One characteristic of humankind by which it distinguishes itself from all other life forms is an insatiable curiosity about the workings of the universe. Societal conditions have not always allowed this curiosity to be indulged; even practical pursuits require wealth and a freedom from manual labour not available to everyone. Theoretical pursuits also demand toleration of investigators whose entire exercise may be mental, and whose results may mean little to the average person. The systematization of such work gives rise both to technology (or technique, as it might better be called) and to science. These are not the same thing, despite being closely linked in the present culture, and the purpose of this section is to give careful consideration to both, to see how they are related, and to investigate some of the scientific and technological influences on society.

First, consider science, the more theoretical of the two. The fundamental premise of any discipline that is to call itself scientific is this:

That is, radical doubters notwithstanding, the doing of science seems to require at least the perception of a systematic reality in order to have something that it can be about. It is usually a simple matter to distinguish disciplines of a scientific type from those of most others. For instance, the creation of music, painting, sculpture, and the writing of poetry or novels are not generally regarded as scientific activities. These pursuits may have rigid rules for some aspects, but need no always, for their practitioners are thought to be free to present ideas and impressions without being bound by anything called reality.

It is also possible, though more difficult, to distinguish science from the humanities (philosophy, languages, literature, etc.) and from the studies of society (sociology, anthropology, psychology, economics, etc.). In the latter group in particular, attempts are often made to apply scientific methods, but this may not be entirely successful. After all, it is not known whether human behaviour is predictable in the same manner as that of things under scientific study is supposed to be (except, perhaps, statistically). Although it is methodology (and not results) that determines whether a discipline is scientific, the method of science assumes an element of predictability, and it is not until some work is done that one knows if this factor is present. A discipline may use the methods of science on the assumption that they are appropriate, but it is only as those methods produce reproducible results that the practitioners gain confidence that they are indeed "doing science," and not something else.

It is easy to assume that scientific methods do or ought to apply to a given field of study. It is more difficult to discover how to make them apply (which is partly a matter of technique). It is harder still to demonstrate that the assumption was correct and the phenomenon being studied can be demonstrated by the methods being used to be predictable. Finally, if some apparently orderly pattern is discovered, these techniques shed no light on the source of the perceived order (or lack thereof.)

The case of the discipline of mathematics is particularly interesting, for its philosophers can take one of two extreme (but not necessarily mutually exclusive) views:

o that mathematical ideas are entirely theoretical and speculative, with no necessary connection to the physical world

or

o that mathematics describes things with a real existence.

To put it another way: Do mathematical ideas come into being the first time someone thinks about them (created by thought), or are they pre-existent (already in the universe) and only being discovered as time goes on? For example, the equation ax² + bx + c = 0 (a, b, c are real numbers with a > 0) can be solved for x by use of the quadratic formula.

Did the quadratic formula exist before it was first written down by a human being, or has it always been inherent in the concept of number?

Although there are some who will hold out for the absolute truth of one or the other of these positions, mathematics actually has both aspects, for while the entities it discusses are on the one hand mental ones, these ideas clearly do on the other hand have some relationship to the physical universe.

o The concept of number is universal and pre-existent. God has always existed in three persons, for example. However, the numerals employed for the communication notation used by humans to express this idea are cultural inventions, not universal truths. Thus, the ideas contained in the assertion that 2 + 2 = 4 are inherent in the concept of number, and are not inventions. However, the notation in which the idea has been written is an artifact, for rather than "two" or "2," one could use "deux" or "II" without changing the meaning.

o The same is true of the meaning of the quadratic formula on the one hand, and any particular way of writing it on the other.

o The use of base ten numerals like 4645 to express the idea of 4000+600+40+5 is probably due to the vast majority of humans having ten fingers with which to begin learning to count. There is no a priori reason why one should not use a system founded on a base of two, three, eight, sixteen, or some other number. Indeed, one does use base twelve (dozens and/or gross) to measure quantities of eggs, buns, or hours, base sixty to measure degrees, minutes, and seconds, and bases two or sixteen inside computers.

o Pythagoras' Theorem on right triangles is true regardless of the way in which it is written out, and it unfailingly categorizes triangles as right triangles or not regardless of what any observer may think or how that observer might write the result down. It is true even if you call the things left triangles.

o Likewise, the interesting observation that the number 1961 reads the same right-side-up or upside-down is entirely a construct of the notation; it has no universal truth in itself. On the other hand, the idea of symmetry that this example illustrates is universal, and can be found wherever some object can be rotated or flipped onto a copy of itself.

o In a broader sense, this example illustrates the universal notion of complementarity found in such pairs of opposite ideas as: left/right, up/down, right/wrong, good/evil--all of which exist independent of the language that describes them.

Similar arguments can be made, not just for number theory, but for statistics, topology, algebra, analysis, discrete mathematics, calculus, transfinite numbers, and set theory. Although many of these ideas have appeared on the human scene recently, the very rationality of their interconnectedness argues that they are in some manner inherent and inevitable (part of an objective reality) and that they will certainly be discovered once one thinks long and deeply enough.

The difficulties in understanding the nature of mathematical statements are compounded by the fact that in all but the simplest cases one must be a mathematician in order to perform its mental experiments. A grade ten student in remedial (general) mathematics once said to me "I know everything there is to know about mathematics already; why should I have to take this course?" The sad fact was that he barely had acquaintance with the multiplication of fractions and had never heard of the aspects of mathematics already mentioned, much less of computational geometry, complex analysis, relativity, probability, combinatorics, or any of their applications--the chasm of his ignorance was unbridgeable.

That is, in this realm "truth" can only get informed consent--can only be understood--if one has sufficient training and experience in mathematical thinking to qualify as a member of the consensus. Not just anyone can comment meaningfully on mathematics, for to grapple with its ideas requires special knowledge and experience. Even among highly qualified mathematicians, embarrassing errors take place. For instance, a proof for a widely accepted theorem is sometimes later shown to be incorrect and either a new proof must be supplied, or the theorem may be shown to be false after all. In one celebrated case in the 60's and 70's a graph theory result was purported to have been proven in published papers by three successive writers, and all three proofs were subsequently shown to be incorrect.

One could summarize by saying that whether mathematical truths are created or discovered by mathematicians, they certainly cannot be discerned apart from the collective experience, training, and beliefs of the mathematical community, and this is not unlike the situation in the scientific community and in other disciplines. Specialized training is required to comprehend the ideas of a discipline.

That is, acceptance of mathematical and scientific results by most people, even those trained in another branch of the discipline, requires some degree of acceptance of the consensus of the expert part of community. This consensus, because it is an interpretation, is not necessarily true absolutely. For example, no matter how much a mathematical model for the first few seconds of the existence of the universe may be consistent with present-day scientific observations, acceptance of the model as a fact is a leap into faith, one that bears a great resemblance to that held by others in an all-powerful creator having made everything in six literal days.

Is Mathematics Certain?

In the latter part of the nineteenth century, a number of logicians showed that the standard methods of logic employed at the time led invariably to fundamental contradictions. For instance, consider the definition:

The barber of Seville shaves all the men of the city who do not shave themselves.

or, similarly

S is the set of all sets that do not include themselves.

Now does the barber shave himself or not? Is S a member of S or not? Unless one sneakily attempts to escape the logical trap by positing that the barber is a woman, a machine, or an alien, either answer leads to a contradiction. The existence of such contradictions introduces an uncertainty into mathematical logic itself, not just into the correctness of part of its consensus. That this uncertainty could not in any way be resolved was shown in 1931 by Kurt Gödel when he showed that no set of axioms used to describe a mathematical system could prove both the consistency and completeness of the system.

Consider, for example, the natural numbers:

N = {1, 2, 3, 4, ...}.

Gödel showed that, on the one hand, any set of axioms (rules) that could be used to prove all true statements about these numbers would necessarily be inconsistent (lead to contradictions like the one above). On the other hand, if the set of axioms is consistent (no contradictions possible) it could never be sufficiently complete to derive all true statements about the system. As Douglas Hofstadter puts it: "In short, Gödel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved." (Hofstadter, p. 19)

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

--Albert Einstein

That is, unless one is willing to use logical tools known to be unreliable (potentially inconsistent), there are always truths about the number system that cannot be proven.

The same principle applies to computing, for one equivalent to Gödel's theorem (the undecidability of the halting problem) demonstrates that no machine can be built using logical systems that can process all problems, or even to determine ahead of time whether they will be successful in the attempt. Simply put, not everything knowable is computable. That is, human beings can know more than finite logical machines, no matter how elaborate those machines may be.

In a similar vein, it is not possible to prove with the rules of human logic that God exists (or that he does not). His existence may be strongly inferred or authenticated from evidence, but not proven in a logical or a priori sense.

This concept of uncertainty is applicable to science as well because Gödel's Theorem applies to all logic when it is applied to infinite systems, and not just to mathematics. Science must also deal with the uncertainty that the closer something is observed the more the very act of observation changes the thing being examined, and so the less accurate the observations are.

Yet, the entrance of such uncertainties into the scientific realm does not create the difficulties for its practitioners that it might for theoreticians. After all, a researcher in some other part of the world with similar equipment needs only to be able to duplicate a reported experiment and obtain essentially the same experimental results within a reasonable margin of error. For science, duplicatible results are the important thing, even where there is no agreement about the interpretation (meaning) of the results.

Indeed, questions about ultimate meaning are not really on the agenda of science, and scientists who speak of them are no longer talking about their own specialities, but about those of others. The Nobel prize winning physicist who goes on the talk show circuit to proclaim there is no God has fewer qualifications to speak on that subject than a theologian does to declaim on gravitational field theory.

It is also important to note that other intellectual pursuits, such as religion, art, poetry, philosophy, languages, mathematics, and so on, while methodologically different, are organized disciplines in the same sense as are the various sciences. Each such field of study constitutes a recognized body of knowledge with its own rules, practitioners, and special methods of interpretation. In fact, one distinguishes a scientific discipline from the others precisely on the basis of its particular intellectual methodology, the rules for which serve both to define what is science, and to determine what are its appropriate fields of study.

A typical elaboration of the scientific method in five steps goes something like this:

1. Observe the universe in question, collecting raw descriptive data.

2. Analyze the data, systematizing and interpreting it.

3. Synthesize a theory (formulate a hypothesis) to explain the data, or develop a model to illustrate it (i.e., create a mental abstraction of the presumed physical reality).

4. Test the theory or model under as many variant conditions as possible to determine the degree of correspondence between the abstraction and the physical universe.

5. Modify the theory or model and re-test it until it agrees with all the relevant phenomena. If a universal consensus is reached on a particular theory it might be promoted to the status of a "law."

This process might be summarized by saying that science is a search for true descriptions of the world by making logical inferences drawn from empirical data. Although this description of the scientific method would probably be accepted (with variations) as a working definition by the vast majority of those who term themselves scientists, one must realize that it is only a close approximation of what science is. A number of cautions must be added to properly explain it.

First, applying the method implicitly assumes the hypothesis is testable. Some are, some are not. Strictly speaking, the latter are not scientific, for if they cannot be tested they can neither be refuted nor verified.

Second, taking a narrow view of this process would exclude mathematics--a discipline that attempts to produce its results by logic alone. Yet mathematics not only provides the language, structure, and tools for systematic investigation, it also has reasons of its own to be applied to the real world. Mathematics is therefore inextricably intertwined with all scientific disciplines. Not only can no science exist without the language of mathematics to describe its investigations, but also the boundaries between applied mathematics and science are quite unclear.

The term "mathematical sciences" has therefore become common today, and few people are unhappy with the tendency to regard these disciplines as more a science than an art. That this acceptance somewhat undermines the working definition of science given above is of little practical importance to most scientists, for an exact definition of the field's overall scope has little effect on their specific work.

Third, whether mathematics is included or not, the definition given has one serious drawback: the tendency of some to regard scientific knowledge as the exclusive form of knowing, and to specifically exclude from the "knowable" category any results obtained by other methods. This philosophy, called logical positivism, asserts that logic combined with the empirical methods of science forms the only possible way to know things. It rejects the conclusions of other methods of enquiry as irrelevant--as not being knowledge.

While this view was widely held in the nineteenth and early twentieth centuries, and some still try to defend it today, it has lost much of its popularity among philosophers. It is now realized that scientists do not operate exclusively within the empirical methods given above. On the contrary, as John Ziman remarks, "they tend to look for, and find, in Nature little more than they believe to be there, and yet they construct airier theoretical systems than their actual observations warrant." (Klemke, p. 35) This may overstate the case somewhat, but it leads to an important observation regarding the "doing" of science.

That is, like everyone else, a scientist brings to the work at hand a framework of ideas about how things ought to work, a set of conceptions about why the world behaves the way it does, and a collection of goals that are regarded as desirable to achieve. All the scientist's work is done within the context of such a world view--it influences every decision and every step. Because of this, theory has a tendency to come before the collection of data and not as a result of it. Consequently, data is often collected under the influence of the theory that is supposed to explain it, and researchers are naturally inclined to reject data that does not fit. This process is not dishonest in any way; rather, it is human nature to observe and interpret the world in terms of what one already believes about it. A shared community world view also lends a consistency to the voice of science as it speaks to matters of public concern.

Fourth, this consensus in a scientific community tends to be remarkably broad, monolithic, and very slow to change. Unfortunately, this very consistency can sometimes hamper objective investigation and make truth harder to discover. The important insights and discoveries in any field often come about because the world view expands or changes to allow people to see things in a new way. Those who make this breakthrough may have a difficult time convincing anyone else to listen because a changed world view often forces people to re-examine matters previously considered settled.

Such "new views" are common in the artistic world. Each generation reflects its world view in its artistic creations, and may fail to communicate with the previous generation through these forms. Rock music, for instance, expresses a direct connection with the emotions, a raw "me-ism" that can be incomprehensible to those who do not share its context. Indeed, it could be argued that it is not the nature of some music to be comprehended--that both its medium and message are entirely emotional.

Examples of such paradigm shifts from the history of science include Galileo's heliocentric model of the solar system, the periodicity of the elements, atomic theory, radioactivity, Einstein's theories of relativity, and quantum mechanics. All were ultimately accepted by the scientific community, but each had difficulty at first due to the radical change in world view required to comprehend the new model.

This explanation is not intended to suggest, as some "new-age" philosophers do, that a new way of looking at things (a new model or paradigm) changes or becomes the underlying reality, a sort of ultimate "man is the measure of all things." Rather, it is to point out that the practice of science does not quite conform to the general view of its philosophy as expressed in the step-by-step scientific method. Scientists do assume an underlying reality, but they interpret or filter that reality, so their results partially depend on the nature of those filters.

Fifth, the results of scientific enquiry are always approximations, subject to reinterpretation in the light of new data that may be more exact, be collected in a different manner, or be interpreted with a different world view. There is also the possibility that new data will overthrow a fraud, a hoax, or a conclusion derived more from wishful thinking than from careful reasoning. For instance, a technician being asked to do radioactive dating of some sample might ask the supplier what range of dates are acceptable, and might not report any test results that fall far outside this range. The non-conforming results are obviously spurious. But, what if there another view that explains them as part of a consistent data set?

Another complication arises from the fact that many scientific workers engage in the building of highly speculative theories with little or no connection to actual data. This habit is particularly widespread among astronomers and others with an interest in the origins and mechanics of the universe as a whole (i.e., cosmologists). Such speculation is healthy, because it tends to open up many new lines for investigation. Scientists must speculate if they are to make any progress at all, for otherwise they will generate no hypotheses and be unable to do anything. However, this necessity of practice does illustrate that the boundaries between science and the more speculative or metaphysical disciplines are not always as sharp as is generally believed.

These observations on the imprecise aspects of science lead back once again to the example of its ally, mathematics, for a better idea of what science is, if it is not just pure logic combined with rigid experimentalism. As mentioned before, mathematics relies on a community consensus of what is "true"--one that is not infallible, but that is at least a reliable determinant of what things are part of the discipline and of what constitutes a properly derived result.

In a way, the existence of this peer consensus is not unlike that of the high-diving or figure-skating judge who holds up a score card after each performance. The consensus of the group (i.e., the average scores) becomes the final judgement on the dive. The determination of what constitutes good science or good mathematics takes more people and a longer time (perhaps generations). It is nonetheless the result of a community examination of the work in question, and is a consensus of its value. It is even possible to quantify this agreement somewhat by counting the number of times a paper or book is cited positively in the bibliographies of later works--the higher the number, the more firm the consensus of worth. In view of the fact that the majority of published "research" papers are never once cited by anyone, those that are quickly distinguish themselves from the rest.

Looked at in this way, science loses none of its empiricism, precision or status--but it is seen as one among many consensual ways at arriving at an agreement about the way the universe appears to work. On the other hand, this view does cause science to lose some of the mystique and exclusivity that it has built up during its 150 year ascendancy over other thinking methods in the West, for this perspective places it in a continuum of disciplines, blurring the edges between it and other concepts of truth-seeking.

The methods of science are also important in a variety of fields, (such as government or economics) where facts need to be gathered and interpreted for the benefit of the decision-making process. Scientific techniques can be invaluable in discovering "what is going on." Of course, subsequent decisions will always depend in part on values and evaluations not provided by the fact gathering process alone.

For instance, as the costs of techniques soar, decisions have to be made about what research to fund and develop, and what to delay or drop altogether. How much goes to AIDS research, how much to computing, how much to developing new fields, to transportation, to the environment, and so on? Such decisions raise political, economic, and ethical questions that scientific investigation cannot by itself answer. Moreover, there is no particular reason to believe that the conclusion about what should be done, when reached by a scientist, is any better or any more logical than the conclusion reached by a politician, or by the general public. Indeed, if a scientist takes pride in the belief that only empirical methods produce knowledge and everything else is erroneous or irrelevant, then the resulting ignorance of other thought processes, disciplines and people is more likely to produce a bad decision than a good one. Knowledge, thinking, decisions, and their consequences are interrelated. Science provides one of many methods of thinking and of obtaining knowledge; it is most effective when integrated with others as well. That is;

One must conclude therefore, that the logical positivists, in seeking to exalt the scientific method as the only road to knowledge, actually restricted its domain and made it less useful than it could have been. In short, it is precisely integration with other ways of thinking that makes scientific methods generally applicable and practical. Such applications of science as well as the relationship between science and technology is the subject of a later section of this chapter. The next section is devoted to placing the theory-making of science into the larger context of a common thinking device.