PART I. Systematic Conceptual Integration: A New Methodology for Cognitive Mathematics
Title: Systematic Conceptual Integration in Cognitive Mathematics: The Integration of Geometry and Topology as a Paradigm Case for a New Universal Methodology
Albert P. Carpenter
In this section, we will outline a new method from the cognitive sciences to mathematics specifically systematic conceptual integration to Cognitive Geometry and Topology. Conceptual integration involves taking two ideas and putting them together to form a third new one. It is nothing new to mathematics. It is quite ancient and has been practiced for centuries. Perhaps most effectively and famously, Rene Descartes integrated Geometry with Algebra to create Algebraic Geometry or Analytic Geometry. It is a common form of cognition that many creative mathematicians continue to do to this day.
What differs in what I will propose is that we formalize this process. We can do this by creating a table or binary matrix for in this case Geometry and Topology as seen below.
Before we continue we should define what we mean by Geometry and Topology. Following other sciences like Chemistry and Biology that study objects like molecules and organisms respectively, we will here say that Geometry and Topology is the study of various objects such as shapes. For example and for simplicity sake, we are going to further define Geometry as, at a minimum, the study of objects such as points, lines, polygons and polyhedra, while Topology studies objects such as knots, links, tori and Mobius strips.
This them enables us to set up a second table or matrix for Geometric Topology as follows:
|Structure and Composition||Knot||Link||Mobius strip||Tori|
- Polyhedral Knot
- Polyhedral Link
- Polyhedral Mobius strip
- Polyhedral Tori
- Polygonal Knot
- Polygonal Link
- Polygonal Mobius strip
- Polygonal Tori
- Linear Knot
- Linear Link
- Linear Mobius strip
- Linear Tori
- Pointal Knot
- Pointal Link
- Pointal Mobius strip
- Pointal Tori
These 16 binary combinations describe the structure and composition of the objects to be created. What we get, for example, in number one is a knot that is constructed with polyhedra and in number two a link that is constructed with, again, polyhedra and so on and so forth through the table. In each case, the geometric object defines the composition of the toplogical structure. Table two defines the categories of objects for Geometric Topology.
An example of the possibilities of this method can be found at this site. These are examples of polygonal topological structures and include; polygonal knots, links, tori and Mobius strips.
We then return to table one and do the same thing for the other three integrated disciplines that we did in table two with the result that we will have 64 categories of objects for all four disciplines in table one. Each of the 64 categories will have at a minimum of 100 objects in its ontology or catalog of objects giving us at a minimum of 6400 objects.
I cannot emphasize enough the significance of defining objects in terms of their structure and composition. Not only does it provide an identity for the objects so defined, but also for the discipline that studies those objects. The result is that both the discipline such as Geometric Topology and the objects it studies are defined in the same terms. In addition, this way of defining objects and knowledge is fully compatible with the material sciences such as Chemistry that defines the identity of molecules in terms of their structure and composition as well.
What are the implications and applications of this new method? The first step will be to create vast mathematical (morphological) databases complete with data sheets. These will then be connected to chemical printers that will build the molecules based on the models in the databases. These, in turn, will be subjected to robotic arrays that will perform chemical and physical experiments on the molecules created by the chemical printers as is already done in robotic chemistry.
The advantages of this new method should be obvious. It is intuitive, simple, powerful, fruitful, based on cognitive process already in place, easy to communicate, and easy to teach.
AND IT IS COMPLETELY REVERSIBLE WITH SYSTEMATIC CONCEPTUAL ANALYSIS!