Chapter 1

An Overview

1.1. Introduction

Mathematical creativity consists in either recognizing that an existing formalism is applicable to the problem at hand or inventing a new one. [Kac and Ulam.]

SOLVING problems involves the use of two types of talent: imagination and skill. In general, the classroom approach to problems tends to emphasize skill because, to a certain extent, skill can be communicated through the knowledge of techniques. However, it is also possible to exercise the imagination through exposure to the analysis of a diversity of imaginatively formulated problems. Formulation seems to depend on the ability (and the knowledge) of the individual to structure problems. The pursuit of solutions is well documented in the mathematical and scientific literature. Considerably much less formal knowledge is available regarding formulation than there is about solution. The only way we have so far to train an individual in modeling is to expose him to a wide variety of problems and to a corresponding variety of models which provide representations of those problems. This establishes the need for a methodological framework for problem formulation.

We should make it clear that this chapter is not intended as a summary of what is covered in the rest of the book. It is designed as our introduction to modeling with some ideas about its philosophy and practice. The material here will be helpful to the student, since it will give him some general ideas about modeling and since it provides a framework or guide in helping him to structure problems. It offers him criteria for testing not only the models in this book to see how they meet the needs of his problems but also his own models as they develop. This chapter is designed to show the variety of thinking about modeling in general, just as the rest of the book is designed to show a variety of models; this is the spirit in which the chapter should be viewed.

1.2. Problems: Identification, Formulation, and Solution

When faced with a problem one first attempts to brainstorm all its relevant aspects, its internal variables, its parameters, and interactions with external factors. Some of the results of brainstorming are qualitative, others are quantitative. The next stage involves a classification of the results of the brainstorming into groups of ideas which belong together. Modeling involves a careful examination of the quantitative components to determine what can be measured and what cannot. The measures of effectiveness of the process under study must be stated and incorporated into a model. In grocery-store queues, for example, the measure of effectiveness is the number of people able to pass through the line per unit of time. These measures define the problem by showing the degree of fulfillment which is possible with the current system and indicating what would be preferred. Often a statistical approach to a scientific problem is the first that comes to mind. Then an estimation model, often formed from a hybrid of statistics and algebra, is used to find a rough way of pursuing the answer.

The problem may be deterministic or it may be subject to chance occurrences and therefore probabilistic. In general a deterministic problem can have a descriptive or a normative setting. In a descriptive framework, equations and inequalities are used to relate the variables of the problem; these equations or inequalities may be algebraic, differential, difference, or integral. One or many solutions of a descriptive framework may be possible. In the normative setting of a problem, there is usually an objective function to be maximized or minimized, subject to descriptive equations or inequalities as constraints. Here one seeks out the optimal from among many possible solutions.

If a problem involves both optimization and probabilities, then the approach would ordinarily require maximizing or minimizing expected utilities after first defining the utility or objective function in terms of the random variables of the problem.

The qualitative description of a problem involves the definition of the objectives being pursued. This is followed by an account of the real-life process to be adopted to attain the objectives. It is essential to identify both controllable and uncontrollable factors in the process. A supermarket can control the flow of goods during its working hours and the number and efficiency of its checkout clerks. However, it has no major control over what a customer may want to buy, although it may control this somewhat by making only certain types of food available.

Analysis of the process requires gathering data and other information. In order to solve a problem it must be decided at the outset how the study is to be conducted. The conclusions and the way they are to be communicated must be developed for persuasion or selling purposes according to whether one is serving as consultant, lawyer, consumer, sponsor, law-enforcer, or decision-maker. Hypotheses are constructed on the basis of the information available, and a model is formed to test the hypotheses. The problem is then analyzed quantitatively through a suitable model which may be based on geometry, statistical analysis and correlation, ratio and proportion, rate of change, input/output (or matrix relations in general), linear algebra, equivalence, ordering preference with weight assignment, probability, optimization with single interest (programming, variational, or control theoretic) or multiple interests (game theory), and graphs and discrete mathematics. In any case, the model may be a first cut at the problem in the form of an estimation of the answer, as an upper or lower bound. Some useful guidelines about problems and problem solving will now be considered.

1.2.1. GUIDELINES ON FORMULATING THE PROBLEM

1. Define the problem and give its history and its causes.
2. State the objectives and the constraints.
3. Are you sure that this is the problem you want to solve? Why do you want to solve the problem?
4. Is there anyone who needs the solution? Are you sure you need the solution? Why?
5. Are there other related problems, perhaps easier, which should be solved first? List them.
6. What is the solution needed for?
7. What effect would it have?
8. How would you implement it?
9. How much would it cost to solve the problem? What are the resources available?
10. How much benefit would there be from the solution?
11. If the problem is ignored, will it terminate over time?
12. Get outside the problem and look at it. Is it significant? What is your vantage point for this judgment?
13. What are the stable solutions of the problem?
14. Can change in law or administration eliminate the problem?
15. Can the problem also be viewed as someone else's problem? Perhaps you can engage his cooperation in modeling a solution?

1.2.2. GUIDELINES ON HOW TO SOLVE THE PROBLEM

1. Does the problem have a solution?
2. Give all alternative solutions: are they exhaustive? How do you demonstrate this?
3. Give an optimal or near-optimal solution.
4. Give an average solution.
5. Give an approximate solution.
6. Start at both ends: the raw data and a hypothesized answer and move toward the middle to develop justification.
7. Start in the middle and move towards the ends.
8. Embed the problem in a larger context and solve it.
9. Abstract the problem to a simpler formulation.
10. Can you derive the solution from that of a related problem?
11. Simulate the problem in search of solution.
12. Construct a working hypothesis.
13. Develop and test the hypothesis.
14. Define the utilities and the payoffs in the process being studied.
15. How sensitive is the solution to changes in the data?
16. What are the invariants of the problem as reflected in the solution?
17. Update feedback of the implemented solution onto the problem under study.
18. Analyze the faulty solutions of the problem to get a better understanding of the preferred ones.

1.2.3. IMPACT OF THE SOLUTION

1. How can you communicate the problem and its solution to others? What is the most effective way to convince different people of your solutions?
2. What is the impact of the solution on people, things, etc.?
3. What people should be involved in implementing the solution?
4. What personnel commitments, organizational structure, and equipment are needed to find the solution and, in particular, to implement it?
5. What happens to the organization after the problem has been solved and the solution implemented?
6. Can the organization solve other problems?
7. How should people be motivated to solve the problem?
8. What are the sanctions on, and threats to, the individuals and organizations involved?
9. What is the moral impact of the problem and its solution on people?
10. Will there be a chain reaction, either creating new problems or solving other problems, as a result of this solution?
11. Are there residual problems which must now be solved?

Knowing that the foregoing comprises major steps in problem solving, take a problem and brainstorm its description and solution with a group. Attempt to quantify the problem. Indicate the type of information needed to make the solution operationally useful.

1.3. Mathematical Models and their Applications

The underlying idea in most of the models presented here is to indicate, in the form of an equation or inequality, (i) an algebraic relationship between variables, (ii) the rate of change of some variables with respect to other variables, and (iii) the sums or integrals of functions in order to obtain cumulative values and to see how they relate to other variables or functions. Naturally, some models differ from these basic types, but the majority will fall into these three classes. The fields of optimization and of stochastic processes each utilize these classes of formulations within a particular framework which is derived from concepts and operations peculiar to their fields. In studying formulations of models it is well to bear in mind that the number of basic ways available for formulation is very limited. The challenge is to ascertain that the given formulation is appropriate to the solution of the problem under study.

Roughly speaking, mathematical models may be divided into two types:

(i) Quantitative or based on the number system. By means of this category of models one attempts to answer questions asking “How many?” or “How much?” It can be used to express relations between elements and properties of systems.

(ii) Qualitative, possibly based on set theory, but not reducible to numbers. With this type of model, one studies relations between systems and their properties. Frequently the formulation of a quantitative model is preceded by a qualitative analysis of the problem under study.

Qualitative mathematical models include the use of axiomatics, set theory, group theory, and graph theory. Quantitative models may in turn be divided into two general categories, continuum (relating to the real or complex number fields), and discrete (relating to the integers). Discrete mathematics frequently involves counting and estimating numbers. Its major branches include Diophantine equations and number theory, optimization in integers and mixed optimization, combinatorial mathematics, certain aspects of graph theory, and geometric number theory. For our purpose, continuum mathematics may be taken as nondiscrete mathematics.

Both categories involve concepts from three major fields: (i) equations and inequalities, (ii) optimization, and (iii) probability and stochastic processes. They can be displayed, as in the following triangle (Fig. 1.1 ).

FIG . 1.1.

The abstract study of these three fields rests mostly in the realm of functional analysis. There is interaction between each pair of these three areas. For example, the field of optimization involves equations and inequalities which describe constraints on the system. Stochastic processes also use equations and inequalities to describe the behavior of systems. Optimization problems are increasingly recognized as occurring in conjunction with probability.

It is not an oversimplification to say that both major types of mathematics, qualitative and quantitative, are concerned with solving problems whether by characterization or by construction. This process may be outlined as follows.

1.3.1. THE PROCESS OF SOLVING

1. A priori bounds. One often begins by establishing the maximum number of solutions a problem can have. If in this process none is found to exist, there is no need to proceed further. If there are several one may decide to pursue a most desirable one in some sense. There is a saying, “There are many ways to skin a cat.” But there is only one way to satisfy thirst, for example, and there are no ways for running up Mount Everest from the bottom in one hour.

2. Existence and uniqueness. Are there any solutions? This differs from the previous step in that one must prove that there is a solution instead of simply putting a bound on the number. Also, there may be no solution. For example, consider the problem of covering a chessboard from which two diagonally opposite squares have been removed. Is it possible to cover it with dominoes, each covering exactly two squares without overlapping? It may easily be seen that there are no solutions to this problem. (Consider the colors of the missing squares and the parity of covering by a single domino.)

3. Convergence. If an iterative instead of a closed form solution method is used, do the iterations converge to the desired solution?

4. Approximations and errors. An iterative process must be stopped. How good an approximation to the solution is given by the last iteration? What is the error incurred by this approximation?

1.3.2. EQUILIBRIUM AND STABILITY

The concept of equilibrium plays a central role in all important modeling whether relating to the solution of an equation, to an optimization problem, or to a stochastic process. We encounter the idea of equilibrium either directly or indirectly depending on the mathematical framework chosen.

It has been well recognized in the development of scientific theories that, to construct a useful model, one must adopt notions around which it is possible to analyze equilibrium and stability. These two ideas are pivots on which analysis revolves. A system is said to be in stable equilibrium if after a small disturbance it tends to return to its original state. It is unstable when a small disturbance tends to move it further and further away from its original state. A ball at the bottom of a salad bowl is in stable equilibrium, but a ball blocked by a pebble on a sloping street is unstable as a small disturbance will release it, never to return. Of course, oscillation is a form of stable behavior. In many social affairs stable equilibria are desirable situations and unstable equilibria are undesirable. For instance, an economic system violently fluctuating between boom and depression is undesirable, whereas one remaining in a well-balanced intermediate position is desirable. There have been several different approaches to political problems utilizing models based on stable equilibria. We will examine some later in this chapter.

There are instances in which stability is a bad thing and an unstable equilibrium is a good thing. To illustrate the first, consider the case of a person caught running between two tigers where the only way to escape is to get closer to one of the tigers. His momentary optimal position is halfway between. It is extremely dangerous for him to stay there, however, so he should make up his mind to escape in the direction of one of the tigers before they close in on him. He needs some extra psychological energy (that is, courage) to escape from this momentarily stable equilibrium.

An example illustrating the desirability of unstable equilibria is that of a small child, placed between two chocolate bars, who has been offered (and constrained) to take only one of them. For a moment he may be in a state of suspended equilibrium, unable to decide on which to choose, but he will soon escape from this unstable equilibrium situation of indecision by some minimal disturbance which attracts his attention to one of them and which makes him rush to that one. More generally, choosing between two good things is usually easy (unstable equilibrium), but choosing between two evils may be very difficult even if indecision is disastrous (undesirable stable equilibrium).

A further generalization of equilibrium is used when analyzing a conflict or competition situation involving several parties. In game theory the concept of “equilibrium point” refers to a situation in which none of the parties has an incentive to change his current strategy as long as the other parties do not change their strategies; the strategies at such a point are called “equilibrium strategies.” It is also possible to classify equilibrium points according to whether they have short-range or long-range characteristics.

The conditions for equilibrium in the case of differential equations are generally obtained by equating to zero the velocity components of the system. The case where the coefficients are also functions of time is a more difficult one and gives rise to the idea of “dynamic equilibrium,” which is a state of equilibrium in a continually changing system. The mode of approaching the equilibrium depends on the roots of the characteristic equation of the system. It is this analysis of the existence and character of the equilibrium that provides understanding of the long-run behavior of a system.

An example of “static equilibrium” is illustrated by a cold car engine which is slow to get started and which takes some time to warm up. During this time it vibrates, coughs, and is exceedingly reluctant to go. After a while this type of activity levels off as the engine warms up. One would never guess now that the car had been so unresponsive in leaving its state of inertia, which is one type of static equilibrium, for another state of static equilibrium, the steady state of operation.

An example of dynamic equilibrium is illustrated by a rigid rod, held at one end by a vertically oscillating pivot. Under some circumstances, the rod will remain upright so long as the pivot continues to oscillate and will fall down and behave like an ordinary pendulum once the oscillations cease. The stability of the system depends on the dynamics (time-varying behavior) of the pivot. Now for a comment on political equilibria.

Some have argued against equilibrium as a goal, saying that nations seeks to resist changes or disturbances, thereby hindering beneficial development. However, establishing a power equilibrium within and between nations, for example, may release in a country resources which would enable the pursuit of economic change. Thus, equilibrium need not be interpreted as an ultimate goal, but rather as a pause to integrate energies and resources in new directions, which may lead away from the initial equilibrium and serve as an incentive toward a new equilibrium. Essentially, we have a path up a mountain with resting places. The rests enable us to walk a steeper route than we could travel continuously.

A useful idea in mathematics is that of a property which remains invariant to change, and hence has the general characteristics of “stability.” It is in this connection that mathematicians have used the idea of a fixed point, which is a convenient idea in modeling. The fixed point has its origins in topology and is amenable to wide interpretation when sets are transformed into themselves or into other sets. If T is a transformation mapping a set X into X , then a solution x 0 of the equation Tx = x , for x belonging to X , is known as a fixed point.

This idea is being used with a measure of success in looking for invariant properties in the social sciences.

1.4. The Modeling Process

There are many ideas to consider in the analysis and modeling of a problem. For example, a problem needs to be identified and background research is necessary before it becomes apparent what the real trouble is. Statistics and other types of information from people involved must be collected. An early estimate should be made of the resources required to study the problem; this involves both financial and problem-solving resources. A cost/benefit analysis might be conducted to determine the tradeoff between investing resources to solve the problem and leaving it alone because its effects are not as costly as its solution would be.

Once one has decided to study the problem, some method of specification or modeling process must be found. Initially, one may rely on the unwritten but widely used scientific principle that any model, no matter how crude, is to be preferred to no model at all. The idea here is that it is generally better to have someone think about a new phenomenon, even if the approach has obvious imperfections from the start. Those who come after can then improve on the simple model. Whatever model one may adopt, the question as to whether a solution to the problem is possible through the use of that model must be answered. People have been known to fall in love with an early formulation of a problem, regardless of whether or not it is valid for obtaining an answer.

In the process of formulation, dependent and independent, random and deterministic, variables need to be distinguished and identified and, as mentioned earlier, an analysis made as to which ones are controllable and which ones are uncontrollable. The parameters of the problem, together with their functional form as they appear in the model and whether they are deterministic or stochastic, must also be investigated. Sometimes, by developing a simple estimation model, a clear plan of attack crystallizes. If one is pressed for an answer, a “first-cut” model, sometimes called a “quick and dirty” model, may be developed. A more sophisticated model may need greater time.

An analytical model, as discussed previously, whether deterministic or stochastic, may not be possible. In that case, a simulation approach may be used instead, sometimes with the aid of a computer. Simulation has become increasingly popular recently. It is often necessary, but sometimes it is used at greater expense where a simple closed form model is available.

The use of ratio and proportion, and of rate of change, is central in setting up relations between variables to form equations and inequalities. After specifying the model and estimating its parameters, criteria are required to verify that the model is a reasonable representation of the problem. This may be tested by using previously collected data to validate the results.

A model should be simple enough to allow data collection and analysis. It should be practical in the sense that it might serve as an aid to implement the solution. Since prediction is focal in most scientific pursuits, most models must be designed with this point in mind. The generality of a model may be an enrichment on a special model developed for a particular problem. However, for the purpose of obtaining a solution, an approach to a problem may be decomposed into smaller models which are then aggregated for an overall result. A model, based on an efficient algorithm that can be used to derive the solution (hopefully without great cost), is preferable to another model from which a solution can only be obtained with great difficulty and considerable compromise and approximation. Algorithms are concerned with those methods of construction for which a list of instructions may specify a sequence of operations that, when followed, lead to the solution in a finite number of repetitions.

When a solution has been obtained, the question of the method and feasibility of implementation must be faced. The structure of the organization involved in this process and any means of making its operation more effective must be considered. The solution of a real-life problem does not stop with the theoretical solution but must be pursued to the point where the recommendations have actually been followed.

We must distinguish between two levels in modeling. There is the single level of penetration model which is short and to the point. There are two types: (a) those utilizing simple tools involving common sense, and (b) those using sophisticated tools. There is also a compound-level model which involves several strata of structure.

One can think of a number of reasons for using mathematical models. For example, models permit abstraction based on logical formulations using a convenient language expressed in shorthand notation, thus enabling one to visualize better the main elements of a problem and, at the same time, satisfying communication, decreasing ambiguity, and improving the chances of agreement on the results. A model allows one to keep track of a line of thought, focusing attention on the important parts of the problem. Models help one to generalize or to apply the results to problems in other areas. They also provide an opportunity to consider all the possibilities, evaluating alternatives, and eliminating the impossible ones. In science, in addition to prediction, they are a tool for understanding the real world and discovering natural laws. Today they are needed to cope with the various complexities of modern life, assisting and improving the ancient art of decision making. In the social sciences, a model can be pivotal in getting people to divulge their value judgments and discuss their view of a problem. In politics, the use of models has been known to provide opportunity to articulate individual feelings about how to approach a problem when such people were not willing or able to do so in other settings. The logic of the model transcends emotion and gets to the heart of the problem.

Suppose we are given a scientific problem which we wish to analyze for greater understanding of a situation and for prediction purposes. Assume that we are successful in formulating a mathematical structure with which we wish to portray faithfully the elements and relations between them in the problem under study. Since the mathematical parts and relations must correspond to the concepts of the problem, we speak of an isomorphism between the model and the problem. Thus, the model and its manipulations are a convenient logical tool for studying the behavior and relation of the elements of the problem.

There are two ways of looking at mathematical models—the local view and the global view, the micro and the macro, as the economists state it, or models which portray processes in the small and models which portray them in the large, as some mathematicians would think of it. It is generally believed that some experience with local modeling is indispensable in giving one a better perspective from which to approach global modeling.

Examples of powerful global theories that are inoperable on a local level are probabilistic models, such as statistical mechanics and queueing theory, which use averages and deviations by studying ensembles of objects and deterministic models of geodesic problems in the calculus of variations, algebraic topology, relativity theory, macroeconomics, and so on.

Local theories may also be very powerful ones; examples are point set topology, network flows with sources and sinks, and the analysis of the local shape of a surface (e.g., surfaces that are locally Euclidean, locally convex, or locally connected). In science, the study of a part of a large structure may enable one to gain understanding of its behavior and the way in which it might influence the whole structure.

Modeling involves a heroic simplification of a problem using the minimum possible number of basic variables in order to come to grips with the essentials. The first attempt usually comprises a stepping stone to more sophisticated elaborations of the model. To build an edifice, one requires a well-planned foundation, without which there could be no sound structure.

The process of model building is painstaking and experimental, involving hypothesis making, trial and error, and considerable innovative daring. In the early stages of modeling, one cannot afford to be too philosophically detailed, to demand a comprehensive statement of the alternative ways the problem can be stated, or to expect that each formulation make sense in all frameworks. The result might be to make it hopeless for structuring. Frequent experience with modeling through class participation indicates that the brainstorming process followed by a first cut at a model can be accomplished on some relatively complicated problem in about one and one-half hours. The results are psychologically impressive. The desired model may require much harder work, but the level of penetration obtained through such participation, and the courage derived from seeing this relatively crude, but fertile, flow of creativity, leaves one with the confidence that he can do the same on his own. Many have gone on to tackle and solve problems which have been discussed in this way, having benefited greatly from the experience.

Mathematical models have been remarkably effective in the physical sciences. However, they have fallen short in providing adequate representations of significant social science problems. Lack of criteria for the measurability of behavioral variables makes the introduction of relevant mathematical concepts difficult in the social sciences. But there have been some interesting breakthroughs leading to applications in areas heretofore regarded as beyond the reach of numbers. (See Chapter 8 )

In the social sciences, modeling has utilized continuous mathematics for the following types of models:

1. A model based on the real numbers, money, length, mass, time, or some other metric as a cardinal scale. We call this type a quantitative-oriented model in the strict sense.
2. A stochastic or probabilistic model.
3. A stability model (frequently useful with ordinal types of utility) whose equilibrium may be either algebraic or geometric.
4. A surrogate model which is designed to portray the interactions of various factors in the absence of a useful way of measurement to obtain numerical answers. A “cardinal scale” is a measure of magnitude which can be found by counting units of measurement. In modeling applications, a “utility function” often serves as a substitute for a metric.

A model incorporates two ideas, the measurement of variables and parameters, and the relationship between these variables.

In a surrogate model, the relationship is displayed but without measurement. Hence, a surrogate model is incomplete. It is difficult to test the relationships it portrays, because means of measurement in this case are not dependable. On the other hand, some people have been able to prove that the solution of a surrogate model remains stable in spite of changes (transformations) of the relations.

Experience with a variety of models applied to conflict resolution has shown that ordinal utility models are more effective in practice than those employing cardinal utility simply because it is easier to apply ordinal scales of measurement to such problems. In the case of equilibria, the judgment of the parties involved in the conflict is used to generate the ordinal utility scale upon which the stability of policies (in a game theoretic framework) is analyzed.

Modeling is frequently used as an aid to decision making, i.e., in making choices among alternatives with the purpose of exercising control, subject to uncertainty in the system. Examples of criteria applied in making a decision are:

(i) obtaining optimal performance;

(ii) developing priorities prescribed by outside constraints;

(iii) humanization by increasing interaction between man and system, leading to learning and feedback to improve performance.

A number of models presented here have their origin in the science of decision making although their number is small compared to the variety of areas represented to which models have been applied.

1.5. Why structuring and solving should be separated

In this book, we focus on formulating problems and very rarely solve them because problem solving requires such considerable skill that it tends to interfere with the free, imaginative style involved in formulation. Too much problem solving creates a spirit of attachment to the particular method used because of the great investment of time and effort in the method. Granted that we formulate problems in order to solve them. However, we often formulate problems in order to decide which one we want to solve. Extensive experience with formulation serves the latter purpose well. Thus, our desire for breadth of scope prohibits giving solutions except in some simple cases.

One outgrowth of the direct attempt to structure a problem is to systematize the division of real-life problems according to their conditions, variables, and parameters, by a classification scheme which simplifies the choice of models. Frequently the choice of a model depends on the experience and taste of the problem-solver, rather than on properties inherent in the problem, though the latter should be the more valid restriction on the choice of a model.

Exposure to diverse models may induce some to wonder whether modeling is the art of contriving purely from the imagination, or whether models are actually dictated by the characteristics of the processes being modeled. For example, the answer to:

Does the external world dictate the choice of axioms, definitions and problems or are these (models) in essence free creations of the human mind, perhaps influenced, or even determined, by its physiological structure? [Kac and Ulam.]

is, we believe, both.

An individual with a technical model needs the ability to persuade others to use it. This process may require time. People might be broken in on it piecemeal and without too much demand on their patience and good nature. The art of letting others see things your way is perhaps one of the most essential attributes of being effective [Boettinger]. Simply because one has an ingenious idea is no guarantee that others, even if they appreciate it, would take the trouble to use it. Effective modeling cannot be decoupled from human nature any more than the brain can live without the body. When viewed in this light the reader will quickly perceive why politics is a high-priority factor in decision making.

Chapter 1—Problems

The following problems are couched in very general terms and are designed to stimulate discussion. You are not expected to produce “answers” but should be able to formulate some ideas on the subjects. You may find it helpful to return to these questions at the end of your course to see if your answers would change.

1. Define the following terms: model, theory, hypothesis, conjecture, postulate, axiom, law, theorem, lemma, corollary, syllogism, deductive argument, inductive argument. Are your definitions adequate? Can you improve them?

2. Given the following two principles about proofs: (a) for someone to know a mathematical theorem he or she must check a proof of it; (b) if a proof of a mathematical statement exists, then, since a proof is rigorous, the statement is absolutely valid. What about a proof by computer which must work out many cases? The computer makes an error somewhere but, if it takes a long time to check an entire proof, is this a valid way of proof? Should we allow a degree of fallibility for proofs? Is 1 + 1 always equal to 2? This sort of statement occurs often in life and one should be able to answer it in general.

3. Theories about nature usually grow and change. Does our understanding of natural law change? How about its applicability in experience: are the effects different?

4. It has been said that deductive thinking by going from cause to effect is linear thinking. How then is man ever to understand and to deal with complex problems which form a network rather than a chain? Is Aristotelean thinking the only way to understand the “real world” or are there other approaches? How effective are these? What are some criteria for evaluating progress with any method of understanding? What is meant by objectivity? Is there an objective reality? How do we know? Some have argued that objectivity is agreed upon subjectivity. What can this possibly mean?

5. In view of the foregoing, what can one say about the absoluteness versus relativeness of models from the standpoint of (a) people, (b) subject matter, (c) the time era in which the work is done, (d) historical accounts, (e) advocacy and politics, i.e., have good friends peddle your theories, (f) the fact that we see the world by the chemistry of our senses and our glands.

6. In view of 5 , now comment about the objectivity of models.

7. Most mathematical models depend on basic “primordial” tools such as arithmetic, algebra, the calculus, and geometry. Have you felt a need for new basic mathematical ideas that could give a better expression for your experience and feelings? Can you describe the feelings and can you give a hint of what kind of new tool?

8. Clarify what is meant by understanding, prediction, control, and planning. Can we plan and modify our methods of understanding?

9. What can one possibly mean by evolving reality? Emergent reality? Do we make things up as we go along or do we discover things as they really are, awaiting us to find them?

10. The systems’ way of thinking considers a number of different ramifications of a problem. Does this differ from the classical approach to modeling? What must we do to deal with problems more realistically?

References

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Kac, Mark and Stanislaw M. Ulam, Mathematics and Logic ; Retrospect and Prospects , A Mentor Book, The New American Library, New York, 1968.

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