|If we push to the more and more subtle aspects of the thing
we are trying to understand, of the problem we are trying to solve, we'll
encounter a "perception" barrier. Usually sooner than later.
That next step is more subtle; it's a slippery step which causes
us to slip back into more-concrete thinking. Someone might even explain
the step but we don't quite "see" it.
"Parameter" has come to mean a limit or boundary; its original, math meaning often becomes a falter point because it's a little more abstract. (It's "...a varable or an arbitrary constant appearing in a mathematical expression..." American Heritage Dictionary.) The science concepts from Newton (17th century) onward are things that take a little (or a lot) of contemplation to get past falter points. They are things we can't just point our eyes at and "see."
The more we learn and understand, the more realize that most of the universe is, in fact, outside our understanding -- like most of the electromagnetic spectrum is outside our perception. Much -- and likely most -- of that "outsideness" is in ways, in dimensions, that we find hard or impossible to grasp.
Very young people sometimes
encounter falter points that adults find amusing. Look at a child's
Click on the eyes to see more.
|The simplest perception -- colorblind individuals falter -- but
so do the rest of us
We have no choice but to resort to rather sophisticated math to even ask, much less answer, questions about color vision if its dimension is greater than the normal human three. Bird color has four to six dimensions. Most people who have two-dimension colorblindness never quite grasp what normal three-cone vision is. They haven't experienced the perception, and the needed extrapolation enters the realm of mathematical abstraction, giving a sense that "it's not real." But anyone with normal 3-cone color vision feels that that color is very real indeed. However, the four, five, or six dimensions of bird color presents to the "normal" human eye and brain the same challenge as normal color presents to the colorblind person. That gives a strong sense of unreality. A bird knows better.
The color perception of birds tells us a lot about our own perception and thinking.
The discoveries of science are the successes in meeting such challenges, challenges of going beyond the evolved perceptions and easy human reasoning.. The principles of science are the abstractions that extend our normal, evolved perceptions. (Of which the first level is the metaphors which are so powerful in our thinkng about the world. The "energy" of engine fuel and the "energy" of food constitute the ancient concept of energy that we use in everyday thought and speech. The "energy" of science is very different, and it's very abstract.)
click on pics to see PBS falter
blue cones - green cones
no red cones
|The simplest physics -- an expert falters
In the most elementary physics of moving objects (dates back to about 1685), the concept of acceleration is the lynchpin of the usefulness of the science. We must be able to distinguish between velocity and acceleration before we can begin to use our knowledge of such things. Both velocity and acceleration are instantaneous values of a ratio in which values may be changing constantly, and so the concept of derivative (calculus) must be understood (although the computational formulas need not be memorized).
A trivially simple, self-answering, question can be used to separate the understanders from the memorizers: "What is the direction, up or down, of the acceleration of a freely bouncing ball at the bottom most point of its bounce, that is, at the instant it's velocity changes from down to up?"
Many classes have answered this question and we can expect 50% correct if they simply guess the answer. Most clsses get 5% correct! Interviews indicate that they had memorized the direction of the acceleration for the topmost point in the bounce.
More revealing was the physics instructor who suggested that the question would be unfair to put on an exam if the answer to the question had not been specifically taught to the class. That instructor was not abstracting the concept of "understanding" which the question was designed to test. He was pushing against a "perception" barrier and it was resisting.
click on ball to see it bounce
|The simlpest logic -- another expert falters.
Many arguments get stuffed with improperly inverted implications -- necessity confused with sufficiency, all confused with some, etc. The slipperiness of the implication concept is demonstrated by P.C. Wason's card selection puzzle. A few people who've never dealt with the p's and q's of an elementary logic course nevertheless get the correct answer immediately and easily. They recognize the abstract relationship as something very common in everyday life.
A revealing example is the professional researcher who had actually published papers on the difficulties of the problem (very popular topic in the 70's in some circles). He claimed that the difficulty was that so very few people had taken courses in logic, and that the abstraction is no more than a construct of academic human minds which doesn't exist in the "real" world, Note that this "constructivist" notion is very popular with the Post-Modern philosophers.
click on cards to read all about it