This is essential reading for anyone interested in any field of science. These days, it seems, the scientific method is either misunderstood, or not known at all by a majority of our population. This essay, by Dr. Robert Howard, not only provides details of the scientific method, but also provides a wonderful backdrop to its development and an encompassing array of information that empowers readers with a strong sense of understanding of the very nature of science. – Steve Woodruff
Essay on the Nature of Science, the Scientific Method, and Measurement
Dr. Robert Howard
CSUN – Geography Dept.
Modern science, that is to say science as we know it today, is really a post-renaissance development. The birth of modern science really begins with late sixteenth – early seventeenth century recognition of the necessity to observe nature first and then from those observations attempt to draw conclusions about nature’s workings. In reality, this is a prescription for that thought process which we refer to as induction or inductive reasoning, in which general statements or conclusions are attained by starting with specific observations. When coupled with Aristotle’s deduction or deductive reasoning (i.e., starting with general statements, propositions or hypotheses and working toward specific conclusions), induction has permitted development of what we call today the scientific method. This is the scientific community’s formula for attempting to determine truth, certainty or validity of propositions. By employing the scientific method western civilization has been able to attempt to understand our Earth and its place in the universe. In so doing, it has also afforded philosophers an opportunity to speculate on, or reconsider humanity’s place in the grand scheme of things.
Four of the most revolutionary advances in our understanding of Earth, and perhaps of ourselves, have come about during the last four hundred years. In first recognizing, and then accepting evidence for these revolutionary theses, western civilization challenged what had heretofore been humanity’s exalted place in the center of the universe as related in the myths of the Judeo-Christian tradition.
First came astronomical observations and conclusions that could be drawn from them. Culminating in the late sixteenth and early seventeenth centuries these conclusions resulted in Earth – humanity’s abode – being displaced from its presumed position in the center of the universe. Philosophically, and by extension, theologically this meant that instead of being the sole purpose for creation of the universe, humanity was reduced to being merely inhabitants of a small planet revolving about an average sized star. In our century we have come to realize that this star, our sun, appears to be far out on a spiral arm of the Milky Way galaxy. Rather than being a special star, our sun appears to be approximately one of 1011 such stars in our galaxy. As if this were not enough of a blow to humanity’s self-importance, it turns out now that our galaxy is one of perhaps 1011 other such galaxies in the visible universe.
Second, during late eighteenth and early nineteenth centuries came a recognition of what John McPhee (1980) has aptly termed “deep time”. This phrase refers to the age of Earth and by extension, age of the universe. “Deep time” was initially a purely philosophical idea based on the outlook (Weltanschaung) of one of geology’s early pioneers (James Hutton, 1729-1797). It eventually became a testable concept that could take on real meaning. Despite some initial false starts, caused in part because radioactive decay was unknown at the time, a fairly accurate age for Earth has been attained. Recognition of Earth’s age-some 4,500 x 106 years-delivered a blow to the myth of Earth’s creation found in the Bible’s Old Testament Book of Genesis.
Third, in mid-nineteenth century and, it might be added, with reverberations still resonating in American society today, Charles Darwin (1809-1882) proposed that rather than being a special creature created in the image of God, humanity has really evolved from lower animals via a process known as natural selection (Darwin, 1859). Although a contemporary of Darwin and considered an independent originator of the idea of evolution, Alfred Russel Wallace (d.1913) did not believe that humans evolved from lower life forms. Instead, he believed that humans were a separate creation. Wallace believed that because humans are so different in intelligence and capacity compared to other animals that only the intervention of the supernatural (in this case God) could account for our existence. Today, biological and paleontological science have shown evolution to be fact. What is still up for discussion, and it might be added, vigorously debated amongst various branches of the life sciences, are the various processes by which evolution might have come about. What we realize today however, is that evolution has taken literally millions of years culminating in the human species and it appears that we have come about by pure accident.
Finally, starting with speculations by the German meteorologist Alfred Wegener (1880- 1930) in the early part of this century and culminating in a scientific revolution by the late 1960’s is the notion that not even our geographic notions are constant. Perhaps the last of humanity’s beliefs in constancy namely, that the continents were fixed in the same places, was shown to be false. Truly, it can be said that the only constant in the universe is change!
A final word here concerning this essay’s organization. It is possible to skip the next section (How did we get to this point?) which is mainly historical and move on to the section following which deals philosophically with the nature of science. This philosophical section is designed to introduce to the reader aspects of science, and in particular what makes a body of knowledge a science, as well as how science operates using the scientific method. While the historical section can be avoided it is best read in order to understand some of the background to all this philosophy. In essence it helps explain why we look at things in certain ways and it introduces my fellow classmates to some key figures in the rise of modern science. In essence this historical section “fleshes out” and provides background for the philosophy. Subsequent sections deal with how science changes its point of view (or undergoes a paradigm shift), and a word about science and society. Finally and very importantly, I want to introduce some of my fellow classmates (particularly the Creationists among them) what do we measure in science, how do we go about setting up a measurement system and then what do we do with data arrived at from these quantitative measures.
HOW DID WE GET TO THIS POINT?
Before beginning any discussion about the nature of science, the scientific method, paradigms and paradigm shifts, or scientific measurement, I believe that it is instructive to understand how we have arrived at this point in time with the philosophical “baggage” that we presently possess. The scientific method is one of western civilization’s greatest contributions to humanity and so my focus in this essay will be exclusively western.
There is a trilogy of Greek philosophers whose names come up continually in connection with development of those reasoning and thinking skills along with ideas that eventually and most tortuously have led us to modern science. These individuals are first, Socrates (ca.470 B.C.-399 B.C.), his student Plato (427 or 428 B.C.-347 B.C.) and finally Plato’s student, Aristotle (384 B.C.-322 B.C.).
Socrates, Plato’s mentor, is best known for taking hemlock poison on order of Athenian authorities. According to his contemporaries he was completely uninterested in all aspects of the natural world. It was said that he was not comfortable when not around people or in a city. Socrates’ interests were entirely within the realm of ethics (the study of right and wrong) and politics. His place here in this essay on science is due to his development and use of a reasoning or arguing process that today we refer to as the Socratic dialectic. This process for ascertaining truth is based on first, development of a statement or thesis. Now for every thesis, one can propose its opposite or its antithesis. Between thesis and antithesis there will inevitably be conflict out of which a compromise results. (Note here, the political scientist in Socrates.) Such a compromise is a synthesis of the best ideas from thesis and antithesis. Now however, this synthesis may be looked upon as a new thesis which can be opposed by its antithesis and so on until we have finally arrived at either truth or the best possible answer as compromise. This method as applied to economic history was used by Karl Marx in formulating communism and is today known as dialectic materialism.
Plato is probably best known for his treaties The Republic which is a study of the just man and the just society. In what was to lead ultimately in a scientific direction, he was also responsible for an historic movement away from Greek gods of Homeric legend. Plato thought that these familiar gods of Mount Olympus were too capricious, too immoral and far too undignified for a Greece that was expanding. During his lifetime a Greek empire, centered first in the Mediterranean basin, was being created through efforts of Philip II of Macedon and later his son, Alexander the Great. Plato thought that such an empire ought to have world-class divinities. He thought that people would recognize the superior dignity of visible gods, and these he envisioned were best thought of as the heavenly bodies. For this reason, Plato wanted Greeks to obtain scientific knowledge concerning periods and motions of the heavenly bodies. This Platonic religious revolution had a two-fold purpose. First, it was to create a new religion using the heavenly bodies as gods. Second, it led to acquisition of scientific knowledge about these bodies through observation. This second purpose meant that mere mortals could not only acquire knowledge of the gods but in doing so they would be able to comprehend godly behavior. The full impact of this scientific revolution would not be fully understood until the seventeenth century (A.D.) with the work of Sir Isaac Newton.
It did not take Greek astronomers long to obtain information on periods and motions of the stars, moon and sun however, five points of light behaved erratically. These five points of light are Mercury, Venus, Mars, Jupiter and Saturn and their erratic motion was termed retrograde movement. While all of the other heavenly bodies moved in geometrically perfect circles these five points of light moved first forward and then backward before moving forward once again. Because of their erratic motion they were termed planets (Greek for vagabond). This erratic motion recalled the capricious behavior of Zeus, Hera and the other gods and goddesses of Homeric legend. Unless it was sufficiently explained it could lead to the failure of Plato’s religious transformation.
After some twenty years of trying Greek astronomers finally resorted to circular reasoning to solve their problem. Their solution was that while the stars, moon and sun appeared to travel in circular orbits about Earth, planets traveled with much more freedom. They are, after all, vagabonds, are they not? Planetary movements, it was thought, are carried out on surfaces of invisible globes or spheres. Globes are as seamless and symmetrical as are circles however, the former are three dimensional. Once this was accepted then the perfect universe was completely understood.
Aristotle, Plato’s student, extended Plato’s religious/scientific scheme. In logical detail Aristotle offered an explanation for why and how Plato’s celestial gods were superior to previous gods as well as all other Earthly things.
Aristotle theorized that the planets, moon, sun and stars were embedded on the surfaces of invisible spheres all of which were concentric with and revolved about Earth. Earth in turn was fixed in the center of the universe. Finally, the universe itself was segregated into two realms. One realm was the celestial while the other was Earthly. To Aristotle, the celestial realm was perfect. It was incorruptible, unchanging, permanent and, in other words, immanent. Stars, moon, sun and planets moved on the surface of invisible spheres, which because of their perfect shape produced the flawless orbits of the heavenly bodies about Earth that were exactly as observed. Spheres holding the planets were permitted a more complex pattern of revolution to account for retrograde motion. Aristotle’s celestial realm he believed, was composed entirely of the Greek element Ether. At that time it was thought that Ether existed only in the celestial realm. It too was incorruptible and could exist in varying densities, thus accounting for different heavenly bodies.
Aristotle believed that the Earthly realm was composed of the remaining four essential Greek elements: Earth, Fire, Water and Air. These Earthly elements existed in accordance with what was believed to be four essential qualities: dry or wet, hot or cold. As examples, Earth is cold and dry; Water is cold and wet, Air is hot and wet, while Fire is hot and dry. The Earthly realm was thought to be corruptible and therefore quite changeable. This pronouncement Aristotle based on observations of what today we would call changes of state. Finally, Aristotle believed that Earthly elements tended to move in straight lines. Philosophically, this is appropriate because lines have a beginning or starting point (birth) and a termination or end point (death). [This is important as such a condition permits historical analysis because here Aristotle is talking about time’s arrow as opposed to time’s cycle, to be expanded below.]
This religious transformation was capped by Aristotle who provided an explanation for heavenly motion. He believed that each sphere was moved by an ethereal wind that was due to the motion of the next outer sphere. The outermost of the spheres was impelled by Primum Mobile, the prime mover or God Himself. All of this explains the significance of astronomy to the Greeks and, it might be added, its handmaiden, astrology.
Once completed by Aristotle, Plato’s religious transformation gave Greeks celestial deities whose behavior was entirely predictable. This introduces the Greek notion of cosmos for the first time; cosmos meaning orderly, beautiful, decent. This new religious theory also satisfied a very western principle known as sufficient reason. This basically means that for every physical effect there must be a rational cause. As an example, today we refer to gravity as the cause of a body falling to Earth. Aristotle would have ascribed this movement as pieces of Earth wanting to be reunited with their primary source and by extension, larger pieces would fall faster because of their greater desire.
What Plato initiated, Aristotle completed. Religion and science (or natural philosophy as it became known in pre-nineteenth century Europe) would be united for centuries to come. Science offered to religion corroboration for existence of a supreme being while religion offered to science a legitimacy it had never had before.
Aristotle is important to the foundation of science because he establishes science as a partner with religion. Aristotle is also important because it is he that introduces logos as opposed to mythos into Greek thinking. Derived from mythos, meaning the word, myths were an important element in Greek culture, as they have been, and continue to be, in virtually all cultures. According to Campbell (1988) myths serve a variety of functions. Some myths are metaphysical, while others are cosmological, social or psychological. Metaphysical myths awaken and maintain a sense of awe, respect and humility in recognition of the ultimate mysteries of life and the universe. Cosmological myths provide an image of the universe and how it works. Social myths validate and maintain the social order and lastly, psychological myths support the “centering and harmonization” of the individual.
In our context here, Aristotle’s cosmology serves as both a metaphysical and cosmological myth. By introducing the notion of logos, in Greek, an account whose truth can be debated and demonstrated (i.e., studied) Aristotle established rules of deductive logic. In this way he founded Greek science on an epistemology
(i.e., a means by which knowledge is acquired) that consisted of formulating premises that were unquestioned and then deducing specific consequences from them.
Western European Views on Nature and Science-Post-Greek to the Renaissance
Rise of Christianity: With passage of time the Greek empire faded from the Mediterranean scene and along with it Greek religion and scientific innovation. Some Greek philosophy was incorporated into the Roman Empire however, it had first to be filtered through Roman culture and its language. In time, Rome too was eclipsed. In western Europe, Christianity rose to dominate the intellectual landscape for over one thousand years. The multitudes of Greek and Roman gods were replaced by the single heavenly God first worshiped by the Jews. In the early years of Christianity, after the fall of Rome around 462 A.D., little was known about Greek scientific thought as Latin was the common language spoken by most educated individuals of Europe. Eventually however, Greek texts were translated into Latin by clerics in various European monasteries. Greek had not been lost, it was read and understood by the educated among Europe’s clerics. It should be remembered that many parts of the Christian Bible or New Testament were in fact initially written in Greek.
The Scholastic Philosophers: By the early thirteenth century European students were being exposed to the Socratic dialectic and Plato’s rhetorical techniques as well as Aristotle’s science and deductive logic. Under the lenient ascendancy of Islam in Moorish Spain, Jewish and Islamic scholars had translated and written commentaries on numerous works of the Greeks to include Aristotle. Eventually, leading European Christian clerics became aware of the Jewish rabbi Maimonides (also known as Moses ben Maimon, 1135-1204) in Moorish Spain and his reconciliation of Aristotle’s cosmology with Hebraic texts that form the basis of Judaism. Also in Moorish Spain, an Arabic philosopher, AverroÃ«s (Abu al-Walid Muhammad ibn Rushd, 1126- 1198), performed a similar function within the Islamic tradition. While he had little impact on Islam itself, through his writings he revealed Aristotle’s ideas on nature to western philosophers, influencing in particular the Dominican monk Albertus Magnus (Saint Albert the Great, 1206- 1280) and his student Thomas Aquinas (1224 or 1225 – 1274). It was in this way that the work of Aristotle eventually became known in medieval Europe. In this context Albertus Magnus wrote: “The sublimest wisdom of which the world could boast flourished in Greece. Even as the Jews knew God by the scriptures, so the pagan philosophers knew Him by the natural wisdom of reason, and were debtors to Him for it by their homage” (as quoted in Guillen, 1995, p.30).
Saint Thomas Aquinas, Duns Scotus and William of Ockham: One of the two great controversies that reverberated through the learned world of the early thirteenth century was a concern with Aristotle’s view of nature, how it was to be examined and how it was to be understood. Thomas Aquinas was to become so embroiled in this controversy that he is best remembered as the scholastic philosopher who attempted to resolve it. [The second great controversy, one that we need not examine here, involved the doctrine of universals; arguments for which no longer are of any importance to us today, save for historical reasons.]
Through AverroÃ«s, Europeans found out that Aristotle was profoundly interested in the natural world and that by studying it there would be no harm done to the soul. With this notion in mind, Aquinas set out to reconcile Aristotle’s scientific ideas with religion. Aquinas believed that humanity united, for better or worse, the City of God with the City of Man. These notions were allegorical constructs of St. Augustine (354 A.D. – 430 A.D.) and the title of his most famous work. Augustine based this work, The City of God, on Plato’s ideas and it was meant to reflect all that is incorruptible and thus holy. The City of God was not a physical place that could be seen with the eye. St. Augustine believed that it was something that existed in the heart and soul of every true Christian. Over time however, this allegorical image of the City of God became associated with the Aristotelian universe beyond Earth. To those who needed a concrete image rather than an allegorical one St. Augustine’s City of God became visible each night in the stars which moved majestically and predictably across the sky. The City of Man on the other hand represents Earthly existence; it is corruptible and a spiritual desert without the City of God. Theologically, these two “Cities” were the incorporation of Jesus Christ’s aphorism directed to his apostle, Peter that he “render unto Caesar that which is Caesar’s, and unto God, that which is God’s”. Aquinas thought of humans as existing at the juncture of these two “Cities” in just the same way as a horizon is the line that separates sky from Earth. As long as humans are vital and alive, and thus not spirits, then both realms must be present in each of us. Therefore both must be dealt with and understood for the sake of salvation. It might be one thing to condemn what is Earthly and worldly but to be ignorant of its power and meaning, its temptations and pitfalls was surely a mistake. The Church in actuality, consciously or unconsciously, continually preached their warnings in recognition of human ignorance of the Earthly.
In all of this Aquinas was drawing attention to the fact that there is a duality that exists in all humans. This duality is the distinction between spirit and nature, or soul and body. At the same time there is also a unity, for where does body end and soul begin? Throughout human life, body and spirit comprise a seamless whole, a joining of apparent opposites. Since body and spirit are joined there cannot be two truths, one of the spirit and one of the body; of religion and of nature; of the City of God and the City of Man. To Aquinas there was a unity of truth and in this he was in opposition to St. Augustine’s view of eternal conflict between the ineffable and the Earthly.
Aquinas attempted to resolve this conflict through the use of reason but in this quest he was opposed by two groups of scholars. First were religious enthusiasts who considered (and, it might be added, continue to do so even today) that reason, use of natural or God-given intellect, is an intruder between humans and their communion with God. These individuals were (and still are) impatient with Aquinas’ attempts to bring them to God along a reasoned path. Second, there was a minority who did not see why natural reason had to bow down before the ruler of the City of God. Was God not also bound to be reasonable? Additionally, where was the evidence that God existed and that He demanded obedience? While evidence for this was lacking, evidence did exist that the world was real and that it required human understanding. During Aquinas’ life the European world was slowly evolving from a rural, primarily agrarian society into one that was progressively urban and mercantile. This second group of scholars had a powerful argument by virtue of the fact that economic, agricultural and social evolution which, however rudimentary, was leading to better lives and therefore ought not be countered by a rejection of history and a return to the Dark Ages. Both of these opposing intellectual groups agreed on the presence of the two truths. The religious group believed in the primacy of religious truth, truth that was the City of God, and a trivial truth inherent in the City of Man. For those opposed to the religious, the primacy of the two truths was reversed. The weight of opposition, unbalancing the argument as it were in one direction or another, disturbed Aquinas’ attempt to resolve this duality by bringing them together and thus ending what he perceived to be the vicious error of the two truths.
Duns Scotus (1265-1308), a Franciscan scholastic heralded the triumph of duality. Scotus wrote that God is absolutely free and this absolute freedom means freedom from reason’s necessity. Aquinas had proposed that what is logically necessary must be necessarily so; Scotus stated that God is not bound by any construct of the human mind and therefore human reason cannot determine God. [Perhaps what we are seeing here are the first glimmerings of theism in the ideas of Scotus and deism in Aquinas (expanded below).]
William of Ockham (1300-1349), another Franciscan monk carried, Scotus’ arguments still further. He wrote that the only real things are singularities or singular entities such as an apple or a man. Universals are only names; they have no existence whatever. Nature consists only of things, and human reason only permits us to encounter them. Nothing that we deduce about things has validity, and particularly what we deduce about the divine. Therefore Ockham says, faith and reason have nothing in common. Each has its own truth. The former is vastly more important for its determines one’s salvation. The latter is concerned only with our creature comforts during our brief existence here on Earth.
What we seeing in these arguments and in the approaches to their solution is that two Roman Catholic holy orders are on opposite sides. Dominicans seem to support the use of rationality and human intellect in resolving the faith versus reason dilemma. Franciscans appear to support the primacy of faith above everything. Irrespective of what reason tell us, it is our faith alone that will lead us on the path to salvation.
The great controversy thus ended rather quietly. Aquinas was unable to reconcile religion and science (in essence the study of the City of Man). As a result, theology prolonged its intellectual dominance for another three centuries. Science was forever to be outside the realm of religion. Thus it was allowed to progress without any of the constraints that it would have been under had Aquinas been able to reconciled it with religion. When in the seventeenth century science made its move and challenged religion the latter was at a disadvantage. While science’s inability to prove the existence of God was not detrimental to science, religion’s failure either to subdue science or to anticipate the effects of its discoveries was detrimental to religion. Ultimately, one truth would prevail and that would be the truth of nature.
What did occur out of all this scholastic argument? I guess it was accommodation of Aristotle’s or Ptolemy’s geocentric cosmology to Christianity. This was accepted because, as a cosmic explanation, it fit into biblical myths as well as being a convenient explanation of what was observable. Heavenly bodies were envisioned to be embedded in perfect crystal spheres that were no longer caused to be moved by ethereal winds but by angels. The Primum Mobile of Aristotle was no longer a Greek or some generic divinity but the Judeo-Christian God. For Jews, Muslim and Christians alike, Aristotle’s cosmology was joined with the great monotheistic religions to form a seamless oneness.
Fourteenth to Seventeenth Centuries: The Black Death of the fourteenth century probably wiped out about one-third of Europe’s population. It hit clergy hardest probably because of their communal living arrangements in monasteries. Recriminations were soon forthcoming from all sides of society. The masses blamed the clergy for not forewarning them or perhaps not interceding with God on their behalf to avert the disaster. Clergy in turn, rebuked the population saying that the plague was God’s retribution for their accepting sinful ways as society inevitably changed. But who could blame a much of western European society for changing in response to population pressures, rural to urban migrations, economic dislocations associated with the rise of mercantilism, agricultural advances and climatic changes. All of these changes went on against a backdrop of religious fervor that had been focused on the Second Coming of Christ as foretold in the Bible. After a thousand years this anxiously awaited event still had yet to occur.
With its clergy decimated by the plague the Church had numerous openings available for priests. As replacements for dead clergy, throughout Europe men who had survived their wives and children flocked to the priesthood because of lucrative positions available. Many of those who joined the priesthood did so for all the wrong reasons. Even Pope Clement VI spoke out against “priestly arrogance given to pomp and neglecting the ways of God” (van Doren, 1991).
The Church in its weakened state was now hit by Martin Luther as he promulgated his 98 theses, nailing them to the door of the cathedral of Worms on the Rhine river in Germany in 1517. Via his theses Luther called for an abandonment of pomp and ceremony (along with purchase of indulgences) and a return to bedrock Christianity via good deeds and an all abiding faith. Thus begins the reformation. Viewed initially as an attack on the Church’s hierarchy from within, via Luther’s excommunication it became a religious attack from without. This attack was supported by petty nobility of northern Germany and eventually Scandinavia-all nations or regions far removed geographically from Rome and thus the immediacy of papal influence. Even England eventually joined in the anti-Rome movement. This however, was more a function of Henry VIII and his dynastic-political needs. The stage was now set for what was to become the great confrontation between science and religion that ultimately leads to the first scientifically inspired revolution in western thought. The revolution started in northern Europe where, freed from ecclesiastical authority and protected by Lutheran nobility, astronomers and others with a bent for explanation of the natural world at variance with Aristotle could flourish.
Sixteenth and Seventeenth Century Astronomers and Philosophers:
Introduction: Astronomy or cosmology of Aristotle and Ptolemy with its inherent immutability had been translated into Hebrew, Arabic and eventually Latin by the twelfth century. As such it formed the basis of western European, Church-sanctioned explanation. The stars moved in their crystal sphere well beyond the planetary spheres. Propelled by angels the spheres made their inaudible heavenly music. The planets also moved in their spheres albeit in much more complex patterns than stars. In the thousand years since Aristotle, quite literally hundreds of astronomers had made observations of the heavens. Each new set of more accurate observations had required continuous and on going modifications to the Aristotelian-Ptolemaic universe. The most significant of these modifications required additions of more and more epicycles to explain the retrograde motion of the planets (i.e., their apparent reversal of direction) as they moved about Earth. This increasing need for epicycles struck many astronomers as both messy as well as an unfortunate necessity given the reigning cosmology. Additionally, there were perturbing questions concerning motion, both its existence as well as its form. To Aristotle and thus medieval astronomers and scholastics the universe’s normal condition was that of no motion, everything should be static (or in a condition termed stasis). The fact that the heavens moved required an explanation. For the medieval world this was the hand of God. Recall, Earth is not only the center of the universe but everything in the universe rotates about Earth. Now it was known because of parallax obtained during repeated observations that the starry sphere was far beyond any of the planetary spheres. Therefore, for the same stars to appear every night required a tremendous speed of rotation for this outer sphere. Explanation for this incredible speed of rotation was that God could do anything He wanted or ordained. Motion itself was another problem. In the celestial realm all motion was circular but in the Earthly, it was straight line. Natural circular motion was never observed on Earth
Nicholas Copernicus (1473-1543): It was only on his deathbed in 1543 that the Polish theologian Nicholas Copernicus finally had the courage to authorize publication of his book On the Revolutions of the Heavenly Orbs. He had been afraid of any religious controversy that his observations and conclusions would cause. From his long years of astronomical observation he became concerned with the overly complex Aristotelian-Ptolemaic theory of the heavens. He wondered whether there might be a simpler explanation. For example, if Earth rotated that would solve a problem by eliminating the necessity for rapid rotation of the starry celestial outer sphere. Complex motion of planets could be solved if Earth revolved about the sun. In studying all of the available ancient Greek astronomical texts he could find, Copernicus discovered that there were Greeks who proposed a heliocentric solar system. It just so happened that in the end Aristotle had won out and his cosmology prevailed. Copernicus’ book did not create the revolution he feared. In the book’s preface, written by a friend, it was emphasized that Copernicus was only advancing an hypothesis that would make the mathematics of astronomy easier. Since a hypothesis was not viewed as fact there was no heresy in what Copernicus had written.
Tycho Brahe (1546-1601): Tycho was in essence the Astronomer Royal to the King of Denmark. He viewed his life’s work as purely observational astronomy in order to correct grossly inaccurate existing astronomical records. In 1572 while making nightly observations he witnessed development of a nova or an exploding star. He published his findings which caused immediate controversy. The Earthly realm was allowed to be chaotic and imperfect and therefore changeable and unpredictable. This accepted, if undesirable situation was due to the Devil, who had disturbed God’s original and perfect world. In the celestial realm of the Christian universe however, immutability was ordained by God and thus appearance of some new celestial phenomenon was not allowed. Clerics in Rome declared that Tycho was in error. This new star could not in fact be new at all. Fortunately, Tycho did not have to fear papal authorities in Rome for he was a Protestant and thus protected by the Lutheran King of Denmark. In 1588 Tycho found himself with less financial support working in Prague with a new assistant, Johannes Kepler.
William Gilbert (1544-1603): Before discussing Kepler, the ideas of William Gilbert, another Protestant scientist need to be introduced. Trained as a physician, Gilbert practiced medicine most of his life. He was however, interested in other sciences and in particular experiments with lodestone. This mineral, known today as magnetite possesses natural magnetism. In studying lodestone, Gilbert determined that Earth behaved as if it were a large magnet. In an enormous leap of imagination he suspected that Earth’s magnetism and its gravity were somehow connected. Under the protection of England’s Protestant monarch, Queen Elizabeth I, Gilbert was free to espouse his ideas as well as argue for the heliocentric theory of Copernicus.
Johannes Kepler (1571-1630): Kepler became Tycho Brahe’s assistant in 1600 and when Tycho died the following year he inherited all of Tycho’s notes and data. Kepler read and had fully accepted the theories of Tycho disputing the crystalline spheres and the movements of planets as well as arguing for a better explanation offered by a heliocentric theory. His greatest contribution to the field of astronomy is to be found in the three laws, still valid today, which bear his name and which he derived using Tycho’s data. These laws resolved problems with epicycles and eccentric motion of planets once and for all. Kepler’s first law states that orbits of planets about the sun are not circular but in fact elliptical with the sun located at one of the two foci of the ellipse. Kepler’s second law of planetary motion asserts that a radius vector drawn between a planet and the sun sweeps out equal areas in equal periods of time. This permits a depiction of changing speeds of revolution of planets as they speed up when near the sun (at their perihelion) and slow down when farthest (at their aphelion). Kepler’s third law asserts that there is a mathematical relationship between periods of revolution of planets about the sun (i.e., time for a planet to make a complete revolution about the sun) and their distance from the sun. This relationships is expressed as Equation 1, where T represents
|T2 α d3 Eqn 1|
period is, d is distance and the Greek letter , means either varies as or is proportional to. [This equation is employed some fifty years later by Isaac Newton in formulating his Law of Gravitational Attraction which was proposed in his Principia published in 1687.] The great unsolved problem in all of this work by Kepler was what caused the planets to revolve about the sun in the first place.
Galileo Galilei (1564-1642): Galileo, an excellent mathematician and a good Italian Catholic, was the first modern man to understand that mathematics can describe the physical world. He is reputed to have remarked that “the book of nature is written in mathematics” (van Doren, 1991). Galileo’s mind worked continuously in mathematical terms. He published the results of his many calculations, some of which had considerable theoretical significance. For instance, he proved that projectiles follow a parabolic path and that pendulums, like Kepler’s planets, also sweep out equal areas in equal times. Galileo also disagreed with the prevailing Aristotelian notions of motion. As long as all of his calculations, theories and resulting publications remained purely theoretical Galileo stayed out of trouble with the Church.
His real troubles with the Catholic Church began in 1609 as a result of building his own telescope. It was greatly improved over those he had seen and with it he immediately set out to examine the heavens. He first turned his telescope on the moon where he was amazed to see significant topography. This did not cause too much of a stir as the moon had never been considered to be pure quintessence (i.e., the fifth Greek element, incorruptible ether) but really a mixture of quintessence and corruptible Earthly materials. Looking farther out into the solar system at Jupiter Galileo discovered its four largest moons. Today these moon are known as the Galilean moons in his honor (Io, Ganymeade, Calisto, and Europa). Galileo noted that, from their changing positions nightly, these moons orbited about Jupiter. His view was the Jovian system could be thought of as a miniature solar system. Finally, Galileo turned his telescope on the sun. There he discovered sunspots whose number and patterns changed continuously. This was one more piece of evidence that the heavens were not immutable.
Summoned before the pontifical court in Rome Galileo brought his telescope and bid its members to view the heavens. They were duly impressed by the sights however, they evidenced no understanding of the implications of what they were looking at. Galileo insisted that they think about and consider the consequences of what their own observations of the heavens were demonstrating to them. Before the court he stated that because of his mathematical powers he could prove that Earth revolves about the sun; that Copernicus was right and Aristotle and Ptolemy wrong; and that there was no such thing as Ether or quintessence. The pontifical court’s view, certainly a reflection of the Church’s long held beliefs and doctrine, was that mathematical hypotheses bear no relationship to physical reality. The court could not and would not see what Galileo was insisting. To “see” Galileo’s view was to negate all order and question the Church’s most deeply held beliefs; in reality their inability to “see” was truly a cultural impediment. St. Augustine’s two cities, allegorical to begin with, had become the only reality. Daily existence occurred in the City of Man, the sublunary world of sights, sounds, smells and sins. The City of God, absolute, immutable perfection, became visible nightly. To call any or all of this into question was to question one thousand years of Church history, doctrine, philosophy and belief. For Galileo, St. Augustine’s City of God held absolutely no religious interest whatsoever. The heavens did hold splendor and promise. It was, however, the possibility of solving its mysteries using mathematics that intrigued Galileo.
The pontifical court opposed Galileo principally because it saw in accepting his science the means by which the Catholic religion was called into question. St. Augustine’s City of God would never be the same if Galileo was right. Galileo fought back because he saw Church interference as restricting the role he envisioned for science to play in human existence, particularly whether scientists should have absolute freedom to speculate about reality. Every one really knew Galileo was right; his hypotheses were certainly better than anyone else’s. But Galileo wanted to go beyond hypotheses (again, recall the Church’s view that a hypothesis bears no relationship to the physical world); he insisted that he could prove using his observations and mathematics that he was right and his hypothesis was true. By extension, this meant that the Church had no authority to describe physical reality. The pontifical court’s view was that if it could not describe physical reality then the Church is reduced to being merely an advisor of souls. That Galileo was forced to recant his views was but a Pyhrric victory for the Church. Ultimately the Catholic Church lost out. They could have allowed Galileo authority in the City of Man while keeping their authority in the allegorical City of God; they demanded authority in both and today they have neither. The Church today has become only an advisor of souls.
Francis Bacon (1561-1626): Bacon was really a politician. He was discredited at the Tudor court and so he turned his considerable intelligence to intellectual pursuits. His two books, Advancement of Learning (1605) and Novum Organon (1620) summarized his inquiries and are his most important contributions to western thought. Bacon fervently opposed Aristotelian deductive logic; calling it a dead-end exercise. He thought that application of rules of deductive logic to initially intuitive premises whose truth was unknown yielded results that, while deductively logical, were often untrue. He greatly preferred “his” inductive method. Following an inductive path the student of nature ascends the “ladder of the intellect” from careful observations of nature to arrive at general conclusions. He felt that natural philosophy (i.e., pre-nineteenth century English science) should rely solely on induction while discarding Aristotle’s deduction. Modern science today combines both forms of logic in what in known as the scientific method (see below).
RenÃ© Descartes (1596-1650): Descartes was an extremely well educated, deeply religious Frenchman whose writings probably did more to undermine the Catholic Church than anyone’s. His life was devoted to searching for certainty, paradoxically by doubting everything. His favorite subject was mathematics because of its certainty; one starts with axioms (statements whose truth is self-evident and therefore accepted without proof) and builds from there. After completing his formal education and after voluminous correspondence with the most well educated minds of Europe he decided to write a book on philosophy. Galileo however, was known to be in trouble with the Church and so Descartes decided to write a different book which he entitled The Discourse on Method. In this book, Descartes discusses his education and how he began to doubt that what he had been taught was true. He continued to doubt until he had reduced everything to one simple conclusion; everything might be doubted but the doubter. The doubter, Descartes, existed because he doubts (Dubito ergo sum, I doubt therefore I am). He proceeded to discover a method of achieving similar certainty in other realms by reducing problems to mathematical form. He did all of this and proved the existence of God and His plan for the universe as a perpetual clockwork mechanism all in twenty five pages!
It was his method that was of utmost importance. He believed that to understand any phenomenon one must rid one’s mind of all preconceptions and then reduce the problem to mathematical form. Once this was accomplished then one employed the fewest axioms to shape the problem. Using analytic geometry (invented by Descartes especially for this purpose) a further reduction of the problem yields sets of equations. The last step is to apply rules of algebra to solve these equations.
Descartes was attacked by numerous authorities for his apparently deistic beliefs. The Church condemned Method even as it wished that mathematics could reduce theology to geometric forms. It could not, unfortunately, for theology deals with the immaterial world, a world that mathematics cannot enter. Mathematics was certainty as theology could now never be again. For one thousand years the immaterial world had been supremely interesting, suddenly it was no longer. This was truly a radical change in the history of western thought. Descartes gave us a way to view the material world, and because his mathematics would not allow us to enter the immaterial, our world ever since Descartes has been one of the material.
Isaac Newton (1642-1727): It is not too far fetched to say that RenÃ© Descartes was the last link in a chain that made Isaac Newton possible. Newton’s intellectual forerunners include William Gilbert, who worked with magnetism and who postulated an Earth-attracting force akin to magnetism; Galileo, who studied the miniature Jovian solar system, and falling objects and measured the force of gravity at sea level; Kepler, who developed his laws of planetary motion based on Tycho’s data; and Descartes, who showed how mathematics could be applied to describing physical reality.
Isaac Newton was truly a genius. He was able to shake off all preconceptions and view phenomena from an entirely different perspective, one which was outside of the prevailing culture and its traditional definitions. He laid the foundation for a paradigm shift of epic magnitude that ushered in the modern scientific age. Along with Gottfried Wilhelm von Leibnitz (1646-1716), he invented the calculus, a new mathematics that permitted him to solve the problems raised by his approach. Descartes’ analytical geometry had been sufficient for dealing with static situations or straight-line motion however, everything in the universe appeared to be in curvilinear motion thus necessitating differential and integral calculus.
In his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy or Principia, its short Latin title) published 1687 Newton defined a world entirely different from that of Aristotle. Motion is completely redefined. In Newton’s new world there is neither a “natural” state of rest nor is there “natural” as opposed to “violent” motion; in particular there is no quintessential motion, that is motion which is naturally uniform and circular. Newton’s redefined world is summed up in his three laws of motion.
Newton’s first law states that a body at rest remains at rest unless acted upon by an external force while a body in motion remains in straight-line motion unless acted upon by an external force. Newton’s second law states that a change in a body’s motion is proportional to the force impressed upon it and always acts in a straight line. This law is expressed mathematically as Equation 2,
|F=ma Eqn 2|
where F is force, m is mass and a is an acceleration. As forces always act in straight lines, Euclidian geometry is sufficient for solution of most problems. Euclidian geometry is not sufficient to explain how a continuous impression of a force upon a body moving in a straight line can make that body move along a curved path. Here Newton is contemplating elliptical paths of planets as they move around the sun. It is because of this problem that he needed to invent the calculus in order to solve equations of motion.
A corollary to Newton’s second law is his Universal Law of Gravitational Attraction (or the force of gravity). Newton gave this law the form shown in Equation 3 where: F is force of attraction, G is the
|F=G(m1m2)/d2 Eqn 3|
universal gravitational constant, m1 and m2 are masses of two objects and d is distance between them. What this laws shows is that the force of gravitational attraction is proportional to the product of the masses involved but inversely proportional to distance between them (i.e., greater the masses, greater the attraction but further they are apart the weaker the force). In deriving this law Newton employed Kepler’s second law that relates the square of the period of a planet’s revolution to the cube of its distance. Finally, Newton’s third law states that for every action there is a reaction of equal magnitude acting in the opposite direction; push on a rock and it pushes back with a force equal to that applied.
In the third book of the Principia, Newton discusses Rules of Reasoning in Philosophy. He states that there are only four rules which must be followed. Rule number one states: “We are to admit no more causes of natural things than such are both true and sufficient to explain appearances.” This is merely a restatement of a long used rule known as Ockham’s razor, “What can be done with fewer is in vain done with more” or, put as simply as possible, simplest explanation is best. Rule number two states: “To the same natural effects we must as far as possible assign the same causes”. Rule number three states: “The qualities of bodies which are found belonging to all bodies within the reach of our experiments are esteemed to be universal qualities of all bodies whatsoever”. This rule sliced through the heart of the scholastic arguments of the thirteenth century concerning the nature of universals. Finally, Newton’s rule number four states: reasoning by induction from observations will yield propositions that are accurate or nearly so until such time as new observations cause us to modify our propositions.
Conclusions: From the very beginning of the sixteenth century through to the very first years of the eighteenth in western Europe the hold of the Catholic Church on philosophy and thought was loosened. Undermining a thousand years of doctrinaire, religion based explanations for natural phenomena was accomplished by a dramatic shift in our understanding of nature. This was brought about by advances in observational astronomy coupled with creation of newer, more powerful mathematical techniques. Ultimately, observation and inductive reasoning leading to explanations, theories or propositions replaced what had been a supreme reliance on Aristotle’s deductive logic. This mode of acquiring knowledge operated from Aristotle’s premises which had been accepted on faith because they seemed to support biblical descriptions of the universe, along with Earth’s and humanity’s role in the scheme of things. It is during this two hundred year period that we see the real birth of modern science.
Theism versus Deism:
These two terms are applied today to connote two very different philosophical approaches to the perception and understanding of Christianity’s God and His role in the affairs of the world. Both theism and deism hold to a belief in God. It is this monotheistic, Judeo-Christian belief that separates these two religious variants from their antithesis, atheism. Probably coming into wide usage in late fifteenth to early sixteenth centuries, especially in Great Britain, it is apparent that these two terms were initially synonymous. Over time however, they drifted apart in meaning. Theism retained the original meaning, connoting religious orthodoxy while deism became a pejorative term used mainly to denigrate those with unorthodox religious attitudes.
Theism: The basic belief of theism is in a God (theos) who is personal and therefore may legitimately be conceived through images drawn from human experience and is worthy of adoration. Most important from our standpoint here are two other notions about God. First, God is ineffable, completely separate from the world and at the same time greater than the world. This differentiates theism from pantheism in which God is, or gods are, part of the world existing as god of the wind, god of the sea, etc. Second, and most critical from our perspective here, is that God is continuously active in the world, intervening whenever He thinks there is a need. Theism holds that God created the universe ex nihilo or from nothingness. The universe God created is imperfect by design and therefore God intervenes from time to time in order to set things right. Why would God create an imperfect universe in the first place? The answers or rationalizations given to this question over the years have been many and varied. However, they can be reduced to two very simple alternative answers. First, God needs something to do. So, to keep from getting bored, He can from time to time intervene to set his universe right. A religious and perhaps only slightly less “anthropomorphic” answer could also be advanced as follows. Although humanity is created in the image of God, because of our expulsion from the Garden of Eden (i.e., the “fall of man”), we are imperfect creatures. The universe therefore is not required to be perfect since it is only a place of transient existence for us. True perfection is to be found in heaven, for that is where God resides, and with the coming of Judgment Day only the worthiest will be selected by God to reside with Him in heavenly perfection.
Deism: As a synonym of theism, deism started out with much the same set of original beliefs. By late eighteenth or early nineteenth century however, the technical metaphysical interpretation of deism had become quite restricted. It had become narrowed to a belief in God as the First Cause. It was He who created the universe and who instituted its immutable universal laws. These laws preclude any alteration as well as divine intervention. Ultimately this led to that outgrowth of Newtonian physics known as the “Clockwork Universe” (Peterson, 1993). Philosophical deists eventually came to to deny God’s immanence altogether while historical deists tended to become critical of the necessity of revelation. John Locke (1632-1704) expressed deistic sentiments when he wrote that religion is not above reason. In his view, in a reasonable religion, however essential supernatural revelation might be to it, must not contradict elementary rational considerations. Locke was really saying that religion might be above reason but cannot contradict it.
The previous pages include sketchy historical material as well as highlights of some of the philosophical arguments that preceded the rise of science. This history and its arguments have actually led to the development of the sciences as we know them today as well as development of the scientific method. While the remainder of this essay can be read without this background, a serious student should look at it and hopefully attain a clearer understanding of the trajectory of science.
THE NATURE OF SCIENCE
Definitions of Science
The Oxford English Dictionary defines science as a branch of study concerned either with a connected body of demonstrated truth or with observed facts systematically classified and more or less fastened together by being brought under general laws. The definition also includes the fact that science has trustworthy methods for discovery of new truth within its own domain. Webster’s Dictionary defines science similarly as knowledge or study dealing with a body of facts or truths systematically arranged and showing operation of general laws. Secondary definitions in Webster’s Dictionary include systematic knowledge of the physical world, or any skill that reflects precise application of facts or principles.
In a philosophical sense, science can be thought of as the study of ideas or phenomena about which universal agreement can be obtained. Universal agreement on the content, meaning and categorization of a body of knowledge might initially appear to be impossible to attain however, as we will see, it really is not. It must be first understood that universal agreement must be obtained not from the entire population of Earth but only among a sciences’ practitioners who, it might be added, generally tend to be skeptical by training or inclination. Universal agreement can be understood in simplistic terms by the following analogy. Suppose numerous scientists, each with the same measurement scale, set out to measure some specified phenomenon. If operational definitions for measurement are accepted by all, then each individual should arrive at the same numerical conclusion concerning the magnitude of the phenomenon under consideration. One should realize that instrumental and operator error will introduce some variability into these measures. Variability however, can be accommodated statistically within the ideal of universal agreement. In fact the necessity for statistical methods rests on a realization that operator and instrumental error are to be expected in any process of measurement. Parenthetically it must be added that a third, and probably most important source of error is introduced by nature itself. In nature there is an inherent variability in virtually any phenomenon that one might consider. As an example, you are at the beach and are measuring wave heights for a study on surf conditions. Would it be reasonable to expect that on any given day every wave arriving at your measurement station will have exactly the same height? Or, even more simply put, is it logical to expect that any given wave will have a uniform height along the length of its entire crest? The answer to either of these questions is a resounding no. A statistical treatment of data is therefore necessitated because of nature’s inherent variability. Further variability due to errors introduced by the measurers themselves as well as by slight differences in their instrument must also be taken into account. We will see below that statistical techniques have been developed to enable us to make inferences based on data that are inherently variable. Regardless of nature’s variability or instrumental and operator errors, universal agreement can still be achieved among practitioners of a scientific field when it comes to measurement of phenomena.
These basic definitions and their variants establish a consensus that science is a body of knowledge consisting in part of systematically arranged facts obtained by observation. The Oxford English Dictionary has a very important addition in that this knowledge can be acquired and its truth or certainty ascertained by a method of logical reasoning. Science’s definition therefore appears to have two parts. First, science is a body of knowledge; and second, science has developed a logical method by which the truth or certainty of its statements and findings may ascertained. The latter is what today we term the scientific method. For this reason we should examine each part of the definition separately. Before we do however, a word is necessary concerning use of certainty as opposed to truth. In logical positivism, a school of philosophy developed by Auguste Compte (1798-1857), the term truth would in effect be used. In logical positivism it is believed that theories can be proven true through induction or repeated empirical tests. Karl Popper, founder of another school of philosophy he terms critical rationalism, has criticized use of the term truth extensively. Popper (1959) argues that observations can never prove a theory because the next observation or set of observations may in fact disprove it. He believes that observations can only disprove theories or falsify them, not prove them. In critical rationalism Popper argues that in fact some theories are true however, we can never know which ones they are. Therefore, instead of showing a theory to be true which, he argues, is impossible and thus a waste of time, the best we can hope for is to demonstrate its certainty. In using the words truth and certainty in the manner in which he does, Popper is really saying that truth is objective, absolute and unknowable. Certainty, on the other hand, is subjective but it can be specified statistically with a particular degree or level of confidence.
Science as a Body of Knowledge
General: In looking first at science as a body of knowledge, realize that I am initially ignoring just how that knowledge is obtained. I am also ignoring ultimately what allows us to call one body of knowledge a science while another is rejected as such. The answers to these two important questions will be tackled subsequently. First however, before we attempt to understand how we recognize a body of knowledge as a science, there are a few ideas that I think ought to be explored. I intend to present these ideas as contrasts through the use of a simplistic device, namely, as dichotomies. Dichotomies such as reason/revelation, empiricist/speculator, true/false, black/white, uniformitarianist/catastrophist, and so forth are often used in argument in daily, scientific or philosophical discourse. In using them, it is important to realize that they may mislead as often as they clarify because they often oversimplify issues. A dichotomy tends to present an issue in an argumentative manner and as an either-or-type of choice. When looking at the philosophy of science as it has evolved historically, this can give one the impression that in the development of a philosophical or scientific position a presently accepted notion was arrived at whole or fully formed and based solely on the superior knowledge of one or more leading individuals in the field. This is generally not the case at all. It is important to realize that science is practiced by human beings who cannot be separated from their cultural milieu. Therefore it must be understood that scientists bring to an argument a variety of culturally determined viewpoints and positions. These are then coupled with their own intellectual capabilities and their own peculiar psychological makeup (i.e., how their mind works). All this means that various viewpoints exist that may be either greatly or only slightly at variance with that particular position which is presently accepted or under question. Additionally, when history of a science is written it is often easier to present the various scientific issues as black or white rather than as the shades of gray which was probably the case. In this way, as Gould (1987) puts it, “whiggish history” can be written presenting cardboard characterizations of the various issues and “villains” who presumably slowed the development of a field by placing impediments in the path of the presently accepted position and its “heroes”. Although being presented as clear cut, dichotomies that I intent to use are really just the end points on a continuum of controversy. [Webster’s Dictionary defines continuum as a coherent whole characterized as a collection, sequence, or progression of values or elements varying from each other by minute degrees. Probably the best and most concrete example of a continuum is the electromagnetic spectrum and especially that portion that the human eye is adapted to perceive, namely, visible light. The visible portion of the spectrum can be broken into its constituent wavelengths to give a rainbow of colors from red to violet. Each color blends into the succeeding color so that it is impossible to tell where for instance, yellow ends and green begins.]
Dichotomies: Initially, at the highest possible level of classification, we can recognize a dichotomy in all human knowledge, endeavor and quite possibly in all human existence for that matter. This primary dichotomy is between what is objective and what is subjective. Objectivity is defined in Webster’s Dictionary as having reality independent of the mind. Science as a body of knowledge is objective because it consist in part of testable, immutable facts, often mathematically precise and subject to replication by others without intervention or interpretation by human emotion or whim. As we shall see below, the scientific method that method scientists use to ascertain truth is designed to be as objective as is humanly possible. The subjective, on the other hand, again defined by Webster’s Dictionary as characteristic of or belonging to reality as perceived rather than as independent of mind. Alternatively, it can also be thought of as being experience or knowledge conditioned by personal mental characteristics or states. Subjectivity then tends to be a function of fashion, whim or emotion and includes those things that change with the currents of human affairs. In spite of humans being its practitioners, sciences tend to be clustered at the objective end of this spectrum while the arts and most other cultural (sensu stricto) fields cluster at the opposite end. Now, all humans have a subjective side to their personalities however, when practicing science, a scientist tries to remain as objective as possible. A further check on the thinking of all scientists is the fact that every other scientist in their field can be thought of as “looking over their shoulder”, criticizing every aspect of their experiment from its design to its conduct to its results to the conclusions drawn from it. In this way, in spite of human subjectivity the body of science remains as objective as is humanly possible.
A word here about scientific objectivity and what it portends for human affairs. Science forces us to confront our deepest and most fervently held beliefs and most cherished desires. It holds these up to the hard cold light of an emotionless objectivity. Because much of what we wish to be true has frequently been shown to be false through scientific investigation, then science becomes for many an enemy of our humanity rather than a means to save us from our subjective often whimsical selves, our own ignorance and our own superstitions. This is especially true for those individuals who, for one reason or another, are predisposed to the subjective.
Occasionally, objective and subjective have been mixed, forming a hybrid “science”, with ruinous consequences for those involved. Two examples serve to illustrate this point. First, in the former Soviet Union, Academician Tromfim Lysenko’s perverse rejection of mainstream genetics, aided and abetted for ideological reasons by Joseph Stalin, stalled Soviet agricultural development. To this day Russian biological science is hamstrung by this several generation failure in real genetic research and is trying to catch up with the rest of the scientific world. Lysenko rejected Mendelian “bourgeois” genetics in favor of his own dialectical materialistic interpretations. This involved his belief in the transmission of acquired characteristics which, in spite of all evidence to the contrary, led to his forcing ideological purity over scientific thinking. A second example is Nazi Germany’s rejection of relativistic physics which, thankfully, resulted in Germany’s failure to develop nuclear weapons during the Second World War. At the insistence of Adolf Hitler modern relativistic physics was rejected as being “Jewish science” and therefore an anathema to Hitler’s Aryan nationalism.
Within the body of science itself there are at least two additional dichotomies that I want to discuss. Irrespective of these dichotomies, what should become clear from a discussion of them is that regardless of how scientists divide themselves (and believe me we do, for there is a very definite “pecking order” within the sciences), its divisions are generally just opposite sides of the same coin. One dichotomy requiring some amplification is that between basic science and technology also known as applied science. These two aspects of scientific endeavor are often looked at as competing with each other for attention. Basic science can be looked at as “discovery science” because it is more interested in posing questions and elaborating on questions that clarify issues. Practitioners of basic science (or basic research) are constantly pushing out and expanding the human frontiers of scientific knowledge. Basic science is best and most fruitfully accomplished by allowing scientists to follow their imagination and by cultivating their intellectual curiosity, again by asking questions. Discoveries made may not have any immediate practical application, but that is not the point of basic science. The new knowledge acquired and developed from these inquiries may remain dormant for years until such time as other aspects of the scientific field catch up, realize the significance of these earlier discoveries and come to understand how they fit into the larger scheme of things. In this way the results of basic research become “scientific capital”; capital that can be used or invested at some time in the future. This intellectual capital often fuels rapid advances in technology. But the important thing that must be kept in mind is that in basic science one never knows ahead of time where one’s investigations will lead. An interesting encounter between Queen Victoria and Michael Faraday sums up this point quite nicely. During a tour of Faraday’s laboratory Queen Victoria finally turned to the scientist and asked him “Of what use is all this?” Faraday, looked at the queen and remarked “Of what use Madam, is a baby?” End of discussion. Because of our perception of the importance of basic science, all scientists bemoan funding cutbacks for basic research by corporate entities or governmental agencies. Granted, scientists are human and they have their pet projects (usually their own research) however, as a collective entity they strongly feel that failure to fund basic research is similar to eating one’s own seed corn; eventually harvests fail and starvation ensues. The analogy being made is that scientific advances will cease unless basic science is funded. Additionally, in the words of Albert Einstein: “As a circle of light expands, so does the circumference of darkness around it.” It is in this way that basic research raises new questions that beg answers and so the entire endeavor is self-driving. On the other side of the coin, we have technology or applied science. This is the endeavor that is generally thought of when most people think of science. Here we are looking at engineering or technological applications of the results of pure science in the production of things that serve humanity. Applied scientists are generally not interested in the framing of questions so much as they are in getting answers to practical problems. When making applications of basic science problems often develop which beg answers. Applied science then can turn back to basic science for the research to frame the proper questions. Answers to these new sets of questions can lead to improvements in present technology as well as lead to still other sets of questions and so on. Now it would be nice and neat if everything worked as laid out in my explanation in a nice, simple, sequential manner. In reality, the operation of basic science and applied science really acts like circuits in parallel. That is to say, both groups of individuals are operating simultaneously and so sometimes it is a little like “which came first, the chicken or the egg?”
A second important dichotomy with which we should come to grips in attempting to understand science is the dichotomy between what many refer to as “pure” science on the one hand and historical science on the other. Phenomena in the universe (matter, energy and their interactions) are studied by looking at their aspects or characteristics. Collectively we could call these properties. Properties can be divided into those that are immanent and those that are configurational. Immanent properties are timeless and unchanging such as scientific laws. As an example, Newton’s Second Law is as true today as it was millions of years ago and it is as true today on Earth as it is on Jupiter. Configurational properties, on the other hand, are historical or time-bound and are characterized by their uniqueness. Examples of configurational properties abound. To illustrate them we could cite da Vinci’s “Mona Lisa” or Beethoven’s Fifth Symphony. In spite of the fact that there have been many great artists and many great works of art or many great composers and many great pieces of music there has only been one Leonardo da Vinci and one Ludwig van Beethoven hence only one “Mona Lisa” and one Fifth Symphony. With these two types of properties in mind we should be able to make some differentiation between “pure” science and historical science. “Pure” science attempts to concerns itself solely with nature’s immanent properties. It attempts to uncover nature’s laws or tries to tie laws together into theories of various complexity. In other words, “pure” science has as its principal aim the elucidating of natural laws which illuminate the way in which nature operates (expanded below). Examples of “pure” science would include mathematics, physics, chemistry, certain aspects of astronomy, certain aspects of biology. There are some however, would might wish to include mathematics within philosophy as the ultimate form of symbolic logic. Historical sciences are those which concern themselves with unraveling the sequencing of historical events (i.e., configurational properties). An historical science is scientific only in as much as it uses immanent properties to understand the configurational. Examples of historical sciences are geology, geomorphology, physical geography, paleontology, archaeology, paleoclimatology, certain aspects of astronomy. If human knowledge were arrayed along a continuum then at the “pure” end, with interest solely in immanent properties and timeless generalizations about nature, we would place mathematics, physics and chemistry. At the opposite end of the spectrum, concerned only with the unique, with configurational properties, we would probably put the arts and humanities. The historical sciences would find themselves arrayed along the continuum somewhere between the ends but closer to the “pure” end of the spectrum of knowledge than to the arts.
Idealized stages of scientific development: A growing body of observations coupled with generalized statements based on those observations and whose truth has been arrived at through a logical method of verification is, say, a new scientific field. We can look at the expected stages in the historical development of this new field by looking at how other fields of endeavor have developed. Initially a new scientific field will usually begin with a descriptive stage. Here everything is new and what is being accumulated are mainly observations which must first be classified and organized. Once a method of systematization is agreed upon (note again, universal agreement) then several different approaches to the field’s problems may be tried. Eventually, the field’s practitioners will settle on an approach that satisfies all or a majority of those concerned. Then they may begin to see or infer relationships between various aspects of the phenomena being studied. Any inferential statement made about relationships between phenomena or observations is termed an hypothesis or theory (more about these terms below). Once a field enters a stage in which theories are being proposed, tested and in some cases not rejected then the field has entered the explanatory stage. At this point the field is connecting itself to other related fields that may be far more advanced. At this time as well, hypotheses put forth should not be at variance with those of related fields. Years of scientific endeavor have shown that hundreds of scientists working on various problems come up with explanations that are all interrelated thus reinforcing the rational character and seamless beauty of nature and our explanations of natural phenomena. Finally, when a field has advanced to the point that it has a good theory base then it can enter the predictive phase of development. This is ultimately the phase that all “pure” sciences strive to attain because they are now in a position to make predictions or pronouncements about the future.
During any of the above phases, scientific fields are constantly oscillating between phases that we might call analysis and synthesis. In an analytical mode a scientific field is generally pushing the frontiers of knowledge. Frequently it acquires more information than it can digest. Eventually then, the field will begin to enter a period of synthesis. During such a period, the field becomes introspective as it incorporates what has recently been learned with what it has already verified. During a period of synthesis, basic science does not come to a halt so that everyone may engage in the work of synthesis. What really happens is that most scientists continue their own research while perhaps a handful of others do the work of integrating and synthesizing to make the “bigger picture”. As they integrate and synthesize new material and develop new theories their work is published so that their colleagues can have a chance to disprove their findings or ideas.
What makes a body of knowledge a science?: Any body of knowledge that wishes to proclaim itself a science must, as an absolute minimum, meet several basic criteria. Just what these criteria are is the crux of the problem. Because by specifying criteria then what we are doing is philosophically circumscribing the limits of what science in general will tolerate as being accepted within its realm.
As a starting point for this discussion of what criteria define science, we can begin with an enumeration of legal parameters that have been used recently in federal court cases and in judicial opinions defining what science is about. While we must examine these criteria from a philosophical standpoint, I think it is noteworthy that, given the increasingly litigious nature of American society today, some reference should be made of the legal basis for recognizing a science. Particularly in the United States, there is a continuous, ongoing assault being conducted against science. This continual skirmishing is carried out by both fundamental religionists who oppose any attempt to modify biblical interpretations or introduce rationality over faith (recall those who opposed Thomas Aquinas) as well as various New Age charlatans or what could be termed “nonsensicalists” for lack of a better word (e.g., channelers, believers in crystal consciousness, believers in UFO’s, believers in UFO abductions, scientologists etc.). The individuals belonging to these groups wish to have you believe that their body of knowledge is scientific when in reality what they propose to have you accept is nothing more than pseudoscience. Typical characteristics of a pseudoscience includes, but is not necessarily limited only to, appeals to myth, a casual approach to evidence (tending to accept only what supports their theories rather than all evidence), irrefutable hypotheses, a refusal or inability to revise their hypotheses and a lack of tentative quality to their conclusions, to name but a few. From time to time, these groups have stepped up their attacks on science, usually without much success. In recent years however, fundamental religionists have pressured some state legislatures into passing laws that require the teaching of aspects of their religious doctrine as science. An example of this is the subject matter collected under the heading of creation science (an oxymoron, if there ever was one).
In a legal ruling which overturned Arkansas Act 590 concerning the teaching of creation science in schools, Federal District Judge William Overton used the following criteria to determine whether or not a body of knowledge qualifies as a science. Judge Overton arrived at his criteria only after receiving testimony from numerous scientists, historians of science and philosophers of science. Judge Overton’s criteria are that a true science is: a) guided by natural law; b) explanatory within the frame of reference of natural law; c) tentative in its conclusions or theories; d) verifiable against nature; and e) falsifiable.
The above are legal criteria because they have been used as standards in court cases. They were however, arrived at in consultation with numerous scientists, historians and philosophers and so they have philosophical basis as well. With this in mind, I think they can profitably be used here.
The first criterion requires that a science be guided by natural law. Science has been practiced for several hundred years, during which time numerous natural laws have been discovered. These laws guide further research and what we find is that new discoveries conform to existing natural law. In those few instances where natural law has failed to explain completely the phenomenon in question, corrections or corollaries have been made to the law. Here I am thinking particularly about how Newtonian physics breaks down as the speed of light is approached. This explains the necessity for relativistic physics (as a corollary to Newtonian physics). What is important here is that natural law is not violated by a body of knowledge claiming to be science.
The second criterion specifies that a science’s body of knowledge is explanatory within the frame of reference of natural law. A scientific field’s body of knowledge must be internally consistent and coherent. A science expands our understanding of natural phenomena. In doing so its revelations about nature and how nature works connect with what we already know from other related sciences as well as leading us to new knowledge. Thus a science’s body of information, laws and theories fits within that grand tapestry of what we know and has been demonstrated. As a corollary, the laws and theories of one branch of science do not contradict fundamental elements of closely related branches. In the formulation of explanatory theories or hypotheses we must always apply Ockham razor. This is a rule of logic formulated by that fourteenth century Franciscan, William of Ockham, and can be loosely stated, “the simplest explanation is best”.
The third criterion is that a science’s hypotheses and theories are all tentative. This might come as a surprise to many people, especially in the twentieth century. We tend to take science and its practical application, technology, for granted and assume that it is all a permanent edifice, albeit painstakingly constructed over years. But at times in the past some branches of science have strayed off onto false paths which ultimately led to dead ends. This has been due to faulty reasoning, imperfect or sloppy observations, or even skullduggery (and this has happened on a few rare occasions). More often, faulty or false theories are based on too few observations or ignore observations that do not fit preconceived notions. Once a false theory or hypothesis is put forth then, what happens next? Fortunately, science is, in the long term, self correcting. By this I mean that when any idea is put forth it must, as a matter of principle, be subjected to close, detailed scrutiny. The job of scientists is to examine critically hypotheses with a view to rejecting them. The scientific method is designed to highlight fallacious aspects of theories. If close examination cannot show a theory to be false then we do not immediately accept the theory. What we do philosophically is just not reject the theory for the present. This tends to maintain a theory’s tentative character. For this reason then all science’s theories must philosophically be considered tentative, that is to say, subject to change. But humans do not like uncertainty. Therefore, if a hypothesis or theory has been continually subjected to testing over many years and not found wanting, then in our minds it tends to be viewed as accepted truth or certainty rather than what it is, a hypothesis that has not been rejected. What is important here is every hypothesis or theory in a science is by definition tentative and thus fully capable of being disproven at any time. What is not tentative in a science are facts, those repeatable observations and experimental results that are capable of being replicated by others. Obviously, facts cannot be tentative because they can be replicated. Facts form the basis for development of a scientific field’s hypotheses and theories. While all theories are tentative, this is not the case with scientific laws.
The distinction between a law and a theory is clear and the two should not be confused. A scientific law merely relates our understanding or interpretation of what happens. A theory or hypothesis, on the other hand, is a tentative statement or explanation of our understanding or perception of how or why events, phenomena, anomalies or observations occur. Theories or hypotheses are based on, or at least do not violate any natural law. Even though theories are based on known, existing scientific laws they may eventually lead to prediction of new laws. Most importantly, theories do not become laws nor do laws become theories, for they are two different things. A series of laws however, may illustrate a theory while new laws may be derived from a theory. It is important here to mention that when we speak of laws in science we are merely expressing in our own human way, as best we can, what we see occurring in nature. Our scientific goal is to attempt to understand how some aspect of nature works. When time and again we observe some cause and effect relationship in trying to understand a phenomenon of interest, then we verbalize a scientific law, which is our way of stating what is happening. Later we may convert measurements of this cause and effect into a mathematical expression of this law. However, when all is said and done, and our law enunciated, it is vitally important to realize that what we have done is express our human understanding, interpretation or perception of what is occurring in nature. Nature is not bound in any way to follow our laws. All we can hope for is that we have perceived or understood nature well enough that we are expressing correctly the relationship implied. If we have observed correctly, then nature will appear to follow our law every time indicating to us that our law is correct. Once a law is promulgated however, it must be remembered that it is a human construct and nature is not obliged to follow it. If we have constructed our law in the wrong way then nature will let us know by not following it.
Lastly, I choose to collapse verifiable and falsifiable and discuss them together. All of a scientific field’s theories are verifiable against nature. Any hypothesis or conclusions drawn from it must be capable of being tested, observed or results replicated. This ties these aspects of verifiability together with the tentative character of scientific hypotheses because they can be modified. Falsifiability means that to test a hypothesis one must be able to falsify it and by doing so disprove it. All hypothesis must be capable of being falsified and thus disproved. Put another way, all hypotheses must be verifiable or testable for if they cannot be tested then they cannot be proved nor can they be disproved. Statements that are taken to be truthful are those that cannot be shown as false.
The Scientific Method
In simplest terms the scientific method is the logical thought process by which truth of a statement, hypothesis or theory may ascertained. Each step in this process follows logically from the preceding step. A general, idealized outline of the scientific method is illustrated in what follows. Initially, we are involved with observing or measuring some phenomenon, anomaly, event or association of interest. Once data are collected, they are then examined for possible relations between them. Upon further examination, say, these observation are thought or presumed to be associated in some particular manner that illustrates an aspect of nature. The next step involves moving from these specifics to the formulation of a more general statement, a hypothesis or theory that explains the how or why of this presumed relationship between observations. This thought process is induction or inductive reasoning. Our postulated theory or hypothesis is merely a tentative testable explanation for the interrelationships we assume to exist between observations. For the time being our hypothesis is assumed to be true. That being the case, then the very next step involves using deductive reasoning (i.e., reasoning from the general to the specific). What we deduce from our hypothesis are consequences that could be expected if our hypothesis is correct. The final step, closing the loop in this reasoning process, involves going back and further examining the phenomenon, anomaly, event or association of interest to see if those consequences deduced or predicted from our hypothesis are in fact met in nature. If consequences deduced from our hypothesis are not met or cannot be found in nature then our hypothesis is rejected. As an alternative, our hypothesis may be modified in some way. If modified, then the scientific method starting with the deductive step begins over again. At any rate, if the deduced consequences are not met by nature, then our hypothesis is considered false and is rejected. If deduced consequences are in fact found in nature then our hypothesis is not necessarily accepted. To put it philosophically, our hypothesis is just not rejected at this time. Technically speaking, scientists are not really in the business of proving hypotheses but rather of disproving and rejecting hypotheses put forth. Therefore any hypothesis or theory that is not immediately disproven is merely designated as not rejected at this time. This underscores the tentative nature of all hypotheses and theories in science. Now scientists are only human and after the passage of a considerable amount of time in which the scientific community fails to disprove and thus reject a theory then that theory begins to take on the aura of a reigning or doctrinal hypothesis that is presumed to be true. A presumption of truth resides in an inability to prove a theory or hypothesis false.
A final word about the scientific method. As outlined above, the method is shown to be a closed loop or cyclical thought process. In actual practice the method is often used in what we could call either a deductive mode or an inductive mode. “Pure” science tends to operate in the inductive mode because it tends to begin with observations and then progress on to hypothesis formulation, experimentation, conclusions and perhaps ending with theory development. Historical sciences, on the other hand, often operate in a deductive mode. Starting with a theory or hypothesis, deducing consequences, experimentation finally concluding with a comparison with nature. In either case the result may lead to theory development, addition of new knowledge to the body of science or insights gained to existing theories.
Multiple Working Hypotheses: In the late nineteenth century, the American glacial geologist Thomas Crowther Chamberlain wrote a marvelous paper that has been reprinted on several occasions. The title of his paper is “The method of multiple working hypotheses” (Chamberlain, 1897). In this paper Chamberlain argues for developing numerous hypotheses whenever a problem arises for which an explanation is sought, needed or desired. By developing several different working hypotheses, an investigator is not overly biased in the same way that he or she might be if only one hypothesis is put forth. With a single hypothesis it might stand a chance of becoming too easily favored by an investigator. Again, scientists are only human and in spite of all their training they may be subject to the same biases or the same ego problems as any other individual. Therefore, in an effort to evaluate a hypothesis in as unbiased a manner as humanly possible, it is best to have numerous of them. In this way no single hypothesis should become favored to the extent that critical evidence against it is ignored.
The body of knowledge that today we call modern science is almost in its entirety a post – renaissance phenomenon. The scientific edifice that has been constructed over the last three or four hundred years is really quite marvelous, having provided modern societies with benefits unimagined even just a century ago.
As an intellectual activity, science is one of the only human endeavors that is trans-cultural, trans-racial, trans-national, trans-ethnic and trans-linguistic. Its methods are practiced and duplicated by every scientist world-wide regardless of whether they even speak the same language. It is also an enterprise whose component parts oftentimes can only be worked on collectively or in collaboration with colleagues who may be in the next laboratory or half way around the world. This body of knowledge that we call science is advanced through the use of experiments whose results are reported in peer reviewed scientific journals to a skeptical audience of other scientists. Controlled experiments are significant to science because they are the source of new data, they are a way of testing hypotheses or our tentative statements of truth and they are capable of being replicated by any other researcher as a check on our methods and results. Peer reviewed journals are important to science because they afford the means of informing the scientific community of new findings as well as a further means of checking and rechecking the ideas and methods being employed in any investigation. Many scientific projects are so large in scale or scope that funding to conduct research on them must be provided by industry or government sources. Applying for grants for financial support is usually a peer reviewed process, thus providing one more additional check on the quality of the science being performed. Finally, and most importantly, the scientific audience is not credulous but quite the contrary it is incredibly skeptical. Credulity is not a trait found in your average scientist. Scientific training ideally teaches the practitioner of science to doubt everything and at all times to keep a strong sense of detached skepticism. Skeptical thinking on the part of scientists means that everything is available for re-evaluation; there is no hypothesis or theory that is sacrosanct. Science has its own error correcting mechanism built in in the form of the skepticism of it members. In this way, even if a false idea were to win acceptance in a scientific field, such a condition would only last for a relatively short period of time. Inevitably error will be uncovered. In all scientists there exists an essential and very necessary tension in their thinking and thought patterns because they must work with two very contradictory mind sets. First, they must have an openness to new ideas no matter how bizarre or counterintuitive they may first appear to be. However, this openness must also be coupled with a ruthless skepticism that requires that all ideas must undergo critical scrutiny.
SCIENCE AND SOCIETY
The rise of religious fundamentalism and New Age pseudosciences in recent years is probably symptomatic of a fundamental disenchantment with science and the overwhelming role it plays in our lives. As mentioned above, science forces us to confront some of our most cherished beliefs. In holding these beliefs up to the cold light of reason, oftentimes they are shown to be false dreams. Science is then blamed for our subjective shortcomings and thus becomes for many the archenemy of our humanity. In reality, science is a very powerful means to save us from our own ignorance and superstitions.
Many people feel threatened by the control science appears to have over their lives. There is a mistrust of science and claims for improvements in life given the Challenger disaster, Three Mile Island, Chernobyl, Bhopal, Love Canal and the like. What I think is the real problem here are the claims of technology or science applied. Recall from our discussion concerning “pure” science versus technology. Scientists are mostly interested in the framing of questions and thus the generating of information and hopefully better questions. It is technology, applied science, if you will, that seeks answers so that applications can be made creating new instruments or devices that are supposed to serve humanity.
It is important to realize that science, in spite of its incredible level of sophistication, has no moral content. It is in fact completely amoral. The power science provides humanity offers no clue as to what use to make of it. For this reason, science has been put to uses from the purely beneficial to the inhumanly perverse and detrimental. In application then science requires guidance from sources external to itself. Input from other aspects of humanity’s intellectual and philosophical endeavors such as ethics and esthetics must provide a leavening influence and a guiding force in science’s application to the problems of humanity.
One last word about science and society’s emotional needs. Science can never answer questions that deal with the nature of the universe before the “Big Bang” nor can it deal with the meaning and purpose of the universe. The very nature of these questions puts them outside the realm of science because they are framed in a way that permits neither observation nor any means of measurement and therefore no way to test them.
THE NATURE OF SCIENTIFIC REVOLUTIONS – SCIENTIFIC ADVANCEMENT: PARADIGMS AND PARADIGM SHIFTS
Prior to the work of Thomas Kuhn (1970, 1977), the idea of a scientific field shifting its entire perspective was not a subject that had elicited much attention. This is probably due to the fact that sciences do not shift their perspective or explanatory basis annually or to suit a fashion; in reality such shifts appear to be a rather rare events. Philosophers of science, and most scientists themselves, would, if pushed for an explanation, probably have explained the phenomenon in terms of the Socratic dialectic. Using the dialectic, the old perspective takes the place of the thesis while the new one becomes its antithesis. After conflict between the two opposing viewpoints, the resulting compromise, the new perspective or basis for explanation, is the final synthesis. Kuhn was one of several philosophers of science who began to devote some considerable thought to this issue during the 1950s and 1960s.
Kuhn started his academic career as a physicist, receiving his Ph.D. in nuclear physics from M.I.T. in the early 1950s. He discovered however, that he was more interested in the philosophy of science and scientific endeavor and so pursued these interests to the exclusion of his work in pure physics. The seed that was to grow into his life’s work was actually planted as a graduate student. While reading Aristotle’s Physica (one of his scientific treaties) and contrasting Aristotle’s discussion of motion with Newton’s discussion in Principia, Kuhn was struck by the total disconnect in meaning and explanation between these two works. Here were two men discussing the same phenomenon, however, it was as if they had started from two entirely different perspectives and, even though they used the same words, those words meant different things to each man. The memory of this lay in the back of Kuhn’s mind until his interests brought him to a philosophical and historical nexus that occurred in physics around the turn of the century. This involved the work of Albert Einstein and his General and Special Theories of Relativity along with Max Planck and Neils Bohr in quantum mechanics. These ideas, counterintuitive though they might be, caused great intellectual turmoil in the physics community. It was as if there had occurred a great disconnect between the certainty of Newtonian physics in the nineteenth century and the great uncertainty caused by these new theories at the start of the twentieth. These changes that occurred in physics and, more importantly, how that scientific field’s practitioners adapted to them, became Kuhn’s abiding interest. He recalled how, as a graduate student, he was struck by Aristotle’s and Newton’s disconnect when discussing motion. He thought he saw in all of this a realization that a scientific field does not change its perspective via use of the Socratic dialectic but through another much more radical means. He developed and published his ideas relating to drastic shifts that scientific fields sometimes undergo in his best known work, The Structure of Scientific Revolutions (1970). It is in this work that Kuhn introduces the term paradigm. Equally important as a paradigm is a phenomenon he calls a paradigm shift. This then is the process Kuhn sees as the means by which scientific advancement occurs.
Paradigms: A paradigm is an overarching theory or “grand theory” that is developed within a particular scientific field. For any scientific field, its paradigm is constructed from facts and observations as well as other theories that are on hand at a particular point in time. Its purpose is to gather together that field’s numerous disparate observations, laws and other hypotheses and tie them into a coherent whole by explaining their interrelationships. By its very nature, as an overarching explanation, a paradigm often becomes a research blueprint for its field of endeavor. A paradigm then is used as a collection of procedures or ideas that instruct scientists implicitly what to believe and how to work. Most scientists do not question their field’s paradigm. In fact most scientists do the normal work of science by formulating and then investigating question that are determined or validated by the paradigm.
Initially, a paradigm must prove particularly successful at explaining relationships between known facts; otherwise it would not have gained credence in the first place. If a paradigm is really true, that is to say it has been developed based on sufficient, necessary and significant facts, then it will do two things. First, it will continue to direct a field’s research, defining the very nature of valid research activities. Secondly, all newly discovered facts will automatically fit into this existing paradigm and thus be explained and connected with all other observations and facts. What happens in development of a scientific field if its paradigm is only partially correct or even entirely wrong? Problems with a paradigm become apparent as newly uncovered facts or observations fail to fit into its framework of explanation. Initially, facts that do not fit a paradigm may be considered as anomalies that may quite possibly be explained later as scientists come to understand more about their field. If a paradigm has only minor flaws, then the self correcting nature of science can accommodate this. Accommodations occur as a paradigm is modified or adjusted sufficiently to explain these anomalies without violating any other part of the paradigm. However, in the case of an incorrect paradigm problems increase becoming significantly greater. The entire basis of a field of scientific endeavor becomes undermined because as more and more research is conducted and its results are made available more and more facts, observations and conclusions drawn from them fail to be explained within the framework of the paradigm. How does a scientific field react to such a state of affairs? Generally, and remarkably, it plods along unwilling to give up its incorrect theoretical explanatory framework because that would involve abandoning something, however flawed, for nothing. Without a paradigm, a scientific field is at a loss. As previously stated, a paradigm provides research direction within a field and without this direction a scientific field can be likened to a rudderless ship drifting aimlessly, a condition intolerable to a scientific community. Stress continues to build up in an existing incorrect paradigm as continuous research adds more and more inexplicable anomalies to the body of scientific facts. The stage is now set for a paradigm shift.
Paradigm Shifts: Generally what has occurred in virtually all of science’s previous paradigm shifts is for some one individual, usually of great intellect, a genius if you will, who is able to step outside the normal way a scientific field has been viewing things. By stepping outside of a paradigm’s defining view, perhaps by having some blinding flash of insight (the true mark of genius?) that allows them to view all the facts including recent anomalies from a different vantage point or perspective, a new way of perceiving sets of relationships emerges. A newly perceived set of relationships often leads intuitively to a completely new paradigm. Although originators of a new paradigm may be of any generation within a scientific community, it is often younger members of that community who enthusiastically embrace a new paradigm. Younger scientists, aware of the old paradigm’s inability to provide necessary overarching explanation, have invested little time in their scientific careers in the old paradigm. There is therefore less for them to abandon. Often the greatest resistance to acceptance of a new paradigm comes from older members of a scientific community. These individuals have invested a major portion of their life’s work in the old paradigm. They may have been involved in efforts to modify that faulty paradigm so that it would fit the anomalies. As more and more modifications are made and fewer and fewer new facts fit the old paradigm they may even have become apologists for this faulty, older paradigm. There is therefore a considerable amount of ego that gets in the way of making a necessary change; scientists are after all human too! Eventually, however, rational behavior takes over, and as each researcher begins to realize that facts they are uncovering are better accounted for under the new paradigm they begin to abandon the old one for the new. Eventually the old paradigm becomes an historical curiosity with a few old holdouts still clinging to it. The new paradigm takes over providing a framework for explanation of existing facts and redefining directions of research and what constitutes valid research.
Conclusions: There are two major conclusion that can be drawn from Kuhn’s work. First, science is not a continual building process. A paradigm shift is eventually forced on a scientific field when its present reigning paradigm, perhaps fashioned out of incomplete information, is shown to be false. The resulting revolution is both destructive as well as creative. The older paradigm is shown to be false, looses favor and is destroyed. In its place however, is a new more powerful one that better explains the facts. Once a paradigm shift occurs, then most scientists adapt to it and continue to conduct their research. The second major conclusion to be drawn from Kuhn’s work is that in a paradigm shift, conversion for most scientists occurs for one of about three reasons. First, there is a sudden intuitive understanding on the part of certain individuals. It is as if a light has just come on and all of a sudden everything is clear to them. Second, for the vast majority of scientists their conversion to a new paradigm is due to the backing it receives from other scientists whose reputations carry the day. Finally, after a paradigm shift is well under way, then students are brought into the picture. They are taught the new paradigm as “truth” in class, laboratory, or field. Once a new generation of scientists is in the process of being educated according to the new paradigm, then the death knell has been sounded for the old one.
The job of most scientists is to conduct research into questions that they have framed based on their imagination. Their research and the question framing process are carried out according to the research blueprint inherent in a field’s paradigm. In this way a paradigm can be likened very crudely to a painting by the numbers. The canvas is laid out and the lines are drawn and the various areas are numbered (this is the paradigm). For the average scientist then, their task is to paint in various numbered areas. This really is what ordinary science is about.
Scientific Heresy: As described above, driving a paradigm shift is the inability of a field’s reigning paradigm to account for all of the facts. Attention may be drawn to a paradigm’s failures and weaknesses in various ways. Two ways of doing it are through heresy and outrageous hypotheses. An interesting discussion of scientific heresy can be found in Goldsmith (1977). In this book, a discussion is presented concerning what has been termed endo-heresy (and its practitioners are thus endo-heretics) and exo-heresy (whose adherents are exo-heretics). Endo- heresy, as its name implies, is an attack on a field’s paradigm from within that scientific field. Endo-heretics are scientists who have been schooled in the old paradigm and have conducted their research under the aegis of the old paradigm. Being part of a scientific field’s establishment, they know what they are speaking about. Even though a majority of scientists may not listen to them initially, they still inspire a certain degree of respect and their criticisms cannot be rejected out of hand. Examples of endo-heretics might include Darwin and Wallace in biology, Lavoisier and Mendeleev in chemistry, Einstein, Planck, and Bohr in physics and Hutton, Lyell, and Wegener in the solid earth sciences. Exo-heretics, on the other hand, are individuals who are outsiders. They are not members of a scientific field’s community. As such they are often not as well schooled in a field’s intricacies and generally their understanding of what a field is about and its problems is overly simplistic. Exo-heretics appear occasionally and these individuals have provided trying times for those scientists to whom has fallen the task of dealing with them. Probably two of the best known exo-heretics in the history and philosophy of twentieth century science are Immanuel Velikovski (1950, 1955) and Eric von DÂŠniken (1970, 1971, 1972). Neither of these individuals has any significant training in the scientific fields in which they have dabbled and about which they have made their pronouncements (astronomy and geology). Numerous other exo-heresies exist in the form of New Age nonsense, creation science, and the current fascination with UFO’s and UFO abductions. Goldsmith (1977) and Sagan (1996) are instructive to read for their reasoned approach to dealing with exo-heretics who often are attempting see myth reestablished as fact.
The value of the outrageous hypothesis was first discussed by Davis (1926). The idea that a scientist can seriously and in good conscience propose a hypothesis that is outside the bounds of “normal” science highlights something that was discussed earlier about the nature of a scientific mind. A scientist must, at one and the same time, have both a completely open mind but also have a mind that is extremely skeptical. An open mind is necessary in order that one does not reject out of hand a hypothesis that may appear at first glance to counterintuitive. – an outrageous hypothesis. At the same time this same scientist must be skeptical of everything, and so this means that the outrageous hypothesis will be put to the test and if found wanting, rejected.
The entire notion of a paradigm shift reinforces the important provisions that science is self correcting and most importantly that all scientific theories are tentative and verifiable. Paradigms, as overarching theories providing explanation for observations and facts within a field, must by definition be tentative. The important thing to be underscored from this discussion of paradigm shifts is that while a theory, hypothesis or paradigm is tentative and subject to change, that change is brought about or required by that theory’s, hypothesis’ or paradigm’s inability to explain existing facts. In a paradigm shift, or in the rejection of any theory for that matter, it is the paradigm or the theory that is rejected, not the facts and observations. They remain as the constant against which all subsequent theories must be measured and rejected or not rejected.
Virtually all fields within science have undergone one or more paradigm shifts at various times during their centuries of development. In physics, a paradigm shift occurred at the turn of this century with relativistic physics replacing classical physics as the basis of explanation for phenomena taking place at or near the speed of light. This new paradigm was necessary because Newtonian physics was no longer capable of yielding satisfactory explanations for what would occur under these special conditions. In biology, paradigm shifts have occurred as knowledge of bacteria replaced notions of spontaneous generation. In perhaps a most important paradigm shift, one whose implications are still reverberating throughout parts of American society today, the notion of evolution replaced earlier, biblical stories concerning the origin of life. These anecdotal and fanciful stories were developed by our ancestors to explain human origins. In the Earth Sciences, an extremely important paradigm shift occurred in the late 1960s and early 1970s. During this period the theory of plate tectonics was formulated, tested, and found to explain observed facts far better than our previous theory. Rejected out of hand when first put forth around the turn of the century by the German geophysicist Alfred Wegener as continental drift, this elegant theory eventually received an adequate testing when our scientific instrumentation reached a higher level of sophistication. This paradigm shift allowed us a much simpler explanation of existing facts and observations concerning Earth. At the same time, it also gave us a blueprint for the way Earth works and thus told us where to look for various phenomena and what type of research to conduct. When we went to look for these phenomena, they were in fact there, enabling us to push the envelope of our knowledge further still.
MEASUREMENT, THE CURRENCY OF SCIENCE
How are Physical Phenomena Measured?-Units of Measure
General Discussion of Measurement: We have just discussed the fact that the Universe is composed of only two basic, measurable properties, energy and matter. In order to define and adequately describe these two properties as precisely as possible, scientists have had to come up with very well defined units of measure. Without these precisely defined units of measure, replication of experiments becomes impossible. Prior to nineteenth century, units of measure were locally defined and tremendous differences existed from place to place. Differences in measurement only lead to confusion and necessity of conversions from one system to another. Without uniformly accepted standards of measure, measurements cannot be replicated easily and thus experimental outcomes are difficult to compare and results be disproved. Today, scientific units of measure are on a world-wide standard known as the Systeme Internationale (SI).
For our purposes here, scientists recognize two specific units of measurement. These are fundamental units of measure and derived units of measure. To arrive at the fundamental units of measure, all possible units of measurement employed were first examined to find their commonalities. Once commonalities were isolated, then fundamental measurement could be distilled down to those minimal elements or components that are absolutely the simplest measures possible. In doing this, seven fundamental units of measure have been recognized and defined using universal standards. These fundamental units of measurement are basic to all other possible units of measure that are then used to describe every possible physical property or phenomenon. A very important point to be made here is that the fundamental units of measure are all based on arbitrary criteria. Though based entirely on arbitrary standards, these fundamental units of measure are very precisely defined. In spite of the fact that they are based on arbitrary standards, measurement is standardized throughout the sciences because every scientist in the world agrees to accept these units of measure. This is a first and very major step in attaining that universal agreement which is part of science’s philosophical definition. Finally, the derived units of measurement are arrived at by combining fundamental units in various meaningful ways as will be illustrated below.
Fundamental Units of Measure: The seven fundamental units include: a) length represented by L (in SI the meter is the unit of length measure); b) mass represented by M (in SI the kilogram is the unit of mass); c) time represented by T (in SI the second is the unit of time measurement); d) temperature (Â°K – Kelvin temperature scale is used in SI wherein 0Â°K or absolute zero is the lowest temperature attainable); e) amount of substance (in SI the gram-mole is the basis of measure); f) electric current (in SI ampere is the unit of measurement); g) luminosity (in SI candela is the unit of measurement).
Two additional measures are often included with these seven fundamental units of measure. These are the radian (used in expressing plane angles) and the steradian (used in expressing angles in solid geometry). The radian is defined as that angle, measured at the center of a circle, which is subtended by an arc along that circle’s circumference equal in length to the circle’s radius. A steradian is the solid angle measured at the center of a sphere and subtending a section on the sphere’s surface equal in area to the square of the sphere’s radius.
Length: To illustrate the first four of these fundamental units of measure we need to start with length. The meter was originally defined as equivalent to one ten-millionth of the distance from pole to equator along the meridian of longitude passing through Paris, France. The fact that it is not is of little significance today. Every scientist in the world now accepts the standard meter length preserved by a bar of platinum-iridium alloy in Paris at the International Bureau of Weights and Measures.
For ease of expression, as measurements get smaller, they may be subdivided into smaller units, allowing us to shift scales of measurement while attempting to continue using whole numbers and avoiding long strings of zeros following or preceding a decimal point. We can illustrate this subdividing using the fundamental SI unit of length, the meter. Whole meter lengths are expressed as meters, but we can also expressed meters in tenths (decimeter = 0.1 m = 10-1 meter), hundredths (centimeter: 1 cm = 0.01 m = 10-2 meter), thousandths (millimeter: 1 mm = 0.001 m = 10-3 meter), millionths (micrometer: 1 Âµm = 0.000001 meter = 10-6 meter), billionths (nanometer: 1nm = 0.000000001 meter = 10-9 meter) or trillionths (picometer: 1pm = 0.000000000001 meter = 10-12 meter). For illustrative purposes we have used length, but we could just as well have used mass or time (or any other fundamental unit of measure for that matter), in which case we would talk of milligrams, milliseconds, nanoseconds, micrograms, etc. For ease of expression of measurement in dealing with larger objects, the meter (or any of the other fundamental units of measure) can be combined in multiples of thousands as kilometers (1 km = 1000 meters = 103 meters), millions as megameters (106 meters) or billions as gigameters (109 meters).
Mass: To continue our illustration of the arbitrary character of these standards of measurement, we look next at mass. The SI unit of mass is the kilogram, which means one thousand grams. A gram is the mass (not weight) of water occupying a volume of one cubic centimeter. First, we should probably discuss the notion that mass is not weight. Recall from Equation 2 (Newton’s Second Law) that force (F) is defined as mass (m) times acceleration (a). Force and weight (Wt) are really the same. What makes them the same is having an acceleration term (a), which for weight is that very specific acceleration namely, acceleration
|Wt=mg Eqn 4|
due to the pull of Earth’s gravity (g). We can therefore rewrite Equation 2 for weight as Equation 4. What then is the difference between weight and mass? The answer must be in the acceleration term or, put another way, gravitational attraction. Mass is really inertia, a property that matter has by virtue of its bulk and also a property that matter has regardless of whether or not there is a gravitational field present. For example in deep space, far from any celestial body, a given volume of matter will have a certain mass by virtue of its bulk. Its weight on the other hand will probably be negligible. There is almost no acceleration applied to the mass because in the absence of any nearby celestial bodies there is practically no gravitational force and thus almost no gravitational acceleration exerted on that matter. Our volume of matter will begin to have weight, but only as it nears some celestial body. Its weight will depend upon the mass of the celestial body it is nearing and the actual distance it is from that celestial body.
But now let us get back to the SI unit of mass. One gram is the mass of one cubic centimeter of water. Why use water? What is the significance of water? It may not come as any surprise to you by now that there is no real significance to the use of water as the basis for our unit of mass. We could just as well have used lead, iron, gold, or even oxygen for that matter. The important thing is that every scientist in the world agrees that one gram is that mass represented by a volume of one cubic centimeter of water. [As a sidelight here, use of water as a basis for our unit of mass can be rationalized somewhat because water is Earth’s most ubiquitous (common or abundant) liquid, it is vital to life and it has the unique property of being able to exist simultaneously as gas, liquid and solid at normal Earth surface conditions.] As a unit of mass, the gram can be subdivided into smaller units such as milligrams (1 mg = 0.001 gm = 10-3 gm), centigrams (1 cg = 0.01 gm = 10-2 gm), etc. It can also be combined into larger units such as megagrams (1 Mg = 1, 000, 000 gm = 106 gm), kilograms (1 kg = 1, 000 gm = 103 gm), etc.
Time and Temperature: Time in SI is represented by the second. The second has been precisely defined as the time elapsing during several thousand vibrations of a cesium atom heated to a specific temperature. Is this standard arbitrary? Once again the answer is yes. We could have chosen any other other atom or any other temperature at which to record vibrations, or we could have chosen a different number of vibrations, all of which are part of the definition of that period of time known as the second. However, once again scientists world-wide have agreed to this definition and therefore, arbitrary though it is, it is precisely defined and therefore applicable to all scientific time measurements. Temperature is one fundamental unit of measure that is not entirely arbitrary. The size of its unit of measure, the degree, is arbitrary; however, the concept of absolute zero is not. Temperature is a measure of the amount of heat energy possessed by a body. It is actually a representation of the speed of motion or the amount of vibration of a mass’ constituent atoms or molecules. It was noted very early on in studies of physical behavior of matter that as temperature was reduced, molecules or atoms making up substances under study moved more slowly. After numerous experiments concerned with this relationship between temperature and amount of energy, it became possible to construct a graph that showed that a point would be reached at which a body’s thermal energy would be zero and the corresponding temperature was therefore labeled absolute zero (0Â°K). Before these experiments defining absolute zero, temperature was measured by arbitrary reference to freezing and boiling points of water. Why use water? What is the significance of water? Once again, it should not come as any surprise to you that there is no real significance to the use of water as the basis for temperature measure. However arbitrary this may be, once again every scientist and engineer in the world has agreed to accept this standard.
The temperature scale used most commonly throughout the world for everyday temperature measurement is the Celsius scale (formerly known as the centigrade scale). On this scale, there are one hundred degrees between water’s freezing/melting point and its boiling point. Water’s freezing or melting point is considered to be zero degrees (0Â°C) while its boiling point is 100Â°C. There is one other temperature scale whose use is confined almost exclusively to the United States. This is the Fahrenheit scale in which water freezes at 32Â°F and boils at 212Â°F. Conversion from Fahrenheit to Celsius is accomplished via a simple conversion formula: C = (5/9)(F-32).
The Kelvin temperature scale, first proposed by William Thomson, Lord Kelvin in 1848, is the scale used throughout physical science. This temperature scale is useful because its degree readings are proportional the amount of thermal energy possessed by that body. Thus when a body has no thermal energy, its temperature will be absolute zero (0Â°K). This also means that a body at 200Â°K is twice as hot as one at 100Â°K, which cannot be said of the same temperatures if measured on either Celsius or Fahrenheit scales. Another important fact is that there is no such thing as a temperature below absolute zero; absolute zero is the coldest it is possible for any object to get. On a thermometer the size of Kelvin and Celsius degrees are the same with absolute zero or 0Â°K equivalent to -273Â°C. Looked at another way, water freezes/melts at 273Â°K (0Â°C) and boils at 373Â°K (100Â°C).
Derived Units of Measure: Derived units of measure are composed by putting together various combinations of fundamental units. Derived units therefore represent all other units of measure used in physical science. Examples of derived units of measure are density, velocity, acceleration, force, momentum, work, energy, calorie, heat of vaporization or heat of condensation, heat of crystallization or heat of fusion, power, etc.
Using dimensional equivalents mentioned above for fundamental units (i.e., L for length, M for mass, and T for time), facilitates showing how derived units are formed. For example, velocity is often given as mph (miles per hour), kph (kilometers per hour), fps (feet per second), or mps (meters per second). What each of these velocity terms has in common is the fact that they all have as their units of measure length per unit of time, which may be represented dimensionally as LT-1 (or L/T). With this in mind, then dimensionally every velocity measure must be in the form of length per unit time or it cannot be a velocity. This same dimensional analysis could also be done for acceleration. Since acceleration is a rate of change in velocity or change in velocity per unit time then dimensionally, it can be given as LT-2 (or L/T2 which is the same as L/T/T). Density is defined as mass/unit volume. Dimensionally this equates to M/L3 or ML-3. In actual measurement this can be in grams/cubic centimeter (gm/cc) or megagrams per cubic meter (Mg/m3). The calorie, a unit of thermal energy measurement, is defined as the amount of heat energy required to raise the temperature of one gram of water by 1Â°C. (I will not define calorie dimensionally here because it involves too much background.) One last example can be cited, this being derivation of dimensions of a force. Using Newton’s Second Law in which F = ma, we can derive the units for force by substituting into the equation the dimensional equivalents for mass (M) and acceleration (LT-2). This yields the fact that forces (all forces) must have the dimensions of MLT-2. The examples chosen here, velocity, acceleration, density, calorie, force, etc., are all derived units of measure based on combinations of the fundamental units. It should be understood that all remaining possible derived units of measure to be found in physical science consist of various combinations of the seven fundamental units of measure.
What is Measured in Physical Science?
Observation is fundamental to virtually all sciences. Much observation can be quantified and therefore measurement is a necessity in science. Oftentimes without measurement there are no verifiable facts nor are there experimental outcomes and thus there is nothing to replicate.
As mentioned above, the universe can be described in terms of its measurable fundamental aspects, space and time. In addition, the universe is composed of only two measurable quantities. These are energy and matter. The fundamental units permit immediate measure of the universe’s fundamental aspects. Time is a temporal framework within which events occur. It is also a means by which events are sequenced. Length allows for the measure of distance area and volume. Turning attention to measurable quantities of the universe, energy can be thought of as that which allows work to be accomplished or performed. (In physics, energy and work actually have the same units of measure, thus illustrating their equality). Matter can be defined initially as possessing mass and occupying space.
Energy: In trying to understand energy it is important to keep in mind the Laws of Thermodynamics. These laws express mathematically the physical relationships that govern the interplay between energy and work. The First Law of Thermodynamics stated in a pithy non- mathematical way that hopefully will make it easy to remember is: “You never get something for nothing”. This means that for any and all work to be performed anywhere in the universe energy must be available-without energy there can be no work accomplished, period. This also illustrates the equality that was mentioned above between energy and work. The Second Law of Thermodynamics, again stated in a pithy non-mathematical manner, is: “No matter how hard you try you will never break even”. This means that if you have 100 units of energy you will not get 100 units of usable work. What this is really saying then is no machine is 100% efficient, for if this were the case, then perpetual motion would be possible. A corollary to the Laws of Thermodynamics is the Law of Conservation. This law states that energy is neither created nor destroyed under normal circumstances (i.e., in a test tube). Obviously, this statement does not apply to nuclear reactions.
While energy is neither created nor destroyed, it may however, be converted from one form to another. Some forms of energy include mechanical, chemical, electromagnetic, and thermal. Mechanical energy may be either potential energy or kinetic energy. Potential energy is that energy possessed by matter by virtue of its position with respect to some reference level. For example an object held above the ground has a certain amount of potential energy with respect to the ground. The higher the object is above the ground the greater is its potential energy. An object gets its potential energy as a result of work that had to be performed on it to elevate it to its position. Kinetic energy on the other hand is energy possessed by matter by virtue of its motion. Let us use the object that was our example for potential energy. That object has a certain amount of potential energy with respect to the ground. If it is allowed to fall back to Earth, then its potential energy will converted into kinetic energy or energy of motion.
Thermal energy is sensible heat, that is energy that can be measured using some heat sensitive device (e.g., a thermometer). Temperature is therefore a reflection of matter’s thermal energy content. Thermal energy (or matter’s temperature) is actually a measure of the average kinetic energy of matter’s constituent atoms or molecules. At the atomic level, matter’s atoms and molecules are in constant motion. They vibrate while fixed in place or they careen around randomly, colliding with their nearest neighbors. How they move depends upon whether the matter under consideration is a solid or a fluid (liquid or gas), respectively. The speed at which the atoms or molecules vibrate or move depends on their thermal energy.
Electromagnetic energy is that form of energy that connects electricity, magnetism, and also thermal energy via electromagnetic waves. As an example we can use the electromagnetic spectrum which is a continuum of wavelengths from the very longest imaginable (e.g., radio waves) to the very shortest imaginable (such as cosmic rays). The most concrete, tangible example of a portion of the electromagnetic spectrum is the rainbow which is created as visible “white” light is broken up into its constituent components (red, orange, yellow, green, blue, indigo, violet) after passing through a prism or a diffraction grating. Electromagnetic waves require no medium for transfer of their energy. Thus, they are able to travel through the void of interstellar and intergalactic space. The energy level or the energy transmitted by an electromagnetic wave is inversely related to its wavelength (i.e., the shorter the wavelength the higher is its energy level). Regardless of wavelength, all electromagnetic waves travel at the same speed-the speed of light (300,000,000 meters/second, which may also be written as 3.00×108 m/sec).
Thermal energy is transferred from a source (hot) to a sink (cold). The actual transfer of thermal energy may occur by one of three means. In solids, because there is continuous contact between atoms or molecules which are also in fixed positions, thermal energy is transferred by conduction. Metallic substances make the best thermal conductors. They also tend to be good conductors of electricity as well. As heat is applied to one end of, say, a metal rod, then the atoms or molecules are heated and begin to vibrate more. As they vibrate faster, some of their vibrational energy is transferred to atoms or molecules farther away from the immediate heat source. Little by little the thermal energy then will be transferred along the rod. In fluids, liquids or gases, heat is transferred by convection via convection currents. Here, because atoms or molecules of fluid are free to move as they gain more kinetic energy, then a portion of the fluid near the source becomes warmer. Warmer fluids expand and thus become less dense than surrounding fluid. They will therefore rise, transferring heat away from the source. In the absence of any medium, thermal energy can be transferred radiantly as electromagnetic waves traveling at the speed of light.
This mention of thermal energy now allows for an important concept in physics to be introduced namely, absolute zero. This temperature is the lowest temperature that is physically possible to attain in nature. At absolute zero, matter’s constituent atoms or molecules have no thermal energy, therefore they do not move or vibrate. There is one other point about absolute zero that is important to mention here because it illustrates a very significant physical fact. All matter that has thermal energy (i.e., has a temperature above absolute zero) radiates or emits electromagnetic energy. The aggregate amount of energy that is emitted or the energy flux (defined as energy/unit area/unit time) varies directly as temperature. This relationship
|J= εσ T4 Eqn 5|
between matter’s temperature (thermal energy) and amount of emitted electromagnetic energy is reflected in the Stefan-Boltzmann Law (see Equation 5). In this equation, J is energy flux (energy/unit area/unit time), ε is the body’s emissivity, σ is the Stefan-Boltzmann constant (5.67×10-8 Jm-2 0Â°K-4, and T is the body’s absolute temperature. Above, it was noted that wavelength of an electromagnetic wave varied inversely as its energy level (i.e., higher energy is transmitted by shorter wavelengths).
|λ(max)=CT-1 Eqn 6|
This relationship between wavelength (and thus energy transmitted) and matter’s thermal energy is reflected in the Wien Displacement Law (see Equation 6). In this equation λ(max) refers to the wavelength of maximum flux intensity while T is absolute temperature and C is a constant. Wavelength of maximum flux intensity refers to that wavelength most frequently emitted by a body. If a body has a perfectly uniform temperature throughout, then every one of its constituent atoms or molecules will emit the same wavelength. In nature, however, temperature may vary slightly from place to place in a body, and so each atom or molecule may emit slightly different wavelengths. The wavelength of maximum flux intensity can be determined from a plot of the distribution of emitted wavelengths.
Matter: Matter has been defined as having mass and occupying space. It can exist in one of four states. State of matter refers to the condition in which the matter exists. The four states of matter are solid, liquid, gas and plasma. Only the first three states of matter are found on Earth, plasma being found only in stars. Matter is composed of atoms or molecules. Atoms have unique construction depending upon which one of the 92 naturally occurring elements they happen to be, from the lightest element, hydrogen to the heaviest, uranium. By construction I am referring to the number of protons or positively charged subatomic particles in the atom’s nucleus and the number of electrons (negatively charged subatomic particles) in the electron cloud which encircles the nucleus. Protons have a weight which has been set at one atomic mass unit (or amu), while the weight of an electron is very close to zero. Every atom of a particular element will have the same number of protons in its nucleus (and thus electrons in its electron cloud), for this is what makes that element uniquely different from all the other elements. Each element has a certain atomic number, this being the number of protons in the nuclei of its atoms. There is a progression then from the simplest element (hydrogen, chemical symbol H) with only one proton in its nucleus to the heaviest element (uranium, chemical symbol U) with 92 protons in its nucleus. A third subatomic particle is also present in the nuclei of atoms. This particle is the neutron, which has a weight of one amu but no charge. By varying the number of neutrons in the nucleus, various isotopes of an element can be created. The total of protons and neutrons in an atom’s nucleus is known as its atomic weight. All of the elements can be arranged in the form of a table in which their complexity (atomic number and atomic weight) increases from upper left to lower right. This tabular arrangement is known as the Periodic Table. Molecules are compounds or aggregates of atoms that are electrically bound to one another. In a compound, the individual properties that might be possessed by atoms of each element present are lost and an entirely new set of properties results. Molecules of the same compound are identical in composition and structure.
Solids can be defined as matter that retains both its shape as well as its volume. A solid is matter occurring in its lowest energy state. Matter’s atoms or molecules do not have enough thermal energy to break the electrostatic bonds that hold them in position while allowing them to vibrate. They may be held in fixed geometric relations with respect to one another in crystalline solids. Alternatively, they may merely be held together in random order as in an amorphous solid. Liquids are in an intermediate energy state (higher than that of a solid but lower than that of a gas). Liquids retain their volume however, they take on the shape of the container in which they are placed. This occurs because a liquid’s constituent atoms or molecules have enough energy to break the electrostatic bonds which would have held them in place thus making them a solid. Each atom or molecule is free to move about randomly. However, they maintain contact with all their nearest neighbors much as marbles in a bag are free to move with respect to one another. Finally, there is the gaseous state. This is the highest energy state occurring on Earth. Gas can be defined as retaining neither its shape nor its volume. It expands to occupy entirely the full volume of any container in which it is placed. Constituent atoms or molecules of a gas have sufficient energy to break contact with or to break away completely from their nearest neighbors. They exist as discrete particles careening around through mostly empty space. They randomly collide with each other and with the walls of their container. These collisions create the pressure which a gas exerts against the walls of its container. This is due to the fact that the greater the kinetic energy of a gas’ atoms or molecules, the faster they move. The faster they move, the more collisions they will have per unit area per unit time with their container’s walls therefore the greater the pressure the gas exerts. Lastly, plasma the highest energy state in nature. Plasma is not found on Earth, only in stars. It represents the highest energy state in which the nuclei of matter can continue to exist. In a plasma, energy levels are so high that electrons have been stripped off atoms and what remains is an emulsion of electrons (negatively charged) and atomic nuclei (positively charged).
Changes of State or Phase Transformations: A change of state (or phase transformation) refers to conversion of matter from one of its forms or conditions into another form. As examples, a state change is said to occur when water freezes, water evaporates, lead is melted, or oxygen is liquefied. Every time a state change or phase transformation occurs, an exchange of energy must take place. This energy exchange occurs between the matter undergoing the phase transformation and the environment external to the matter (the matter’s surroundings). Where energy is supplied from and where it goes to depends upon the nature of the state transformation. If a phase change is being made from a lower to higher energy state (e.g., solid to liquid or liquid to gas), then energy needed to bring about the transformation is supplied from the environment external to the matter. In other words, the matter’s surroundings loose or gives up energy. Conversely, if a phase change is being made from a higher to a lower energy state (e.g., gas to liquid or liquid to solid) then energy is supplied by matter undergoing the phase change and is therefore returned to the surrounding environment. In other words, the environment external to the matter gains energy. In this way the Law of Conservation is not violated. Energy is neither created nor is it destroyed, it is merely exchanged between different parts of nature. During any phase change, the temperature of matter undergoing transformation remains constant. This constant temperature is either the matter’s melting or freezing temperature or its vaporization (boiling) or condensation temperature. While having different numerical values for every substance, these two temperatures are nevertheless unique constants for each substance. They are in fact part of that array of characteristics that are referred to as a substance’s physical properties. The actual amount of energy that must be supplied to or removed from matter to cause a state change is also a unique characteristic of each substance and therefore is also one of a substance’s physical properties.
Let us examine some simple, every day phase changes to illustrate the points made so far. The matter we will use for illustrative purposes here is water (H2O). We use water because it is ubiquitous on Earth, it is a common substance so everyone knows something about it, and it is necessary for life. Additionally, water has many unique properties because of its molecular construction. We can illustrate the solid to liquid transformation by looking at melting of ice (solid water). This phase change requires that the environment external to the ice provide energy necessary for melting. This energy will be used to break down electrostatic bonds between each water molecule. These bonds hold all ice’s water molecules in fixed rigid positions so that they will form that crystalline solid we call ice. Thermal energy that is provided to ice from the surrounding environment is called heat of fusion. Again, the actual numerical value for heat of fusion is different for every substance and is one of that substance’s many physical properties. For solid water, its heat of fusion is 80 calories/gram (or cal/gm) at 0Â°C (its melting point). If the process just discussed is now reversed so that the state change is from liquid water (at a higher energy state) to its solid form (ice, at a lower energy state), then energy must be removed from the liquid and returned to the surrounding environment. This energy is now referred to as the heat of crystallization. For water, crystallization (or freezing) occurs at 0Â°C (its freezing point), and its heat of crystallization is 80 cal/gm. Taking this one step further, let us look at what happens when liquid water’s temperature is raised to 100Â°C. If at sea level, then at this temperature water will boil. When water boils, even though energy is being added from the surrounding environment, liquid water’s temperature remains constant at 100Â°C. The heat energy being added to boiling water is energy required to make the state change from liquid water to water vapor. This energy, when parceled out to each water molecule, allows those molecules to move so fast that they are able to break away from their nearest neighbors thus leaving lots of empty space between them. No longer in constant physical contact with their neighbors, much of the volume they occupy is now empty space and thus the water now exists as that gas we call water vapor. As a gas, water vapor consists of individual, discrete molecules of water careening around just like the oxygen and nitrogen molecules that make up the atmosphere we breathe. Heat energy used to make this phase transformation is called heat of vaporization late. For water, at 100Â°C, its heat of vaporization is 540 cal/gm. Energy required for this phase change from liquid water to water vapor is large and because the energy must come from water’s surroundings, then the effect of this phase change is to cool the surroundings. When water has evaporated, the heat of vaporization that brought about the state transformation is “locked up” or held in reserve in the now gaseous water molecules. This energy, is referred to as latent heat. To reverse this process and thus lower the energy level of water vapor, water molecules begin to aggregate or “clump” together. This “clumping”, termed condensation, forms water droplet that can be seen with the unaided eye. These water droplets may appear as mist or fog, but the important point is that the droplets are no longer water vapor (gas) but droplets of liquid water. The water molecules are now at a lower energy level than they were as water vapor. When this change of state occurs, energy is released from condensing gas back to the surrounding environment. Heat energy released is known as heat of condensation and for water vapor at 100Â°C it heat of condensation is 540 cal/gm. Because this heat energy is release back into the surrounding environment, condensation is a warming process.
Two final point should be made here. First, water is unique in that it may be turned into water vapor at temperatures well below water’s normal boiling temperature of 100Â°C. Your own experience should inform you of this fact. For example, when you perspire, water is secreted by your body onto the surface of your skin. That perspiration evaporates, or is vaporized, at a temperature well below 100Â°C. What occurs under these circumstances is that your body must provide more than 540 cal/gm to get that moisture to vaporize (i.e. water’s heat of vaporization is dependent upon temperature). At 10Â°C (50Â°F) the heat of vaporization for water is 590 cal/gm, while at 0Â° (32Â°F) it is 600 cal/gm. Since your skin temperature is somewhere around 35Â°C (98Â°F) then the heat of vaporization you need to provide to your perspiration should be between 590 and 540 cal/gm. The physiological significance of perspiration’s evaporation is that because vaporization requires heat to be supplied from the water’s (perspiration’s) surroundings, that heat comes from your body, thus cooling your body down. If you think about it, when do you perspire most? Normally when you have been working hard physically. This physical work produces heat that your body must get rid of, and the way it does so is via perspiration. The second point that should be made here concerns boiling. Just what occurs during the boiling process? First it is important to know that boiling is more than just a function of temperature. We tend to think of water boiling at 100Â°C, however, if you have ever been backpacking high in the mountains and tried to cook a three-minute egg you find it is still raw or only partially cooked after three minutes. But, you note, the water has been boiling. Then why isn’t the egg cooked? The answer lies in the fact that, yes, boiling is a function of water obtaining its heat of vaporization but atmospheric pressure also plays a role in the boiling process. Boiling actually occurs when vapor pressure (pressure caused by water gas) in an incipient vapor bubble equals ambient pressure. Once this is attained, then boiling commences. To try to visualize this, imagine a small volume of liquid water. It consists of millions of water molecules all moving randomly but nevertheless each molecule is still in physical contact with each of its nearest neighbors. This might be envisioned by imagining a bag of marbles. If each marble represents a water molecule then they are all in contact with their nearest neighbors but still free to move if you change the shape of the bag. As this small volume of water is heated, added heat energy is parceled out to each water molecule. Those nearest the heat source will generally receive more heat than those further away. If several molecules in close proximity receive that quantity of heat that will increase their kinetic energy to the point that they can move so fast that they break away from all their nearest neighbors then a vapor bubble can form. When the pressure inside this incipient vapor bubble is equal to ambient pressure (i.e., pressure external to the boiling water) then boiling commences. Because this vapor bubble is less dense than its surrounding water, it rises to the surface and bursts, releasing its water gas. A vapor bubble rises to the surface of boiling water for exactly the same reason that a helium filled balloon rises when you let it go. The explanation for this buoyancy is summed up by Archimedes Principle, which states that a body immersed in a fluid is buoyed up (pushed up) with a force equal to the weight of the volume of fluid displaced.
Types of Measurement
When scientists observe phenomena, they generally try to do it quantitatively. It is through quantitative measurement of attributes (as opposed, say, to qualitative description), that observations made by one scientist can be replicated by others in their ongoing attempt to disprove or check on the work of their fellow scientists. Measurement can reveal much about the phenomena under examination. The actual type of measurement that can be made on a particular attribute of a phenomenon also may limit the kind of mathematical analysis that can ultimately be accomplished with the data collected. This in turn may limit or qualify conclusions that may be drawn from a study or experiment.
Nominal: The nominal type of measurement is the weakest that can be employed for it severely limits the nature of statistical analysis that can be made with data obtained. It is not a type of measure that is necessarily chosen by an investigator; however, it may be the only one available or possible. Nominal types of measures are necessitated when all that can be accomplished quantitatively is the counting of objects. In census counts of rock types, minerals present in rocks, insect types, tree types, colors, shapes, etc., nominal measurement is all that can be employed. In using this type of measurement, one would specify ahead of time the classes of phenomena to be counted. For example, one might specify the various types of rocks to be counted or the classes of insects or plants to be enumerated. Once counting is accomplished, then tallies of each class can be made and that which has the highest population or greatest frequency specified. In statistics, that measure of central tendency of a data set (a sample) which is the most frequently observed or that class having the highest total count is known as the mode, modal value, or modal class. With nominal scales of measurement, the only statistically significant piece of information obtainable is modal class. Not much else can be done with such data mathematically and therefore it is the weakest type of measurement. For this reason nominal measurement is to be avoided if at all possible when creating a research design.
Ordinal: Ordinal scales of measure are employed when all that can be accomplished quantitatively is a ranking of phenomena. In ranking, sometimes actual numerical intervals between various ranks are unknown. Probably the best example of an ordinal scale of measurement is found in geology. In the study of minerals, one of the physical properties that often must be determined before a mineral can be identified with any certainty is its hardness. The scale that is used for ranking hardness is known as Mohs Scale. This scale ranks minerals on a range from one to ten, with the mineral talc having a hardness of one (talc being extremely soft) up to diamond which has a hardness of ten (diamond being the hardest of naturally occurring minerals). Being able to array objects or phenomena in an orderly manner allows one the determine the mid-point in a collection of data. In statistics, the middle class, middle point or middle value in a spread of data is known as the median. With an ordinal set of data then, median is all that can be determined.
Interval: Interval types of measure are warranted when data collected can be graphed because the interval between values is known and measurable. Interval data permit virtually all statistical techniques to be used (provided that all other necessary statistical qualifications are met). Data measured can be tabulated and mode and median values determined. Additionally, two other powerful statistical measures can be calculated, the sample mean or average value and standard deviation. The latter is a measure of the spread or dispersion of the data about the mean value.
Ratio: Ratio types of measures are the most powerful and the ones ideally to be aimed for when establishing a research design or setting up an experiment. With ratio measurement all of the measures of central tendency (mean, median, mode) and dispersion (standard deviation) can be calculated. In ratio measurement, not only are order and interval known but there is also an absolute zero value. This means that significant comparisons can be made between various observations. For example, using Kelvin temperature scale it becomes possible to specify that 200Â°K is twice as hot as 100Â°K. The same cannot be said using the Celsius temperature scale because the zero value on such a scale is arbitrarily set at the freezing or melting point of water.
After measurements have been made in the field, say, on some particular phenomena of interest, then a scientist returns to the laboratory or office for the next step in investigation. In a laboratory, a scientist may attempt to replicate what has been observed in nature through a series of experiments. Under controlled conditions, a scientist is better able to observe relationships by eliminating confusion introduced by interactions between many variables some of which may play little or no part in the phenomena of interest. After performing experiments, the data acquired in the laboratory can be compared with the “raw data” collected in the field to see how close the two data sets happen to be. If they are widely divergent, then most likely the scientist eliminated in his or her experiment one or more of the variables that happen to be important but for the sake of the experiment were not initially assumed to be significant. Under these circumstances experiments must be repeated this time incorporating the eliminated variables. When experiments are finally, completed data sets are once again compared. If they do not diverge or are not “too” divergent, then they can be examined for possible mathematical relationships. Alternatively, a scientist may return to the office directly from the field and begin to examine immediately the “raw data” just collected for possible mathematical relationships.
In either of these examples a scientist will be looking for relationships that may be expected to exist between various variables. One way of examining data to see if there are relationships between variables is to begin by plotting data on graph paper. The type of graph paper chosen plays a role in helping a scientist determine what types of mathematical relationships may be inferred to exist between variables measured.
Constant interval: A constant interval scale can be illustrated by a meter stick or a yard stick. It is marked off in equal units of length (either centimeters and millimeters or inches). The integers 1, 2, 3, …., n are all equally spaced apart along the length of the stick. This type of scale with equal units throughout its length is the simplest form of scale that can be used for measuring or plotting. A constant interval scale is also known as an arithmetic scale. If you used a set of dividers to measure off equal units of length along a constant interval scale, you would find that there is a constant difference between values encountered. For example, if you set your dividers initially from 0 to 3 and then walk them along the length of your measuring stick you would read 3, then 6, then 9, then 12, and so on. The difference between 12 and 9 is 3, that between 9 and 6 is also 3, hence the term constant difference. Constant difference is one of the properties of an arithmetic progression and constant interval scales are examples of arithmetic progressions.
Constant ratio: A constant ratio scale is one in which units are not marked off in equal distances apart. A constant ratio scale is also known as a logarithmic scale. Spacing between integers is variable, decreasing very rapidly as numbers increase in value. On a constant ratio scale, equal units of length maintain a constant ratio with one another. If you use your set of dividers again, this time to measure off equal units of length along a constant ratio scale, you would find that between values encountered there would be a constant ratio. For example, if you were to set your dividers initially from 1 to 3 and then walk them along the length of your measuring stick you would first read 3 as the highest number, then 9, then 27, then 81, and so on; even though your divider spacing is not changed. Constant ratio is one of the properties of a geometric progression and therefore a constant ratio scale is an example of a geometric progression.
Figure 1: Example of arithmetic
graph paper and the two possible
arithmetic mathematical models.
Graph paper and graphing: Plotting data obtained from field observation or laboratory experiment on graph paper is one of the visual means of examining the data to see whether or not there are relationships between the measured variables. This is done by taking variables and plotting them in bivariate (two variable) combinations. In bivariate relationships, one variable is either known or assumed to be a causal variable while the second is then known or assumed to be the effect. The cause variable is the independent variable because its values vary independently. The effect variable is the dependent variable because its values depend upon those of the independent variable. By convention, independent variables are plotted on the horizontal or x-axis which is also known as the abscissa. Also by convention, dependent variables are plotted on the vertical or y-axis which is termed the ordinate.
Figure 2: Example of semi-logarithmic
graph paper and the two possible
exponential mathematical models.
We have discussed above the two types of graphic measurement scales-constant interval and constant ratio. These two scales can now be combined to form various types of graph paper. The most common type of graph paper and the simplest to understand is one that utilizes constant interval scales in both the vertical and horizontal directions (see Figure 1). This type of graph paper is known as arithmetic graph paper. A second type of graph paper utilizes a constant interval scale for the horizontal or x-axis while employing a constant ratio scale for the vertical or y-axis. With this graph, paper the independent variable changes in a constant
interval manner while the dependent variable changes in a constant ratio manner (see Figure 2). This type of graph paper is known as semi-logarithmic graph paper. We could reverse the axes on semi-logarithmic paper and thereby have the vertical axis as constant interval while the horizontal is constant ratio. This third type of paper is known as logarithmic graph paper. You do not need to purchase special paper for this, merely rotate semi-logarithmic paper 90Â°. A fourth type of graph paper is available in which both vertical and horizontal axes are constant ratio scales. This type of paper is referred to as log-log graph paper (see Figure 3). These four types of graph paper described and illustrated here are not the only types available. There are numerous others that have been designed to serve special needs. These four however, are probably the most frequently employed.
Figure 3: Example of log-log graph paper
and the two possible power function
A straight line on a graph – its significance: We know from elementary geometry that a straight line is determined by two points. So ideally, when an investigator plots data on graph paper, he or she is looking for the collection of data points to line up as if they all set on a straight line. If data points do line up in a straight line, then the type of graph paper that the data are plotted on tells the investigator what type of mathematical relationship the variables follow. For example, if the plotted data are linear on arithmetic graph paper, then the relationship is an arithmetic one (see section below). If however, the data are plotted on arithmetic paper and the plot is curvilinear, then the variables are not related arithmetically. The next step would then be to plot the same data on semi-logarithmic graph paper. If the plot is linear on the semi-logarithmic paper, then the relationship is an exponential one (see section below). However, if the data once again follow a curvilinear plot on semi-logarithmic paper then the relationship is not exponential. The last step would be to plot the data on log-log graph paper. If they plot as a straight line, then the
General: Mathematics, in the words of Galileo “is the language of nature” and therefore it generally forms the basis for much of the physical sciences. Physical sciences in turn form much of the basis for explanation in historical sciences. In this way mathematics is fundamental in one way or another to virtually all sciences. Once data are collected and graphed in order to determine the nature of their mathematical relationship (see Figures 1 to 3), then they are often put into some type of mathematical form (i.e., an equation). If data are grouped two variables at a time, then their initial form is known as bivariate. In writing an equation which suggests the relationship between two variables, the mathematical forms discussed above are employed. These mathematical relationships between variables can be direct or inverse, arithmetic, exponential, logarithmic or power function.
The reason for producing an equation from data in the first place is to facilitate prediction. One of science’s most important tasks, prediction is accomplished using an equation by substituting various values for the independent variable and calculating (i.e., predicting) a corresponding value of the dependent variable. This mathematical technique only applies however, within the range of the data for the variables as measured from field observations or from experiments.
Data derived from field observations or even the most carefully controlled laboratory experiments will exhibit a certain amount of variability. This variability is manifested in a scatter of data points that appears when data are first plotted on graph paper. Recall from discussions above that we are interested in straight-line relationships because a straight line on a particular type of graph paper tells us the kind of mathematical relationship existing between variables. This variation. as already mentioned, has three sources. They are operator error, instrumental error and probably the most important source of variability is that which is inherent in nature. Natural variability arises for many reasons, however, a major cause is probably due to the fact that we are looking only at a bivariate (or even trivariate, or four variable or higher) relationships. In actuality there are many variables which interact to give us any natural phenomenon and so by looking only at two or three or so variables, and in essence ignoring the rest, we are not “seeing the whole picture” but merely a cartoon or approximation of this phenomenon. Seeing only the approximation is the source of much or our error which we look at as variability inherent in nature. If it were possible to know, define, and then measure absolutely every variable involved in any natural phenomenon, then this natural variability would disappear. Our inability to do this means that nature’s maddeningly perverse variability will always be with us.
We can deal with this variability by treating data statistically. Statistical treatment means using techniques that enable us to examine a plot of scattered data points on a graph, still produce an equation that shows the mathematical relationship between the variables, and use this equation to predict values of the dependent variable from known values of the independent. Now, it is not my intent in this short essay to teach a course in statistics nor is it my intent to write a book on statistics. What I want to do here is define some terms and examine some aspects of the philosophy that allows us to treat data statistically. If you have had an applied statistics course, this should be a review. If you have had no statistics, then consider this but the barest of introductions.
Mathematical Relationships: Using the graphs in Figures 1 through 3, three different mathematical models or equations of specific form can be written. These equations express the mathematical relationship between the independent and dependent variables. Figure 1 depicts two possible arithmetic expressions on arithmetic graph paper.
|Y= aX+b Eqn 7a Y= -aX+b Eqn 7b|
One relationship is direct (Line 1) while the other is inverse (Line 2). Equations for these two arithmetic relationships are found as Equations 7a and 7b. In these two very simple equations (and in all other equations that follow), X is the independent variable, Y the dependent variable, a is the mathematical slope of the line, and b is the Y-axis intercept. The mathematical slope of the line is found by looking at the rate of change, symbolized by the Greek letter Δ, in Y with respect to X (i.e., ΔY/ΔX = a). Where a is positive (a > 0) then a direct relationship is indicated. When a is negative (a < 0) the relationship is inverse. The Y-intercept is the value Y becomes when X is zero.
Figure 2 shows semi-logarithmic graph paper the two possible exponential relationships. Again, Line 1 indicates a direct relationship while Line 2, an inverse. The mathematical expression for an exponential relationship has
|Y= beaX Eqn 8a InY= In b + aX
Y= be-aX Eqn 8b InY= In b – aX
the form of Equations 8a and 8b. The symbols used in these two equations have the same meaning as those in Equations 6a and 6b. The only addition is the symbol e which is the base of the natural (Napier) logarithms (e = 2.17..).
Lastly, the two lines on the log-log graph paper in
|Y= bXa Eqn 9a InY= In b + a InX
Y= bX-a Eqn 9b InY= In b – a InX
Figure 3 are those for the two possible mathematical expressions of power functions. The form of a power function is given by Equations 9a and 9b. Once again, all symbols are as previously defined. Since there is no zero on a logarithmic scale, then Y-intercept, b, is the value the dependent variable assumes when the independent variable is unity (i.e., X = 1).
Deterministic versus Statistical Approaches: In my early mathematics courses, mathematical models were developed to express various bivariate relationships in a deterministic manner. Mathematical determinism means that for each value of the independent variable there is one and only value for a dependent variable. In higher order equations, the number of possible values for a dependent variable increases; however, the number of possible values are fixed, finite in number and this is known ahead of time. If we were to try this deterministic approach to describing nature and natural phenomena we find very quickly that it is far too simplistic. We already know that nature is inherently variable and thus there exists an indeterminate range of values for the dependent variable for each value of the independent variable. This necessitates a statistical approach. What mathematical determinism does do however, is provide a foundation, a beginning point that can be used to build on in order to develop a statistical approach.
Statistical Approach: The most significant step that must be accomplished in treating data mathematically actually begins well before obtaining the data. This first step is development of a research design. The design of one’s experiment, the how of data gathering, requires thoughtful consideration of the step-by-step procedures that will be used. Protocol is a term sometimes used for research design, especially in experiments involving humans or other animals. As part of a research design, equipment employed must also be considered and any necessary operational definitions must be worked out. An operational definition means defining a variable and then specifying exactly how it will be measured. This ensures that data will be internally consistent for the experiment. Additionally, it means that a data set will be capable of replication by other scientists if they know exactly how it was obtained.
Another part of a research design is is the sampling plan. This plan lays out exactly how a scientist is to acquire data and at the same time insure that the sampled data is random. In randomly sampling for data collection a scientist is trying to eliminate as much bias as is humanly possible. This should allow eventual mathematical manipulation of data to provide as true a picture of reality as is possible.
Once data are finally collected, the first step in statistical procedure is to look at the data’s distribution. A distribution in statistics, in extremely simplistic terms, refers to the shape of a plot of the data collected on a graph of magnitude (actual data values) versus frequency of occurrence of those values. Most of us are acquainted (some even obsessed) with the normal distribution – the bell-shaped curve. There are many other possible frequency distributions for data other than the normal. Those distributions other than normal require different statistical treatment and handling. Before beginning statistical analysis, data distributions should be checked. This can be done using a statistical technique known as a chi-square (Χ2) test. In this test, a data set’s distribution is checked against a specified distribution (usually a normal distribution). The chi- square test permits a determination to be made, within a certain level of significance, that a null hypothesis of either following or not following the specified distribution is rejected or not rejected.
For simplicity’s sake here we will assume that data collected, after being subjected to a chi- square test, are shown to be normally distributed. This will allow us to to use parametric statistics. Parametric statistical techniques are powerful procedures that require a data set whose distribution is normal. Parametric statistics are therefore distribution dependent. Alternatively, if our data set did not show a normal distribution and could not be normalized, then applicable statistical techniques are less powerful and are known as nonparametric statistics. Nonparametric statistical techniques are distribution free, that is to say, they do not depend on the nature of the distribution.
Assuming a normal distribution, the first parametric technique to be used is generally correlation, obtaining what is termed the Pierson product moment correlation coefficient (r). Correlation, a technique that is also found in nonparametric statistics but with a coefficient having a different meaning, measures the degree of statistical association between variables. Pierson’s product moment coefficient has a range from zero to unity, both positive and negative (i.e., -1 < r < +1). Positive values indicate a direct relationship while negative indicate an inverse. A coefficient of unity (positive or negative) indicates perfect relationship. To see how much of the statistical variability in the dependent variable is “explained” by the independent variable, a coefficient of determination (CD) can be calculated by squaring the Pierson product moment coefficient (i.e., CD = r2).
Once degree of association between variables has been ascertained, the next step is to write an equation for the relationship between variables so that we can predict values of the dependent variable for selected values of the independent. The procedure for developing an equation for a data set is termed regression analysis. Basically, in regression analysis what is being calculated from the data set are the values for a and b (slope and Y-intercept, respectively) found in Equations 3 through 5. These values for the regression line’s slope and Y-intercept will give the equation for that line which is a “best fit” to the scatter of data points. By “best fit” is meant that the line will be positioned so as to minimize the scatter of points about it. Once this is accomplished, then the standard error of the estimate can be calculated. The standard error is to a regression line as a standard deviation is to a mean.
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