In Constructive and Informal Mathematics, in Contradistinction to Any Empirical Science, there are Non-Trivially True Statements with the Predicate of the Current Knowledge in the Subject

Apoloniusz Tyszka

doi:10.20944/preprints202307.2063.v9

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preprints.org > computer science and mathematics > logic > doi: 10.20944/preprints202307.2063.v9

Preprint Article Version 9 Preserved in Portico This version is not peer-reviewed

In Constructive and Informal Mathematics, in Contradistinction to Any Empirical Science, there are Non-Trivially True Statements with the Predicate of the Current Knowledge in the Subject

Apoloniusz Tyszka

Version 1 : Received: 27 July 2023 / Approved: 28 July 2023 / Online: 31 July 2023 (10:54:27 CEST)
Version 2 : Received: 2 August 2023 / Approved: 3 August 2023 / Online: 4 August 2023 (07:45:49 CEST)
Version 3 : Received: 14 August 2023 / Approved: 15 August 2023 / Online: 15 August 2023 (08:49:43 CEST)
Version 4 : Received: 29 August 2023 / Approved: 30 August 2023 / Online: 31 August 2023 (03:53:31 CEST)
Version 5 : Received: 5 September 2023 / Approved: 6 September 2023 / Online: 7 September 2023 (05:01:30 CEST)
Version 6 : Received: 20 September 2023 / Approved: 21 September 2023 / Online: 22 September 2023 (05:14:58 CEST)
Version 7 : Received: 9 October 2023 / Approved: 10 October 2023 / Online: 10 October 2023 (10:05:52 CEST)
Version 8 : Received: 17 October 2023 / Approved: 18 October 2023 / Online: 19 October 2023 (04:49:53 CEST)
Version 9 : Received: 4 December 2023 / Approved: 6 December 2023 / Online: 6 December 2023 (12:11:37 CET)
Version 10 : Received: 14 January 2024 / Approved: 16 January 2024 / Online: 16 January 2024 (06:42:23 CET)

A peer-reviewed article of this Preprint also exists.

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

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.

Keywords

constructive algorithms; constructive mathematics; current knowledge in a scientific discipline; current mathematical knowledge; informal mathematics; known algorithms

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.

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Comments (1)

Comment 1

Received: 6 December 2023
Commenter: Apoloniusz Tyszka
Commenter's Conflict of Interests: Author

Comment: I added Example 4. I changed keywords and the title. I shortened the abstract.

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