Tyszka, Apoloniusz
(2023)
The predicate of the current mathematical knowledge substantially increases the constructive mathematics what is impossible for the empirical sciences.
[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, 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. We assume that the current mathematical knowledge is a finite set of statements which is timedependent. Nicolas D. Goodman observed 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 nontrivial 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. 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.
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: 
28 Jul 2023 17:55 
Last Modified: 
28 Jul 2023 17:55 
Item ID: 
22337 
Official URL: 
http://applicationsoflogic.uni.wroc.pl/XXVKonfe... 
Subjects: 
Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics 
Date: 
27 July 2023 
URI: 
https://philsciarchive.pitt.edu/id/eprint/22337 
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