ARTICLE | doi:10.20944/preprints202307.2063.v9
Subject: Computer Science And Mathematics, Logic Keywords: constructive algorithms; constructive mathematics; current knowledge in a scientific discipline; current mathematical knowledge; informal mathematics; known algorithms
Online: 6 December 2023 (12:11:37 CET)
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 theorem of any mathematician from past or present belongs to K. The set K exists only theoretically. 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. In every branch of mathematics, the set of all knowable truths is the set of all theorems. This set exists independently of K. Algorithms always terminate. 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-trivial statements on decidable sets X⊆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.