Science in the Eyes of a Scientist

By Mark Perakh

Posted on November 3, 2001
Updated August 11, 2003

Contents

1. Introduction

2. Features of good, bad and pseudo-science

3. The language of science

4. Definitions

5. The building blocks of science

6. Methods of experimentation

7. Data

1. General discussion of data

2. Reproducibility and causality

3. Ceteris Paribus

4. Errors

5. Complex systems

8. Bridging hypotheses

9. Laws

10. Models

11. Cognitive hypotheses

12. Theories

13. Science and technology

14. Science and the supernatural

15. Conclusion

In many articles on this site, especially those discussing books and papers which are devoted to the most recent incarnation of the creationism, disguised as the so-called "intelligent design theory," or which purportedly assert the harmony between science and the Bible, the question will repeatedly arise of what constitutes legitimate science and how to distinguish it from either bad science or pseudo-science. In this paper I intend to discuss the above question in general, thus laying foundation for the subsequent discussion of that question in particular books and articles.

Obviously, this distinction is possible only if a certain definition of science has first been agreed upon. Unfortunately, it seems that a rigorous definition of science is hard to offer, and perhaps even impossible, for reasons which may become clear from the following discourse. In a non-rigorous and most general way, science can perhaps be defined as a human endeavor consciously aimed at acquiring knowledge about the world in a systematic and logically consistent manner, based on factual evidence obtained by observation and experimentation.

As it should become clear from the following discourse, it is hard to consistently adhere to the above definition of science because the latter seems to inevitably encompass elements beyond the limitations of that definition.

The subject of this article is traditionally discussed within the framework of the philosophy of science. My intention is to look at the problem from a different standpoint – that of a scientist rather than of a philosopher. I envision this as a glance from within science rather than from a philosophical outside. Such an approach entails considerable difficulty. Whereas philosophers may legitimately avoid discussing the specific nuts and bolts of scientific work, a glance from within requires analyzing specific details of the process of scientific inquiry and this may pose a serious obstacle to non-scientists trying to understand my thesis. I will have, on the one hand, to avoid the Scylla of engaging in arcane specifics of the modern highly sophisticated scientific methodology and, on the other hand, the Charybdis of oversimplification by limiting myself only to the most general notions and thus actually becoming a dilettante philosopher.

I don't know if I can succeed in such an act of balance on a thin wire between the two approaches and probably will not be able to eschew the trap of becoming an amateur philosopher anyway.

I also realize that it may be simply impossible to analyze the essence of science as though all of its branches were characterized by the same set of features. Science is very complex and multifaceted so an analysis which is quite adequate for some fields of science may happen to be irrelevant in other fields. My approach will be relevant first and most to such fields as physics, chemistry, engineering disciplines, and biology, while many aspects of my discourse may be less applicable to medical science, and even less to history, esthetics, literary criticism and the like.

One of the first questions in the discussion of science is what is the driving force of science. While different scientists may be driven by different motivations, I submit that the main reason for the existence of scientific research is the curiosity which seems to be inherent in human nature. This statement does not imply that every man and woman necessarily possesses such curiosity. On the contrary, the vast majority of people lack it. As the great Russian poet Aleksandr Pushkin said (he meant only the Russian people, but his words can probably be applied to humanity as a whole) "We are lazy and not curious." However, there has always existed a fraction of people differing from others in that they are driven by insatiable curiosity. Apparently, some ethnic groups, at different historical periods and for unknown reasons, had a larger fraction of such people than others.

It seems that historically attempts to satisfy the curiosity in question were made in three distinctive ways. These are religion, philosophy and science.

(As any simple scheme, the above classification does not fully represent all aspects of the human drive to understand the world. For example, the above three-part scheme leaves out such important forms of human endeavor as art. However, this scheme is convenient for the purpose of the main subject of this article).

At the dawn of civilization, when humankind's knowledge of the world was extremely limited by the narrow experience within a small locality, the curiosity about the surrounding world could be satisfied only by means of religion. For example, until recently, the Indian tribes in America, as well as native tribes in Africa and Polynesia, all had certain religious systems, but none of them had developed either philosophy or science.

Religion is a way of explaining the world which is not based on evidence but largely on imagination and fantasy. The religious explanation of the world does not require either factual evidence or logic. The holy books of the world's largest religions not only are incompatible with each other but are also each full of internal inconsistencies and descriptions of improbable events, but this has in no way impaired their impact on large masses of people. Although religions often borrowed concepts from each other, most of them claim the monopoly on truth. They have an enormous power of survival despite any evidence against their tenets.

As the boundaries of the explored world expanded, the next steps in the attempts at explaining the world were philosophy and science. It is hard to establish a definite chronological sequence of their emergence. In some cases science preceded philosophy. Indeed, well developed science is known to have existed in the ancient Babylon and Egypt while philosophy does not seem to have left signs of existence in that remote past. In other cases philosophy and science seemed to emerge simultaneously, often being for a while indistinguishable from each other. Whatever simple scheme can be suggested, it necessarily will be incomplete and contradict at least some features of the complex relationship between the three ways of comprehending the world I listed above.

Like religion, philosophy does not necessarily require a factual evidence (although it may) but, unlike religion, philosophy requires logic. Unlike religion, philosophy rarely claims the monopoly on truth (although some philosophical theories did just that – an example is the Marxist philosophy).

Finally, as more factual evidence was accumulated about the world, science took over as a way to gain knowledge in a systematic matter, trying to base its conclusions both on factual evidence and logic.

In the course of millennia science underwent multiple modifications, for many centuries remaining indistinguishable from philosophy. The modern science which is quite distinctive from philosophy developed largely during the few most recent centuries.

In a certain sense, the relation between science, philosophy and religion can be represented, necessarily simplifying the actual situation, by three concentric circles. The inner circle envelopes scientific knowledge. Its radius is constantly increasing, as more and more knowledge is accumulated. Beyond that inner circle of scientific knowledge is the concentric circle of a larger radius, the area between the two circles representing the domain of philosophy. Philosophy reigns where science has yet not acquired sufficient knowledge, thus leaving room for speculations limited only by logic. Finally, an even larger concentric circle surrounds the two mentioned circles – that of science and that of philosophy. This is the domain of religion where neither science nor philosophy has so far managed to offer good explanations of the world. The inner circle – that of science – is continually expanding at an ever increasing rate, as science accumulates more and more knowledge. This expansion of science pushes philosophy into a gradually narrowing domain where it still has the freedom of logical explanations not based on rigorously collected factual data. This, in turn, pushes religion even further to the margins of the world's comprehension. The enormous power of survival obviously inherent in religions is due not to their gradually shrinking explanatory abilities but rather to their ability to satisfy the emotional needs of man.

We have to distinguish between "good" and "bad" science as well as between genuine and pseudo-science. While these two dichotomies are of different nature, sometimes it is difficult to distinguish between bad science and pseudo-science.

Bad science differs from good science in that it employs the same legitimate approach to inquiry as good science but at some stage of that inquiry fails, often inadvertently, to follow the path of an objective and/or logical process and goes astray in its conclusions. Generally speaking, good science serves the progress of science itself and of its derivatives in engineering, medicine, and technologies of various kinds whereas bad science is rather promptly discarded. Since bad science is usually recognized rather swiftly, it is often just a nuisance and is rarely so harmful as to seriously impede the progress of good science.

A case of bad science can be exemplified by the story of the so-called cold fusion. The discovery of cold fusion was announced by two groups of scientists, of which one consisted of professors of electrochemistry Martin Fleischman and B. Stanley Pons and their associate M. Hawkings, and the other was a group headed by Professor Steven Jones.

Whereas I never heard of Pons, Hawkins, and Jones before they announced their discovery, I was acquainted with Professor Fleischman. In the seventies, we both served as members of the Council of the International Society of Electrochemistry (of which Fleischman later served as president). I also twice visited the University of Southampton where Fleischman was a professor of electrochemistry, and gave a talk there. Since Fleischman's research was in a field rather remote from my own, I was not really familiar with his scientific production, but I knew of his reputation as a serious electrochemist. Therefore, when I first heard about cold fusion I was both greatly impressed and puzzled. I was impressed by the mere claim that two electrochemists, using an experimental set-up which only could be characterized as primitive, reportedly achieved a revolutionary result of immense importance. I was puzzled because that result seemed rather unlikely (in particular because these researchers did not observe emission of neutrons from an alleged nuclear fusion process, and, moreover, their description of the experimental procedure seemed to be too vague.) On the one hand, it seemed hardly probable that experienced scientists would have made such an extravagant claim without having meticulously studied the claimed phenomenon, and on the other hand, it seemed equally hardly probable (although not impossible) that their claim was true. Soon, several groups of researchers were busy with verification of the sensational cold fusion reports. Most of them rather rapidly came to the conclusion that the cold fusion effect could not be reproduced. The sensation died and most of the researchers discarded the cold fusion report as bad science. (Despite the disappointment by the evident irreproducibility of the initial results by Fleischman and Pons, some research in that direction is still going on, and regular conferences devoted to that subject are being held; although no actual cold fusion has been reported so far, these efforts may be useful if we remember a saying attributed to Rutherford. This famous physicist was once asked why he allowed one of the researchers in his lab to conduct an obviously hopelessly futile study. Reportedly, Rutherford answered: "Let him do it. He hardly will find what he is looking for but maybe he'll find something else which happens to be useful.")

Pseudo-science, on the other hand, is an endeavor which essentially is not real science but something else disguised in quasi-scientific clothes for the sake of a certain agenda, usually having nothing to do with science and often hostile to science. It may be quite detrimental to genuine science because of its ability to successfully disguise itself as genuine science and thereby forcing on science an unnecessary defensive effort requiring the spending of valuable human resources needed for the progress of genuine science. A little later in this article I will suggest some criteria for discriminating between genuine and pseudo-science and suggest appropriate examples.

It seems useful to distinguish between the language of science (and definitions as a part of it) and its building blocks.

The language of science is overwhelmingly mathematical. First, this statement can justifiably be viewed as a simplification. The interaction between science and mathematics is neither straightforward nor one-dimensional. Second, an adequate analysis of mathematics as a predominant language of science would require a preliminary analysis of what is a language. This would, though, lead well beyond the scope of my topic. Third, this statement may be disparaged as allegedly dismissing non-mathematically expressed knowledge as non-science. Of course, it is just a question of semantics. Is psychology a science? Is history a science? Surely, these and many other fields of inquiry are legitimately listed among sciences even though mathematics has so far found only a limited room for itself within those sciences. (A different story is that mathematics finds its way into those sciences via the ever increasing penetration of computers and artificial intelligence.) What is more important, the path these sciences follow seems to repeat that traveled by physics, chemistry, and biology. Centuries ago, actually before Johannes Kepler and Galileo Galilei, physics used little mathematics and was largely a descriptive science, much like psychology or history are today. It took time for the marriage of physics and mathematics to occur, but when it happened, the impetus for the development of powerful theories of physics was enormous. "Mathematics is a language," said the great American physicist Josiah Willard Gibbs. Modern mathematics is so multifaceted that it is probably more proper to speak about mathematical languages in plural.

Mathematics is, though, more than a mere language (or languages). It is a compressed logic immensely facilitating a theoretical comprehension of experimental facts. Without mathematics, contemporary physics would not exist. Hence, while saying that the language of science is mathematics, this is fully applicable to the physics of today but also, probably, to the psychology or history of tomorrow.

The mathematical language is overwhelmingly used in science not because it is more powerful than the polymorphous natural vernacular. The opposite is true – the polymorphous natural language is, generally speaking, more powerful. The use of mathematical language in science is dictated not by its power but by necessity, as will be discussed in more detail when we turn to the concept of definition. Actually, whereas mathematical language immensely facilitated and accelerated the development of science, toward the middle of the 20th century it has become evident that the scientific study of the so-called large or poorly-organized systems (which will be discussed in subsequent sections) is sometimes impeded by a strict adherence to the rigorously enforced use of a uniform mathematical language. The sharp demarcation between natural language with its inherent ambiguity of terms and the rigorous uniform mathematical language started gradually diffusing.

Hence, although the mathematical language is still the prevalent, largely preferred and often simply necessary way of scientific discourse, polymorphous language gradually finds its way into that discourse, a phenomenon which is often viewed simply as a modification of the mathematical language itself. Indeed, mathematics constantly develops new tools (such a chaos theory, non-standard logic, and the like) which help it catch up with the needs of physics and other disciplines in refining their mathematical representation.

The question of why mathematics is applicable in physics puzzled many scientists and philosophers alike.

Physics is a discipline in which mathematics is so intertwined with the body of that science that often it seems hard to see where physics ends and mathematics starts. Physics and mathematics have historically developed, constantly exchanging ideas and methods. The greatest mathematicians often happened to be also outstanding physicists and vice versa. Isaac Newton developed some of the most fundamental theories in physics and also invented a revolutionary part of mathematics – differential calculus. Leonard Euler made enormous contributions both to mathematics and to physics. New mathematical methods which seemed to be of a highly abstract nature (like functional analysis) swiftly found their way into specific fields of physics while methods developed to tackle specific problems of physics (like vector algebra and then vector calculus originally invented in hydrodynamics) very soon were generalized to become chapters of abstract mathematics.

There is something fascinating in that mysterious usefulness of mathematics in physics despite the obvious difference in their treatment of objects. Here is a simple example. We all know that matter has a discrete structure. It consists of tiny particles. It is impossible to visualize these particles since their behavior is enormously different from the behavior of the macroscopic bodies we see, hear, taste, smell and touch. Whatever these particles really are like, we know that matter is not continuous. However, we routinely apply in physics an abstract mathematical concept of infinitesimally small quantities, like infinitesimally small electric charge and the like. This mathematical trick works very effectively despite its obvious contradiction of our knowledge of the discrete structure of matter. The concept of an infinitesimally small charge is contrary to the reliably established facts of the discrete nature of electric charge.

I believe the described discrepancy between mathematics and physics does not undermine the extreme usefulness of mathematics in physics for the same reason that the use of physical models does not undermine the study of real objects. One of the following sections will discuss the use of models in science in detail. Preempting that discussion, we may note that models differ from the real objects in that the models share with the real objects they represent only a small fraction of properties while ignoring the overwhelming majority of the latter. A good model shares with the real object only those few properties which are crucial for the problem at hand. For example, in Newton's celestial mechanics the model of a planet is a point mass. It has only one property common with a real planet – its mass. Replacing a real object with its model, while introducing a certain, usually insignificant imprecision, enables scientists to tackle otherwise enormously complex problems. Similarly, applying an abstract mathematical concept such as infinitesimally small charge, is similar to using a model in which the discrete structure of the real electric charge is ignored as inconsequential for the problem at hand. Since those properties of an electric charge which are crucial for the problem at hand are accounted for in the theory assuming a continual structure of the electric charge, the powerful mathematical machine works very well, immensely facilitating the analysis of the behavior of real charges.

Therefore the contradiction between abstract mathematical concepts and the real properties of the objects of study in physics is more of philosophical than of practical significance.

Still, the amazing usefulness of mathematics in physics seems to be a vexing problem for curious minds. It has so far not received a commonly accepted interpretation, although more than once various explanations have been suggested.

I believe that the fact of mathematics being the language of science can be understood through a rather simple and general idea. One of the requirements for an approach to be genuinely scientific, which is not limited to science alone, but is equally valid for any discourse which may be viewed as reasonable, is to be logical. Logic itself cannot serve to distinguish between correct and incorrect statements, because logic is simply the proper path from a premise (or premises) to a conclusion. The logical conclusion is only as good as the premise. However, if the premise is correct the conclusion reached via a logical procedure is reliable. Therefore it is imperative to use logic at every stage of a scientific inquiry. Mathematics is the most powerful and the most concise form of logic. Naturally, science had to resort to mathematics as an extremely powerful tool of logical discourse.

Leaving this philosophical problem aside, we may assert, that, generally speaking, mathematics is the proper language for science.

In many of the following articles, the question will arise of what constitutes a proper definition of the concepts under discussion (see, for example, A Consistent Inconsistency, sections on probability and complexity). Therefore it seems appropriate to provide first a general discussion of what the features of proper definitions are and which types of definitions can be acceptable in a scientific discourse.

Obviously, a fruitful scientific discourse is possible only if certain definitions are agreed upon. Proper definitions are therefore of the utmost importance and introducing them requires the maximum possible care.

It seems a platitude to say that a definition is essentially an agreement regarding the meaning of a term.

In the everyday discourse, we are rarely concerned with the precise definition of words used in the vernacular. The vagueness of the vernacular is actually its advantage, supplying flexibility to the discourse and enlivening the latter. The vagueness of the vernacular is also connected to its redundancy, which facilitates communication. Redundancy, providing a certain amount of excessive information (in the layman's sense of that term), helps to compensate for the possible distortion of the message which is due to the unavoidable noise accompanying the transmission of a message.

Scientific terms, on the other hand, usually require precise definitions and this is one of the features which distinguish scientific discourse from an everyday chat.

A term may have quite different definitions depending on the branch of science in which it is being utilized. For example the term vector in mathematics and its applications in physics means a quantity fully determined by its magnitude and direction (as opposite to a scalar which is determined by its magnitude alone, without any reference to direction). On the other hand, in pathology the same term, vector, means an organism, such as a mosquito or tick, that carries disease-causing microorganisms from one host to another. This example illustrates the utmost importance of rigorous definitions of concepts used in each field of science.

Science is a collective enterprise. It is impossible without communication within the community of participants in that enterprise. Communication is predicated on the acceptance of certain common definitions of concepts of which the edifice of a particular science is composed.

Formulating a definition of a concept is a logical procedure which, although it may vary in particular details, is subordinated to certain general rules common for all the variations of definitions. One of the most salient features of a definition is that it presupposes a certain preliminary knowledge relevant to the subject of definition. No concept can be defined unless the meaning of certain underlying concepts has already been agreed upon. If the definition of a concept is attempted in the absence of some underlying relevant knowledge, such an attempt will not provide a legitimate definition, but at best a quasi-definition which will necessarily be ambiguous and logically deficient.

A corollary to the last statement is that every field of science starts by introducing a seminal concept for which no real definition can be given. The word "starts" in that sentence implies a logical rather than chronological order of steps in building up a field of science.

There are several possible classifications of definitions.

One such classification is based on the logical structure of a definition. According to that classification, three types of definition can be distinguished. They are a) Deductive definition; b) Inductive definition, and c) Descriptive definition.

A deductive definition has the form of a triad. It necessarily comprises three elements which are: a) a general concept which encompasses a certain set A of elements and which is assumed to be commonly understood within the appropriate circle of the participants of a discourse; b) a particular concept which is the target of the definition and which encompasses a certain subset B of elements of the set A, and c) qualifiers which specify those features of the elements of subset B distinguishing the latter from all other elements of set A not included in subset B. Obviously, for a deductive definition to be meaningful, the participants of the discourse must have a preliminary understanding, assumed to be identical for all participants of the discourse, first of the general concept which determines set A and second of the qualifiers characterizing subset B.

A deductive definition also usually contains sentential elements, either explicitly or implicitly, in the form of "such.... that," "that (those)...which (who)" and the like. These sentential elements can be omitted but they are nevertheless implied.

For example here is a deductive definition of the term "colt."

A colt is such a horse that is a) young, and b) male.

In this definition the general concept is that of "horse." It is assumed to be known to all participants of the discourse. If it were not known, the definition would make no sense as it would define the target - "colt" – through an unknown concept requiring its own preceding definition. By the same token, the qualifiers, of which there are two in this case (young and male) are assumed to be commonly understood by all participants of the discourse, otherwise the definition of a colt would be useless. All horses, regardless of their being colts or not colts (i.e. fillies, geldings, stallions and mares) are elements of set A, while only those horses which meet the qualifiers (i.e. are male and young) are elements of subset B.

Here is another example.

Daughters of Ann and Andrew are those of their kids who are female.

I have intentionally chosen an example which sounds primitive in the extreme because, despite its simplicity, it is representative of a deductive definition.

According to the goal of a definition, deductive definitions can be of three basic types, a) phenomenological, b) taxonomic, and c) explanatory definitions. The distinction between these three types of definitions may sometimes be quite clear, while in some other cases this distinction may be blurred. For example, a taxonomic definition may be at the same time also either phenomenological or explanatory. Also, a definition may be viewed as either phenomenological or explanatory depending on the requirements to the degree of its explanatory power.

The distinction in question is determined not as much by the contents of a definition as by its goal.

A taxonomic definition is always a member of a set of "parallel" definitions all of which contain the same general concept but differ in qualifiers. For example, the definition of a colt discussed above is taxonomic, as it is a member of a set, including also definitions of filly, mare, stallion and gelding.

From another standpoint, deductive definitions can be either qualitative or quantitative.

Let us review an example of a definition taken from the elementary course of physics: Semiconductors are those materials whose electric conductivity increases with temperature.

In this case set A comprises all materials whereas subset B comprises only semiconductors, while the qualifier is a certain feature of the behavior of the elements of subset B which distinguishes them from all other materials – namely the temperature dependence of their electric conductivity. This definition belongs to the phenomenological type. It determines the target of definition by specifying a certain behavior of elements of subset B without providing an explanation of why they behave as they do. Assume now that we need a definition of "semiconductor" which is of explanatory type, i.e. containing a reference to the mechanism underlying the behavior of conductivity described in the phenomenological definition. We discover that the mechanism in question is not the same for all semiconductors, so that there are three distinctive classes of semiconductors, in all of which electric conductivity increases with temperature, but the underlying mechanisms of that phenomenon are different. Therefore, in order to provide an explanatory rather than a phenomenological definition of a semiconductor, we have to first introduce taxonomic definitions for each of the three classes of semiconductors. One such class is the so-called intrinsic semiconductors (the two other classes are the donor-type or n-type semiconductors, and the acceptor-type or p-type semiconductors). A taxonomic definition of the intrinsic semiconductors can be given as: An intrinsic semiconductor is such a semiconductor that is not doped by either donor or acceptor elements. (The term doping means deliberately inserting into a material A small amounts of another material, B, which alters the properties of A in a desired manner). Now, as the concept of an intrinsic semiconductor has been taxonomically defined, we can give an explanatory definition of that concept: Intrinsic semiconductors are such materials in which the energy gap between the conduction and the valence energy bands is narrow.

This definition is better in that it specifies the intrinsic property of elements of subset B which predetermines their behavior. However this definition, while explanatory, is only qualitative. A still better scientific definition would be quantitative. It would numerically specify the meaning of the term "narrow."

If either a general concept encompassing set A or the qualifiers distinguishing subset B from the rest of set A are unknown, the triad-like deductive definition is impossible.

The first type of situation is common if a seminal concept of a science is to be chosen. Since at the starting point of a new field of science there are yet no defined concepts in the arsenal of that science, its starting concept cannot be deductively defined. In this case the first element of the triad is absent, hence the triad cannot be logically constructed. A good example is the concept of energy in physics.

Energy is the most fundamental concept in physics. It is also its most seminal concept (not in a chronological but in a logical sense). In a certain sense, physics is a science about energy transformations. The law of energy conservation is the most fundamental basis of the science of physics. However, since physics has no concept which is more general than energy, the latter cannot be deductively defined.

Of course, one can find in textbooks on physics, especially those published many years ago, any number of alleged definitions of energy, but most of them are logically deficient and are not real rigorous definitions but rather attempts at explaining the meaning of that concept without really defining it in a logically consistent way. Some of those quasi-definitions are plainly absurd. One can see in some old textbooks such quasi-definition of energy as one asserting that energy is "the ability of a body to perform work," which is a meaningless statement, appealing to the laymen's understanding of the term "work."

The famous physicist Richard Feynman, in his excellent course "The Feynman Lectures on Physics," prudently avoided attempts to define energy in a rigorous way, providing instead an explanation of the term energy as "a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes." (R. P. Feynman, R.B. Leyton and M. Sands, "The Feynman Lectures on Physics, Addison-Wesley Publishing Co, 1963, v.1, p. 4-1). This is a descriptive definition which cannot be replaced by a rigorous triad because we lack any concept which is more general than energy, so the vague term a certain quantity is utilized to avoid the unsolvable problem of defining the seminal concept.

Another case when the rigorous triad-like deductive definition is impossible is when no qualifiers are known. In such cases we know the general concept and know that, in principle, the target of the definition is a sub-category within that general concept, but we don't know what distinguishes that sub-object from other sub-objects within the same general concept. For example, let us say John and Mike are identical twins. Obviously they are two different persons, both being, though, sons of Andrew and Mary. However, we don't know how to distinguish between John and Mike so we can't provide a triad forming the proper definition of either who Mike is or who John is. Any attempt to provide a definition, say, of Mike, will result in a tautology which, although necessarily true, is of no substance (for example, an assertion that Mike is he of the twins whose name is Mike). At best, such a definition can be of what can be called "accidental" type, which is not really useful because it refers to some irreproducible (accidental) qualifiers and therefore cannot be applied under conditions which are in any way different from the specific situation used for generating such qualifiers. For example, such a definition can sound like: Mike is that son of Andrew and Mary who on Tuesday, January 3, 1978 opened the door of their house when we rang the bell. Although that definition formally is in the form of a deductive triad, its qualifier is of an accidental type and cannot be used to distinguish between the twins at any time other than Tuesday, January 3. 1978.

While a deductive quantitative explanatory definition is the most desirable scientific definition, quite often we have to satisfy ourselves with different types of definitions. The reason for that is not necessarily the principal impossibility of formulating a deductive triad, but often simply the history of the development of a certain range of concepts. If an experimental work generates a set of data, this often provides the basis for formulating an inductive definition of a new concept which becomes indistinguishable from a specific type of law of science. (We will discuss the concept of laws of science in detail a little later, but at this point we can note that a law of science which results from a process of induction is not equivalent to those laws which constitute parts of a scientific theory. Inductive laws, which are often no more than inductive definitions, belong to the phenomenological type. They state facts without explaining them.)

Inductive definitions which spell out a law can often be characterized also as relational definitions – they state a certain relation between two or more concepts. For example, assume a scientist studied the behavior of the metal Gallium at various temperatures. Repeating the tests many times, she found that at a pressure of 10⁵ Pascal, pure Gallium melts at 302.5 Kelvin. She postulates that her data, consistently showing that result, are not accidental but reflect a law. She formulates a law: "Pure Gallium melts at 302.5 K if the pressure is 10⁵ Pascal." Her postulate resulted from the procedure of induction wherein she proceeded from a set of individual events to generalizing it as a law. Her formulation defines a law and therefore is also a definition. It is a relational definition as it establishes a relation between a certain event (melting of Gallium) and two factors – temperature and pressure. It is also a phenomenological definition since its states a fact without explaining its nature.

Even if an inductive definition is not stating a law, the path to such a definition is similar to the procedure of scientific induction wherein the results of observations or experiments repeated many times are generalized, postulating that the data in question are the manifestation of a law of nature. We will discuss the transition from a set of observational or experimental data to a postulate of a law in detail in a subsequent section. At this point, let us simply illustrate the idea of inductive definition by an example.

Imagine that a zoologist whose name is Johns went to explore jungles in, say, some part of Africa, and discovered there a number of animals hitherto unknown. Let's say all these newly discovered animals happened to be various kinds of cats – cat M, cat N, cat L, etc. Each of these cats differs in certain characteristics from other cats, but all of these cats have certain features in common distinguishing them from cats found in other parts of the world. The zoologist suggests a name for this newly discovered subfamily of cats, say Johns's cats. Johns provides a detailed description of each of the kinds of John's cats – M, N, L etc. He has then to provide a definition of the new subfamily of cats as a whole. He defines them as follows: "Johns's cats is a subfamily of cats comprising cats M, N and L." He may also provide a description of the features which are common for all Johns's cats, but are absent in all other cats (for example, all Johns's cats may have a peculiar shape of their ears, which is not observed in any other cats). This is an inductive definition, generalizing subcategories into a broader super-category.

Inductive definitions, by their nature, are open definitions, in that the list of their elements can be expanded if new elements are discovered which can be included into the definition. On the other hand, deductive definitions are closed in that they encompass all possible elements which can be included in the definition.

The lack of a rigorous definition, agreed upon by the participants of a discourse, may render an otherwise sophisticated discourse too vague to be genuinely scientific and may deprive even a very ingenious argument of its evidentiary value.

The principal building blocks of science are

1) Methods of experimentation,

2) Data,

3) Bridging hypotheses,

4) Laws,

5) Models,

6) Cognitive hypotheses,

7) Theories.

A warning seems to be in order. The above list may create the impression that science can be presented by a neat straightforward scheme, as a combination of clearly distinguishable building blocks. No such neat scheme exists. The above listed building blocks of science can overlap, intersect and emerge in an order different from the above list. However, the division of the body of science into supposedly separate building blocks, besides providing convenience in analyzing the extremely complex and multifaceted form of human endeavor we call science, also reflects the real composition of science, even though the boundaries between its constituents may sometimes be diffuse.

We will discuss the above concepts in detail. Before doing that, note that for each of the above elements of good science there is a corresponding element of bad science, i.e. bad methods, bad data, bad hypotheses, bad laws, bad models, and bad theories. On the other hand, pseudo-science differs from genuine science not just because one or several of the above building blocks can be characterized by adding the term pseudo (which, of course, is possible in principle) but because pseudo-science typically lacks legitimate data. Since data are that part of science which provides evidence, the absence of real data means the absence of evidence which would support the hypotheses, laws, models and theories. A hypothesis, law, model, or theory suggested without the real supporting data is typical of pseudo-science.

The statements in the preceding paragraph require clarification of what distinguishes good data from bad data and from wrong data. This question will be discussed in the section on data.

Whereas bad science is usually swiftly recognized as such, pseudo-science sometimes features a strong power of mimicry, disguising it as genuine science. The litmus test enabling one to distinguish genuine science from pseudo-science is in looking for the data underlying the hypotheses, laws, models, and theories of the supposed science. Quite often, discovering no data, i.e. no evidence supporting those hypotheses, laws, models and theories, serves as an indication that we are dealing with pseudo-science, regardless of how sophisticated the hypotheses, laws, models and theories in question may seem to be and regardless of how eloquently they are presented.

Review a few examples of pseudo-science. Probably the most egregious example of a pseudo-science which had tragic consequences on an enormous scale is that of Marxism. Created in the 19th century Europe mainly by Karl Marx and Friedrich Engels, it so successfully disguised as science that it won a large number of adherents despite its obvious contradiction with facts. Marxism comprised several parts which can be roughly defined first as a philosophy of the so-called dialectical materialism, second as the alleged historical analysis of society's development (the so-called historical materialism) and third, as the Marxist political economy. The dialectical materialism was an eclectic philosophical theory combining elements of Feuerbachs's materialism and Hegel's dialectics. Neither Marx nor Engels contributed much to the development of their predecessors' philosophical ideas but they reformulated those ideas in the simplified form of a neat set of simple statements which could be much easier comprehended by non-philosophers (such as a vulgarized version of Hegel's principles of the transition of quantity into quality or negation of negation, etc). (It is interesting to note that a frequent feature of pseudo-scientific theories is that they often suggest a neat and simple scheme allegedly representing complex reality). The core of Marxism was though their theory of the class struggle whose essence they themselves succinctly expressed by the maxim that the history of mankind is the history of the struggle of classes. That theory reduced all the complexity of human history to one factor – the struggle of economic classes. Marx and Engels did not seem to be worried by the fact that their representation of history was completely contrary to the histories of such countries as, for example, China or India, and reflected even the history of Europe only in a rather strained way. Such absurdities as interpreting religious wars as the results of the struggle of economic classes did not seem to make the creators of "scientific Marxism" pause and allow for the role of any factors besides the class struggle in the history of the mankind. Their extremely narrow concept ignored a multitude of various factors that affected the history of the human society. In other words, they arbitrarily chose a very narrow subset of data from the much wider set of available data in order to fit the data to their preconceived theory. This is a classic example of pseudo-science, whose main fault is the absence of a sufficient scope of data so that the theory is built accounting for only a deliberately selected tiny fraction of factual evidence. Since, however, Marxism had all the appearance of a scientific theory, its predictions won wide popularity, with one of the consequences being the bloody revolution in Russia. Although the actual revolution and its consequences quite obviously did not fit the blueprint predicted by Marxism, the latter acquired the status of a godless religion, believed in with a fanatical stubbornness despite its obvious futility and the blood bath it caused. Indeed, like in religions, one of the persistent claims by Marxism was that it was "omnipotent because it was true."

Another example of a pseudo-science is Lysenko's pseudo-biology which was imposed by decree of the Soviet communist rulers as the only true biology compatible with Marxism-Leninism. Lysenko's theory was a complete hogwash with no basis in factual data whatsoever. Its imposition also had tragic consequences, as many genuine scientists were arrested and perished in the Gulag because they tried to refute Lysenko's pseudo-science by pointing to the numerous data which it plainly contradicted.

One more, this time rather comic, example was the allegedly revolutionary theory of viruses suggested in the late forties in the USSR by a semi-literate veterinary physician Boshian. According to that theory, which was officially approved by the communist party, viruses constantly convert into bacteria and vice versa. Of course, Boshian's theory was based on fictional data. When, after many unsuccessful attempts, a commission of scientists finally gained access to Boshian's secret laboratory, all they found, instead of samples of the alleged virus-bacteria colonies, was plain dirt.

Other examples of pseudo-science include the so-called "creation science," with its arrogant distortion and misuse of facts, as well as its more recent and more refined reincarnation under the label of the "intelligent design theory."

This theory, promoted by a large group of writers, including many with scientific degrees from prestigious universities and with long lists of publications, and propagated on various levels of sophistication, has all the appearance of scientific research, as it offers definitions, hypotheses, laws, models, and theories like a genuine science. What is, though, absent in the "intelligent design theory," is evidence. No relevant data which support hypotheses, laws, models, and theories could be found in the papers and books by proponents of intelligent design, only unsubstantiated assumptions. Therefore it can justifiably be viewed as pseudo-science. I will also touch on this subject in the section on the admissibility of the supernatural into science, as well as in other articles on this website.

The statement that methods of experimentation are the engine of the progress of science seems to be quite trivial Whereas this statement is especially transparent in physics, chemistry, biology and in engineering sciences, it is valid as well for any other science even if it may sometimes be not as obvious as it is for the listed fields of inquiry.

When Galileo built his telescope, this provided an enormous impetus to the development of astronomy. When the Hubble telescope was put in orbit, astronomy underwent another powerful push forward. When Antony van Leeuwenhoek built his microscope, it revolutionized biological science. When the first electron microscope was built, it again revolutionized several fields of science, leading to the amazing modern achievements like actually seeing and even manipulating individual atoms.

These well known examples are only the tip of the iceberg, because the progress of science is pushed forward by the multiple everyday inventions of various ingenious methods of experiment or observation.

In thousands of research laboratories scientists whose names are usually unknown to anybody except for the narrow circle of their colleagues, every day apply their ingenuity and inventiveness to creating new, ever more subtle methods to question nature. Without these often extremely ingenious experimental methods and highly sophisticated experimental set-ups there would be no progress of scientific penetration into objective reality, which often guards its secrets very jealously.

There are two paths experimental science takes to progress.

One path is the ever increasing capabilities of devices designed as versatile tools of scientific inquiry and of engineering or medical development, such as electron microscopes, telescopes, and electronic devices (signal generators, potentiometers, recorders, oscilloscopes, etc).

The other path is the design of unique experimental contraptions designed for specific experiments and measurements.

These two path intersect at many points, as designing a specific new experimental contraptions is often mightily assisted by the availability of the more advanced tools of a versatile type.

Whereas inventions such as a microscope or radio transmitters and receivers justifiably gain wide publicity, many amazingly ingenious methods invented by scores of not-very-famous scientists usually remain unknown except for a narrow circle of researchers working in that particular field of science.

Here is a typical example. In the late seventies, at one of the foremost research institutions in the USA, a group of highly qualified and trained scientists was developing certain types of magnetic memory for computers. In one part of that study a magnetic wire was pulled at a certain speed through a tube filled with an electrolyte, and certain electrochemical processes on the surface of the wire were investigated. The researchers encountered a problem: as the electrolyte was pumped through the tube, the friction between the electrolyte and the walls of the tube caused the pressure's drop along the wire and this distorted the data. Considerable effort was invested in order to somehow eliminate or at least alleviate the described effect, but the effect persisted. Then a guest scientist arrived who has never before dealt with the experiment in question. At the very first meeting where the detrimental effect of the pressure drop was discussed, the guest scientist looked at the diagram of the tube and the wire moving along the latter, and said: "Why, instead of pumping the electrolyte through the tube, just eliminate the pump, make a rifle-type spiral groove on the inner surface of the tube and put the tube into rotation. The groove will push the liquid with a pressure which will be automatically uniform all along the wire."

This is a very simple example of how a fresh look at a problem may suddenly solve a seemingly difficult problem in a very simple way, but this simple example is typical and exemplifies those thousands and thousands of small and sometimes not very small inventions and innovations which occur every day in numerous research laboratories.

The unstoppable development of experimental methods, from designing giant accelerators of elementary particles to small improvements in measuring and observational methods, is what underlies the progress of science.

Therefore, the statement that a revolution in science starts when the experimental technique proceeds to the next digit after the decimal point reflects an important aspect of the progress of science.

a) General discussion of data

In the words of the famous Russian physiologist Ivan Pavlov, facts are the bread of science. The term data is synonymous with facts. It is also synonymous, in a certain sense, with evidence. Without data there is no science, only pseudo-science. In order to function as evidence, data must be reliable. There are two main types of data, observational and experimental. Both are equally legitimate, the difference being in that observational data are usually obtained in a passive manner, by merely recording whatever nature deigns to display by itself, whereas experimental data are gained as the result of a deliberate procedure of a planned research in which specific conditions are artificially created to force nature to display answers to specific questions. The demarcation between observation and experiment are not quite sharply defined since observation may be a natural part of a designed experiment.

Observation, and even some primitive form of experimentation, may happen to occur in a completely non-scientific activities. In a popular comic Russian poem for children, a Russian orthodox priest counts ravens sitting on trees, just for the heck of it. It is an observation and it is accompanied by a measurement, but this does not make the priest's activity scientific. Scientific observation or experimentation, while driven mainly by curiosity, always has an either conscious or subconscious purpose – to establish facts which may shed light on the intrinsic structure and functioning of the real world.

A crucial element of a scientific experiment is measurement. Measurement makes data quantitative, thus enormously enhancing the data's cognitive value. There is a theory of measurement which teaches us about the precautions necessary to ensure the reliability of the measured quantities and about the proper estimate of unavoidable errors of measurement. This theory is beyond the scope of this essay, but we have to realize that, on the one hand, the data obtained via a properly conducted experimental procedure are reasonably reliable, but on the other hand they are always true only approximately and cannot be relied upon to an extent exceeding that determined by the margin of a properly estimated error.

Consider some examples.

Astronomy is a science wherein observation is by far more prevalent than experimentation. A classic example of highly valuable observational data is the tables of the planets' positions compiled by Tycho de Brahe over the course of many years of painstaking observations accompanied by meticulous measurements. These tables, after they wound up in the possession of Johannes Kepler, served as the data the latter used to derive his famous three laws of planetary motion. We will discuss later the transition from data via hypotheses to laws, and from laws to theories. My point now is to illustrate how reliable data, gained via a proper observational procedure accompanied by measurement became a legitimate part of a scientific arsenal.

A classic example of the experimental acquisition of data is the discovery, in the early years of the 20th century, of superconductivity by a group of researchers in Leyden, the Netherlands, headed by Heike Kamerlingh Onnes. These researchers systematically measured the electric resistivity of various materials at ever lower temperatures. In a specially designed experiment, samples of material were gradually inserted deeper and deeper into a Dewar flask on whose bottom there was a puddle of liquid helium, while their electric resistance was measured. Analysis of the obtained data led to the conclusion that at certain very low temperatures the electric resistivity of certain materials dropped to zero. This unexpected result seemed contrary to the common understanding of electric resistance prevalent at that time. However, data take precedence over theories. The scientists of Leyden, however puzzled and astonished by their data, came to believe in those data's reliability and thus announced the discovery of superconductivity. Their data were then reproduced by other scientists and a new law asserting the fact of superconductivity was formulated. The formulation of a law did not mean a theory of superconductivity was forthcoming. It took half a century until a theory of superconductivity was developed. Neither the law in question nor the theory of that phenomenon would be possible without first acquiring reliable data on the behavior of superconducting materials, these data acquired in a deliberately designed specific experiment.

As we will see, the path from data acquisition to a law can be quite arduous and prolonged. It involves steps necessarily requiring imagination and inventiveness, because no law automatically follows from data. This is even more true for the path from data, via law, to theory. However, only those laws and theories which stem from reliable data are constituents of genuine science. (The question of what constitutes reliable data will be the subject of subsequent sections wherein the reproducibility of data and the problem of errors will be discussed.)

I would like to illustrate the last statement by an example. In 1994, three Israelis, Witztum, Rips, and Rosenberg (WRR), published a paper in the mathematical journal "Statistical Science." In that paper they claimed to have discovered in the Hebrew text of the book of Genesis a statistically significant effect of the so-called "codes." According to WRR, the codes in question contained information about the names and dates of birth and death of many famous rabbis who lived thousands of years after the book of Genesis was written. If WRR's claim were true, its only explanation could be that the author of the book of Genesis knew the future. This would be a powerful argument for the divine origin of that book. The paper by WRR caused a prolonged and heated discussion. As a result of a thorough analysis of WRR's methodology, the overwhelming majority of experts in mathematical statistics concluded that the WRR's data were obtained in a procedure which in many respects was contrary to the rules of a proper statistical analysis. In other words, the community of experts concluded that WRR's claim stemmed from bad data. Therefore, WRR's work was rejected as bad science. It was not, though, originally rejected as pseudo-science, because they based their law on certain data to which they applied a statistical treatment (although the latter was partially flawed as well). However, despite the scientific community's almost unanimous rejection of WRR's work as based on unreliable data, two of them (Witztum and Rips) as well as a small circle of their adherents, stubbornly continued to insist that they made a genuine discovery, thus converting their theory from bad science to pseudo-science. Because the story of the alleged Bible code is quite educational with regard to how pseudo-science appears and persists, it is discussed at length in a separate article (see B-Codes Page).

On the other hand, claims such as those by Immanuel Velikovski have been rejected from the very beginning not as bad science but rather as pseudo-science because his claims did not stem from any data but only from arbitrary assumptions. In a book published in 1950 and titled "Worlds in Collision" Velikovski offered a whole bunch of wild theories allegedly explaining many mysteries that contemporary science could not explain. For example, one of his suggestions was that when, according to the biblical story, Yehoshua (Joshua) stopped the sun in the sky, the earth indeed stopped its rotation. Another hypothesis by Velikovski postulated a near-collision of Venus and Mars with Earth, thus allegedly explaining numerous biblical miracles. Of course, there are no data whatsoever which would serve as evidence for such theories, therefore they were justifiably relegated to pseudo-science. While Velikovski acquired substantial notoriety and was compared in non-scientific publications to Newton, Einstein, and other great scientists, no scientific magazine accepted his papers because they, while plainly contradicting Newtonian mechanics, did not offer a shred of evidence which would support his claims.

b) Reproducibility and causality

To be legitimately useful in science, data must meet several requirements, one of which is reproducibility. Neither the reputation of the scientists claiming certain experimental results nor the impressive appearance of their data seemingly conforming to the strict requirements of a properly conducted experiment are sufficient for the data to be accepted as a contribution to science. Data become a part of science only after they have been reproduced by other scientists. Indeed, the demise of the cold fusion (at least as it stands for now) was due precisely to the fact that other groups of researchers could not reproduce the data claimed by Pons-Fleischman and Jones. Likewise, the data claimed by Witztum and Rips could not be reproduced by other scientists, which was the main reason their theory was rejected by the scientific community.

The requirement of reproducibility is based on the assumption of causality as a universal law of nature. This assumption presupposes that reproducing certain experimental conditions must necessarily lead to reproducing the outcome of the experiment. This supposition is of course a philosophical principle to be accepted a priori. This principle has an ancient origin. It was already discussed in detail by Aristotle who introduced the concept of a hierarchy of four causes, the so-called material causes, the efficient causes, the formal causes, and the final causes. In more recent times, the principle of causality known as the principle of determinism was formulated by Laplace, and has been universally accepted in science for at least two centuries.

The advent of quantum mechanics seemed to have shattered that principle. The very fact that many outstanding scientists were prepared to abolish a principle that had been a foundation of experimental science for so long testifies against the claims by adherents of the so-called intelligent design theory who assert that scientists are dogmatically adhering to "icons" of metaphysical concepts rather than keeping open minds.

However, the interpretation of quantum-mechanical effects as a breach of causality is by no means unavoidable. Rather, it testifies to the insufficient understanding of submicroscopic processes wherein causality is allegedly absent.

First of all, even in the case of quantum-mechanical events, causality is obviously present as long as the macroscopic manifestations of those events are observed. Recall the example of the alleged absence of causality, namely experiments with microscopic particles passing through slits in a partition. If only one slit is open, on the screen behind the partition a diffuse image of the slit is observed. If, though, two slits are open, the image on the screen is a set of fringes. On macroscopic level, the outcome of the experiment is reliably predictable and fully consistent with the principle of causality. Indeed, the image on the screen is always the same if only one slit is open, and can be easily reproduced at any time, any number of times, anywhere in the world. If two slits are open, the picture on the screen is different from the case of only one opened slit, but it is reliably predictable and can be easily reproduced at any time anywhere in the world. Hence, causality is present as long as the macroscopic effects of the experiment are the issue. A question about the validity of causality is raised when the details of the event on a microscopic level are considered.

If an electron is a particle, it cannot pass simultaneously through two slits. Therefore we could expect that in the case of two opened slits, the image on the screen would be two separate diffuse images of slits rather than a whole set of fringes. Indeed, how can an electron passing a particular slit "know" whether the other slit is open or not? However, the behavior of individual electrons is different, depending on whether the other slit is open or not.

Thus, the results of the described experiments are sometimes interpreted as indicating that the same microscopic conditions lead to different outcomes, depending on the variations in macroscopic conditions. On the microscopic level, as this argument goes, the same conditions may result in different outcomes thus causality is absent.

Quantum mechanics tells us that the explanation of the described phenomena is in that electrons are not particles like those we can see, touch, etc. They are very different entities, which under certain conditions behave like waves rather than like particles. It does not seem possible to visualize an electron, because it is unlike anything we can interact with by means of our senses in our macroscopic world. Of course, it is well known that macroscopic waves (for example, sea waves) can very well pass several openings simultaneously. Therefore, the two slit experiment is not really an indication of the absence of causality on a microscopic level, but rather an indication of our inability to visualize a subatomic particle.

Richard Feynman said that nobody understands quantum mechanics. To interpret this statement, we have first to agree what the term "understanding" means. We can't visualize an electron, and in this sense we may say that we don't understand its behavior. However, we can reasonably well describe the electron's behavior using the mathematical apparatus of quantum mechanics, and can successfully predict many features of that behavior. In that sense, we can say that we indeed do understand quantum mechanics.

Since we can't visualize the intrinsic details of the electron's behavior, we are actually uncertain of whether, on the microscopic level, its behavior is indeed non-deterministic, or we simply don't know what causes this or that seemingly random path of the electron. And since we are uncertain, there is no reason to assert that the principle of causality breaks down at the microscopic level.

Review another situation wherein causality is often claimed to be absent. If we have a lump of a radioactive material, we can easily measure the rate of its atoms' decay. This rate is a constant for a specific material. For example, we can experimentally find that, in the course of a prolonged experiment, the fraction of the atoms which decayed every second was, on the average, 0.01. Thus, if initially the lump consisted of N₀ atoms, one second later the number of atoms of that material was N₁< N₀, one more second later, the number of atoms was N₂ < N_1, etc, whereas the ratios N₁/N₀, N₂/N₁, etc, averaged over the duration of the experiment, were 0.99. This result is well reproducible, thus, again, on the macroscopic level causality seems to be intact. However, if we discuss the phenomenon on microscopic level, the question arises of why at any particular moment of time a particular atom decays while other atoms do not. All atoms within the lump of material are, from a macroscopic viewpoint, in the same situation. There is a certain probability of an atom's decay, which is exactly the same for every atom in that lump of material. However, at every moment some atoms decay whereas others do not. Therefore, the argument goes, causality does not seem to be at work in that process. All atoms are in the same situation, with the same probability of decay, but only a certain fraction of them decays without any evident reason which would distinguish the decaying atoms from those still waiting their turn.

The described problem of the possible "hidden parameters" affecting microscopic processes is the subject of a rather heated discussion among scientists, wherein no consensus has so far been reached. For example, Feynman maintained that an explanation referring to "hidden parameters" is impossible, because nature itself "does not know in advance" which atom will decay at which moment, or which path a particular electron will choose at a given moment. With all due respect to the views of the brilliant scientist Feynman, I find it hard to accept his position. I express here my personal view of the problem.

In my view, when we assert that all atoms are in the same situation, this is true on macroscopic level. We have insufficient knowledge of the situation of each individual atom on the microscopic level. The deficit of knowledge forces us to discuss the situation in probabilistic terms. Probability is a quantity whose cognitive value is determined by the amount of knowledge about the situation at hand. If we knew exactly the intrinsic details of the situation each atom is in, we could, instead of probabilities, discuss the matter in terms of certainties. Rather than abolish the principle of causality, the much more natural interpretation of the described process of atoms' decay is that there are definite causes of individual atoms decays at this or that moment of time, but we simply do not possess sufficient knowledge of the details of the process and therefore cannot predict which atom will decay at which moment. Our lack of knowledge does not necessarily mean the decay of this or that particular atom occurs without a specific cause.

Actually, an analogous situation can be observed on a macroscopic level as well. This happens, for example, in the so-called Keno game in Las Vegas casinos. In that game, players choose a set of several numbers out of a table containing, say, 49 numbers. The winning set is determined by a machine in which a pile of small balls is constantly mechanically shuffled in a shaking plastic vessel. Each ball bears a number. Balls move up and down the pile, constantly exchanging their positions, some of them pushed up by the rest of the balls, some others sinking deeper into the pile. Every couple of minutes, one of the balls is pushed high enough up from the pile and rolls out of the vessel. The number on that ball becomes part of the winning set. This is a chance process in which each ball has the same probability of rolling out at any given minute. This situation, albeit on a macroscopic level, is rather similar to the atoms' decay. At any moment of time, a particular ball rolls out while the rest of the balls remain within the vessel, waiting for their turn to get out. Why a particular ball happens to get out, but not any other ball, despite all of them having the same probability of being pushed out? We don't have a sufficient knowledge of the intricate web of interactions between the balls in the constantly shuffled pile, therefore we resort to probabilistic approach. However, if we knew precisely all the intricate details of the multiple encounters of those fifty or so balls in the shuffling machine, we would be able, instead of a probabilistic estimate, to predict precisely which ball will roll out and when. The lack of detailed knowledge, while forcing a probabilistic approach, does not mean that the observed occurrence – the choice of particular ball through a chance procedure - did not have a definite cause making the fallout of that particular ball at that particular moment inevitable while for no other ball there existed a similar cause to get pushed out.

The atoms' decay occurs on a microscopic level, and we have much less knowledge about the intrinsic conditions within and between the decaying atoms that we have in the case of the balls in a Keno game. This by no means must be interpreted as an indication that the acts of decay have no definite causes. The principle of causality can be preserved, even accounting for the quantum-mechanical effects.

Whatever interpretation of causality is preferred, if there is no causality, there is no science.

c) Ceteris paribus

The next question to be discussed in relation to data as a foundation of science is the so-called principle of ceteris paribus. This Latin expression means "everything else equal." The idea of that principle is as follows: Imagine that in a certain experiment the behavior of a quantity X is studied. This quantity is affected by a number of factors, referred to as factors A, B, C, D, etc. Some of those factors may be known while some others may be not. From a philosophical standpoint, everything in nature is interconnected and therefore the number of factors affecting X is immensely large, so we never can account for all of them. However, an overwhelming majority of that multitude of factors have only a very minor effect on X and therefore, in practical terms, most of them are of no consequence for the study at hand. Still, there is a set of factors whose effect on X cannot be ignored if we want to gain meaningful data from our study.

The principle of ceteris paribus is a prescription regarding how to conduct the study. Namely, according to that principle, the process of research must be divided into a number of independent series of experiments. In each series only one of the factors - either A, or B, or C, etc., has to be controllably changed while the rest of the factors must be kept unchanged. For example, if A is being changed, B, C, and D have to be kept constant, and the behavior of X has to be recorded for each value of A. Then another series of experiments has to be conducted wherein only factor B is being changed and X recorded while A, C, D, etc., are kept constant.

Of course, the application of the described procedure is based on the assumption that factors A, B, C, etc., are independent of each other, so that it is possible to change at will any one of those factors without causing all the rest of those factors to change simultaneously.

There are two problems inseparable from the above approach. One is related to the errors of measurement and the other to the very core of the ceteris paribus principle, i.e. to the above mentioned assumption of the factors' independence.

The problem of errors stems from the fact that no measurement can be absolutely precise and accurate ( the difference between precision and accuracy is defined in the theory of measurement and is beyond the scope of our discourse). In particular, even if factors A, B, C, etc,. are independent of each other, it is impossible to guarantee that while one of them is being changed, the rest of them remain indeed absolutely constant, because the values of B, C, D, etc., while in principle they are supposed to remain constant, cannot be measured with absolute precision and accuracy. Moreover, the values of factor A itself, which are supposed to be changed in a controlled way, are never guaranteed to have precisely the desired values, and, finally, the target of the study, X itself cannot be measured with absolute precision and accuracy.

The problem with the principal foundation of the ceteris paribus methods stems from the fact that in many systems, factors A, B, C, etc., are in principle not independent of each other. Hence, as factor A is being changed, it may be impossible in principle to suppress the simultaneous changes of factors B, C, D etc, which are supposed to remain constant. Such systems are sometimes referred to as "large" or "complex," or "diffuse," or "poorly organized" systems and their study necessarily can be conducted only beyond the confines of ceteris paribus.

Philosophically speaking, all systems are complex, so that factors A, B, C, etc., are always to a certain extent interdependent. However, the extent of their interdependence may be insignificant to a degree which makes the assumption of ceteris paribus reasonably substantiated.

For example, assume we study the behavior of ice/water in the vicinity of 273 K, which is, of course, 0⁰ C or 32⁰ F. The factor we change is temperature. Besides temperature, the behavior of ice/water is affected by a multitude of other factors. Most of them are insignificant and can be ignored because they have only a very minor effect. However, at least two factors have a pronounced effect – pressure and the purity of the water. In this case it is relatively easy to ensure the condition of ceteris paribus. We can keep a sample of ice in a vessel where pressure is automatically controlled and kept at a predetermined level, regardless of changes in temperature. Likewise, the purity of ice can be maintained at a constant level at all values of temperature investigated. In such a study we find that ice containing, say, not more than 0.01% impurities, at a pressure of about 10⁵ Pascal, melts at 273 K.

d) Errors

We have to distinguish between various types of errors. Sometimes errors stem from a basically wrong approach to scientific inquiry. It may be due to an insufficient understanding of the subject by a researcher, which results in an improperly designed experiment. It may be due to a preconceived view or belief, or to a strong desire on the part of a researcher to confirm his preferred theory. The researcher may subconsciously ignore results which are contrary to his expectations and inadvertently choose from the set of measured data only those which jibe with his/her expectations. Errors of these type, while by no means uncommon, should be viewed as pathological; they produce wrong data, which are a version of bad data, and are one of the constituents of bad science. An example of such bad data is the already discussed case of the alleged Bible codes.

However, even in the most thoroughly designed experiments, errors inevitably occur despite the most strenuous effort on the part of a researcher to avoid or at least to minimize them. These honest errors are due to the inevitable imperfection of experimental setups and to the equally inevitable imperfection of the researcher's performance. Let us discuss these honest errors in experimental studies.

Until recently, a common notion in the theory of experiment was the distinction between systematic and random errors. While this distinction is certainly logically meaningful and often can be made quite rigorously, lately it has become evident that on a deeper level the demarcation between these two types of error is rather diffuse. Moreover, the error can often be viewed as either systematic or random depending on the formulation of the problem. For example, an error can be viewed as systematic if it occurs systematically provided we always conduct measurements in a specific laboratory A. However, the same error can be viewed as random if we have a choice among many laboratories A, B, C, etc., and choose laboratory A, to conduct the measurement on a particular day, by chance. The deeper analysis of that question, as well as of the distinction between precision and accuracy of measurements, which is a proper subject of the theory of experiment, is beyond the scope of this paper, whose subject is the general discussion of the nature of science.

The question of experimental errors is, though, related to the problem of causality. If data are a legitimate component of science only if they are reproducible, the immediate question is: What are the legitimate boundaries of reproducibility? It is an indisputable fact that the expression "precise data," if interpreted literally, is an oxymoron. If an experiment is repeated many times, in each run the measured data will be to a certain extent different. There is no way to ensure the absolute reproducibility of data. The conventional interpretation of that fact is not that it negates the principle of causality but rather that in each experiment there are uncontrollable factors which change despite the most strenuous effort on the part of the experimenter to keep all conditions of the experiment under control.

Therefore the principle of causality is not a direct product of the experimental evidence but a metaphysical principle borne out by the total body of science. Science is based on the assumption of causality without which it simply would make no sense.

Therefore the inevitable errors of measurement are something science has to live with, making, though, a very strong effort to distill objective truth from the chaos of measured numbers. Data which are a legitimate foundation of laws and theories are therefore the result of a complex process in which wheat is to be separated from chaff and regularities have to be extracted from the error-laden experimental numbers thus building a bridge from data to laws. The bridge from data to laws will be discussed in subsequent sections.

It seems appropriate to discuss at this juncture of the discourse two points related to the manner in which an experiment is conducted.

One of these points is calibration. Calibration is a very powerful tool for ensuring the meaningfulness of an experiment, wherein the effects of many factors, which are extraneous to the subject of the study and whose contribution can only mask the phenomenon under study, are summarily neutralized in one step. For example, when Cavendish performed his famous experiment in which he, in his words, "weighed the earth," he needed to measure the attraction between lead balls. The force of attraction to be measured was determined by the angle of rotation of a rod to whose ends the lead balls were attached. The rod hung on a string, so that the elastic forces in the string created a torque which counterbalanced the forces of gravitational attraction. That torque was approximately proportional to the angle of the rod's rotation. Instead of calculating the forces which would cause a certain angle of rotation, Cavendish simply calibrated his contraption by applying known forces to the rod and measuring the correspondent angle of rotation. He obtained the "calibration curve," which incorporated all known and unknown factors that could affect the angle of rotation thus eliminating in one step – calibration – most of the sources of error. Calibration in various forms is routinely used in scientific experiments whenever possible, thus substantially eliminating many sources of error.

Another point to discuss is the difference between the discrete and the continual methods of recording experimental data. Each of these two methods has both advantages and shortcomings.

If a discrete recording is employed, the target of study - X - is measured for a set of discrete values of the controlled parameter A, while all other parameters affecting X are kept constant. Factor A is given values A₁, A _{2 ,}A ₃ etc, and at each of those values of A, X is measured, yielding a set X₁, X₂ , X₃ , etc.

As a shortcoming of such method, it is often asserted that there is never certainty regarding the values of X between the measured points. However close the values of A at which X is measured are, there are always gaps between those values of X, and to fill those gaps the researcher has an infinitely large number of choices. It is often expressed as an assertion that there are infinitely many curves which can be drawn through the same set of experimental points, regardless of how close those points are to each other. (Of course, a more accurate assertion should be that the curves may be drawn through the margins of error surrounding each experimental point).

While the above assertion is certainly correct in an abstract philosophical way, it rarely has serious practical implications, as will be discussed in the section on the bridging hypotheses.

The mentioned shortcoming of the discrete method of experiment seems to be eliminated if a continual method is employed instead. In this method, the controlled factor A (or often several controlled factors A, B, C, etc.) are being continually changed at a certain rate, while the values of the target of the study, X, are simultaneously and continually recorded.

An obvious advantage of this method is the substantial saving of time and effort necessary to accumulate information about the phenomenon under study. The wide availability of sophisticated recording equipment in the 20th century made continual recording the favorite method of experimental work in physics, chemistry and other fields of study.

Since the method of continual recording generates continuous curves rather than sets of experimental points, it may seem to eliminate the arbitrariness involved in drawing curves through discrete experimental points. Unfortunately, this supposed advantage of the continual method is illusory. The illusion is related to the problem of a system's response time to a perturbation, i.e. to an abrupt change of parameters. If any parameter affecting the system's behavior is abruptly changed, it sets in motion a process of the system's transition to a new state, this process proceeding with a certain finite speed. If a system was in a state wherein X had a certain value X₁ while factor A had the value A₁, and at some moment of time t₁ parameter A is changed to A₂, the system's measured property X does not instantly jump to a new value X₂, but rather undergoes a transition to the new value which takes a certain time ("relaxation period"). Therefore the experimental curves reflecting the functional dependence of X on A will have different shapes depending on the rate at which A is being altered by the experimental set-up. In a discrete method of experiment normally the time interval between consecutive measurements of X is sufficiently large for X to complete the transition to the new equilibrium value. In continual recording, depending on the rate of change of A, the transition of X to the new equilibrium value is often not yet completed as A already shifts to new values. It is well known that, for example, the loops of magnetic hysteresis are very different depending on the rate of the magnetizing field's change.

In connection with the question of experimental errors, the notion of noise seems to be relevant. The term noise entered science from information/communication theory. That branch of science studies the transmission of information (of "signals") which is inevitably accompanied by noise in the elements of the transmission channel. The noise is superimposed on the useful signal and has to be filtered out if the signal is to be reliably interpreted. The term noise gradually gained a wider use, referring to any unintended extraneous input to the experimental setup. Normally, to extract useful data from the experimental output the desired data have to be distilled from the overall output by excising noise. However, more than once noise happened to be the source of scientific discoveries. For example, when, at the end of the 19th century, physicists studied the processes in a cathode ray tube, these processes were always accompanied by emission of a certain radiation from the walls of the tube. Some researchers did not notice these rays, others did but viewed them just as noise and ignored them. Then Wilhelm Conrad Roentgen, who specialized in the study of dielectrics, and was generally highly meticulous in his research, looked at the "noise" as a phenomenon interesting in itself. He very thoroughly studied the phenomenon which his predecessors dismissed as just a nuisance and thus became the discoverer of that important type of radiation which he called X-rays.

Another well known example of how a phenomenon which was viewed as annoying noise led to an important discovery is that of the background cosmic radiation discovered in 1965 by Arno Penzias and Robert Wilson.

Generally speaking, if an experiment yields results which could be predicted, this adds little to the progress of science. The matter becomes really interesting if the results of an experiment look absurd. If a researcher encounters an absurd result, which persists despite efforts to clean up the experiment and to eliminate possible sources of error, there is a good chance the researcher has come across something novel and therefore interesting. The absurd outcomes may often be attributed to noise, but not too rarely that noise carries information about a hitherto unknown effect or can be utilized in an unconventional way for a deeper study of the phenomenon.

Let me give an example from my personal experience.

Many years ago I studied the process of adsorption of various molecules on the surface of metallic and semiconductor electrodes. One of the methods of that study was the measurement of the so-called differential capacity of the electric double layer on the electrode surface. A couple of my students participating in that research built a setup enabling us to measure the differential capacity at various values of the electric potential imposed on the electrode. After having worked with that setup for several weeks, they became grossly frustrated by their inability to eliminate the instability of the measured capacity. Each time they changed the value of the electrode potential, the differential capacity started shifting and they had to wait for many hours and sometimes days until a new, uncertain equilibrium seemed to set in. They viewed this unpredicted "instability" of the measured differential capacity as annoying noise preventing them from performing the desired measurement. They complained to me about their frustrating experience and asked for advice - how to get rid of it. I remember that moment when a kind of a sudden light exploded in my mind and I shouted, "Lads, it is wonderful! We got an unexpected result.< The capacity creeps – and this means that if you accurately measure the curve of capacity versus time, this curve will contain plenty of information about the kinetics of the adsorption process!" I set out to develop equations reflecting the relaxation process of the capacity. A new method of scientific study we named potentiostatic chronofaradometry was thus born. Instead of trying to eliminate the supposed noise and routinely measure the supposed equilibrium values of the differential capacity, now my students concentrated on measuring those very relaxation curves which they angrily considered to be annoying noise. Plenty of information about the kinetics of the adsorption/desorption process was extracted from that "noise," assisted by the equations derived for that process.

e) Complex systems

For over 200 years, one of the underlying hypotheses of science, often unspoken, was that every system which is a subject of a scientific study is "well organized," in the sense that it was always possible to separate a few phenomena or processes of similar nature, dependent on a limited set of important factors, in such a way that the system's behavior could be studied by isolating factors one by one, finding the functional dependencies between pairs of factors, and attributing to those functional dependencies the status of laws. In the 20th century the described approach started encountering serious difficulties. In many systems the separation of various factors turned out to be impossible. Science encountered the ever increasing number of what is referred to as "large" systems, or "poorly organized systems," or "diffuse systems," etc., reflecting various aspects of such systems' behavior. Such systems cannot be legitimately studied under the ceteris paribus approach because it is impossible to separately and controllably vary any one parameter affecting the system without causing a simultaneous variation of other parameters.

It should be noted that the concept of a "large" or "poorly organized" system is not equivalent to the concept of stochastic systems. The latter were well known and successfully dealt with in 19th century physics. An example of a stochastic system is a gas occupying a certain volume. It consists of an enormous number of molecules (for example, at atmospheric pressure and room temperature 1cubic centimeter of gas contains about 10¹⁹ molecules). The behavior of each molecule is determined by the laws of Newton's mechanics. However, because of the immense number of molecules, it is impossible to analyze the behavior of gas by solving the equations of mechanical motion for each molecule. Therefore, 19th century science came up with very powerful statistical methods which enabled it to analyze the behavior of a gas without a detailed analysis of the motion of each individual molecule. (There are theories according to which a system already becomes stochastic, i.e., not treatable by studying the behavior of its individual elements, if the number of these elements exceeds about 30.)

However, stochastic systems such as gases behave stochastically only on a microscopic level. On a macroscopic level they can be easily treated using the ceteris paribus approach. Macroscopic properties of a gas, such as pressure, temperature, volume, etc., can be very well studied by isolating and controllably changing these properties one by one while keeping the rest of the properties constant. This is possible because the immense number of elements of such systems does not translate into a large number of macroscopic properties. A gas containing quadrillion quadrillions of microscopic molecules, all of them identical, macroscopically is characterized only by a few parameters, such as pressure, temperature and volume.

On the other hand, the "large," or "poorly organized" systems are characterized by a very large number of macroscopic parameters, many of which cannot be individually varied without simultaneously changing some other parameters affecting the system.

Historically, one of the situations whose study mightily served the realization of the ineliminable interdependence of various parameters in a "poorly-organized" system is emission spectral analysis. A good description of the problems encountered when trying to interpret the data obtained via emission spectral analysis was given in a book "Teoriya Eksperimenta" (The Theory of Experiment) by Vasily V. Nalimov (Nauka Publishers, Moscow 1971; in Russian).

In that process, a sample of material to be studied is placed between two electrodes into which a high voltage is fed. The breakdown of the gap between the electrodes causes a spark discharge wherein the temperature reaches tens of thousands of Kelvin. The density of energy at certain locations on the electrodes reaches a huge level. An explosive evaporation occurs, followed by an equilibrium evaporation. A cloud of evaporated matter is created between the electrodes in which simultaneous processes of diffusion, excitation and radiation take place. Besides, oxidation/reduction processes occur on the electrodes' surface, and diffusion of elements occurs in their bulk, dependent on the temperature gradient, phase composition of electrode material, defects etc. All these processes are of a periodic nature, on which a one-directional drift is imposed by the electrodes' erosion. Attempts to analyze the described extremely complex combination of phenomena by means of separating various factors one by one turned out to be in vain. The system is too complex and therefore its analysis forced scientists to give up attempts to proceed using the ceteris paribus approach.

To study the "large" or "complex" systems, the science of the 20th century developed two main approaches. One is the multi-dimensional statistical approach and the other is the cybernetic (or computer-modeling) approach.

Since this article is not about the theory of experiment but about a general discussion of the nature of science, I will discuss the above two approaches very briefly and only in rather general terms.

The statistical approach was to large extent initiated by the British mathematician R. A. Fisher. It is often referred to as multi-dimensional mathematical statistics (MDMS). Essentially, MDMS is a logically substantiated formalization of such an approach to the study of "large" systems wherein the researcher deliberately avoids a detailed penetration into intricate mechanisms of a complex phenomenon, resorting instead to its statistical analysis using multiple variables.

The application of MDMS requires solving a number of problems involving the appropriate strategy of an experiment - the proper choice of the essential parameters - since accounting for too many parameters may make the task beyond the available intellectual and computational resources, while accounting for too few parameters may reduce the solution to a triviality. The effectiveness of MDMS has been drastically improving along with the availability of ever more powerful computers and software. Statistical models of complex systems have been successfully employed for a multitude of problems which could not be tackled by the traditional methods of direct measurement.

The cybernetic approach, whose origin can be attributed to Norbert Wiener, is in some respects very different from the statistical one, although it also has become really useful only with the advent of powerful computers. While the method of multi-dimensional statistics and that of a computer modeling are principally different, both are successfully applied, sometimes to the study of the same "complex" system.

The cybernetic approach makes no sense as long as only simple "well organized" systems are studied. In such systems one-dimensional functional connections fully determine the system's behavior and the problem of control (management) is moot. For example, the motions of planets are fully determined by Kepler's laws. On the other hand, control of the behavior of a "poorly organized," or "large" system, which is quite challenging, can be fruitfully approached via a cybernetic model. (Various meanings of the term model will be discussed in the section on models in science).

One of the frontiers of modern science is the field of artificial intelligence which exemplifies the cybernetic approach to a "complex" system (human intellect) by using a computer model of the latter.

Biological systems are mostly "large" or "poorly organized" in the above formulated sense.

Another example of a "large" system is the economy of any country.

In every country there is a multitude of factories, farms, companies and individuals each pursuing its/his particular economic interests, All these elements of the economic system interact among themselves in an immensely complicated web of transactions, changing every moment so that this enormously complex system is in a constant flux whose trends are usually not obvious. It is impossible to account for each and all features of that immensely complex game, not only because of its sheer size and the huge number of its constituent elements, but also because of its unstable character. Before the available data about the state of economy have been digested and interpreted, they have already changed. No computer, however powerful, can follow all the nuances of the economic game. Therefore the economy cannot be in principle "scientifically managed" as it supposedly was in the allegedly "socialist planned economy" of the former USSR. This was one of the reasons for the abysmal failure of the "socialist economy," which was no more scientific than it was really socialist. The actual economic system in the former USSR was "state capitalism," with all the drawbacks of an extreme form of a monopolistic capitalist system but without the advantages of a free enterprise. The attempts to sustain a planned economy allegedly based on a scientific analysis of the resources, demands and supplies, resulted in a seemingly controlled but actually chaotic economy wherein stealing and cheating became the only possible means of survival for the powerless slaves of the state.

Pretending to maintain a scientifically planned economy, as Marxist theory required, actually resulted in a system that was contrary to elementary basics of science.

The economy of the modern world functions in a cyclic manner, following its own poorly understood complex statistical laws wherein its oscillatory motions have an immense inertia, so that attempts to steer it in this or that direction (for example, by regulating the interest rate or changing the tax laws) have only a marginal effect on the waves of prosperity and recession replacing each other as if by their own will.

Large (or diffuse) systems require for their analysis specific approaches, and 20th century science has provided these approaches in the two above mentioned ways.

Whatever method is used to acquire data, either the traditional multi-step measurement under the ceteris paribus principle, or the application of multi-dimensional statistics, or a computer model, or any other approach, the acquisition of reliable data is an ineliminable step in the scientific method. It is possible to apply the scientific approach without strict definitions or without good hypotheses or without good models or without good theories (although omitting any of the listed components would seriously impair the quality of the scientific study, often making it bad science). It is impossible to have any science, good or bad, without reliable data. In the absence of data, the endeavor, however cleverly disguised and eloquently offered, can only be pseudo-science.

Perhaps, the title of this section should more appropriately be "Bridging Hypothesis," in the singular, because this hypothesis is essentially always the same. This hypothesis postulates the objective existence of a law. It constitutes a "bridge" from the raw data to the postulated law.

In the case of "simple" or "well organized" systems which can be studied under the ceteris paribus condition, the bridging hypothesis appears in its most explicit form, while in the case of "large" or "poorly organized" systems it can often be hidden within the complex web of the data themselves.

Imagine an experiment under the ceteris paribus assumption wherein a set of values of a target quantity X has been measured for a set of values of a parameter A, so for each value of A₁, A₂, A₃, etc., corresponding values X₁, X₂, X₃,etc., have been measured and also the margin of error ±Δ has been estimated so that every value of X_i is believed to be within the margin of X_i ±Δ.

For the sake of example, assume that the measured values of X are found to increase along with the increase of the corresponding values of A. Every set of X and A, however extensive, is always still only a selected subset of all possible values of X and A.

Reviewing the sets of measured numbers, we try to discern a regularity connecting A and X. To do so we have necessarily to assume that the set of actually measured numbers indeed reflects an objective regularity. There is never unequivocal proof that such a regularity indeed objectively exists, we have necessarily to postulate it before formulating a law.

More often than not such a hypothesis is implicitly present without being explicitly spelled out. In scientific papers we, quite usually, see statements introducing a law, without mentioning the bridging hypothesis. The researchers reporting their results routinely assert that, for example, "as our data show, in the interval between A₁ and A_k, X increases proportionally to A." This statement is that of a law. In fact, though, such statement of a law might not be legitimately made without first postulating the very objective existence of a law connecting X to A. The bridging hypothesis according to which the observed sets of values of X correspond to the selected sets of values of A, not as an accidental result of an experiment conducted under a limited set of conditions but because of an objectively functioning law, must necessarily precede, if often subconsciously, the statement of a law.

The hypothesis in question, which is necessarily present, most often implicitly, in the procedure of claiming a law, is never more than a hypothesis and cannot be proven. Fortunately it happens to be true, at least as an approximation, in the vast number of situations, thus constituting a reliable building block of science. In fact, if sets of As and of Xs seem to match each other according to some regularity, more often than not such a regularity does indeed exists, although sometimes it also

Of course, the bridging hypothesis is not applied if no law is postulated. A researcher may obtain sets of values of the measured quantity X corresponding to a set of values of a parameter A which may look like a display of a regularity, but for various reasons she may reject the bridging hypothesis and avoid an attempt to spell out a pertinent law. The reasons for that may be, for example, some firmly established theoretical concepts making the law in question very unlikely, or serious doubts regarding the elimination of some extraneous factors which could distort the data thus creating the false appearance of a regularity, or any number of other reasons.

More often, though, a scientist may be just uncertain in regard to the choice between accepting or rejecting the bridging hypothesis rather than flatly rejecting it. Often such uncertainty is based simply on the insufficiently complete data. The researcher is uncertain whether or not the available sets of data do indeed properly represent the entire multiplicity of possible data.

If that is the case, the proper remedy is to resort to a statistical study of the phenomenon in question. The proper tool for such a study is the part of mathematical statistics called "hypotheses testing."

The procedure involves introducing two competing hypotheses, one called the "null hypothesis," and the other the "alternative hypothesis." The null hypothesis is the assumption that the available set of data does not represent a law. The alternative hypothesis is in this case a synonym for what I called the bridging hypothesis, i.e. the assumption that the available set of data reflects a law. As a result of a proper statistical test, the researcher compares the likelihood of the null hypothesis vs. the likelihood of the alternative hypothesis. The hypothesis whose likelihood is larger, is accepted, while the other hypothesis is rejected. This choice is never assured to be ultimately correct, since discovery of additional data may change the likelihood of the competing hypotheses. In this sense, laws of science are usually referred to as tentative.

Fortunately, the rigorous self-verification inherent in a proper scientific procedure makes a vast number of laws of science work very well despite their "tentative" character. In the next section we will discuss the laws more in detail.

Whereas the bridging hypothesis discussed in the previous section is simple and uniform in that its gist is simply in assuming that the set of experimental data reflects a law, the specific formulation of a law itself is quite far from being simple and uniform. While in every scientific procedure the same bridging hypothesis is present, this in itself contains no indication of how the particular law has to be spelled out.

Formulating a law is not a mechanical process of stating the evident behavior of the studied quantities. It necessarily involves an interpretive effort on the part of the scientist, who makes choices among various alternative interpretations of the assumed regularity which have to be extracted from arrays of numbers defined within certain margins of error.

In the simple case of a functional dependence between two variables, the scientist is confronted with a set of experimental points which seem to display a certain regularity, often rather nebulous because of inevitable experimental errors. Hence, besides the bridging hypothesis which simply postulates the very existence of a law, the scientist must then postulate the specific law itself.

Consider an example. In Coulomb's well known law of interaction between point electric charges, the force of interaction is presented as being inversely proportional to the squared distance between the charges.

The inverse proportionality of the force of electric interaction to the square distance between point charges is, however, not a direct conclusion from the experimental data. One of the reasons for that is the inevitability of errors occurring in every procedure of measurement. If the dependence of Coulomb's force on the distance between the point charges is measured, it is assumed that a) while the distance is being changed, the interacting charges are indeed being kept constant, b) that no other charges which could distort the results are anywhere close to the measuring contraption, c) that the distance itself is measured with sufficient accuracy, etc. All of this is, of course, just an approximation. Let us imagine that in the course of an experiment wherein Coulomb's force was measured, the distance between the supposedly point charges was given the following values (in whatever units of length): 1, 2, 3, 4, 5, 6, the total of six points at which the above values were determined with an error not exceeding, say, ±5%. This means that when the distance was assumed to be 2, it could actually be anything between 1.9 and about 2.1, and similarly for every other value of the distance. Imagine further that the measured Coulomb's force (in whatever units of force) happened to have the following set of values, averaged over many repeated measurements: 1097, 273.87, 121.64, 68.97, 43.12 and 31.01. Repeating the measurement many times, the experimentalist evaluated the margin of error for the force to be ± 10%. Reviewing the set of numbers for the values of the measured force, the researcher notices that these numbers are rather close to a set of numbers obtained by dividing the maximum force of 1097 (measured for the distance of 1) by the squared distances. Indeed, dividing 1097 by the squared values of distances 2, 3, 4, 5, and 6, one obtains the following set: 274.25, 121.89, 68.56, 43.88, 30.47. Every number in the experimentally measured set differs from a corresponding number in the second, calculated set, not more than by 10%, which is within the margin of error of that experiment. Although none of the measured numbers exactly equals the numbers calculated upon the assumption that the force drops inversely proportional to the distance, the scientists postulates that the difference between the measured and calculated sets is due to experimental error and that the real law, hidden behind the measured set of numbers, is indeed the inverse proportionality between the force and the squared distance.

What is the foundation of that postulate? If instead of the power of 2, the distance in the same formula appeared with the power of, say, 2.008 or 1.9986, such a formula would describe the experimental data as well as Coulomb's law, where the power is exactly 2. Of course, the precision and accuracy of the measurement can be improved so that the margin of error is substantially reduced. However, it can never be reduced to zero. Therefore, the best experimental data cannot provide more than is inherent in them. With a much improved technique we may be able to assert that Coulomb's force changes inversely proportional to the distance to the power being, say, between 1.9999999987 and 2.0000000035, but we can never assert, based just on the experimental data, that the power is exactly 2. We have to postulate that the power is exactly 2 rather than any other number between 1.9999999987 and 2.0000000035.

Such a postulate is often made on non-scientific grounds. It is often based on some metaphysical consideration, some philosophical principle, or such criteria as simplicity, elegance and the like. We actually have no scientific grounds to prefer the power of 2 to, say, the power of 2.0000986345, since both numbers equally well fit the experimental data. We choose 2 not because the data directly point to that choice but because this choice seems to be either simpler (i.e., using Occam's razor), or more elegant, or more convenient, or maybe just favored by a particular scientist for purely personal reasons.

Of course, the difference between 2 and, say, 2.0000897 has no practical consequences, so that we can safely use the formula wherein the exact power of 2 has been postulated, even though we may not be absolutely confident that it is indeed the exactly correct value.

Furthermore, even the principal form of the law – the power function - is not logically predetermined by the data. There are many formulas differing from the simple power function which would yield numbers within the margin of error of the experimental data. For example, it is always possible to construct a polynomial expression whose coefficients are chosen in such a way as to make the numbers calculated by that expression match the measured data within the margin of the experimental error. Hence, not only the value of 2 in the formula of Coulomb's law, but even the form of the law itself is the result of a scientist's assumption. From the above discussion it follows that the laws of science are necessarily postulates. We don't know that the distance in Coulomb's law must be squared, we postulate it.

The choice of a postulate is limited by two considerations. One is the requirement not to contradict experimental data, so we may not arbitrarily assume that the distance in the formula in question must have the power of, say, 2.75, because it is contrary to evidence. The other requirement is not to contradict the overall body of scientific knowledge. Otherwise any number which is within the margin of the experimental error has, in principle, the same right to appear in the formula of law, and the choice of 2 instead of, say, 1.99999867 is a postulate based on metaphysical grounds.

The application of some mathematical apparatus such as, for example, least square fit, does not principally change the situation because the output of the mathematical machine is only as good as is the input.

The above discourse may create an impression of arbitrariness of the laws of science. Some philosophers of science think so. Are the laws