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Tyszka, A. Constructive Mathematics with the Predicate of the Current Mathematical Knowledge. SSRN Electronic Journal 2024, doi:10.2139/ssrn.4710446.
Tyszka, A. Constructive Mathematics with the Predicate of the Current Mathematical Knowledge. SSRN Electronic Journal 2024, doi:10.2139/ssrn.4710446.
Tyszka, A. Constructive Mathematics with the Predicate of the Current Mathematical Knowledge. SSRN Electronic Journal 2024, doi:10.2139/ssrn.4710446.
Tyszka, A. Constructive Mathematics with the Predicate of the Current Mathematical Knowledge. SSRN Electronic Journal 2024, doi:10.2139/ssrn.4710446.
Abstract
This is a shortened and revised version of the article: A. Tyszka, Statements and open problems on decidable sets X⊆N, Pi Mu Epsilon J. 15 (2023), no. 8, 493-504. The main results were presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, see http://applications-of-logic.uni.wroc.pl/XXV-Konferencja-Zastosowania-Logiki-w-Filozofii-i-Podstawach-Matematyki. We assume that the current mathematical knowledge is a finite set of statements which is time-dependent. In every branch of mathematics, the set of all knowable truths is the set of all theorems. This set exists independently of our current scientific knowledge. Nicolas D. Goodman observed in Synthese that epistemic notions increase the understanding of mathematics without changing its content. 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 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 from a particular context.
Keywords
composite numbers of the form 22n + 1; constructive algorithms; current mathematical knowledge; decidable sets X N; epistemic notions, informal notions, known algorithms, known elements of N; primes of the form n2 + 1; primes of the form n! + 1; primes of the form 22n + 1
Subject
Computer Science and Mathematics, Logic
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Apoloniusz Tyszka
Commenter's Conflict of Interests: Author