The math of intuition

We all manipulate things around us in order to build, tweak or tune (hopefully not to destroy!). How are we able to do it efficiently? We believe in two things. We know how a variable affects the output, irrespective of the value of the other variables. For instance, if we strike a coin harder, it will move further irrespective of the weight of the coin. Think how jeopardized we would be, if this relation was different for a heavier coin vs. a lighter one. The other, we use monotonic relations: increasing a variable would either increase or decrease the output. Again, if this wasn’t true, it would be hard to know how to move the coin further, by striking it harder or otherwise. 

Reproduced from Wikipedia: We implicitly assume that the mass, 
length and gravity ‘independently’ influence the time of oscillation.

Fortunately, these beliefs are correct for a whole lot of things and thus we easily manipulate. In mathematical terms, this is called coordinate-wise monotonicity, the weakest structure we implicitly assume the world has. It is our inductive ‘bias’ to understand the world. Those who are able to grasp this bias are called ‘logical’ (Piaget!). 

We recognized this in 2006. This applied beautifully also to the case of analog circuits, which designers magically designed circuit by intuition – aha – by co-ordinate-wise monotonicity! We think this should be taught in a methodical way.

Some incomplete and semi-complete reports on this from 2006-2007 when I was part of Dr. Una-May O’Reilly’s group at MIT: