The predicate of the current mathematical knowledge increases the scope of mathematics what distinguishes mathematics from other fields of study

Tyszka, Apoloniusz (2023) The predicate of the current mathematical knowledge increases the scope of mathematics what distinguishes mathematics from other fields of study. [Preprint]

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Official URL: http://applications-of-logic.uni.wroc.pl/XXV-Konfe...

Abstract

This is an expanded 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, pp. 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. Nicolas D. Goodman observed that epistemic notions increase the understanding of mathematics without changing its content. We show that the predicate of the current mathematical knowledge increases the scope of mathematics. This distinguishes mathematics from other fields of study. We assume that the current mathematical knowledge is a finite set of statements, which is time-dependent. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω⇒P(n^2+1)⊆[2,(((24!)!)!)!]. We heuristically justify the statement Φ. This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. The distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known inspires statements on decidable sets X⊆N that refer to the current mathematical knowledge on X.

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Item Type:

Preprint

Creators:

Creators	Email	ORCID
Tyszka, Apoloniusz	rttyszka@cyf-kr.edu.pl	0000-0002-2770-5495

Keywords:

composite numbers of the form 2^{2^n}+1, constructive algorithms, current mathematical knowledge, decidable sets X⊆N, epistemic notions, informal notions, known algorithms, known elements of N, primes of the form n^2+1, primes of the form n!+1, primes of the form 2^{2^n}+1

Subjects:

Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics

Depositing User:

Apoloniusz Tyszka

Date Deposited:

21 Jul 2023 14:58

Last Modified:

21 Jul 2023 14:58

Item ID:

22328

Official URL:

http://applications-of-logic.uni.wroc.pl/XXV-Konfe...