On The Mathematical Foundation of The Theory of Inventing

Swiss psychologist Jean Piaget used to conduct the following experiments. He would show children of less than 5 years old two identical pieces of plasticine

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and ask: "Which one has more plasticine ?" They would naturally answer: "They are the same." Then he would elongate a bit one of the pieces
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and would ask the same question. The children would answer that the piece on the right has more plasticine.

- Why ? - Piaget would ask them.
- Because it is longer, - children would reply.

But when they grew up a bit, they were not already so sure. At times they still claimed that the piece on the right has more plasticine because it is longer but at other times they claimed that the piece on the left is bigger because it is wider.

When the children turned 5 years old they were already able to resolve this contradiction. They would start to realize that despite elongating made the right piece longer, it also made it thinner. They would start invariably claiming that the both pieces of plasticine contain the same amount of the matter.

They resolved the contradiction (the right has to have more plasticine because it is higher and less plasticine because it is thinner) not by separating the contradictory requirements and not by their uniting1 but by compensating the increase in one parameter by the decrease in the other. It appears that resolving contradictions by applying a pair of mutually compensatory actions is the most general way of contradictions resolution.

Suppose we face a contradiction: object has to have property P and has to not have property P. In other words, it has to be made (or chosen) to have property P and has to be made (or chosen) to not have property P. "Has to be made or chosen" is nothing else but action, which we denote as L. In order to resolve the contradiction we have to compensate it. However, direct compensation with the help of anti-action L-1 is impossible because

L-1•(L•a) = a,

where a is our object. Anti-action simply negates the action and leaves the object intact, which is not what we want. We want to obtain a new object, which would not have the problem.

In case of Piaget experiments children compensated increase in length of the piece of plasticine by decrease in its width. As is known, length and width are conjugate characteristics. There are many other of them. Let u and u* be conjugate characteristics of object a. Let's denote this fact as <u|a|u*>. Setting u to x amounts to acquiring property P by a. Let us denote this action as L • u. Then applying compensatory action L-1 to the conjugate characteristic u* resolves contradiction. In other words, contradiction is resolved by transition from <u|a|u*> to <L•u|a|L-1•u*>:

<u|a|u*>   =======>  <L•u|a|L-1•u*>

Unlike children in Piaget experiments inventors never compensate decrease in one dimension by increase in other dimension. It is not an inventive approach. They may compensate decrease in length of an object by increasing the number of such objects but never would compensate it by increasing its width. Consider, for example, a contradiction: an object has to be short and has to be long. A typical solution is to make object short but employ many of them instead of one long. What differs such conjugate characteristics as length and width from such conjugate characteristics as size of object and number of objects ? The difference is that whereas length •width remains invariant under physical transformations of object (such as squeezing), size_of_one_object • number_of_objects never remains invariant under inventive transformations. (For gaps or overlaps between smaller objects may be needed for them to work together, etc.)

Physics studies invariant transformations of objects. Its laws can be derived from existence of certain invariants (see e.g. Landau and Lifshitz course of theoretical physics 2 for details). Inventing, on the contrary, is such an activity which seeks transformations that break invariants and create something new.

The reader is referred to the latest version of the toll paper "Mathematical principia of the theory of heuristic methods" for further details.