A Distinctive Family of Errors Part II: At the Edge of Human Comprehension

By Phil Pennington

The author is a member of O4R and a physicist who has worked at Hughes Semiconductors and on the physics faculty of PSU. He has a special interest in problems of education and the misunderstanding of physics.

There is an edge to human comprehension. An edge where the "obvious" goes perpetually unobserved, where the very simple can be very difficult. We know evolution has left information hidden from us. It is difficult, however, for us to even imagine the occult world beyond human perception, although we know other creatures have been endowed with perceptions beyond our own: polarization perception of bees; magnetic field detection of birds and bacteria; electric field perception of sharks; six-component color vision of birds; ommatidia vision, rather than focused images, of insects; sound wave imaging of bats and cetaceans; etc. Science has opened human perception beyond its evolutionary edge, and then has pushed beyond these primitive, non-human perceptions, ultimately to the most profound and mysterious perception - that of the abstract.

     Science works at this edge of human comprehension, constantly trying to make knowledge lying further and further beyond the edge somehow intelligible. Progressively, seventeenth and eighteenth century science corrected misinterpretations of everyday experience. By the twentieth century science was pushing far beyond human perception, using subtle insights and sophisticated mathematics. Newton's three laws of motion, unnoticed through all of human history, corrected errors of interpretation of everyday experience and laid the foundation for the sophisticated insights of the Einsteinian universe that followed.

     Pseudoscience often ambitiously reaches far outside the edges of human comprehension, but without the corrections and sophisticated insights of science. We can better understand science and pseudoscience by understanding how insights at the edge of human comprehension succeed - and fail. Knowledge at this edge has properties which go pervasively and persistently unnoticed and unattended. Think about the following observations. They are not as trivial as they may at first seem:

   While the winter solstice is December 20, the earliest sunset occurs on December 9 and the latest sunrise on January 2 - obvious, yet unobserved.

   Around February 4 the graph of Portland's daily record low temperatures shows a sudden discontinuity, a whopping 30 degree rise, which goes unnoticed even by TV weather forecasters.

   Many TV weather broadcasters and sunscreen ad writers believe temperature influences the ability of sunlight to cause sunburn. It does not (see Pro Facto, Fall 1995).

   Most physics students who learn Newton's first law of motion, "every body will persist in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it," nevertheless believe that motion implies a force. Their belief obviously contradicts the principle they have learned.

   Most physics students can solve simple problems involving force and acceleration, yet cannot correctly answer the question, "What is the direction, up or down, of the acceleration of a freely bouncing ball at the bottommost point in its bounce, that is, at the instant its velocity changes from down to up?" The correct answer (up) is obvious when one has a working understanding of acceleration, since the question answers itself in its last phrase.

   Said on Star Trek, "If we don't restore power in 20.7 seconds our orbit will decay and we will crash into the planet." Said of the roller coaster with six and a half seconds of free-fall, "When you go up the hill gravity will take over eventually and you'll just hang there for six and a half seconds, then you'll fall back down and end up where you started." Said of rocket propulsion, "According to the laws of physics, you can't have motion without an opposing force." These statements are inconsistent with Newton's first, second and third laws of motion, in that order.

   Most dictionaries and many physics textbooks define energy with "energy is the capacity to do work," but elsewhere define free energy as "total energy minus energy unavailable for doing work." An obvious contradiction.

   Wason's card puzzle is persistently answered wrong. While for puzzle-solvers the solution seems obvious, for non-solvers the points are persistently missed.

   "I could care less!" is commonly said when "I could not care less!" is meant. Or, "The ship will destruct in T minus five minutes!" when "The ship will destruct in five minutes!" is meant. Negation of negation is simple; and while the statements obviously say the opposite of what is intended, the error often goes unnoticed.

   An educational TV program on perception depicted one-component color vision (total colorblindness) with black and white photography, but two-component color with a blue monochromatic picture. Clearly the production team did not understand dimensionality (degrees of freedom) of color vision.

     The odd 24-day offset from the earliest sunset to the latest sunrise gives us a window into the same knowledge that led to Newton's laws of motion. Newton's insights were simple, but also subtle. In some sense they are obvious; but the qualities that kept them unobserved before Newton, keep them largely unobserved today. Most students of elementary physics see them as abstract, academic, not real, useless. So perceived, these concepts cannot be fully comprehended: they will be unseen when they are encountered in the real world.

     Today we have technical advantages for observing the odd behavior of sunset and sunrise times that Newton could never have dreamed of - television, digital watches and easy access to precision time standards. Our path of reasoning should go something like this: The offsets are the same every year. They occur at the summer solstice, but with half the offset. The effect, therefore, could not be due to the influence of the other planets or some other random effect. The effect's symmetry rules out refraction by the atmosphere, an explanation given by one TV weather forecaster who didn't notice refraction would make the effect symmetrical around noon. Between December 9 and January 2, both sunrise and sunset are getting later. This means the time halfway between sunrise and sunset - essentially solar noon - is getting later, so daylength, as measured from noon to noon, is changing. Daylength is determined by both the earth's rotation and the earth's orbit. We can dismiss rotation once we time rotation through observation of a star - or with our knowledge of the forces that would be required to change the earth's rotation velocity. This leaves us with the realization that sunrise and sunset times tell us interesting things about the orbit of the earth, including the fact that it lacks circular symmetry.

     Newton and Kepler had to start with less obvious facts than these! So the next January when you notice your drive home is getting lighter but your drive to work is the darkest of the year, ask yourself, "What other simple observations might lead to 'obvious but unobserved' - potentially important - knowledge?"

     Were Newton's laws of motion so different from ordinary insight? And, if so, how? Newton's first law of motion required recognizing that many potential influences, including the force of friction, must be sorted through and eventually extrapolating to the unattainable limit of no friction. Jean Piaget gave children experimental apparatus to direct them to this discovery. The children were asked to predict the motion of marbles rolling across surfaces with various friction: carpet, linoleum, glass, etc. Piaget was studying the normal, genetically guided development of reasoning. In essence he was examining where evolution has taken our abilities to know the world, and how has it done so. About five percent of the children recognized Newton's first law. Physics professor, Jack Clement, taught that law and tested students for their understanding. About five percent understood; the rest only learned procedures for solving related textbook problems. The majority were "learners," not "seers," and persisted in the erroneous belief that "motion implies a force." Clement taught, Piaget did not; and yet, the outcomes were essentially the same. An obvious, but often unnoticed, implication is that teaching and learning can be irrelevant to important aspects of understanding. Newton's insights were first steps leading outside the edge of human comprehension. They are insights, not particularly teachable, that must be "seen." Usually they are not seen, and understanding remains - unknowingly - pre-Newtonian, pre-scientific. Newtonian physics is obvious, but unobserved: simple, but difficult.

     The faulty definition of energy and the difficulty of Wason's card puzzle both come from a simple, but subtle, property of multi-element interactions: the distinction between necessity and sufficiency.

     The energy of common usage refers to our needs when we feel fatigue, hunger or deprived of sleep. The most salient property of such energy is that it is not conserved. Intertwined in this common usage is "the capacity to do work," a nineteenth century concept that tries to quantify colloquial energy. The energy of science is one element necessary, but not sufficient, for getting work out of a system. It is unfailingly conserved. Saying "energy is the capacity to do work" is like saying "a vegetable is a potato." There are other kinds of energy, including "the energy unavailable for doing work." Physics textbook authors who define energy as the capacity to do work are not distinguishing between mutually inverse implications. They are confusing necessity with sufficiency.

     Wason's card puzzle, likewise, lies at the edge of human comprehension because of this simple, but subtle, principle of the relationship of logical implication, "if A then B." The most common error in card selection is to see the relationship only as "vowel and even number go together," in which case, the answer seems to be to turn over the vowel and the even numbered card. Many people go no further. Others, however, will systematically look at all possibilities and ask more questions: What about the even numbered card? So what if there is a vowel on the other side? They will, finally, turn over the vowel and the odd numbered card. The required reasoning straddles the edge of human comprehension. Wason's card puzzle exemplifies the distinction between "learning" and "seeing." Solving the puzzle follows a logical, obvious path, but for many the "obvious" remains unobserved, unreal. The answer might be learned while remaining unseen.

     Are the "seers" more intelligent? The question itself is illogical because it assumes that intelligence is scalar, i.e. that it has one component. Multi-component measures are most often misinterpreted as having properties of one-component measures. Physics textbook authors who insist upon defining energy as "the capacity to do work" are no doubt highly intelligent in the academic, IQ, repeat-on-tests-what-you've-been-told measure of intelligence. An intuitive sense for the relationship of logical implication, however, seems to be at least somewhat independent of this one-dimensional measurement of intelligence. In fact, learning ability is probably irrelevant. Using a learned response to Wason's problem could indicate a failure to understand, just as the person who must learn that the top light on a traffic signal is red probably has colorblindness and isn't really seeing red. A good indication that an individual is actually "seeing" implication relationships is when that person notices the puzzle demonstrates the importance of falsifiability (a la Karl Popper).

     Understanding the distinction between necessity and sufficiency is a requirement for understanding the interplay of multiple, interacting influences. Some physicists seem to believe that because the basic laws of physics are necessary for a complete description of anything, those laws are sufficient for a complete description. They search for that "simple set of differential equations that describe, explain and predict everything." New-age pseudoscientists see that the basic laws of physics are insufficient to describe and predict the really important human issues and declare the laws to be unnecessary. "Science is only one alternative reality," they respond. Both groups are confusing necessity and sufficiency.

     Mathematics often baffles. Newton's second law of motion, F = ma, states that the acceleration of an object is proportional to the net force acting on it; the constant of proportionality is the object's mass. Proportionality and vectors baffle (force and acceleration are both vectors). Vectors are generally not recognized as having multiple, inseverable parts. Simple derivatives also baffle. Newton's second law is a proportionality of two vector quantities, one of which is a derivative. Probably less than 15% of first year physics students can recognize and apply Newton's second law in unfamiliar situations.

     When one understands proportionality, one understands huge numbers of dollars - millions, billions, and trillions - because one prorates appropriately; one understands conversions of quantities expressed in different units; and one understands physical quantities expressed as constants of proportionality, such as electrical resistance, viscosity, and capacitance - a big step in understanding physics.

     The components of a multicomponent measure, such as color, are as inseverable as the two sides of a sheet of paper. An isolated, single component (and any rank ordering by the measure) is oversimplification, and usually nonsense. Velocity, acceleration, color, intelligence, monetary value, the value of a human being, etc. are correctly seen only as inseverable multiple elements. Although we usually associate rank order with any measurement, unique rank order is a property of one-component measure only. Anyone who understands multi-component measure will take umbrage at rank ordering people by intelligence, value or worth. "All men are created equal" recognizes the worth of one man cannot be unambiguously declared greater than another. Cost/benefit analysis divides a multi-component cost by a multi-component benefit: mathematical nonsense. The division conceals biases and arbitrary choices because both multicomponent measures must be oversimplified to single-component measures.

     Perhaps 95% of students do not learn as their instructors intend they should. The instructor expects certain simple basic principles to be understood and applied to simple lifelike situations. Instead, the majority of students learn ritualistic solutions to textbook-type problems. These students learn much, but "see" very little. And when they look at the world around them, they do not recognize the relationships to what they have learned. When students do recognize a principle, very little learning is needed. When the principle is seen, it becomes useful. There is an important difference between learned knowledge and understood knowledge.

     The difference between learning and understanding is exemplified by the totally colorblind person who must learn the color names to go with every object encountered. That's a lot of learning compared to learning the names of the colors, as is done by color-seeing individuals. Additionally, two colors that are obviously different to the color-seeing person can look identical to the colorblind individual. Arguments that they are identical will not convince the color-seeing person. In this case, it is not correct to say one view is "correct" and the other "incorrect." Both are correct, one is more complete and, therefore, more useful than the other.

     Misinterpretation of multiple, interacting elements, influences and variables creates this distinctive family of errors. It is oversimplification that searches for "The Cause" when there are many causes.

     If we recognize the prevalence of multiple elements we will recognize the need for the word "parameter" and not be so baffled by it that we substitute it when we mean "perimeter" (a limit or boundary). We will be more resistant to deception like that of advertising which substitiutes one glowing statement for another, one more real but with less appeal. Nor will we use "all" for "some" as in, "Doctors recommend ..." when it was only the two doctors working for the manufacturer. We will sense the need for the unique word "unique" (meaning one and only) and not accept its popular corruption which uses "unique" to mean unusual, thus confusing one with several.

     If we recognize that concepts are constructed of multiple elements, we should recognize that the concept of "truth" has multiple elements. We need the whole truth: sufficiency of knowledge is as important as its validity. We need nothing but the truth: no deceptions, by self or others, and no goofs. We especially need the many-faceted correspondence with the world that gives knowledge its ability to help us anticipate the outcomes of our interactions with the world. Therein lies the lies the power of knowledge.