An interesting puzzle for those who mastered their high-school physics:

What determines the order in which several split peas fly off a smooth turntable as its angular velocity increases from zero to fast enough to send all the peas flying?

How did we get here?
Traveling from egocentrism toward science
A Case Study

Study of human rationality must eventually encounter the work of Jean Piaget. Not necessarily his interpretations; he missed a couple of things. But he understood that human rationality is a product of biological evolution.

Piaget watched human rationality under construction by observing children develop from infancy to adulthood. He saw coherent and orderly changes in reasoning. He was asking, "Where has evolution been taking us in our abilities to know the world, and how does it go about it?"
"Now I admit nature can't improve upon man: We're probably the supreme being,"
said Floyd Dominy,(1) expressing his version of "What I see is what exists."
This is genetic epistemology. It studies developments. It, like physics, is often something other than what it seems. Most of Piagetís followers usually seek something different: "How do we learn?" they ask. They often have little interest in epistemology and even less in genetics.

"What I see is what exists."

"Egocentrism" plays a central role in Piagetís work. For example, Piaget asked children to draw a picture as seen from anotherís viewpoint. The youngest drew the picture from their own viewpoint; older children drew it correctly. Such egocentrism makes understanding the beeís knowledge of polarization of light, the sharkís knowledge of electric field, and the birdís six-factor color very difficult. (A recent PBS program suggested that perhaps only mankind has color vision: all other living creatures are colorblind, they suggested.)

Expanding awareness comes with expanding elements of reasoning. For example, Piaget poured water from a short, broad beaker into a tall, skinny beaker and asked if the quantity of water was then less, the same or greater. Only the older children realized that it must be the same. He squashed a sphere of clay to make its diameter larger and asked the same question. Only the older ones knew it must be the same. And then Jan Smedslund came along and squashed the clay and dropped it in water to check whether the volume of the clay changed by observing how much the water level rose. But Smedslund surreptitiously removed a piece of clay before dropping it in the water the second time. Children at an age where the logic was just developing might rationalize with the arguments typical of younger children: "Itís flatter, so it must be smaller." However, some of the older children jumped off their stools and insisted that the experimenter open his hands and turn out his pockets. They knew! "Buzz-saw certainty(2)."

Piaget didnít pay much attention to the earliest developments, such as color vision (first few weeks), stereopsis (10-19 weeks), and even language (pretty much in place by age 2-4 yrs). He paid a lot of attention to our notions of imagining another personís viewpoint, conservation, reversibility, classification; these develop around ages 5-8. Such notions get placed into a system of loosely coordinated cognitive elements that become our intellects.

Whatever evolutionary developments lead us to scientific understanding must be the most "advanced." These develop after about age 10-12. They lead to "buzz-saw certainty" of such relationships as complex negations of negations, and the multi-element logic you learn in logic courses with Venn diagrams and pís and qís of Boolean algebra. They are, roughly speaking, kinds of logic upon logic.

"Everything is relative."

Since Piaget was an ardent admirer of Einstein, his epistemology was "constructivist." He saw the abstractions of science much as did the researcher who saw Wasonís card selection puzzle as so difficult for most people "because implication doesnít exist in the real world but is only an abstract, academic construct." He did not see p-implies-q as a real-world entity everywhere about us all, as did one physicist who immediately solved the puzzle on first seeing it and then explained how and why most people would probably give a wrong answer and how the puzzle exemplifies very common logic errors.

Relativists and constructivists usually see things like color as being more real than more abstract entities, like energy and other concepts of physics. Einstein is seen to have supported this view by showing that "everything is relative." Einstein actually assumed the very opposite in order to obtain a consistent theory of electromagnetic interactions relatively free of arbitrary subjectivity. In fact, human color is actually an arbitrary selection, by evolution, of a three-dimensional shadow of the infinite dimensions of wavelength distribution of light. Energy-and much else in physics-is as free of arbitrary or subjective selection as any knowledge we have and is so because it is abstract. Abstraction has power to see into the reality beyond egocentrism.

Constructivism thus turns us away from a really valuable insight.  It turns us away from seeing our simple perceptions as a useful exemplar for the "higher" human information skills.  It keeps us from "seeing" a more coherent-whole picture of a wide range of observations.  The critical age feature of "Piagetian developments" was assailed from the beginning on the grounds that children could learn behaviors at ages before the critical age for the development. The developmental quality of elements of reasoning was thereby seen as invalid. But thatís like denying a distinction between color vision and colorblindness on the grounds that a colorblind person can learn color names to go with every object encountered.

Many rejections of distinctions between learning and development crumble into mind dust when examined with this exemplar.

Physics is simple but subtle . . . subtler than you might think.

In one of Piagetís studies(3) of the development of the reasoning we need for understanding science (they used mostly physics problems), this situation was presented to children ages 6-15:

Three metal balls of different weights are placed on a disc at three different distances from its center.  The disc is rotated faster and faster until the balls roll off the disc because of centrifugal force.  The problem is to predict in what order they will leave their initial positions and why.
This is similar to the split-pea-on-a-turntable problem with metal balls in place of the peas. But when a sphere, unlike a hemisphere, sits on a turntable, friction cannot keep the sphere from rolling. The instant the turntable starts to rotate, a sphere will start to roll. The only way Piagetís experiment can work as described is for some kind of depression or hole to hold the ball within its edge. However, read through the description and you will find no mention of any such holes.

But letís assume Piagetís rotating disc has the needed holes. What would determine the rotational speed at which a ball leaves its hole and rolls off? This is a fairly simple application of Newtonís laws of motion, a problem one might find at the end of the chapter in any first year physics text, high school or college.

The answer is appropriately simple:

Consider the moment just before the ball rolls out: the upward force on the ball by the edge of the hole is no longer distributed around the edge but is now concentrated at the point where the ball will roll out.
As the angular velocity, w, increases, the direction of the force exerted on the ball by the edge of the hole moves from vertical toward the horizontal.  The ball rolls out of the hole when that line passes through the center of the ball.  That value of w is wflee.

where g is the acceleration due to gravity, r is the distance from the axis of rotation to the center of the hole, and q is the angle the ball must roll up to escape the hole. The mass of the ball is not a factor: anyone who understands that mass is irrelevant to trajectories of bodies in free fall should intuitively see that mass must similarly be irrelevant here.

Piaget and coworkers saw the physics differently.

They invoked "centrifugal force," OK if you understand Newtonís laws, fraught with pitfalls if you donít. Then they "derived" a "formula" for the centrifugal "force," F = mw2r, and stated:

...since the speed of the disc is constant with the initial acceleration, the subject need isolate only the factors m and r . . . a ball is displaced sooner in direct proportion to its weight and later in inverse proportion to the distance from the center . . . A heavy ball placed at a point nearer the center may move at the same time as a lighter one closer to the periphery.
Their initial phrase is incomprehensible. After that, they confused mass and weight, a distinction essential to the correct analysis of this problemĖbut a pervasive and persistent point of misunderstanding in Newtonís laws. And, while failing to identify mass as irrelevant, they completely missed one of the two critical influences, the geometry of the ball sitting in the hole. Piaget and his coworkers clearly did not quite understand Newtonís laws of motion, and they failed to sort the relevant from the irrelevant.

One child may have tried to interpret the problem more correctly than did the experimenters: (Experimenter), "What determines the result?" (Child), "The size and the holes a little; no only the size because the holes are all the same; and the force with which the ball is thrown off." (This is the only indication-buried in one of the detailed protocols-that the holes were in fact there.) The experimenters apparently guided that child away from a correct analysis.


Human "rationality" is the application of biologically evolved elements of information processing. Some, especially those we call "perceptions," are firmly developed for most people. Others, such as those which start developing at about 5-7 yrs, are "perceptions of abstractions" that nearly everyone "sees" with considerable ease-conservation, classification, transitivity, and negation, e.g.
Finally, come a more advanced set of "insights"-such as complex negation of negation, Boolean relationships, statistical relationships, extrapolation to limits, and ratio & proportionality-that usually develop to a much lesser clarity of "perception." These give us insights into multi-element interactions; they let us sort out whatís relevant from what is not, let us systematically construct alternatives to what actually exists, and show us logical contradictions that we otherwise miss. This results in science distinguishable from pseudoscience.  That distinction can be very sharp.  The insights may take some hard work.

Physics is simple but subtle. Subtler than Piaget realized. Subtler than most graduates of elementary physics courses realize. Subtler than many authors of science textbooks realize.  Subtler than believers of pseudoscience realize. Physics is seldom what it seems at first.

And physics is the simplest of modern science.

Richard Feynman, describing all of seventeen shelf-feet of science textbooks he reviewed:
 ďEverything was written by someone who didnít know what the hell he was talking about, so it was a little bit wrong, always!  And how are we going to teach well by using books written by people who donít quite know what they are talking aboutÖThe books were lousy.  They were false.  They were hurriedÖThey were teaching something they didnít understand, and which was in fact uselessÖĒ (4)

  1. In Cadillac Desert (after the book by Marc Riesmann), PBS-TV (1997). MORE
  2. See Martin Gardner's "Buzz Saw Puzzle" as described on this Web site.
  3. B. Inhelder  & J. Piaget,  Growth of Logical Thinking, from Childhood to Adolescence, Basic Books (1958), pp 210 - 223.
  4. R. Feynman, Surely Youíre Joking, Mr. Feynman, Bantam Books (1985), pp 262-276.