## Incommensurate

Next, I went back and color coded the labeled gaps. In this figure, I’ve put lines at the bridged gaps indicating the use of a new Back in 2009 or so a reader of this blog asked me to define the term incommensurate. I’ll do that again here.

I’ll start with a graph from S. Arbesman’s The Half-Life of Facts. That graph was a surprise to me. It displayed the results of fifty or so experiments about temperature. Some of the experiments intersected with other experiments. Other experiments were parallel to the existing experiments. I’ve drawn a graph here showing the same kinds of things.

The darker lines are the results of a regression of data contained by the light gray rectangle. Each rectangle represents a single experiment and its replications.

Where the lines intersect, we can call those results commensurate. They result from what Kuhn called normal science. The experiments were designed differently, but reflect a single theory. The measurements within a single experiment reflect a particular apparatus. Changing the apparatus would give you another experiment with potentially different results.

Where the lines don’t intersect, we can call those results incommensurate. I’ll point out the gaps in the next figure. These gaps reveal an inadequacy in the current theory.

This graph can show us all of the experiments at once. But, that covers up things that would be revealed better in an animation. We don’t know, from this graph, when a particular result showed up. If we attended to the temporal aspects of the underlying data, we’d be able to see other gaps. The experiments characterized the gaps across the ranges and domains of several experiments.

ln this figure I’ve highlighted the continuities, the intersections, with red squares. I’ve assumed that all of these intersections exist. The results of one experiment, in the top left, is shown in blue.  I’ve assumed that this experiment was incommensurate and that the experiments that intersect with it did not exist at the time. The experiment that connected it to the chain of experiments to its right happened later.

The experiments shown with red lines are still incommensurate. They exhibit gaps with those experiments to their right. At the bottom right, three experiments exhibit continuity with each other, but exhibit a gap with both the other experiments above and to their right, and the other experiments to their left.

Normal science looks like a well connected network. Extending the range and domain of the existing theory is the job of normal science. A single regression would result in a decreasing function. Where the details differ from that single regression, we have an opportunity for clear functional differentiation.

Each of those commensurate experiments enables continuous innovation that extends the life of a category after the discontinuous innovation gives birth to the category. The technology adoption lifecycle is driven by improvements in a technology’s price-performance curve or S-curve. It is the price-performance curve that delivers on the promises made when the technology was sold and purchased. The demanded performance gets easier and easier to deliver and the range and domains of the underlying experiments expand.

In the next figure,  I’ve circled the discontinuities, the gaps, the incommensurate experiments. We won’t pursue experiments to bridge the gaps labeled G and H. We won’t try moving to G, because we can already read that temperature. We might want another way to take that measurement. We could develop a pass-fail thermometer where we are just interested in knowing if we have to make a bigger effort to get a numeric reading. Then, jumping that gap would make sense. The gap H just hasn’t been worked on yet.

Next, I went back and color coded the labeled gaps. The black rectangles show the range and domains involved in bridging a given gap. Bridging a gap requires new theory. The gap at A is from the experiment represented by the  blue line to the experiment on the right. The gap at E can bridge to any of three branches on the right. Any one branch will do. Continuous paths can get you to the other branches. Think step functions. The gap at F actually gaps with a collection of experiments to its right. The gap at B bridges two large subnets. Bridging this gap is critical. The gap at D can bridge to the left or the right. Either will do. Again, paths exist to get to and from the left and right side.

In this figure, I’ve put lines at the bridged gaps indicating the use of a new parameter that enables us to bridge the gaps. These parameters are labeled p and q. Their use was described in a new theory. The dark purple lines demonstrate how a continuous path through the network resolves a branch in resolving the gap.

The gaps E and A were resolved via parameter p and the network flow. The three gaps at F were resolved by parameter p as well. The gap at B was resolved by the solution to the gap at F. The gap at G continues to be ignored. The gap at D and C was resolved via the parameter q and network flows. The gap at H, again, ignored.

In these experiments basic research has showed us where our opportunities lay. It has delivered to us the incommensurate seeds of categories, and the commensurate life blood of new growth (dollars, not populations) to lift us slightly from the swamps of the margins from our nominal operations.

Another Explanation

The simplest explanation of what incommensurate means is that every theory is a water balloon. A theory can only hold so much of what it manages to contain. When you want more than a theory can deliver, when continuous improvements run out, you need a new trick to combine two water balloons. Have fun with that.