Chapter 16, Origins and Limitations
of Rules and Patterns

Introduction

A rule, law or agreement may say that when one event happens, another event should also happen or may also happen. Most physical and legal theories, if not all, use rules which are approximately correct. The rules are like all human discoveries and creations; some are more reliable than others. The formulation of laws and rules and agreements by people leads to the chance of error and incompleteness. Even with uncertainty, once rules or laws or agreements have been stated, we can use them tentatively, to reach conclusions or judgments. Locating the weakest links in our reasoning gives us a chance to strengthen or replace them.

The question of what rules to accept, use or change, and how much confidence we can have in them is often discussed. This question is debated or negotiated at various levels in mathematics, philosophy, politics, business and religion. We think or dream of what might be. We speculate. Then we need to test to see what parts, if any, of our speculations are correct. We correct what we can and speculate again. Knowledge here comes from an approximation, or a sequence of approximations, some better than others.

We find and obtain rules to obey or ignore from at least three sources:

1. Private Agreements. The first source occurs in deals between brothers and sisters, or between business people. These implication rules suggest that if you do this, then and only then will I do that. These implications are agreements or promises. The agreements and commitments here can be one-way or two-way. They may be written or spoken. People get upset whenever such rules are not obeyed or not understood.
2. Public Laws in Society. A second source is given by rules or laws used to say what is acceptable in society. These govern in part our behavior. Such laws say what we should or should not do. They may even specify penalties or punishments for disobeying them. A rule that is not enforced, or is enforced weakly, is often ignored or forgotten.
3. Physical Laws. A third source of rule occurs in technology, mathematics and science. These record or state our observations of nature and the patterns it follows. They may describe what has been seen. They record human experience. Examples of the latter are provided by the recipes for cooking and operating instructions for machines. Reliable and carefully followed procedures give reproducible results. Further, recipes and reliable patterns can be joined together to suggest more recipes and patterns of behavior.

These three sources of rules or patterns are discussed next.

Private Agreements, Rules from

Private agreements involve one- and two-way commitments. We have to be aware of the difference between the two. They provide one of the sources of rules to be formed and argued or talked about. Agreements may say that this and that will be done. The agreements may say that an action will be done by one person or party, if or when another party does something else. The agreements may involve time limits. Agreements between two parties try to define and control what will be done in a manner satisfactory to both. Each party to a contract may try to get the most possible from the other. Whether this is fair or not depends on the situation. Some people like to negotiate from strength. In any event, each would-be party to a possible agreement needs the ability to read and understand rules and obligations in the agreement precisely.

A private agreement or business contract may be broken into a sequence of mutual obligations (clauses): If you do this, I will do that and then that. Then you will do this. A weakness of business contracts lies in the failure to agree in advance to changes forced by circumstances beyond the control of one or both parties. Disagreement or arguments can be avoided if possible stopping and withdrawal points and penalties are agreed to in advance.

The abilities to think, write and read carefully are all needed in negotiating and handling business deals, and the one-way and two-way implication rules which occur in them. Broken or misunderstood promises keep courts and mediators busy with arguments about rights and wrongs. Damages or disappointments may result from a failure to keep or understand a written or oral agreement.

The ability to understand and read exactly agreements, contracts, and in particular the obligations and duties which they create or generate, is most important. Without this ability, disappointments, false expectations and perhaps penalties appear. So to avoid future disputes and disappointments, care has to be taken to see that the wording of an agreement is understood by all. Absence of this care in rushed deals results in false expectations and disappointments. Broken commitments and broken laws may lead to a court or a place of arbitration for arguments over losses and damages.

When disagreements are not settled out of court, people on opposite sides of an argument will put forward opposing sets of reasons. The aim is to show that the other side is wrong, at least in part. Here each side may argue from different positions and for different judgments. In the arguments, reasons and implication rules may be chained together deductively. Several reasons may be given to support one idea or conclusion. Each side of a debate will have different accounts of what happened or of what was originally agreed.

Public Laws in Society

Writing rules and regulations is a human endeavor. Certain laws and principles may appear self-evident or clear. They may need to be written for everyone to be aware of their existence. What is obvious or clear for one is not always obvious and clear to another. Laws it seems, are needed to control or restrict our otherwise uncivilized (meaning unruled) behavior.

Laws, regulations, rules and guidelines are written and sometimes enforced by governments and corporations to say what should happen in various circumstances. Some laws and regulations are enforced by means of penalties. Stiff penalties are applied for breaking them. As a result, these laws and regulations are respected. Laws and regulations which are not strongly enforced become in effect voluntary guidelines. The latter can be ignored. Weakly enforced and uninforced laws and regulations are all cosmetic.

With human-made rules

in law, ethics and mathematics

and in regulations, we can hope but we cannot verify in advance that they will not conflict in all possible situations. Some situations may not have been envisioned or foreseen when the laws or rules in question were written. When we are dealing with human-made laws there is a danger of them or their consequences conflicting.

Physical Laws,

Observation of, Testing them and Conjectures

In communities, rules and laws may be debated and then written to control our behavior. In technical knowledge, rules and patterns we may use or follow are subject to less debate, but they are (or should be) tested to see in what circumstances they hold. Here rules and patterns are recognized and discovered rather than written. But in technical knowledge, we meet the main problem of inference, that is, inductive thought. It is the problem of finding and/or identifying reliable rules and patterns. Here rules and patterns to describe how the physical world behaves are proposed and then tested. Those that fail are rejected while those that pass are only partially confirmed. Here some rules and patterns appear more reliable than others. Not all is certain.

Empirically based rules may say or suggest that whenever a first event or situation is seen or done, a second event occurs. The rules describe links between different parts of nature. Why these rules or patterns are obeyed is sometimes unknown. They are only descriptions. They are only observations - patterns that have been seen.

Rules and theories for physics, mathematics, medicine and machines are empirical. Empirical knowledge comes from experience and discoveries. This includes trial and error β€” the first time we try to do or find something, we typically make a mistake. Correction of the mistake or mistakes can lead to a method or recipe that can be repeated and followed again.

Our collective experience and knowledge of physical situations and patterns are described and summarized through the statement of laws, recipes, instructions and formulas. The recipes may say when one event occurs, another event will also occur. This experience may tell us how to control nature, wisely or not. The recipe you follow in cooking a meal provides an example of this. Experience may suggest or predict the result of an ill-conceived action or inaction.

The physical sciences contain different theories (recognized patterns) for different circumstances. Some theories may overlap. Some may be approximately true. Some will be better than others. Being approximate, many theories are incomplete. The theories contain unsolved problems to keep us humble about the extent of our knowledge. Unsolved problems illustrate the limitations of each theory, and the need for more work to be done. Comprehension of a theory may be based on accounts of what has been tried (and has perhaps failed) along with examples which work and illustrate its concepts.

Accidental Patterns Again

Suppose we are given a rule which involves the idea that every time a first situation A occurs. We can be sure that the rule is correct provided we can check that every time the first situation A happens, that the second situation B also occurs. This checking is possible (feasible) if we are reading a story. In the story the rule may be seen to always hold. So it has been verified. But, except for written stories describing past or fictional events, we cannot check that each time the first situation A occurs, the second situation B happens. An observation

Every time the first situation A happened, the second situation B also occurred

represents history. Forcing the first situation A to occur will not make the second situation B occur if the pattern is accidental. For example, suppose two children have gone to school each day, for the past three weeks. This behavior or pattern gives no guarantee both will show up every day of the following week. The behavior of the observed children need not be governed by this pattern. The explanation of the children's attendance at school lies in a home life or health status which we have not seen. The pattern seen might have been accidentally established.

Observing the pattern

Every time the first situation A happened, the second situation B also occurred

in a given circumstance or setting C suggests the implication rule if A then B might hold in the circumstance C. Seeing that this pattern holds several times builds confidence in the reliability or truth of this implication rule. We may become very confident in the suggested implication rule β€” and perhaps take it for granted β€” but we cannot be sure. We cannot conclude for certain from observation or experience, that this implication rule is never disobeyed. In contrast, observing the occurrence of situation A once without the situation B in the given circumstance C shows that the rule can be disobeyed.

Confidence in Patterns

Rules Reliable or Not?

The question of what is not accidental needs to be examined. The concept of a controlled or reproducible situation provides an answer. In rule-based reason and processes or methods, we gamble. We build theories on implications we assume or hope are not disobeyed in the circumstances of interest. A theory here is given or suggested by a chain of implications and assumptions. Our confidence in the conclusions provided by these chains depends on the confidence we have in the chain of implications, assumptions, suggestions and approximations leading to them. As in arithmetic, a false step may lead to a false conclusion.

Confidence is obtained for some implication rules or patterns through the idea of reproducibility and repeatability in controllable situations. Confidence may also be obtained from prediction-based tests. How all this is done will be described below.

Technology and the rules for the operation and manufacture of devices and products in it, are all based on the pattern of reproducibility. We rely on this reproducibility in our everyday lives: plumbing, electricity, automobiles, radios, airplanes, toasters, ovens, furnaces, etc.

Cooking illustrates the confidence building aspect of reproducibility. By following instructions and putting together ingredients carefully we can make certain meals repeatedly. Anyone else following the same instructions with enough care can also prepare or reproduce the same meal or dish.

Carefully recorded experience in chemistry gives reproducible methods for making, remaking and classifying different substances. The same substances can be produced again and again by a chemical process provided the recipes for its production are followed with enough care. The amount of care required depends on each recipe. Further, empirical experience in physics and engineering gives formulas and suggestions for modeling and controlling various situations. This experience is cumulative β€” if recorded in some form by means of a written statement about a device or a procedure which worked or failed. This gives a verifiable, reproducible science or technology. Recording or remembering what ideas failed gives paths to avoid and questions to ask.

The Scientific Method

Control and a Scientific Method

Reproducibility and repeatability form the basis of our daily technology. We look for a regularity – a repeatable pattern. Then we rely on it. Moreover, once a regularity is found, variations of it are tried in the hope of finding an improvement. A reproducible event gives a situation which can be controlled and then experimentally disturbed.

Inductive and Empirical Reason

The observation of regularity provides a basis for empirical, inductive reason. Here patterns which appear to be reliable are extracted from experience or trial and error. The use of these patterns in chains of reason then provides examples of deductive reason. But uncertainty in the patterns cast doubt on the conclusions obtained. Not all is certain, but some patterns appear to be reliable. Confidence in them comes gradually.

Review Question – A Hint of the Contrapositive:

For a reliable rule which says that when a first situation occurs, so does a second, what can you conclude when the second situation does not occur? (Hint: See the first logic puzzle in the chapter Implication Rules or all of chapter The Contrapositive to find the answer.)

A Scientific Approach to Cooking

In cooking and other situations, when we do not do anything differently, nothing different results or nothing extraordinary results. The situation is reproducible. When we modify some recipe, instruction or procedure, a new result may be produced. We can be fairly sure that whatever we did or changed made the new result appear. To be more confident of this, we could describe the reproducible situation in writing and describe what happens with and without the change. Then we could ask someone else to follow this description. If other people can obtain the same result(s) as us, without further instruction from us, the change we have made has caused [2] another repeatable and reproducible process.

[2] There is an assumption here.

Controlled Situations and Exploratory Changes

When repetition of a sequence of actions leads to one result and no other, a controllable situation has appeared. Again, this is like cooking. Following a recipe carefully enough leads to the same result each time β€” the reproducible meal. Further, after the recipe is seen to work, we can ask what happens if one step in the recipe or sequence of actions is changed. This can lead to more reproducible results (or reproducible disappointments or disasters). In this manner desirable and not desirable recipes and implication rules for cooking can be found and tested.

In the physical sciences and in technology, circumstances which can be repeated and changed (perturbed) give opportunities for finding and experimenting with reproducible results. Reproducible results are possible in those controllable situations which almost repeat themselves, or can be repeated by us. Rules which say what should happen in repeatable situations can be tested. Just set up the situation (or wait for it). Then do your test.

Reaction to Failed Tests

If the desired results are not obtained when we follow a known recipe or procedure, we then look for

1. an incorrectly described or followed step in the recipe,
2. a malfunction in the equipment,
3. an incorrectly measured ingredient,
4. an ingredient polluted by a foreign substance, or
5. a factor not previously considered.

The foregoing may identify a remedy or leave a puzzle.

Cause and Effect

When a reproducible operation is under way or running, nothing unusual happens. If we introduce a disturbance or do something to affect the process, we may see a departure from the ordinary or usual behavior. So we should strongly suspect that the departure is (most likely) due to the disturbance. This suspicion is tested and confirmed if the departure is repeated (several times) whenever we make the disturbance re-occur, or we may be fooling ourselves β€” always a possibility.

Through trial and error, we may look for a disturbance producing or causing a variation in behavior which we want to keep. This disturbance can then be made a normal part of the process. A new controlled situation results. Experimentation and fine-tuning of the process at hand can continue.

When introducing a variation in the operation of a mechanism, care needs to be taken to ensure the variation is the only one that is done. When two disturbances are introduced simultaneously, one unknown to us, we may think a variation in behavior is due to the disturbance we saw. But the variation could be due to the other disturbance or to the fact that both disturbances occurred simultaneously. Pattern recognition is not always straightforward.

Confidence

Confidence in old and new rule-based processes may be built through repeatable and reproducible experiments and observations. Experiments and their results are accepted if they can be reproduced and repeated by others besides their inventors. For this, the inventors or first discoverers of a phenomenon must carefully record the method, art or recipe used to get their results. With no such record lengthy or cryptic, results are subject to argument and doubt.

Making Theories and Predictions

Rules (and suggestions) can be linked together to get or suggest further rules. That is, combining a rule which says when a first situation A occurs so will a second B to another rule which says when the second B occurs so will a third C gives a new rule: when the first situation A occurs, so will the second B and the third C.

By combining reliable and not so reliable implication rules together, we can make predictions. The chains of reason by which the predictions are made are called theories. When a prediction fails, at least one part of the chain of reason leading to it is uncertain. Knowing which parts of the chain are the most certain and which are the least may suggest a new course of action: informed trial and error. In contrast, when a prediction succeeds, we may become confident or overconfident in the chain of reason which suggested it.

Proposed implication rules can be shown to be false. They can never be shown to be true. As a result, rule-based reason depends on implications rules and assumptions supposed or pretended to be true and reliable.

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Prediction versus Self-Delusion

In physics, amazingly long chains of implications and approximations are used for predictions. Because of approximations or suspect implications, the reasoning is unsure. The predictions need to be tested. A prediction gives a value for a number or quantity. The value could be one of many. If the observed value is far from a predicted value, a step or two in the creation or derivation of the prediction must be wrong. The theory needs repair. Confidence in the predictive method/theory is increased when the predicted value and the observed value are close.

In physics, confidence grows in a theory (one long chain of reason) if it correctly predicts one result, not seen before. In physics, those making the predictions are the theoreticians while those making the observations are the experimenters. [3]

[3] A division of labor has occurred between scientists making predictions (the theoreticians) and those doing the experiments (the experimenters).

Implication rules and theories become more trusted (or useful) when predictions based on them are seen to be true or very close to the observed values. Successful theories are hard to find. But when a theory gives a good prediction, variations of the theory may be tried to get other predictions.

Avoiding Self-Delusion

A theory can predict a future event. If the event occurs, confidence in the theory grows. A theory can also suggest a value or values for a quantity which has already been measured. This suggestion is like a prediction of a future value of a quantity provided the measured values were not used in the construction of the theory.

In contrast, if past measurements are used in the construction of a theory, the agreement of the theory with the measurements on which it is built is no surprise. It is expected for the sake of consistency. A theory that did not agree with the values used in its construction would be inconsistent.

Confidence only grows in a theory when it fits and predicts observations that are not part of its construction. Here the prediction of future measurements instead of matching past measurements provides a more reliable test. It is less susceptible or prone to self-deception. Not all theory makers are logical and some illogical theory makers can be partially correct. Self-deception is to be avoided.

Chaos

Does it equal unpredictable and uncertain situations?

Chaotic situations make observation and pattern spotting difficult, perhaps impossible. In situations which cannot be controlled, there is no observable normal or stable state of affairs to which the system returns after any disturbances. In such situations, reproducibility of results is not seen. There is too much movement for any stable pattern to emerge. Each situation is not repeated nor seen again by an observer.

Knowledge is most certain in dealing with machines and bureaucracies where behavior is repetitive, controllable and reproducible. The rules for their operation are firm and rigid. Less certain knowledge appears in uncontrollable, and non-reproducible situations. Think of economics or weather systems. These are examples of uncontrollable circumstances. Rules describing their behavior may never be found.

What can we do in uncontrollable irreproducible situations? The answer perhaps is to look for patterns. These may provide some control. Chaos is reduced each time a reliable pattern is spotted and confirmed.

Statistical Inference and It Limitations
Chapter 16

A statistic is a number or function which depends on the data collected or observed. It provides one window, a narrow one perhaps, on the data.

In controllable situations where we can repeat processes and procedures, patterns can be observed and tested. In the study of situations not fully controlled, counts and measurements may be made and collected. Then statistical computations are done to find patterns and characteristics which may be reliable. Here chance and probabilistic estimates are used to recognize or judge whether observed or imagined patterns of behavior hold. All this belongs to the art of statistical inference.

There is a true art to statistical pattern identification. Unfortunately, many people apply its methods without fully understanding them. If you engage in statistical inference, please use only the concepts which you fully understand, and when in doubt, don't. The further description of statistical inference is left to other books.

Scandal and Hype

In colleges and universities, I have seen students with insufficient mathematical background (a) run and rerun statistical programs in order to compute fashionable but ill-understood numbers; and (b) from these estimate the significance or reliability of a pattern. The uncertainty here, coupled with an incomplete understanding of how the numbers and measurement were handled or interpreted, invites skeptism. Statistical inference has its limitations. The blind application of this art in any discipline is a scandal. It leads to error.

Beyond this, politicians and bureaucrats sometimes use the many ways in which numbers and measurements can be described and reported to select those perspectives most favorable to their cause β€” hype, hype, hype, hooray with numbers. There is a classic 1954 book How to Lie With Statistics by D. Durf which describes these matters further. It is published by Norton and Company (ISBN -0-393-31072-8). A more recent work with a similar theme is Use and Abuse of Statistics by W. J. Reichmann, 1961, Pelican Books (ISBN 0-14- 02-0707-4). Both books were mentioned in the chapter Deception.

End Notes if not a Review

1. In a rule which suggests that whenever a first situation is made to happen, a second situation will follow, the first situation is called a possible cause of the second. The second situation is also said to be a possible effect or consequence of the first.

2. Human made rules or models for nature's behavior suggest or describe patterns without explaining why they occur. Science and technology are mixtures of facts, guidelines and recipes. Some parts are certain or almost certain. Other parts are less certain. The empirical approach to knowledge tries to identify those repeatable, reproducible processes: processes that work, accidentally or otherwise.

3. Scientific and technical knowledge can be viewed as a collection of theories or recognized patterns and recipes (implication rules). Details accompanying such rules should say when they do or don't apply – the range of applicability. Knowledge of this range can be unclear. Our knowledge of the physical sciences forms both a collection of recorded patterns or recipes for solving some problems and a collection of unsolved problems.

4. The unsolved problems (or mysteries) say or indicate that more work is required. We humans have discovered many skills and techniques, wonderful or not. In any area of application, only a few of these skills are pertinent, that is, applicable. In any area of application, further skills or techniques are often required. In technical areas, we find two kinds of knowledge: a knowledge of processes that work and a knowledge of processes that don't. So more work is required on them. Whether or not this work is feasible always remains to be seen.

5. Technical knowledge is based on repeatable and reproducible methods, along with some trial and error from deliberate experimentation (sometimes accidents) to find them. As human beings, we can spot or imagine patterns. From them, we try to predict what will happen.

6. Creativity and subjectivity (guesses, past knowledge and experience, guidelines/assumptions) are involved in deciding what chains of implications to form or investigate. Once a chain of implications with an interesting result or conclusion has been discovered, the result or conclusion and how it was obtained can be shown to others. The path to such a result or conclusion can then be repeated by others. Mathematics, engineering, science, chemistry, cooking, computer science, all these disciplines follow this pattern of discovery and repetition or reproducibility. Reporting how a conclusion or goal was obtained or missed is a result. It is a result which informs how something was done (or missed). We can learn from the experience and the errors of others and ourselves.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science