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We have two fundamental ways in which we can obtain scientific knowledge: the empirical-inductive process and the axiomatic-deductive process. The first of those, in which we infer the laws of nature from observations of the real world, provides us with what we normally think of as the Scientific Method. The second, in which we use formal logic to deduce rules from axioms, seems suitable only for use primarily by mathematicians. We seem to have two perfectly separate processes, one rooted in actual observation of Reality and the other lying in the realm of fantasy, but, in fact, we can conceive the history of physics as a slow evolution of one into the other.

The Scientific Method comes more or less naturally to us, even if scientists took some time to formalize it. We start with language, which we acquire as children and which gives us categories of objects suitable for study. For example, the word "bird" gives us the category of all living things with wings and feathers. As we grow older we become familiar with different varieties of birds, the varieties based largely in our minds on differences in size and color of the feathers. We know of sparrows, robins, pigeons, blue jays, crows, eagles, and others and we know certain facts about them. We know, for example, that they all fly. Indeed, flight is the chief connotation of the word bird and we are thus inclined to think that all birds can fly. But then we remember that chickens and turkeys have wings and feathers and that farmers can keep them in open pens precisely because they can't fly. We may even be familiar, through travelogues, with ostriches, penguins, kiwis, and other exotic birds that can't fly. We thus modify our thought and think that some birds can fly and some can't. On the other hand, we know something else about birds: we know that they reproduce their kind by laying and incubating hard-shelled eggs. Robins do it, pigeons do it, ducks do it, and hawks do it, but chickens and turkeys do it too, as do ostriches, penguins, and kiwis. Indeed, no one has ever reported a bird that does not lay eggs, so we feel justified in asserting that all birds lay eggs.

As we learned in grammar school, the Scientific Method consists of discerning a pattern in some set of observations, accepting that pattern as an hypothesis, making further observations in order to test the hypothesis, and then drawing conclusions based upon the results obtained from those new observations. Thus, flight is the pattern that we discern in our observations of common birds, so we hypothesize that all birds fly. Further observations make us aware of chickens and other flightless birds, leading us to the conclusion that our hypothesis failed the test and that the statement that all birds fly is false to Reality. On the other hand, our hypothesis about the eggs passed the test and we conclude that the statement that all birds lay eggs is true to Reality.

That procedure conforms to the ideas presented by Francis Bacon at the beginning of the Seventeenth Century. Bacon expressed his main idea, that "we are not to imagine or suppose, but to discover, what Nature does or may be made to do", in his science-fiction novel "The New Atlantis". His description of a technologically advanced society on the previously undiscovered island of "Bensalem" led directly to the formation of England's Royal Society, which was, and still is, meant to be a repository for scientific knowledge like "Solomon's House" that Bacon conceived for his fictional Bensalemite Merchants of Light. It was Bacon, after all, who promoted science in the most effective way possible by telling us that "human knowledge and human power come to the same thing".

But we cannot study all phenomena as easily as we study birds and other phenomena of the natural world. Reality offers us more subtle and abstract phenomena. How can we apply the Scientific Method to them? If we cannot find the observations that we want, then we must contrive them; that is, we must design and perform an experiment. For me the best example of that modification of the Baconian Scientific Method was provided by Hans Christian ěrsted, a professor of physics at the University of Copenhagen, in his discovery of the relation between electric current and magnetic force.

(Note: pronounced correctly, the Danish/Norwegian letter ě sounds like the "o" in neighbor.)

ěrsted was teaching the University's course in electricity in July 1820, when he made his discovery. Too often writers present this incident as a purely accidental discovery, an example of the serendipity to which the Scientific Method opens us. In fact ěrsted carefully contrived his experiment, basing it upon a well-conceived hypothesis. Prior to conducting his experiment, he reminded his students of Benjamin Franklin's discovery, some seventy years earlier, that lightning is a kind of electric spark. Further, he reminded them of travelers' reports that in the presence of thunderstorms the needles of their compasses twisted and turned erratically. From those facts he had inferred that some feature of thunderstorms produced a magnetic force and he had guessed that the force came from the electric current and the incandescence associated with lightning that comprised that feature. (Later experiments would eliminate the incandescence from ěrsted's hypothesis, leaving only the electric current.) Thus, he contrived to bring only those features into his classroom, leaving out the thunder, the wind, and the rain. He passed the electric current from a voltaic pile through a thin platinum wire and then he and his students watched as the needle of a compass that he had placed under the wire swung around and settled into an orientation pointing perpendicular to the direction in which the wire ran. The discovery astonished people and it would have astounded them even more had ěrsted applied a fairly simple analysis to his results (see either Appendix V or ěrsted's Missed Opportunity in The Map of Physics) and thereby discovered electromagnetic induction as well.

ěrsted shows us the very stereotype of the serious scientist. He certainly followed Bacon's admonition and refused to imagine or to suppose what Nature does. In essence he interrogated Nature directly, through observation and experiment. We can see readily that a large part of his success in discovering the magnetic effect of electric currents came from his commitment to the empirical-inductive method. We have more difficulty seeing that commitment as the reason behind his failure to discover electromagnetic induction, which Michael Faraday actually discovered by accident in August 1831. As strange as it may seem, Bacon had it wrong: we can, indeed, imagine, even at the levels farthest removed from our experience of the world, what Nature does and come to a correct understanding of Reality in that imagining. Some forty plus years after ěrsted made his discovery, James Clerk Maxwell became the first person to make a significant discovery as a result of such imagining.

In 1861, as Maxwell strove to combine the theories of electricity and magnetism into a single theory, he noticed that Ampere's law seemed to lack something. In 1821, inspired by ěrsted's discovery, Andre Marie Ampere had reasoned that if an electric current could exert a magnetic force, then it would also respond to a magnetic force, so two electric currents would thus exert magnetic forces each upon the other. On the basis of that idea he carried out elaborate extensions of ěrsted's experiment to determine the mathematical description of the forces exerted between currents as well as between a current and a magnet. By the time the results came down to Maxwell, in the 1850's, Faraday's concept of the forcefield (one of his "aids to the imagination") had enabled physicists to replace the clumsier description of forces between currents with a simpler description of the magnetic field generated by one or more currents (as a kind of prelude to calculating the force that the said current or currents would exert upon some other current or currents). The simplified version of Ampere's law tells us (as it told Maxwell) that if we consider a closed loop and the surface that it bounds, then the net magnetic-field-times-distance product calculated around the loop will come out proportional to the net amount of electric current passing through the surface.

Maxwell's problem came from the simplest manifestation to which that law applies. He imagined a long straight wire passing through the center of a circle. If a current flows through the wire, then the product of magnetic field and distance calculated on the circle has a very simple mathematical description. This fact remains true to Reality even if we do as Maxwell did and imagine cutting the wire at some distance from the center of the circle and attaching to the broken ends of the wire metal plates that face each other without touching. In that case we will have an electric current of the alternating variety, one sloshing to and fro in the wire as electric charge builds up on the metal plates and then forces its way back down the wire. Nonetheless, at some instant we will have a current flowing in the wire and, consequently, a magnetic field on the circle. Up to this point you have very likely envisioned the surface bounded by the circle as a flat disc: after all, our desire to give a simple example of Ampere's law implies the simplest geometric form. Maxwell started with that simple image and then imagined so deforming the surface, as if he had made it of rubber or soap film, that it passed through the gap between the metal plates and, thus, did not touch the wire at all. In that altered image he still had a magnetic field on the circle, but the electric current no longer penetrated the surface, so the situation no longer satisfied Ampere's law.

Rather than dismiss Ampere's law as false to Reality (competent scientists had inferred that law from well-designed experiments, so he didn't have that option available to him), Maxwell modified it by adding a new term to the electric current in the mathematical description. He noticed that as the electric charges flowed into and ebbed from the metal plates the electric field that they generated in the gap between the plates changed, strengthening or weakening in time with the pulse of the current. Indeed, the rate of change in the electric flux across the gap was, at any instant, proportional to the electric current in the wire, so Maxwell added that rate of change to the electric current in Ampere's law.

Neither Maxwell nor any of his contemporaries ever carried out in a laboratory the experiment that Maxwell had conceived. Using only his imagination, Maxwell discovered a thitherto unsuspected fact about electromagnetism. Nonetheless, for all it's fantastic origin, scientists used the result of that imaginary experiment in their theorizing. Maxwell combined his version of Ampere's law with Faraday's law of electromagnetic induction and came up with an equation that describes electric and/or magnetic fields propagating through space in the manner of waves. Further, that equation contains a formula for calculating the speed of propagation in terms of electrostatic and magnetostatic quantities that scientists could and did measure in the laboratory: the calculated speed, as Maxwell put it, "agrees so exactly with the velocity of light calculated from the optical experiments of M. (Monsieur Armande Hippolyte Louis) Fizeau that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." That was strong evidence in favor of Maxwell's hypothesis, but the real (in Baconian doctrine) proof and verification came in October of 1886, when Heinrich Hertz first succeeded in generating and detecting electro-magnetic waves in his laboratory.

What enabled Maxwell to obtain scientific truth from a fantasy? Though he most likely did not work out his hypothesis so formally, he could have made his discovery through classical logic; in particular, through a device known as a syllogism. Devised by the Ancient Greeks, the syllogism gives us a means of reasoning that resembles a game of dominoes. Just as each domino displays two patterns of dots, so each logical statement (or proposition) connects a subject to a predicate, thereby displaying two terms. And just as we can legitimately adjoin two dominoes only where they have identical patterns of dots, so we can only adjoin two propositions where their terms match each other identically. Thus the classical example of a syllogism (two propositions joined to produce a third proposition) gives us the first domino as "Sokrates is a man", the second domino as "all men are mortal", and yields the third domino as "Sokrates is mortal". We have a guarantee of the truth of that new proposition from the fact that we have no mutually exclusive alternative to the second proposition; that is, we have no domino that says "some men are immortal". As long as we have that kind of exclusively true statement, one that uses words like "all" or "only" or "none", we can use the syllogism to generate a new true statement.

So look again at Maxwell's reasoning. First, he removed from his imagination everything except what was absolutely necessary to his experiment; a broken wire in empty space. Next he established his first domino, a simple statement of Ampere's law: this magnetic field is generated by whatever penetrates this surface. Then he moved the surface and generated his second domino: this surface is being penetrated only by the flux of a changing electric field. His third domino, the conclusion of the syllogism, was thus: this magnetic field is being generated by the flux of a changing electric field. Finally, he added the generalization of that statement to Ampere's law.

But the syllogism belongs to the axiomatic-deductive method of obtaining knowledge. As our primary example of knowledge obtained by deduction from axioms we have Euclid's plane geometry. Starting with a set of self-evident axioms, Euclid showed Humanity how to deduce theorems about figures that a person can draw using only a compass and a straightedge. Euclidean geometry gives us a purely mental construct that we can represent in drawings. Maxwell did something similar, except that he did not begin with self-evident axioms. Ampere's law certainly is not self-evident: scientists had to infer it from observations and experiments. Maxwell's method, then, comprised a kind of hybrid between the empirical-inductive method and the axiomatic-deductive method.

Can we go beyond that hybrid and use the pure axiomatic-deductive method to obtain knowledge that turns out to be true to Reality; that is, can we figure out the laws of Nature without looking at Nature? The Greeks believed that we can, but they failed to develop a physics that's true to Reality. In the first half of the Seventeenth Century Rene Descartes tried again and he also failed. But we have much more scientific knowledge today than the Greeks or Descartes had. We are like the fellow assembling a jigsaw puzzle while looking at the picture on the box's lid: we can cheat more effectively. And because we can cheat, we know that the answer to the question is yes. Albert Einstein obtained the theory of Special Relativity through an almost purely Rationalist method. True, Einstein's two postulates were still not fully self-evident axioms, but neither were they obtained from observation or experiment. And from those postulates Einstein, like a modern Euclid, deduced all of the theorems pertaining to the science of uniform relative motion.

In his deductions Einstein used a technique that he called a Gedanken-experiment (a thought experiment), a curiously whimsical method seemingly nothing like Euclid's. In a typical Gedanken-experiment Einstein would make certain that he had the essentials of his subject clear in his mind and then he would wrap the experiment in some fanciful imagery. He would imagine passengers on a train traveling near the speed of light trying to set their watches, for example, and he would deduce from this little skit the way in which relative motion distorts time. Apparently the introduction of familiar images in fairy-tale form eased the reasoning without invalidating its result. In the course of this series of essays I will show you such fantasies and explain how they lead to a valid description of the relation between space and time.


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