematics and systematic procedures for constructivizing classical theorems.

Abstract: This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ① (this symbol is called grossone). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts (Gutman et al. in Found Sci 22(3):539–555, 2017; Gutman and Kutateladze in Sib Math J 49(5):835–841, 2008; Kutateladze in J Appl Ind Math 5(1):73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics.