Two Subdivisions of Mathematics

Mathematics can be subdivided as:

Mathematical Notation

Mathematical notation can be subdivided as:

  1. expressing a problem in mathematical notation
  2. manipulating the notation, reducing the expression of the problem.

Pencil and Paper

Mathematical notation was invented for use with pencil and paper.

Mathematical notation is a 2D notation, not a 3D notation.

Time Variance

Manipulation of mathematical notation works mostly because time-varying variables have been expunged from the notation (“side-effect free”).

Removing time variance from the notation is an approximation. It works only in domains where time is not a major factor.

[Note that other approximations, like finite-element analysis, can give more accurate results in specific domains that have time variance.]


A layer is only as complicated as one wishes to make it.

The complexity lies in the amount of detail required to express the inputs and outputs of the layer.

One can tune the complexity of each layer by using different notations and approximations in each layer, instead of trying to encompass all variables in a single layer.

Complexity is “in the eye of the beholder”. Mathematical notation appears complicated to some (most), but appears simple to those who have steeped themselves in the notation.

Who is the Audience?

Regarding mathematical notation, an interesting question to ask is:

Is the notation meant to be used as communication between two mathematicians, or, between a mathematician and a non-mathematician (lawyer, graphics designer, construction worker, etc.)?


Complexity comes from trying to address too many details at any one layer of a design.

Successful complexity-management is the art of finding a group of inter-related details and relegating them to a layer.


Engineers map notation from one domain into another.

For example, in construction, Engineers use mathematics (from the Architects, from textbooks, etc.) and produce blueprints that are understood by construction workers.2

Engineers filter out details.

  1. Actually, this can be subdivided even further. There is experience in using the notation and there is experience in thinking about problems (which are then mapped to this notation).  ↩︎

  2. One doesn't usually find Engineers performing actual construction work. Engineering is usually done before the construction begins. On–site Engineers oversee that the work follows the specifications. Engineers might alter the specifications to better match reality as work progresses.  ↩︎