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.
Highlights
Many mathematical concepts went through a long process before they got their modern abstract definition
Let V be a set of coloured objects as follows (Figure 1): Rik Verhulst: A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers
Later on we will show that we can start from a partition of a set as well as from a surjective mapping of a set onto another to define new notions
Summary
Many mathematical concepts went through a long process before they got their modern abstract definition. We will use this ‘machine’ for creating totally new concepts in an abstract way, for example bidecimal numbers of different kinds. Let V be a set of coloured objects as follows (Figure 1): Rik Verhulst: A ‘Machine’ for Creating Mathematical Concepts in an Abstract Way, Bidecimal Numbers. The composition g of f and b is a surjective mapping from T onto M We use this three-step ‘machine’, equivalence relation, partition and mapping, for creating mathematical concepts. Later on we will show that we can start from a partition of a set as well as from a surjective mapping of a set onto another to define new notions. The equivalence relation R connects the objects with the same shape in subsets of V All these subsets form a partition of the set V. We can attribute names or symbols to the equivalence classes and represent them in a new set M, e.g. (Figure 3): 3. Examples of Well-known Concepts
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
