• View
• Edit
• History
• Print

Research program

Map of Math: Affine, Projective, Conformal, Symplectic Geometry

Map of Areas of Mathematics

• Based on journal classification
• Based on the Companion
• Dualities

Axioms of Set Theory - Visualizations - Restructurings

• 6 layers in the cortical column

Four classical Lie families - Four levels of geometry and logic

Polytopes - Choice frameworks

Classical Lie families

Investigation: 4 classical Lie families, 4 geometries, 4 choice frameworks.

One of my goals is to be able to make a map of how mathematical subjects, concepts and objects become relevant. Such a map would systematize existing mathematics, identify overlooked mathematics, and show the directions in which math can evolve in the future.

I started by organizing the subjects listed in the Mathematics Subject Classification by trying to show which areas depend on which other areas. Acknowledging my general ignorance, I was able to draw several conclusions.

As expected, there do seem to be two major areas, algebra and analysis. The capstone of math seems to be number theory, which makes use of tools from all of math. Lie theory seems especially central as a bridge between algebra and analysis.

Surprisingly for me, geometry seems to be a well spring for math. I studied algebraic combinatorics as "the basement of math" from which I thought mathematical objects arose. Geometry thus seemed rather idiosyncratic. But from the map it seems that geometry is a key ingredient in math, in terms of its content, perhaps in the way that logic is, in terms of its form.

I then tried to improve my map by adding more detail. I used the graphic editor yEd. This simply yielded a spaghetti diagram. However, I am hopeful that ultimately it should be possible to discover principles for making a meaningful map and collaborating with others to make it a comprehensive resource related to Wikipedia and MathStackExchange/MathOverflow.

Collecting and analyzing such examples could be a collaborative effort. Here is a database I made of almost 200 examples of figuring things out in math.

I used my philosophical structures to systematize the recurring patterns. This yielded the following diagram. I would like to sharpen the results.

The lower half of the diagram grounds the mathematical thinking which is pre-systemic. The upper half of the diagram grounds that which takes place within a mathematical system.

In particular, I am interested in understanding, intuitively, the cognitive foundations for the four classical Lie groups/algebras. I have been learning about the classification through the Dynkin diagrams. But that does not explain intuitively the qualitative distinctions. So instead I have been working backwards, from the Cartan diagrams, trying to understand concretely how to imagine the growth of a chain (how it ever adds a dimension via an angle of 120 degrees) and the possible ways that chain might end.

I am encouraged that I myself have made some mathematical discoveries by focusing on these questions. I have thought a lot about the regular polytopes which the Weyl groups are symmetries of. In particular, I was able to come up with an interpretation for the -1 simplex and a novel q-analogue of the simplex.

I am also seeing how the polytopes can be thought to arise by a "center" which ever generates vertices (for simplices), pairs of vertices (for cross-polytopes), planes (for hypercubes) and "coordinate systems" (for demicubes). This type of process is very relevant for my theological ideas, see: God's Question: Is God Necessary? In particular, I think about the "field with one element" as being interpretable as 0, 1 and infinity.

Discover Cognitive Foundations for the Classical Lie Groups/Algebras. It is surprising that in mathematics there is a small collection of structures which seem most rich in content. This is a point that Urs Schreiber keeps returning to. Thus one task is to make a list of such structures and try to relate them with a map, and indeed, understand how they fit in a map of all math. In particular, John Baez and others have pointed out that the classical Lie algebras ground different geometries. I would like to learn the basics of affine, projective, conformal and symplectic geometries so that I could understand how they relate to the four classical groups.

The center of a regular polytope <=> God

The totality of a regular polytope <=> Everything

Perspectives

One place to look for the cognitive foundations of mathematics is to develop models of attention, for example, in terms of category theory.

6 visualizations and 6 paradoxes

Cortical minicolumns

10 axioms of set theory

The axioms of Zermelo Frankel set theory (except for the Axiom of Infinity) and the Axiom of Choice are all present in the above system and so I would like to work further to clarify their role. Of special interest to me, currently, is to study the four concepts (in orange) that seem to ground logic but also geometry. These methods apply the concepts of truth (argument by contradiction), model (solving an easier version), implication (working backwards) and variable (classifying the problem).

4 plus 6 generators of the Poincare group, the symmetries for quantum field theory

Map of Math

Another way to build a map is to use the tags from MathOverflow. The idea is to make a list of the, say, X=100 most popular tags, and also to make a list of the most popular pairs of tags, where pairs are created for any two tags that are used for the same post. In the map, for each popular tag, I would show a link to its most popular pair, and also include, say, the most popular 2X links overall.

Study the Process of Abstraction. Thus it is important to study the process of abstraction. One approach is to try to describe, in an elegant way, a theory that is practically complete, such as the geometry of triangles in the plane. Norman Wildberger's book and videos are very helpful for this. It may be that a matrix approach might be insightful. Having stated a theory it may be possible to see in what directions it develops further.

Another approach is to identify classic theorems in the history of mathematics and consider how abstraction and generalization drove them to arise and develop further.

I would like to learn more about the kinds of equivalences in math - I know that Voevodsky, etc. have studied that deeply - and draw on that and perhaps contribute.

Algebra and geometry

• adjunction: tensor product (geometry) - homset (algebra)
• Comparing the price of a sandwich {$\frac{4}{3}\frac{2}{3}=\frac{8}{9}$} as illustrating that algebra is thinking step-by-step.
• Geometry as homogeneous space, uniform choices.