PART I. Intradisciplinary Systematic Conceptual Integration (Synthesis) and Analysis: 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!
Thus far, we have considered intradisciplinary conceptual integration as in between disciplines within Mathematics. In the next section (number 2: Interdisciplinary Conceptual Integration), we will look at an example of interdisciplinary conceptual integration. In that part, we will look at the conceptual integration of Geometry and Chemistry as a paradigm case for the integration of Science and Mathematics.
PART II. Interdisciplinary Conceptual Integration
Title: The Systematic Conceptual Integration of Cognitive Mathematics and Science: The Conceptual Integration of Geometry and Chemistry as a Paradigm Case for a New Universal Methodology
Albert P. Carpenter
In the first lecture, we looked at the intradisciplinary integration of Geometry and Topology using a new method called Systematic Conceptual Integration and defined it in terms of structure and composition.
In this, the second lecture, we will look at the application of this new method to the conceptual integration of Geometry and Organic Chemistry. It is to be understood as an interdisciplinary integration between science and mathematics.
We return again to our table as before only this time we replace the terms Geometry and Topology with Geometry and Organic Chemistry.
|Operator: Integration||Organic Chemistry||Geometry|
The resulting integrations are:
- Geometric Organic Chemistry
- Organic Chemical Geometry
Before we proceed we should again define what these disciplines are. Chemistry will be understood as the study of molecules; in this case, organic molecules such as mono carbon, bicarbon, cyclopentane and cubane and Geometry as the study of shapes; specifically points, lines, polygons and polyhedra.
So now we can set up our second table focusing this time on Geometric Chemistry:
|Geometric Chemistry-Structure and Composition (Geometric Composition)||Examples of Molecular Structure|
|Line||Ethane, Ethelene and Acytelene|
|Polygon||Propane, Butane and Pentane etc.|
|Polyhedron||Cubane, Pentaprismane and Hexaprismane etc.|
The result that we get are 4 categories for Geometric Chemistry. We are then in a position to apply these results to the second table we found in the first lecture for Geometric Topology as follows:
|Composition and Structure||Knot||Link||Mobius Strip||Tori|
|Ethane, Ethylene andAcetylene||5||6||7||8|
|Propane, Butane and Pentane etc.||9||10||11||12|
|Triangular Prismane, Cubane and Pentaprismane etc.||13||14||15||16|
- Ethane, Ethylene and Acetylene Knot
- Ethane, Ethylene and Acetylene Link
- Ethane, Ethylene and Acetylene Mobius strip
- Ethane, Ethylene and Acetylene Torus
- Propane Knot etc.
- Propane Link etc.
- Propane Torus etc.
- Propane Mobius strip etc.
- Triangular Prismane Knot etc.
- Triangular Prismane Link etc.
- Triangular Prismane Torus etc.
- Trianguale Prismane Mobius strip etc.
We then again return to Table 2.1 and place the results of table 2.3 in the grid ( no. 2) for Geometric Organic Chemistry. This provides the structure for the discipline and organizes its objects.
Once more, the structure and composition of the object structures and organizes the discipline.
As in Part I., we create integrated databases that connect with chemical printers and so on and so forth.
I am especially indebted to George Lakoff at U.C. Berkeley, Mark Johnson at the University of Oregon, Mark Turner at Case Western Reserve University and Gilles Fauconnier at U. C. San Diego were it not for their collective works this methodology and its results would never have come to fruition.