(The Cognitive Development of Humankind.)
Piaget has outlined the stages of cognitive development of children, adolescents, and adults, in the individual sense. Could we combine this with an outline of the collective cognitive development of humanity as a whole? This would correspond mainly to the evolution of science and other fields of knowledge. This process is not only unfinished at present, but very probably will never be finished, as we cannot expect to have access to the complete comprehension of all the truth about everything.
The brain and the sense organs are our means for at least partial comprehension of reality, but it is important to understand that they did not evolve primarily for that purpose; their evolutionary value is for knowing just enough about the environment to ensure survival. The thirst for abstract knowledge (scientific curiosity) is a later addition, but one which we value greatly.
The Piaget stages of individual cognitive development include the sensori-motor stage (up to 18 months of age), the pre-operational stage, the stage of concrete operations, and the stage of formal operations.
During the sensori-motor stage, we imagine that the infant perceives the world as an ever-changing colorful kaleidoscope, without making much sense of it; there must be confusion, but also much pleasure and wonderment. There is probably no sharp distinction between the self and the world, no appreciation of the flow of time, no constancy of objects or the expectation that objects or persons persist and may return. However, all this is largely conjectural, as we have no direct memory of early childhood (the memory mechanisms have not yet matured), and probably overstated, since we know that the infant does from a rather early age recognize its mother or primary caregiver and bonds or im-prints quite tenaciously to her.
The next stage, the pre-operational, is the stage when certain fundamental “schemata” (Piaget’s term) first develop and give the surrounding world some order and shape. It is important and interesting to realize that these schemata are already generalizations from experience, akin to early scientific laws. One of these is the schema of permanent objects: not only do certain groups of visual sensations group together to define a perceived object (something that hangs together), but also it persist when we are not looking at it and we can see it again when we look in that direction or return to that room. If the object is a person, like the mother or the nurse or a sibling, the person may walk away but then return. (The child has some considerable anxiety about the reappearance of loved persons; hence the educational and reassurance value of playing peek-a-boo and later hide-and-seek.) Philosophers who later throw doubt on the permanency and persistence of the objective world when we are not perceiving it go against this primitive schema of a very young child, which is deeply ingrained in the adult, and so meet much resistance to this notion.
Another sensori-motor stage schema is the concept of time and temporal succession: that of one event come before or after another. The arrow of time is not necessarily a “given” in nature, but is probably a biological necessity, as will be explained later. To the concept of temporal succession is soon added the schema of causality: that an event does not merely occur after another in time, but that it occurs because of the action of that earlier event. The schema of causality is also deeply ingrained, being formed so early in the child’s mind, and difficult to erase later, when certain philosophical doubts may arise.
It is already evident that these early schemata are not derived deductively or proved with certainty, since they can later be put in doubt by mature philosophers. This again emphasizes the fact that these schemata serve primarily the function of making some sense of the world and introducing some order into sense experience in order to facilitate survival, not primarily to discover the true nature of ultimate reality.
During the pre-operative stage, the somewhat older child is especially busy acquiring language; not only the sounds (both the production and the understanding of them), but also the formation of concepts that are intrinsically linguistic in nature. Thus children learn to use symbols and understand thought. They form the idea that other persons think like them. Even before that, they get to distinguish sharply between inanimate objects and persons, and the need to interact quite differently with each. And they discover those interesting entities that are certainly not objects, but not quite like persons — animals like dogs, cats, other pets, and for the fortunate children, farm animals. At this age, plants are usually still regarded as inanimate, mainly because they do not move. The fact that they grow like animals is realized only later.
However, there are still many things that the pre-school child does not know; some of them may surprise parents and older siblings. One of most often cited is that the child at this stage does not know that the volume of a liquid such as water is conserved, i.e. that pouring water from a tall thin vessel into a broad shallow one preserves the same quantity of water. The child does not know, until taught in the higher grades of elementary school, that mass is conserved. And ideas of what causes things to move are not at all well formulated.
When a child matures to the stage of concrete opera-tions, he or she is usually already in school and at the “latency plateau” of late childhood in emotional development (a stage which Erikson terms “industrious” or which could be called “thoughtful”). While at the sensori-motor stage, learning was mainly due to the maturation of the infant’s primitive nervous system (still getting wired up), and at the pre-operational stage changes were mainly due to experience (defined as an active interaction between the natural and social environment and native intelligence), at the stage of concrete operations the main factor is the cultural transmission of knowledge from the older to the younger generation, i.e. formal teaching and learning. The child learns not only in school from teachers, but also at home from family members, and in play groups from other children. These days children also learn at an early age from television (not always desirable things, e.g. violence), and then increasingly from books, as soon as their reading skills be-come adequate enough to make reading enjoyable.
At this stage, children learns how to classify and order objects and concepts and how to handle and manipulate numbers. They know the notions of space and time, the logic of classes and relations, and they form elementary ideas of geometry and physics. This is all apart from also learning and accumulating a lot of facts about the world, such as geography and knowledge of nature, and a bit of history. But it is children’s ideas about elementary physics that we want to focus on, since we are tracing the cognitive development of scientific ideas among humans.
Normal children have a great deal of natural curiosity about the world in which they live, quite apart from the practical aspects. They are always asking “why”, questions which many adults cannot answer, to their embarrassment. In that sense, every normal child is a scientist. Many of the early ideas that children form turn out later to be wrong, but this is quite normal in science, where hypotheses are being formed and perhaps later overthrown and superseded by others. This inductive reasoning process begins quite early and spontaneously.
Let us illustrate the first ideas about the physical world which usually arise. One is that application of a force is needed to make things move from a state of rest. This seems to follow naturally from the experience of using one’s muscles to move an object along the floor, such as a toy wagon. Another early physics idea is that heavy objects fall faster than light objects. Again, early observation of a hammer and a feather falling seems to confirm this. Our child scientist is not being foolish in formulating these first hypotheses, only inexperienced in carrying the observations and experiments to a more sophisticated level. We can call this the stage of pre-Newtonian (or pre-Galileo) physics. Adults who have never studied physics probably still believe in these pre-Newtonian ideas, and they get along quite well, showing that this stage of physical knowledge is quite adequate for coping with the world.
Sometime during Junior High or regular High School, children and adolescents enter the stage of formal operations, which upgrades the skills and knowledge of the previous stage to higher and more refined levels. They learn the basic operations for handling both numbers and language: addition, subtraction, multiplication, division and fractions in arithmetic, grammar, syntax, and composition in verbal and literary skills. They can handle the logic of hypothesis-formation, induction and deduction, and propositional logic, combinatorial lattices, group structures, and the explanation of classes and categories. While during the stage of concrete operations, they probably knew already how to add and multiply, and they certainly knew how to construct a grammatically correct sentence (Chomsky thinks that this knowledge is innate), during the stage of formal operations they learn WHY these processes are correct and why they work and should be used. The various concepts and logics that were implicit are now formulated explicitly. Once this is done, it is possible to build on these foundations such higher structures as combinatorics, infinite series, calculus, and computer programming, to name just a few.
While maturation was needed to confirm and consolidate the sensori-motor development, and experience continued to build the pre-operational stage, and social transmission operated at the stage of concrete operations, the process which finally consolidates the stage of formal operations (being able to operate consciously at the abstract level) is called “equilibration” by Piaget, who sees it as a form of self-regulation by which these new modes of thought are consolidated and become a steady habit.
Let us go on to higher stages of physical knowledge about the world. From this point on, this kind of knowledge is no longer essential for survival, it becomes a pure search for truth. Not all children or adults engage in it, only those highly interested and motivated. And since this knowledge is recorded in books and journals, it is cumulative; it becomes formal science. Also from this point on, we can regard further development as collective for humanity as a whole rather than individual for each scientist or truth-seeker.
The next stage after pre-Newtonian physics is the phy-sics of Newton and Galileo. It replaces the statement that force is needed to maintain the motion of a material body to the statement of Newton’s two laws of motion: that a body is at rest or in a state of uniform motion if no force is acting on it, and that a force is proportional to acceleration or change in the direction of motion (not to the velocity, but to its rate of increase — its first derivative). This is somewhat counter-intuitive when a student first encounters it, and it takes the student some time to internalize it. Those of us who have done this a long time ago tend to forget this initial feeling of surprise and discomfort.
The first law, which expresses the equivalence of being at rest and moving uniformly in a straight line (and shows us that there are no physical means of distinguishing these states) is already a statement of relativity; this is where relativity begins — a kind of a pre-Einsteinian relativity. In the second law, we find out, to our surprise, that the reason why we have to push a toy wagon along the floor is only the overcome the force of friction. Newton’s laws of motion force us to re-think a lot of fundamental concepts which we thought we knew.
The other notion, that heavy bodies fall faster than light bodies, is also superseded, as was shown 3 centuries ago by Galileo Galilei. The reason why the feather seems to fall more slowly than the metal hammer is air resistance, again something akin to a frictional force. And so we learn to view things more abstractly, not centred in our particular situation of rough floors and dense air, but in an imaginary world of a frictionless smoothness and an airless vacuum. It is more remote from ordinary life experience, but it is closer to the truth; though, as we shall see, not the whole truth by any means.
After the stage of Newton-Galileo physics, things become somewhat more complicated, branching off in four different directions, without as yet any obvious integration. This started to happen during the 19th century with the development of thermodynamics and then statistical mechanics, and continued at the beginning of the 20th century, when Einstein introduced first the theory of special relativity and the theory of general relativity, Planck introduced quantum theory, and Schrodinger and De Broglie added wave mechanics to quantum mechanics. In our own days, new theories of complexity are emerging: self-organizing struc-tures and chaos. These almost simultaneous developments in what I perceive as four different directions represent cognitive revolutions in our ways of thinking about the physical world which parallel in their far-ranging implications those earlier upheavals in thinking from concrete to formal operations or from intuitive to Newtonian physics. I will try to explain each of the four different directions in turn.
Thermodynamics is the science that deals with conversions of mechanical energy into heat and vice versa. Where it differs basically from ordinary dynamics is that the conversion of mechanical to heat energy is not reversible, the way mechanical energy changes are (e.g. potential to kinetic energy and back in a pendulum), as long as friction is absent, since friction converts some of the kinetic energy into heat. In other words, while mechanical processes can be run equally well forwards and backwards without losing anything, thermodynamic conversions cannot. True enough, the sum total of energy is conserved, according to the first law of thermodynamics, but some of the energy, when converted into heat (especially at lower temperatures) becomes degraded, less useful, energy, which cannot all be converted back. This fact is embodied in the second law of thermo-dynamics, which states that entropy (a measure of degraded energy) always increases in spontaneous processes, and in any processes in closed systems.
It becomes clear why this is so when we consider the detailed molecular picture of the process by means of statistical mechanics. Heat also consists of motions, like the large-scale motions of e.g. a pendulum, but with heat energy this involves the random motions of very large numbers (like 10 to the 23rd power) of individual molecules, each moving in ways not coordinated with the motions of the other molecules. It then becomes increasingly improbable, as the number of molecules increases, that their random motions would ever line up again to move the whole object in one direction. Thus the total conversion of heat to mechanical energy becomes very highly improbable, though not in principle im-possible.
What this irreversibility does is to introduce the arrow of time, i.e. time’s unidirectional flow. Now this is certainly an intuitive notion which we would not resist, since for us too time flows in only one direction; in fact, it is the time-reversibility feature of classical Newtonian physics which should be counter-intuitive to us, but we are not always made aware of this feature when we first encounter elementary physics in school.
The idea that some processes are irreversible is also familiar to us from everyday experience: water flows downhill, not up, unless pumped; heat flows from the heated end to the cold end of a metal rod; when a sugar solution in one compartment is put in contact with pure water in another compartment by removing the partition between them, the sugar molecules move from the solution into the water until the sugar concentrations are equalized; and so on.
So basically we have no cognitive problems with thermo-dynamics and statistical mechanics, but may be troubled by another problem. If with time nature tends to move toward increasing entropy, which means increasing disorder in the motions and patterns of the molecules, then where does the observed order in the universe originate? First of all in such structures as crystals (recall the beauty of a snowflake!), and then in living cells and organisms? Evolution too is a natural process, and what happened here to the law of increasing entropy?
Well, snowflakes and crystals are not so difficult: there are forces of attraction between molecules, essentially electromagnetic forces, which by a balance of attractions and repulsions position the molecules in regular arrays. These are possible and stable at low enough temperatures, where the random heat motions are not vigorous enough to disrupt the regular order. With increasing temperatures, all crystals eventually melt or vaporize, into phases (liquids and gases) which are increasingly disordered. It is essentially a matter of a tug of war between the forces which try to create order and the heat motions which disrupt order, and at higher temperatures the heat motions always prevail.
The explanation for living cells and organisms is very different, and involves another of the four directions which developed from Newtonian physics which I am trying to discuss. Suffice it to say here that living structures are open, not closed, systems, and that they do generate entropy in quite large amounts (we might call it pollution), but they export it to the environment, creating a high amount of order internally, like a temporary reverse eddy in a river which as a whole flows in the opposite direction.
The second change from Newtonian (classical) physics which we need to explore is the theory of relativity. As already stated, Newton’s first law of motion already announces the equivalence of the state of rest and the state of uniform rectilineal motion, and calls them experimentally indistinguishable by any laws of mechanics. Yet later scientists theorized that perhaps one could tell the difference by measurements of light (electromagnetic radiation), since light was thought to be wave motion in an all-pervading medium called the ether, which could be considered to be in a state of absolute rest.
But when Michelson and Morley in a famous experiment tried to measure the velocity of motion of the Earth through the luminiferous ether, they failed; Earth seemed to be at rest in the universe, which did not make any sense in the light of post-Copernican astronomy; moreover, it seemed to be at rest at different points of its orbit around the Sun, which made even less sense. The only possible conclusion was that there was no ether — and what is truly revolutionary, that the velocity of light is the same regardless of the state of motion of the observer, and that it is in fact the maximum velocity possible for anything to travel in the universe. These are the postulates of Einstein’s theory of special relativity, and they are highly counter-intuitive from the word go, and even more so in some of its conse-quences.
These consequences involve such notions as time flowing at different rates for observers in different states of motion, length measurements shrinking at high speeds when observed by someone not moving at this speed, but not shortening for those who do move with the measuring stick; and particles such as electrons increasing in mass as they approach the speed of light. A consequence of the latter is that matter and energy are interconvertible, and a derivation of the equation that E = Mc squared.
Considerable mind-stretching is required to internalize these new concepts, and you may wonder why we never see them operating in everyday life. The answer is that in ordinary life nothing moves at anywhere near the speed of light, except of course light, but there is no observer riding on the light beam. So Newtonian physics is quite satisfactory for ordinary purposes, and in fact is a special case (an approximation) derivable from Einsteinian physics when the velocities of motion are only very small fractions of the velocity of light.
The special theory of relativity deals only with motions at uniform velocities in straight lines. To generalize this, Einstein came out a few years later with the general theory of relativity. It could deal with accelerating (or slowing-down) motions and curvilinear motions, by linking this to gravitation (large massive bodies attracting each other; small ones do too, but there the force is too weak to measure). The overall conclusion was that the presence of matter introduces curvature into space, especially noticeable if it involves a very massive object like a star. Mo-tion of a planet around a star is then curvilinear (elliptical in general), because of this space curvature, not because of a special force called gravitation. (But actually both are equally valid ways of regarding what is happening.) In this case, the reason why we do not observe the operation of the laws of general relativity in ordinary life is because we do not normally deal with very large objects such as stars.
We are beginning to get the idea that the real world is very strange indeed, and the only reason why we normally see it as simple and “commonsense” is because we live in conditions where nothing moves too fast and nothing is too large — and, as we shall see in the next section, nothing is very small. Of course, the real reason why we normally think that the world is simple is that we have become accustomed to our everyday experiences and they BECOME commonsense because of that.
The third amendment to classical physics comes from the world of the very small, from atoms and sub-atomic particles, electrons and photons. The two mutually related theories here are quantum mechanics and wave mechanics. First Planck formulated the theory that light comes in little bundles called quanta — somewhat like particles, thought it is basically waves. (Actual Einstein made the first discovery here too, as in several other unrelated fields.) Then others discovered that electrons and other particles of matter has properties of waves, though they are primarily particles. Thus both radiation (like light) and matter took on the same two aspects, as both waves and particles, depending how you decided to measure them? How can they be both? This boggles the mind, and while there are several possible explanations, none of them are in consonance with common sense views. Now they are not particles and waves AT THE SAME TIME, they can be either depending on the measurement.
From this again stem many weird phenomena: e.g. the uncertainty principle (Heisenberg) according to which you can determine the exact position of an electron only if you give up on knowing much about its momentum, and if want to pin down the momentum, you will remain relatively ignorant about its position. Or, the famous two-slit experiment (a thought experiment really), in which a single electron goes through two slits and produces interference patterns with itself as a wave would. Which slit did it really go through? You are not supposed to ask. Or another famous purely thought experiment, in which a cat is either dead or alive depending on which slit the electron went through, and the poor cat’s state of being remains indeterminate until an observer looks inside the box; it is not just that the observer does not know if the cat is alive or dead until he looks — there would be nothing unusual about that — but the cat does not “know” either; it is in limbo between life and death until the observation “collapses the wave function” of uncertainty.
Do we meet quantum phenomena in ordinary life? Well, actually we can observe some in the macro world, like the weird properties of liquid helium near absolute zero, or in superconductors in which electric current can flow forever without resistance, or in the spectra of elements where the emission or absorption lines represent quantum jumps of electrons in an atom from one orbit to another, or in a so-called tunnelling microscope, in which an electron seems to go through a barrier in a “forbidden” transition. (Actually it may vanish and another electron may appear on the other side; but since electrons are “fungible” and not individually identifiable, that is the same as tunnelling through.) Yet all these phenomena appear only in laboratories or in high-tech applications, and ordinary people can get along quite well without internalizing the weird notions of quantum mechanics.
Finally, the fourth direction away from classical physics, which is more recent, comes from studies of very complex phenomena, such as living structures. It started out with Prigogine’s studies of so-called dissipative structures, the simplest ones not alive, but physical systems like water heated from below in a shallow pan forming convection patters, or chemical systems where cyclic reactions made the mix turn blue at exact alternating time intervals. Dissipative structures (not a very inspiring name) are defined as being open systems far from thermodynamic equilibrium, which maintains themselves at low entropy by a continuing inflow and outflow of materials and energy, like primitive metabolism. Such systems can form apparent temporary exceptions to the second law of thermodynamics. Another (more inspiring) name for them is self-organizing or homopoietic structures.
Further studies of complex structures involve chaos theory, which focuses on non-linear phenomena long ignored by physicists simply because they did not know how to solve the equations involved. Non-linear phenomena include such things as turbulent flow of liquids (frequently seen in nature in white-water mountain streams) and smoke rings; but it also involves such practical concerns as weather prediction. Weather is a non-linear system, and as such shows so-called instability, i.e. being very sensitive to the precise starting conditions and to very small disturbances. The common phrase is that weather shows the “butterfly effect”: if a butterfly flaps its wings in Thailand, it may rain in Kansas. It is this complexity and sensitivity that makes wea-ther prediction so difficult.
Another example of chaos theory occurs in the mathematical iteration of certain rather simple non-linear equations, e.g. the logistic equation which predicts population fluctuations in animal populations. At relatively low values of a parameter indicating the annual rate of increase, there is a single “attractor” (a value to which the iteration results converge); at higher values there are two attractors, and the population oscillates between two values. At still higher values the attractors split or bifurcate again and we have four attractors; then in rapid succession there are 8, 16, and so on, until above a certain critical value (which surprisingly is the same for different equations!) there is an infinite number of attractors and we say that chaos has set in. The population can now go to any value at all, even zero (signifying extinction), all because the rate of increase was too high!
Chaos phenomena are actual quite common in ordinary life, but they often go unnoticed. Fractal shapes belong here: these are irregular outlines like seashores which show fluctuations or roughness at all scales, from fairly large bays and promontories to grains of sand that stick out on a beach. The connection to living systems? It has been conjectured that life thrives at the boundary between order and chaos. This concept should still be better defined and refined.
It is time to recapitulate. This is probably best done by means of diagrams. In Figure 1, the Piaget stages are represented by a shaft leading up to Newtonian science, and from this then branch out the five other directions: special relativity for the very fast, general relativity for the very large, quantum mechanics for the very small, thermodynamics and statistical mechanics for the very numerous, and chaos and self-organization for the very complex. The overall shape of this model in three dimensions would be somewhat like a child’s windmill — five blades on a stick.
Figure 2 shows a head-on view of the windmill. For the sake of simplicity, special relativity has been left out, because it is included within general relativity. We now have something like a mandala, with classical physics at the centre. The axis from thermodynamics to complex systems represents the axis along which the arrow of time manifests itself, though for different reasons. The axis from relativity to quantum theory represents the scale from the very large to the very small, like the “Cosmic Zoom” cartoon film by Norman McLaren from the National Film Board of Canada.
Eventually other directions radiating out from classical physics may be found. For example, what laws apply at very high density, as in neutron stars or black holes? Possibly we will never be able to find out.
The big defect in all this is that no one has figured out how these different theories can be reconciled with each other. Scientists, including Einstein, have tried to formulate a Grand Unified Theory (GUT) combining general relativity and quantum theory, so far without success. Even this would not be the Theory of Everything that we really crave. We may have to be satisfied with a series of partial views without being able to put them together.
The real regret is that it is so difficult to internalize the knowledge presented even in the partial views, because they run so strongly against common sense. My ideal aim would be not only to arrive at GUT, but to develop a “gut feeling” for it.