In Constructive and Informal Mathematics, in Contradistinction to any Empirical Science, the Predicate of the Current Knowledge in the Subject is Necessary

Abstract

We assume that the current mathematical knowledge K is a finite set of statements from both formal and constructive mathematics, which is time-dependent and publicly available. Any formal theorem of any mathematician from past or present forever belongs to K. Ignoring K and its subsets, sets exist formally in ZFC theory although their properties can be time-dependent (when they depend on K) or informal. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivially true statements on decidable sets X \(\subseteq\) \(\mathbb{N}\) that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X . This and the next sentence justify the article title. For any empirical science, we can identify the current knowledge with that science because truths from the empirical sciences are not necessary truths but working models of truth about particular real phenomena.