On contradiction between universality and usefulness of mathematical notation in modeling system transformations

Y. B. Karasik,
Thoughts Guiding Systems Corp.,
Ottawa, Canada.
e-mail:karasik@sympatico.ca

Every invention is the result of a series of inversions and transpositions. They are not always easily discernable and it might appear that it is not true. But a thorough examination always reveal them.

Uncovering inversions and transpositions requires presenting the idea of invention in such a notation where they transpire. Consider, for example, the invention where a set of electrodes, each of which is made of a unique metal, is replaced by a set of electrodes, each of which is made of an alloy of these metals. What inversions or transpositions are involved here ? The verbal statement of the idea does not reveal them. The idea has to be modelled mathematically with the help of some notation to see the transpositions. In this case the model is as follows. For the original electrodes the following statements hold:

  1. EACH ELECTRODE is made of ONE metal
  2. THE NUMBER OF DIFFERENT ELECTRODES in the set is MORE THAN ONE
For the electrodes proposed in the invention, the following statements hold:
  1. EACH ELECTRODE is made of MORE THAN ONE metal
  2. THE NUMBER OF DIFFERENT ELECTRODES in the set is ONE
It is easily seen that both the previous art and the proposed invention are described by the same phrases (up to the transposition of words "ONE" and "MORE THAN ONE"). This transposition is the core of the invention. The system transformation that takes place can be graphically presented as follows:

Let us define the following vector like multiplication:

(x1 ? y1, ..., xn ? yn) × (a1, ..., an) = (x1 a1 y1, ..., xn an yn),

where sign ? in the first vector means a placeholder to be filled with the corresponding value from the second vector. Then the idea of the above invention in this notation can be represented as the following system transformation:

(x1 ? y1, x2 ? y2) × (a1, a2) ⇒ (x1 ? y1, x2 ? y2) × (a2, a1)

Unfortunately, this notation has a very limited scope of applicability and is not suitable to represent the ideas of many other inventions. On the other hand it is possible to devise more universal notations, which correctly outline the transformations taken place in many inventions, but fail to represent them as inversions or transpositions. I will present such a universal notation in a separate article. Thus, a contradiction exists: the more universal the notation to represent system transformations is, the less it capable to represent them as series of inversions and transpositions. It would be a great achievement to devise such a notation, where this contradiction is overcome.

This issue of the journal is devoted to presenting notations that are at the opposite ends of the spectrum of the contradiction.