The predicate of the current mathematical knowledge substantially increases the constructive and informal mathematics and why it cannot be adapted to any empirical science

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dc.contributor.authorTyszka, Apoloniusz
dc.contributor.organizationHugo Kołłątaj University, Kraków, Polanden
dc.date.accessioned2023-08-16T10:35:31Z
dc.date.available2023-08-16T10:35:31Z
dc.date.issued2023
dc.description.abstractThis 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 as a formal science. This content remains also unchanged when the current mathematical knowledge leads to new conjectures. 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 from a particular context.en
dc.identifier.citationPi Mu Epsilon Journal 15 (2023), no. 8, pp. 493-504.en
dc.identifier.urihttps://open.icm.edu.pl/handle/123456789/22820
dc.language.isoen
dc.publisherPi Mu Epsilonen
dc.rightsDozwolony użytek*
dc.subjectconstructive algorithmsen
dc.subjectcurrent mathematical knowledgeen
dc.subjectepistemic notionsen
dc.subjectinformal notionsen
dc.subjectknown algorithmsen
dc.subjectknown elements of Nen
dc.titleThe predicate of the current mathematical knowledge substantially increases the constructive and informal mathematics and why it cannot be adapted to any empirical sciencepl
dc.typearticleen
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