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."
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
Protanopic color is a two vector.
Total colorblind color is a one vector,
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
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.
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
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:
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.
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.
TO THE BEGINNING OF "SEE LIKE A BIRD"