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Epistemological considerations about mathematical concepts

José Luis Usó Doménech (Department of Applied Mathematics, University of Alicante, Alicante, Spain)
Josué Antonio Nescolarde-Selva (Department of Applied Mathematics, University of Alicante, Alicante, Spain)
Hugh Gash (Faculty of Humanities and Social Sciences, Institute of Education, Dublin City University, Dublin, Ireland)


ISSN: 0368-492X

Article publication date: 18 May 2021

Issue publication date: 5 January 2022




Categories (particular (P) and general (V)) constitute a bipole with epistemological implications. The mutual categorical implication of this bipole is embodied in ordinary notions. It follows that a concept because it forms an element of concrete, sensible-rational, practical-theoretical activity has to unite the two inseparable poles, the general and the particular. If the concept of a physical quantity is abstract in relation to the physical object, it is concrete in comparison with mathematical quantity. This product of a secondary abstraction covers the background of physical qualities to extract the pure number, legitimately named abstract number. Both kinds of numbers are mutually exclusive: either the numbers are attached to a unit name and the number is concrete or nothing is attached and the number is abstract. However, in addition to their coordination in extension, they involve each other in comprehension: in fact, the pure number is the general pole V and concrete numbers form the particular pole of the dialectical concept of number K. The purpose of this paper is to provide a model for epistemological issues that arise in the context of meaning, concepts and use of words.


A dialectical theory of the binomial comprehension-extension of mathematical magnitudes.


The findings provide an objection to the traditional deductive order being also true in mathematics, and also that the reverse order cannot be considered as characteristic of mathematics, but show dialectic as universal. This opens the way to the special scientific deduction (mathematical, physical, biological, etc). going from the general to individual.


The structure of the mathematical concepts is elaborated.



Usó Doménech, J.L., Nescolarde-Selva, J.A. and Gash, H. (2022), "Epistemological considerations about mathematical concepts", Kybernetes, Vol. 51 No. 1, pp. 95-115.



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