Tyszka, Apoloniusz
(2023)
The predicate of the current mathematical knowledge increases the scope of mathematics what distinguishes mathematics from other fields of study.
[Preprint]
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. 493504. The main results were presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, see http://applicationsoflogic.uni.wroc.pl/XXVKonferencjaZastosowaniaLogikiwFilozofiiiPodstawachMatematyki. 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 timedependent. 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.
Item Type: 
Preprint

Creators: 

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://applicationsoflogic.uni.wroc.pl/XXVKonfe... 
Subjects: 
Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics 
Date: 
20 July 2023 
URI: 
https://philsciarchive.pitt.edu/id/eprint/22328 
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The predicate of the current mathematical knowledge increases the scope of mathematics what distinguishes mathematics from other fields of study. (deposited 21 Jul 2023 14:58)
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