Mathematics as a Numerical Language
作者： Errett Bishop
DOI: 10.1016/S0049-237X(08)70740-7
关键词: Interpretation (logic) 、 Algebraic topology (object) 、 Negation 、 Intuitionism 、 Constructive analysis 、 Constructive 、 Probability theory 、 Algebra 、 Meaning (philosophy of language) 、 Computer science
摘要: Publisher Summary This chapter discusses the role of mathematics as a numerical language. Constructive describes or predicts results certain finitely performable, albeit hypothetical, computations within set integers. Brouwer's intuitionism contains elements that are extremely dubious; free choice sequences and allied concepts admit no ready interpretation. The content intuitionistic is diluted by over-reliance on negativistic techniques. negation in predictive philosophically secure, if negative statements having exist. provides examples from probability theory, algebra, elementary algebraic topology. Elementary topology should be constructive, but definition singular co-homology groups gives trouble. most urgent foundational problem constructive concerns meaning implication. Constructivists have accepted definitions mathematical connectives quantifiers, implication particular.
elsevier.com
sci-hub.se 参考文章(4)
Von Kurt Gödel, ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES Dialectica. ,vol. 12, pp. 280- 287 ,(1958) , 10.1111/J.1746-8361.1958.TB01464.X
Gabriel Stolzenberg, Uniform approximation on smooth curves Acta Mathematica. ,vol. 115, pp. 185- 198 ,(1966) , 10.1007/BF02392207
J. F. Koksma, Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins Compositio Mathematica. ,vol. 2, pp. 250- 258 ,(1935)
G. Kreisel, Functions, Ordinals, Species Logic, Methodology and Philosophy of Science III. ,vol. 52, pp. 145- 159 ,(1968) , 10.1016/S0049-237X(08)71192-3