Discontinuity of understanding

It may be widely believe that knowledge progresses gradually, in small steps, in gentle increments, or in slight gradations.  It may be not so, even for us as individuals.  New knowledge or understanding often begins with insights. 

When you gain an insight, and it is true, it becomes applicable and remains so from then on.   This changes one’s problem solving game. This insight provides a new tool or rule, but importantly  it creates a discontinuity in your problem solving methods.   So, we might ask whether insights can come gradually?  In some cases, probably yes, though examples are difficult to furnish.  The emergence of infinity, often credited to Cantor, took centuries of dancing around the edges by philosophers.  The germ theory of disease so often attributed to Pasteur was anticipated almost with the invention of the microscope.  On the other hand plate tectonics seemed to arise by a simple insight by a single person, Alfred Wegener, in 1912 and took a mere half century to achieve acceptance.   In many cases the insight may incubate in a collective or individual  for some time before emerging in full bloom. 

A few questions.  Must an insight be associated with a particular problem?  Probably.  Can one simply arise out of the blue and then find its application somewhere? Probably not.  Are we always receiving insights and then forgetting them just as quickly? Absolutely yes.  Are some insights wrong?  Yes, and in great numbers.

Knowledge and understanding, it seems, progress not as a continuous evolution, but in discrete steps.  Moreover, it is the solution of a problem that drives insights.