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Some important themes pervade science, mathematics, and technology and appear over and over again, whether we are looking at an ancient civilization, the human body, or a comet. They are ideas that transcend disciplinary boundaries and prove fruitful in explanation, in theory, in observation, and in design.

This chapter presents recommendations about some of those ideas and how they apply to science, mathematics, and technology. Here, thematic ideas are presented under four main headings: systems, models, constancy and change, and scale.Top button



Any collection of things that have some influence on one another can be thought of as a system. The things can be almost anything, including objects, organisms, machines, processes, ideas, numbers, or organizations. Thinking of a collection of things as a system draws our attention to what needs to be included among the parts to make sense of it, to how its parts interact with one another, and to how the system as a whole relates to other systems. Thinking in terms of systems implies that each part is fully understandable only in relation to the rest of the system.

In defining a system—whether an ecosystem or a solar system, an educational or a monetary system, a physiological or a weather system—we must include enough parts so that their relationship to one another makes some kind of sense. And what makes sense depends on what our purpose is. For example, if we were interested in the energy flow in a forest ecosystem, we would have to include solar input and the decomposition of dead organisms; however, if we were interested only in predator/prey relationships, those could be ignored. If we were interested only in a very rough explanation of the earth's tides, we could neglect all other bodies in the universe except the earth and the moon; however, a more accurate account would require that we also consider the sun as part of the system.

Drawing the boundary of a system well can make the difference between understanding and not understanding what is going on. The conservation of mass during burning, for instance, was not recognized for a long time because the gases produced were not included in the system whose weight was measured. And people believed that maggots could grow spontaneously from garbage until experiments were done in which egg-laying flies were excluded from the system.

Thinking of everything within some boundary as being a system suggests the need to look for certain kinds of influence and behavior. For example, we may consider a system's inputs and outputs. Air and fuel go into an engine; exhaust, heat, and mechanical work come out. Information, sound energy, and electrical energy go into a telephone system; information, sound energy, and heat come out. And we look for what goes into and comes out of any part of the system—the outputs of some parts being inputs for others. For example, the fruit and oxygen that are outputs of plants in an ecosystem are inputs for some animals in the system; the carbon dioxide and droppings that are the output of animals may serve as inputs for the plants.

Some portion of the output of a system may be included in the system's own input. Generally, such feedback serves as a control on what goes on in a system. Feedback can encourage more of what is already happening, discourage it, or modify it to make it something different. For example, some of the amplified sound from a loudspeaker system can feed back into the microphone, then be further amplified, and so on, driving the system to an overload—the familiar feedback squeal. But feedback in a system is not always so prompt. For example, if the deer population in a particular location increases in one year, the greater demand on the scarce winter food supply may result in an increased starvation rate the following year, thus reducing the deer population in that location.

The way that the parts of a system influence one another is not only by transfers of material but also by transfers of information. Such information feedback typically involves a comparison mechanism as part of the system. For example, a thermostat compares the measured temperature in a room to a set value and turns on a heating or cooling device if the difference is too large. Another example is the way in which the leaking of news about government plans before they are officially announced can provoke reactions that cause the plans to be changed; people compare leaked plans to what they would like and then endorse or object to the plans accordingly.

Any part of a system may itself be considered as a system—a subsystem—with its own internal parts and interactions. A deer is both part of an ecosystem and also in itself a system of interacting organs and cells, each of which can also be considered a system. Similarly, any system is likely to be part of a larger system that it influences and that influences it. For example, a state government can be thought of as a system that includes county and city governments as components, but it is itself only one component in a national system of government.

Systems are not mutually exclusive. Systems may be so closely related that there is no way to draw boundaries that separate all parts of one from all parts of the other. Thus, the communication system, the transportation system, and the social system are extensively interrelated; one component—such as an airline pilot—can be a part of all three.Top button



A model of something is a simplified imitation of it that we hope can help us understand it better. A model may be a device, a plan, a drawing, an equation, a computer program, or even just a mental image. Whether models are physical, mathematical, or conceptual, their value lies in suggesting how things either do work or might work. For example, once the heart has been likened to a pump to explain what it does, the inference may be made that the engineering principles used in designing pumps could be helpful in understanding heart disease. When a model does not mimic the phenomenon well, the nature of the discrepancy is a clue to how the model can be improved. Models may also mislead, however, suggesting characteristics that are not really shared with what is being modeled. Fire was long taken as a model of energy transformation in the sun, for example, but nothing in the sun turned out to be burning.

Physical Models

The most familiar meaning of the term "model" is the physical model—an actual device or process that behaves enough like the phenomenon being modeled that we can hope to learn something from it. Typically, a physical model is easier to work with than what it represents because it is smaller in size, less expensive in terms of materials, or shorter in duration.

Experiments in which variables are closely controlled can be done on a physical model in the hope that its response will be like that of the full-scale phenomenon. For example, a scale model of an airplane can be used in a wind tunnel to investigate the effects of different wing shapes. Human biological processes can be modeled by using laboratory animals or cultures in test tubes to test medical treatments for possible use on people. Social processes too can be modeled, as when a new method of instruction is tried out in a single classroom rather than in a whole school system. But the scaling need not always be toward smaller and cheaper. Microscopic phenomena such as molecular configurations may require much larger models that can be measured and manipulated by hand.

A model can be scaled in time as well as in size and materials. Something may take so inconveniently long to occur that we observe only a segment of it. For example, we may want to know what people will remember years later of what they have been taught in a school course, but we settle for testing them only a week later. Short-run models may attempt to compress long-term effects by increasing the rates at which events occur. One example is genetic experimentation on organisms such as bacteria, flies, and mice that have large numbers of generations in a relatively short time span. Another important example is giving massive doses of chemicals to laboratory animals to try to get in a short time the effect that smaller doses would produce over a long time. A mechanical example is the destructive testing of products, using machines to simulate in hours the wear on, say, shoes or weapons that would occur over years in normal use. On the other hand, very rapid phenomena may require slowed-down models, such as slow-motion depiction of the motion of birds, dancers, or colliding cars.

The behavior of a physical model cannot be expected ever to represent the full-scale phenomenon with complete accuracy, not even in the limited set of characteristics being studied. If a model boat is very small, the way water flows past it will be significantly different from a real ocean and boat; if only one class in a school uses a new method, the specialness of it may make it more successful than the method would be if it were commonplace; large doses of a drug may have different kinds of effects (even killing instead of curing), not just quicker effects. The inappropriateness of a model may be related to such factors as changes in scale or the presence of qualitative differences that are not taken into account in the model (for example, rats may be sensitive to drugs that people are not, and vice versa).

Conceptual Models

One way to give an unfamiliar thing meaning is to liken it to some familiar thing—that is, to use metaphor or analogy. Thus, automobiles were first called horseless carriages. Living "cells" were so called because in plants they seemed to be lined up in rows like rooms in a monastery; an electric "current" was an analogy to a flow of water; the electrons in atoms were said to be arranged around the nucleus in "shells." In each case, the metaphor or analogy is based on some attributes of similarity—but only some. Living cells do not have doors; electric currents are not wet; and electron shells do not have hard surfaces. So we can be misled, as well as assisted, by metaphor or analogy, depending on whether inappropriate aspects of likeness are inferred along with the appropriate aspects. For example, the metaphor for the repeated branching of species in the "tree of evolution" may incline one to think not just of branching but also of upward progress; the metaphor of a bush, on the other hand, suggests that the branching of evolution produces great diversity in all directions, without a preferred direction that constitutes progress. If some phenomenon is very unlike our ordinary experience, such as quantum phenomena on an atomic scale, there may be no single familiar thing to which we can liken it.

Like any model, a conceptual model may have only limited usefulness. On the one hand, it may be too simple. For example, it is useful to think of molecules of a gas as tiny elastic balls that are endlessly moving about, bouncing off one another; to accommodate other phenomena, however, such a model has to be greatly modified to include moving parts within each ball. On the other hand, a model may be too complex for practical use. The accuracy of models of complex systems such as global population, weather, and food distribution is limited by the large number of interacting variables that need to be dealt with simultaneously. Or, an abstract model may fit observations very well, but have no intuitive meaning. In modeling the behavior of molecules, for instance, we have to rely on a mathematical description that may not evoke any associated mental picture. Any model may have some irrelevant features that intrude on our use of it. For example, because of their high visibility and status, athletes and entertainers may be taken as role models by children not only in the aspects in which they excel but also in irrelevant—and perhaps distinctly less than ideal—aspects.

Mathematical Models

The basic idea of mathematical modeling is to find a mathematical relationship that behaves in the same way the system of interest does. (The system in this case can be other abstractions, as well as physical or biological phenomena.) For example, the increasing speed of a falling rock can be represented by the symbolic relation v = gt, where g has a fixed value. The model implies that the speed of fall (v) increases in proportion to the time of fall (t). A mathematical model makes it possible to predict what phenomena may be like in situations outside of those in which they have already been observed—but only what they may be like. Often, it is fairly easy to find a mathematical model that fits a phenomenon over a small range of conditions (such as temperature or time), but it may not fit well over a wider range. Although v = gt does apply accurately to objects such as rocks falling (from rest) more than a few meters, it does not fit the phenomenon well if the object is a leaf (air drag limits its speed) or if the fall is a much larger distance (the drag increases, the force of gravity changes).

Mathematical models may include a set of rules and instructions that specifies precisely a series of steps to be taken, whether the steps are arithmetic, logical, or geometric. Sometimes even very simple rules and instructions can have consequences that are extremely difficult to predict without actually carrying out the steps. High-speed computers can explore what the consequences would be of carrying out very long or complicated instructions. For example, a nuclear power station can be designed to have detectors and alarms in all parts of the control system, but predicting what would happen under various complex circumstances can be very difficult. The mathematical models for all parts of the control system can be linked together to simulate how the system would operate under various conditions of failure.

What kind of model is most appropriate varies with the situation. If the underlying principles are poorly understood, or if the mathematics of known principles is very complicated, a physical model may be preferable; such has been the case, for example, with the turbulent flow of fluids. The increasing computational speed of computers makes mathematical modeling and the resulting graphic simulation suitable for more and more kinds of problems.Top button




In science, mathematics, and engineering, there is great interest in ways in which systems do not change. In trying to understand systems, we look for simplifying principles, and aspects of systems that do not change are clearly simplifying. In designing systems, we often want to ensure that some characteristics of them remain predictably the same.

Stability and Equilibrium

The ultimate fate of most physical systems, as energy available for action dissipates, is that they settle into a state of equilibrium. For example, a falling rock comes to rest at the foot of a cliff, or a glass of ice water melts and warms up to room temperature. In these states, all forces are balanced, all processes of change appear to have stopped—and will remain so until something new is done to the system, after which it will eventually settle into a new equilibrium. If a new ice cube is added to the glass of water, heat from the environment will transfer into the glass until the glass again contains water at room temperature. If a consumer product with a stable price is improved, increased demand may cause the price to rise until the expense causes the number of buyers to level off at some higher equilibrium value.

The idea of equilibrium can also be applied to systems in which there is continual change going on, as long as the changes counterbalance one another. For example, the job market can be thought of as being in equilibrium if the total number of unemployed people stays very nearly the same, even though many people are losing jobs and others are being hired. Or an ecosystem is in equilibrium if members of every species are dying at the same rate at which they are being reproduced.

From a molecular viewpoint, all equilibrium states belie a continual activity of molecules. For example, when a bottle of club soda is capped, molecules of water and carbon dioxide escaping from the solution into the air above increase in concentration until the rate of return to the liquid is as great as the rate of escape. The escaping and returning continue at a high rate, while the observable concentrations and pressures remain in a steady state indefinitely. In the liquid itself, some molecules of water and carbon dioxide are always combining and others are coming apart, thereby maintaining an equilibrium concentration of the mild acid that gives the tingly taste.

Some processes, however, are not so readily reversible. If the products of a chemical combination do not readily separate again, or if a gas evaporating out of solution drifts away, then the process will continue in one direction until no reactants are left—leaving a static rather than dynamic equilibrium. Also, a system can be in a condition that is stable over small disturbances but not large ones. For example, a falling stone that comes to rest part way down a hill will stay there if only small forces disturb it; but a good kick may free it to start downhill again toward a more stable condition at the bottom.

Many systems include feedback subsystems that serve to keep some aspect of the system constant, or at least within specified limits of variation. A thermostat designed to regulate a heating or cooling system is a common example, as is the set of biological reactions in mammals that keeps their body temperatures within a narrow range. Such mechanisms may fail, however, if conditions go far outside their usual range of operation (that is what happens, for example, when sunstroke shuts down the human body's cooling system).


Some aspects of systems have the remarkable property of always being conserved. If the quantity is reduced in one place, an exactly equal increase always shows up somewhere else. If a system is closed to such a quantity, with none entering or leaving its boundaries, then the total amount inside will not change, no matter how much the system may change in other ways. Whatever happens inside the system—parts dissolving, exploding, decaying, or changing in any way—there are some total quantities that remain precisely the same. In an explosion of a charge of dynamite, for example, the total mass, momentum, and energy of all the products (including fragments, gases, heat, and light) remains constant.


Besides the constancy of totals, there are constancies in form. A Ping-Pong ball looks very much the same no matter how it is turned. An egg, on the other hand, will continue to look the same if it is turned around its long axis, but not if it is turned any other way. A human face looks very different if it is turned upside down, but not if it is reflected left for right, as in a mirror. The outline of an octagonal stop sign or a starfish will look the same after being turned through a particular angle. Natural symmetry of form often indicates symmetrical processes of development. Clay bowls, for example, are symmetrical because they were continually rotated while being formed by steady hands. Almost all land animals are approximately left-and-right symmetrical, which can be traced back to a symmetrical distribution of cells in the early embryo.

But symmetry is not a matter of geometry only. Operations on numbers and symbols can also show invariance. The simplest may be that exchanging the terms in the sum X+Y results in the same value: Y+X = X+Y. But X-Y shows a different kind of symmetry: Y-X is the negative of X-Y. In higher mathematics, there can be some very subtle kinds of symmetry. Because mathematics is so widely used to model how things in the world behave, symmetries in mathematical descriptions can suggest unsuspected symmetries that underlie physical phenomena.

Patterns of Change

Patterns of change are of special interest in the sciences: Descriptions of change are important for predicting what will happen; analysis of change is essential for understanding what is going on, as well as for predicting what will happen; and control of change is essential for the design of technological systems. We can distinguish three general categories: (1) changes that are steady trends, (2) changes that occur in cycles, and (3) changes that are irregular. A system may have all three kinds of change occurring together.


Steady changes are found in many phenomena, from the increasing speed of a falling rock, to the mutation of genes in a population, to the decay of radioactive nuclei. Not all of these trends are steady in the same sense, but all progress in one direction and have fairly simple mathematical descriptions. The rate of radioactive decay in a sample of rock diminishes with time, but it is a constant proportion of the number of undecayed nuclei left. Progressive changes that fit an identifiable mathematical form can be used to estimate how long a process has been going on. For example, the remaining radioactivity of rocks indicates how long ago they were formed, and the current number of differences in the DNA of two species may indicate how many generations ago they had a common ancestor.


A sequence of changes that happens over and over again—a cyclic change—is also familiar in many phenomena, such as the seasonal cycles of weather, the vibration of a guitar string, body temperature in mammals, and the sweep of an electron beam across a television tube. Cycles are characterized by how large the range of variation is from maximum to minimum, by how long a cycle takes, and by exactly when its peaks occur. For the daily cycle in human body temperature, for example, the variation is about a degree; the cycle repeats about every 24 hours; and the peaks usually occur in late afternoon. For the guitar string, the variation in movement is about a millimeter, and each cycle takes about a thousandth of a second. Cycles can be as long as the thousands of years between ice ages, or shorter than a billionth of a second in electric oscillators. Many phenomena, such as earthquakes and ice ages, have patterns of change that are persistent in form but irregular in their period—we know that it is their nature to recur, but we cannot predict precisely when.

The extent of variation during a cycle can be so great as to disrupt the system, such as when vibrations crumble buildings in earthquakes, or can be almost too small to detect, apparently lost in the random activity of the system. What is random and what is regular, however, is not always obvious from merely looking at data. Data that appear to be completely irregular may be shown by statistical analysis to have underlying trends or cycles. On the other hand, trends or cycles that appear in data may sometimes be shown by statistical analysis to be easily explainable as being attributable only to randomness or coincidence.

Cyclic change is commonly found when there are feedback effects in a system—as, for example, when a change in any direction gives rise to forces that oppose the change. A system with such feedback that acts slowly is likely to change a significant amount before it is nudged back toward normal; and when it gets back to normal, the momentum of change may carry it some distance in the opposite direction, and so on, producing a more or less regular cycle. Biological systems as small as single cells have chemical cycles that result from feedback because the products of reactions affect the rates at which the reactions occur. In complex organisms, the feedback effects of neural and hormonal control systems on one another produce distinct rhythms in many body functions (for example, in blood cell counts, in sensitivity to drugs, in alertness, and even in mood). On the level of human society, any trend is eventually likely to evoke reactions that oppose it, so there are many social cycles: The swing of the pendulum is evident in everything from economics to fashions to the philosophy of education.


Knowing the exact rules for how parts of a system behave does not necessarily allow us to predict how the whole system behaves. Even some fairly simple and precisely defined processes, when repeated many times, may have immensely complicated, apparently chaotic results. Most systems above the molecular scale involve the interactions of so many parts and forces and are so sensitive to tiny differences in conditions that their detailed behavior is unpredictable.

In spite of the unpredictability of details, however, the summary behavior of some large systems may be highly predictable. Changes in the pressure and temperature of a gas in equilibrium can often be predicted with great accuracy, despite the chaotic motion of its molecules and the scientist's inability to predict the motion of any one molecule. The average distribution of leaves around a tree or the percentage of heads in a long series of coin tosses will occur with some predictable reliability from one occasion to the next. Likewise, prediction of the average behavior of members of a group of individuals is likely to be more reliable than prediction of any individual's behavior. Climate is fairly predictable, even if daily weather is not.

Many systems show approximate stability in cyclic behavior. They go through pretty much the same sequence of states over and over, although the details are never quite the same twice: for example, the orbiting of the moon around the earth, the human cycles of sleep and wakefulness, and the cyclic fluctuations in populations of predator and prey. Although such systems involve interplay of highly complex influences, they may persist indefinitely in approximating a single, very simple cycle. Small disturbances will result only in a return to the same approximate cycle, although large disturbances may send the system into very different behavior—which may, however, be another simple cycle.


The general idea of evolution, which dates back at least to ancient Greece, is that the present arises from the materials and forms of the past, more or less gradually, and in explicable ways. This is how the solar system, the face of the earth, and life forms on earth have evolved from their earliest states and continue to evolve. The idea of evolution applies also, although perhaps more loosely, to language, literature, music, political parties, nations, science, mathematics, and technological design. Each new development in these human endeavors has grown out of the forms that preceded them, and those earlier forms themselves had evolved from still earlier forms.


The fins of whales, the wings of bats, the hands of people, and the paws of cats all appear to have evolved from the same set of bones in the feet of ancient reptilian ancestors. The genetic instructions for the set of bones were there, and the fins or paws resulted from natural selection of changes in those instructions over many generations. A fully formed eye did not appear all of a sudden where there was no light-sensitive organ before; nor did an automobile appear where there had been no four-wheeled vehicle; nor did the theory of gravitation arise until after generations of thought about forces and orbits.

What can happen next is limited to some extent by what has happened so far. But how limited is it? An extreme view is that what happens next is completely determined by what has happened so far—there is only one possible future. There are two somewhat different reasons for doubting this view. One is that many processes are chaotic—even vanishingly small differences in conditions can produce large differences in outcomes. The other reason is that there is a completely unpredictable random factor in the behavior of atoms, which act at the base of things. So it seems that the present limits the possibilities for what happens next, but does not completely determine it.


Usually, people think of evolution of a system as proceeding gradually, with a series of intermediate states between the old and the new. This does not mean that evolutionary change is necessarily slow. Intermediate stages may occur very rapidly and even be difficult to identify. Explosions, for example, involve a succession of changes that occur almost too rapidly to track—whether the explosions are electric as in lightning, chemical as in automobile engines, or nuclear as in stars. What is too rapid, however, depends on how finely the data can be separated in time. Consider, for example, a collection of fossils of fairly rare organisms known to have existed in a period that lasted many thousands of years. In this case, evolutionary changes that occurred within a thousand years would be impossible to track precisely. And some evolutionary change does occur in jumps. For instance, new biological developments do not arise only by successive rearrangement of existing genes but sometimes by the abrupt mutation of a gene into a new form. On an atomic scale, electrons change from one energy state to another with no possible intermediate states. For both the gene and the electron, however, the new situation is limited by, and explicable from, the previous one.


Evolution does not occur in isolation. While one life form is evolving, so are others around it. While a line of political thought is evolving, so are the political conditions around it. And more generally, the environment to which things and ideas must respond changes even as they are evolving—perhaps impeding, or perhaps facilitating, their change in a particular direction. For example, abrupt changes in a long-steady climate may lead to extinction of species that have become well adapted to it. The economic hardships in Europe after World War I facilitated the rise of the Fascists who instigated World War II. The availability of recently developed mathematical ideas of curved spaces enabled Einstein to put his ideas of relativity into a convincing quantitative form. The development of electricity fostered the spread of rapid long-distance communications.Top button



The ranges of magnitudes in our universe—sizes, durations, speeds, and so on—are immense. Many of the discoveries of physical science are virtually incomprehensible to us because they involve phenomena on scales far removed from human experience. We can measure, say, the speed of light, the distance to the nearest stars, the number of stars in the galaxy, and the age of the sun, but these magnitudes are far greater than we can comprehend intuitively. In the other direction, we can determine the size of atoms, their vast numbers, and how quickly interactions among them occur, but these extremes also exceed our powers of intuitive comprehension. Our limited perceptions and information-processing capacities simply cannot handle the whole range. Nevertheless, we can represent such magnitudes in abstract mathematical terms (for example, billions of billions) and seek relationships among them that make sense.

Large changes in scale typically are accompanied by changes in the kind of phenomena that occur. For instance, on a familiar human scale, a small puff of gas emitted from an orbiting satellite dissipates into space; on an astronomical scale, a gas cloud in space with enough mass is pulled together by mutual gravitational forces into a hot ball that ignites nuclear fusion and becomes a star. On a human scale, substances and energy are endlessly divisible; on an atomic scale, matter cannot be divided and still keep its identity, and energy can change only by discrete jumps. The distance around a tree is much greater for a small insect than for a squirrel—in that on the scale of the insect's size there are many hills and valleys to traverse, whereas for the squirrel there are none.

Even within realms of space and time that are directly familiar to us, scale plays an important role. Buildings, animals, and social organizations cannot be made significantly larger or smaller without experiencing fundamental changes in their structure or behavior. For example, it is not possible to make a forty-story building of precisely the same design and materials commonly used for a four-story building because (among other things) it would collapse under its own weight. As objects increase in size, their volume increases faster than their surface area. Properties that depend on volume, such as capacity and weight, therefore change out of proportion to properties that depend on area, such as strength of supports or surface activity. For example, a substance dissolves much more quickly when it is finely ground than when it is in a lump because the ratio of surface area to volume is much greater. A microorganism can exchange substances with its environment directly through its surface, whereas a larger organism requires specialized, highly branched surfaces (such as in lungs, blood vessels, and roots).

Internal connections also show a strong scale effect. The number of possible pairs of things (for example, friendships or telephone connections) increases approximately as the square of the number of things. Thus, a community ten times as large will have approximately a hundred times as many possible telephone connections between residents. More generally, a city is not simply a large village, since almost everything that characterizes a city—services, work patterns, methods of governance—is necessarily different from, not just larger than, that in a village. Systems sometimes include so many interconnected components that they defy precise description. As the scale of complexity increases, eventually we must resort to summary characteristics, such as averages over very large numbers of atoms or instants of time, or descriptions of typical examples.

Systems of sufficient complexity may show characteristics that are not predictable from the interaction of their components, even when those interactions are well understood. In such circumstances, principles that do not make direct reference to the underlying mechanisms—but that are not inconsistent with them—may be required. For example, the process of scouring by glaciers can be referred to in geology without reference to the underlying physics of electric forces and crystal structure of minerals in rocks; we can think of the heart in terms of the volume of blood it delivers, regardless of how its cells behave; we can predict the likely response of someone to a message without reference to how brain cells function; or we can analyze the effects of pressure groups in politics without necessarily referring to any particular people. Such phenomena can be understood at various levels of complexity, even though the full explanation of such things is often reduced to a scale far outside our direct experience.Top button

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