# Mathematical model of problem solving process

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

When facing new concepts one first recognizes the old elements. Most stop here. For them there is nothing new under the Moon. Others proceed with learning new elements. Then old elements, which they already recognized in the concept, start looking not completely old but also novel to a degree. At the same time what seemed completely new starts looking not quite so. The process continues on and on until the correct mixture of the new and old in the concept is grasped and the concept is completely digested and internalized.

This iterative process of attaining the correct perception and understanding of a concept can be illustrated as follows:

Similarly, the process of problem solving is an iterative process of converging to a solution by alternately taking one of the dual points of view and transforming the problem from the taken standpoints. Suppose that the dual points of view are quality and quantity of something. Then such an alternate process looks as follows:

Generally, problem solving requires taking a look at a problem not from one pair of dual standpoints but from many such pairs. Thought not only alternately moves along the sides of one dual pair but also jumps from one pair to another one. Here is an example involving three dual pairs:

Thus, the problem solving process involves transitions between opposite sides of a duality, moving along the sides, and transitioning to other dualities. That is why it can be viewed as a train of dual transitions.

I anticipate some people saying there is nothing new in the concept. Such people only view everything through the glasses of what they already know. If they know Kelly's repertory grids they will try to recognize/find them in the above concept. If they know Hegel's dialectics and progression along the helix, they will definitely see them too. If they know TRIZ they will see technical contradictions here.

It does not matter how much in common (if any at all) the concept has with what they see. They still will try to recognize what is familiar to them. This is an unavoidable first step in cognition of anything, as stated in the first paragraph of this article. The issue is how to make them move beyond the first step.