International Journal of Discrete Mathematics

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A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers

Received: 7 April 2021    Accepted: 18 May 2021    Published: 26 May 2021
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Abstract

In mathematics, the creation and definition of new concepts is the first step in opening up a new field of research. Traditionally this step originated from intuition by a process of observation, analysis and abstraction. This article will show a general method by which most of the common notions of number theory, geometry, topology, etc., can be introduced in one and the same particular way. Therefore, we only need some of the tools of naïve set theory: a set of terms to which we apply an equivalence relation. This equivalence relation induces a partition of the terms with which we can consequently associate new concepts. By using this method’s ‘machine’ in an abstract way on arbitrary sets of terms we can create new notions at will, as we will show in this article, for instance, for bidecimal numbers of different kinds. The fact that we reverse the usual procedure of intuition before abstraction, doesn’t mean that we only create esoteric objects without any meaning. On the contrary, their abstract nature precisely provides our imagination with many possibilities for several interpretations in models in which they become useful. So, for example, we can use our bidecimal numbers to define elementary transformations on a cylinder or on a pile of tori.

DOI 10.11648/j.dmath.20210601.13
Published in International Journal of Discrete Mathematics (Volume 6, Issue 1, June 2021)
Page(s) 15-22
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Equivalence Relation, Partition, Mapping, Model, Bidecimal Number of the First Kind, Bidecimal Number of the Second Kind

References
[1] Bochner S. (1965) The Role of Mathematics in the Rise of Science. Princeton University Press.
[2] Cohen P. (1966) Set Theory and the continuum Hypothesis. W. A. Benjamin.
[3] Cohn P. M. (1974) Algebra, Volume 1. John Wiley & Sons.
[4] Eves H. (1964) An Introduction to the History of Mathematics. Holt, Rinehart and Winston.
[5] Flegg G. (1993) Numbers. Their history and meaning. Barnes & Noble Books.
[6] Hardy G. H. (1960) An introduction to the theory of numbers. Oxford University Press.
[7] Kline M. (1972) Mathematical Thought from Ancient to Modern Times. Oxford University Press.
[8] Lang S. (1980) Algebra. Addison-Wesley.
[9] Martin G. E. (1975) The foundations of Geometry and the Non-Euclidian Plane. Springer-Verlag.
[10] Verhulst R. (1982) Begripsvorming in de Wiskunde. Bidecimale getallen. VWNL-CAHIER, Vereniging voor Wis- en Natuurkunde Lovanienses, nr. 09.
[11] Verhulst R. (1992) The universe of the coloured arrows, alias the Papygrams. Bulletin de la Société Mathématique de Belgique, Tijdschrift van het Belgisch Wiskundig Genootschap, vol. 4, nr. 2 serie A.
[12] Verhulst R. (2006) In de ban van wiskunde. Het cultuurverschijnsel mathematica in beschaving, kunst, natuur en leven. Garant: 24–42.
[13] Verhulst R. (2019) Im Banne der Mathematik. Die kulturellen Aspekte der Mathematik in Zivilation, Kunst und Natur. Springer Spektrum: 10–28.
[14] Verhulst R. (1982) Nomograms for the calculation of Roots. International Colloquium on Geometry Teaching. ICMI.
[15] Verhulst R. (2020) Recursive formulas for root calculation inspired by geometrical constructions. The teaching of Mathematics Vol. XXIII, 1, 35–50.
[16] Wilder R. (1952) An introduction to the Foundations of Mathematics. John Wiley.
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    Rik Verhulst. (2021). A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers. International Journal of Discrete Mathematics, 6(1), 15-22. https://doi.org/10.11648/j.dmath.20210601.13

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    Rik Verhulst. A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers. Int. J. Discrete Math. 2021, 6(1), 15-22. doi: 10.11648/j.dmath.20210601.13

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    AMA Style

    Rik Verhulst. A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers. Int J Discrete Math. 2021;6(1):15-22. doi: 10.11648/j.dmath.20210601.13

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  • @article{10.11648/j.dmath.20210601.13,
      author = {Rik Verhulst},
      title = {A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers},
      journal = {International Journal of Discrete Mathematics},
      volume = {6},
      number = {1},
      pages = {15-22},
      doi = {10.11648/j.dmath.20210601.13},
      url = {https://doi.org/10.11648/j.dmath.20210601.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20210601.13},
      abstract = {In mathematics, the creation and definition of new concepts is the first step in opening up a new field of research. Traditionally this step originated from intuition by a process of observation, analysis and abstraction. This article will show a general method by which most of the common notions of number theory, geometry, topology, etc., can be introduced in one and the same particular way. Therefore, we only need some of the tools of naïve set theory: a set of terms to which we apply an equivalence relation. This equivalence relation induces a partition of the terms with which we can consequently associate new concepts. By using this method’s ‘machine’ in an abstract way on arbitrary sets of terms we can create new notions at will, as we will show in this article, for instance, for bidecimal numbers of different kinds. The fact that we reverse the usual procedure of intuition before abstraction, doesn’t mean that we only create esoteric objects without any meaning. On the contrary, their abstract nature precisely provides our imagination with many possibilities for several interpretations in models in which they become useful. So, for example, we can use our bidecimal numbers to define elementary transformations on a cylinder or on a pile of tori.},
     year = {2021}
    }
    

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    JF  - International Journal of Discrete Mathematics
    JO  - International Journal of Discrete Mathematics
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    AB  - In mathematics, the creation and definition of new concepts is the first step in opening up a new field of research. Traditionally this step originated from intuition by a process of observation, analysis and abstraction. This article will show a general method by which most of the common notions of number theory, geometry, topology, etc., can be introduced in one and the same particular way. Therefore, we only need some of the tools of naïve set theory: a set of terms to which we apply an equivalence relation. This equivalence relation induces a partition of the terms with which we can consequently associate new concepts. By using this method’s ‘machine’ in an abstract way on arbitrary sets of terms we can create new notions at will, as we will show in this article, for instance, for bidecimal numbers of different kinds. The fact that we reverse the usual procedure of intuition before abstraction, doesn’t mean that we only create esoteric objects without any meaning. On the contrary, their abstract nature precisely provides our imagination with many possibilities for several interpretations in models in which they become useful. So, for example, we can use our bidecimal numbers to define elementary transformations on a cylinder or on a pile of tori.
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Author Information
  • Department of Mathematics, KdG University of Applied Sciences, Antwerp, Belgium

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