On justification in mathematics
Abstract
In this article the problem of justification of mathematical axioms (in the context of traditional standpoints in the philosophy of mathematics) is discussed. Stress is laid on the methodological analysis, which concerns the notion of “justification” itself. Concrete choices, known from mathematical practice are not discussed here.
In the process of formulating an axiomatic theory, the problem of the choice of the appropriate axiom system and of the justification of this choice emerges. In particular, the following problems are connected with it:
(1) The problem of the relation between the concept of “justification” and “truth” of mathematical sentences (when the classical definition of truth is assumed).
(2) The problem which criteria of justification can be considered appropriate, and whether the problem of justification is well-posed.
(3) The problem, whether these criteria can be applied only to axioms, in the process of constructing an axiomatic theory, or also to independent sentences (after their metamathematical status has been settled. In that case, extending a theory T by an independent sentence φ or ¬φ cannot be justified by a formal proof.)
(4) The problem, whether the choice of a particular justificatory procedure is motivated philosophically; in particular, whether the problem of justification is considered well-posed.
These questions are analysed in the context of classical philosophical standpoints in the philosophy of mathematics, such as: (1) strict formalism; (2) Hilbert's formalism; (3) mathematical instrumentalism; (4) intuitionism; (5) Quine's realism; (6) Gödel's realism. The standpoint of the “working mathematician” is also discussed.
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