Objectivity over Objects: A Case Study in Theory Formation

• Published in: Synthese (Dordrecht) Vol. 128; no. 3; pp. 245 - 285
• Main Author: Hauser, Kai
• Format: Journal Article
• Language: English
• Published: Dordrecht Kluwer Academic Publishers 01.09.2001; Springer; Springer Nature B.V
• Subjects: Axioms; Determinacy; Epistemology; History of science and technology; Intuition; Large cardinal properties; Logic; Logic and calculus; Mathematical objects; Mathematical problems; Mathematical sciences and techniques; Mathematical set theory; Mathematical sets; Mathematics; Objectivity; Ontology; Philosophical axioms; Philosophy; Set theory; Translations; Truth; New York; United States > US; Axiomatics; Methodology; Ontology; Objectivity; Theory; Science philosophy; Set theory; Justification; Mathematics; Natural science; Hypothesis; Truth
• ISSN: 0039-7857; 1573-0964
• DOI: 10.1023/A:1011994619998

The most singular event in the modern development of set theory was Cohens discovery of the method of forcing and the subsequent realization that the majority of the fundamental problems in the subject are unsolvable on the basis of the standard axioms.1 One reaction to the independence phenomenon has been to question whether those problems are legitimate mathematical problems in the first place. [...]some of the traditionally unsolvable problems can now be considered as solved. [...]questions about the existence and nature of mathematical objects are considered exclusively in the context of the discussion of truth.6 In other words one is aiming toward the solution of outstanding questions of set theory on the basis of various kinds of evidence for and against their truth, and not by relying on a particular picture about mathematical objects. [...]epistemology takes priority over ontology, and the primary epistemological concept is evidence rather than truth. [...]for any given sentence exactly one of and must be true according to our initial bifurcation of all statements into true and false ones. [...]the moral of the incompleteness theorems is that the highly transfinite concept of objective mathematical truth must not be confused with that of demonstrability.7 Admittedly, Gdels examples for undecidable sentences were somewhat artificial, but incompleteness248 KAI HAUSERbecame a practical concern for mathematics after the independence of Cantors continuum hypothesis (CH) from the commonly accepted axioms for set theory (Zermelo Fraenkel with the axiom of choice, ZFC) had been established.8 Subsequently a great number of questions in set theory as well as other branches of mathematics have turned out to be undecidable on the basis of the standard axioms.