Scalar-limited concept
of
MEASUREMENT
When we were young, we developed the ability to recognize that if John is bigger than Joe, and Joe is bigger than Jerry, then John is bigger than Jerry.  That was a big step that leads to some very useful understanding of simple rank orders.  Mathematicians call it "transitivity."

But transitivity is only a first step toward a much more powerful concept of number and measurement.  It is, however, a step that takes us up to the edge of easy human comprehension.  The next steps are much more difficult.  The mathematics is quite recent; late 19th century, in fact.  (Calculus comes from the 17th century.)

Color is a vector

Human color is a three (component) vector.
Bird color is a four, five, or six vector.
Spectrographic color is an infinity vector.
Reflectivity is a rank-two, order-infinity tensor...tensors!??  See it–IN DEPTH–here............tensors!??  See it–IN DEPTH–here.

Protanopic color is a two vector.
Total colorblind color is a one vector, a scalar.

Only scalar measure leads to transitivity and (unique) rank ordering.

Nevertheless, we almost automatically interpret all measure as scalar.  If it can be measured, it can be  rank-ordered, we intuitively sense.  Relationships are transitive.  The rank order is unique.

Our intuition is oversimplification; very great oversimplification.

But we might also sense that "some things cannot be measured."  This feeling—perhaps it's only a vague feeling—might be the beginning of awareness of things beyond immediate human comprehension, of things beyond the oversimplification of our intuition.  (It might not, of course—how unfortunate!)

That sense of unmeasureability, far more often than not, is our sensing of  measure more complex than scalar: it doesn't lead to unique rank order, which is virtually all there is to our idea of "measure."  But we err in thinking that means it "can't be measured."   It can be measured — as something just a little beyond the edge of easy human comprehension.  Most measures have multiple components.  Like color.

In physics:

  • Speed is scalar; velocity is a vector.
  • Position is a three vector; but the laws of physics aren't self consistent until a fourth component is mixed into the description to give the space-time four-vector.  It's a quaternion, a very special mathematical entity. 
  • (Many more examples might be added to the list.)


Outside physics:

  • IQ is a scalar; intelligence is more complex.
  • Cost is scalar; value is more complex.
  • (This list can be extend almost indefinitely: add your favorite examples.)


Logical errors result when a scalar measure is substituted for a multicomponent measure.  For example:

Cost-benefit analysis usually divides one multicomponent measure by another.  Division by most multicomponent measures has no mathematical meaning: it simply is not allowed.  (There are a couple of exceptions which allow, for example, division by a space-time vector and division by an alternating current value.  These are division by a quaternion and a complex number, respectively.)  Before the economist can calculate a cost-benefit ratio, he or she must first squeeze the many components of cost and benefit into a couple of scalars. This assures oversimplification and opens wide the door  to personal bias. 

Intelligence is almost universally understood as something by which people can be rank ordered.  Nevertheless, ability to wield language with great skill and ability to easily recognize improperly inverted implications seem to be virtually independent.   J. P. Guilford finds about 17 dimensions in the space of human intelligence. 
 


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