Mathematical-methods.

A list of methods.

  • Semantic.

    The basic vocabulary which has been introduced for an understable description of the theory of the (E) question.
  • The-intrinsic-method.

    The "intrinsic method" is a mathematical tool which has been developed to analyze how linear systems can be decomposed in any three-dimensional space.
  • The-extrinsic-method.

    The (E) question generates polynomial forms of degree two. This gives the opportunity to compare these forms with Taylor developments.

First steps.

Let suppose that we focus our scientific curiosity on the concept of “division” and that we must explain it to someone else. We would probably start with a simple question like: “What do you find when you divide six by three?” Most of the spontaneous answers will be: 2. But why? And the next answer would certainly come: “It’s natural!”. And, as a matter of facts, 2, 3 and 6 are natural numbers with which we are seemingly customized to count.

Unfortunately for those who are looking for the simplicity, and as one teaches it in schools, there are a lot of other categories of numbers: relative natural: Z = {-…, -2, -1, 0, 1, 2, …}, fractional: Q ={…-5/17, …, 9/253, …}, real: R, complex: C, etc. Therefore, the next question is: “Why did nobody give the following answers to my question: 6 = ½. 3 + 9/2 or 6 = 2,3. 3 - 0,9?” These answers would also have been correct because, my fault, I haven’t precise an important parameter in my initial question: “What do you find when you divide six by 2 and your answer must be a fractional number (respectively a real number, etc.)?” So: I should have first given information on the set in which the solution(s) must be.

What do we state next? The answers are, most of the time, far to be unique; and what else? Any correct answer is not only made of one number, but of a pair of numbers which we may generically label (the main result, the remaining part).

The trivial answer accompanying the answer when we stay in N = {0, 1, ….} is misleading our thoughts on the topic because it gives us the sensation that all responses contain only one number. The correct answer to my initial and quasi-naïve question when the solutions must be in N should have been the pair: (2, 0). It is just because there is no remaining part that we didn’t realize the duality of the answer.

These basic, but in some way fundamental elements of a discussion about the concept of division, can now be extrapolated. We may ask if we can divide six by a vector, for example? Or we can consider any vector and ask if we can decompose it with a matrix? These interrogations would slowly drive the discussion into the direction of the concept of torsion or an abstract generalization of it.

These first paragraphs, here, are only a short presentation of what is waiting for you on this website if you want to deepen the mathematics illustrating these preliminaries; they are the essence of the theory of the (E) question (alias TEQ).

At the very beginning, it looks like some useless preoccupation and the readers may ask why an old man is joking with this thematic. In the end, you will discover how one can reach the deepest thoughts with this and touch the most advanced today knowledge in theoretical physics.

© Thierry PERIAT.