The Map of Physics Revisited

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    Philosophers give the name Rationalism to the doctrine that holds as true to Reality the idea that our most certain and significant knowledge comes from reason, not from sense experience. Both Descartes and Leibnitz held that every fact in the world entails every other fact and Hegel believed that we could start with the bare concept of being and deduce the whole of the laws of Nature. I believe that the Map of Physics fulfills those beliefs, because it gives us, in essence, a flowchart showing how we can deduce all of the laws of physics from the observation that we exist.

    Imagine the following scenario. Someone has given you an intricately carved bas-relief, handed you a slab of damp clay, and asked you to create in the clay an impression that matches precisely the pattern of the bas-relief. The obvious thing to do is to press the clay against the bas-relief and then to peel it off. If we see the bas-relief as a model of the Universe, then what you have done is analogous to the operations of empirical Science: in devising theories from observations of Reality scientists are, in essence, pressing the clay of mathematics against the bas-relief of the world and its workings. But suppose now that upon being handed the clay you had turned your back on the bas-relief, picked up a stylus, and, like some mad Sumerian scribe, pressed a pattern into the clay as you saw fit. If that pattern turned out to match the bas-relief precisely, then there would be no doubt that you possessed the deepest knowledge and understanding of the bas-relief. Equivalently, if we were to deduce the laws of Nature from the laws of mathematics and some small set of axioms, in much the manner that the Classical Greeks deduced the laws of plane geometry, then we would definitely possess the deepest knowledge and understanding of the Universe.

    We have two fundamental ways in which we can use reason to obtain scientific knowledge: the empirical-inductive process and the axiomatic-deductive process. The first of those, in which laws of nature are inferred from observations of the real world, is what we normally think of as the Scientific Method. That's the method that's analogous to pressing the clay against the bas-relief. The second, which involves the use of formal logic to deduce rules from axioms, is used primarily by mathematicians. That's the method that's analogous to pressing patterns into clay without looking at the bas-relief. The two methods seem to be perfectly separate processes, but the history of physics can actually be seen as a slow evolution of one into the other.

    The Scientific Method actually 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 images of those varieties based largely in our minds on differences in size of the animal and color of its 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 are flightless. 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 accords with 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 the "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 not all phenomena are so easily studied as are birds and other phenomena of the natural world. Some are more subtle and abstract. 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.

    ěrsted was teaching the University's course in electricity in July 1820, when he made his discovery. Too often this incident is presented as a purely accidental discovery, an example of the serendipity to which the Scientific Method opens us. The fact is that 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 it was 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. It was an astonishing discovery and would have been even more so had ěrsted applied a fairly simple analysis to his results and thereby discovered electromagnetic induction as well.

    ěrsted could be 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 his commitment to the empirical-inductive method is a large part of the reason for his success in discovering the magnetic effect of electric currents. What is harder for us to see is that it is also 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 was wrong: it is possible to imagine what Nature does and to be correct in that imagining. The first person to make a significant discovery as a result of such imagining was James Clerk Maxwell.

    In 1861 Maxwell was working with the theories of electricity and magnetism, trying to combine them into a single theory, and he noticed that something seemed to be missing from Ampere's law. Inspired by ěrsted's discovery, Andre Marie Ampere had reasoned that if an electric current could exert a force upon a magnet, then electric currents acted as magnets, so an electric current could also exert a force upon another electric current. To test 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). What the simplified version of Ampere's law tells us (as it told Maxwell) is that if we consider a closed loop and the surface of which it is the boundary, then the net magnetic field calculated around the loop will be 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 magnetic field 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 metal plates that face each other without touching. In that case the electric current will be of the alternating variety, 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 there will be a current flowing in the wire and, consequently, there will be a magnetic field on the circle. Up to this point you have very likely been envisioning the surface bounded by the circle as a flat disc: it is, after all, implicit in our desire to give a simple example of Ampere's law. What Maxwell did was to start with that image and then to imagine so deforming the surface, as if it were made 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 there is still a magnetic field on the circle, but the current no longer penetrates the surface, so Ampere's law is not satisfied.

    Rather than dismiss Ampere's law as false to Reality (Ampere had inferred his law from well-designed experiments, so that option was unavailable), 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 flowing 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. Nonetheless, they 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 can be measured 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 was it that enabled Maxwell to obtain scientific truth from a fantasy? Though it's highly unlikely that he worked out his hypothesis so formally, the fact is that its creation could have been made through classical logic; in particular, through a device known as a syllogism.

    In his "Metaphysics" Aristotle declared that "The firmest of all first principles is that it is impossible for the same thing to belong and not to belong to the same thing at the same time in the same respect." In that proposition Aristotle was stating together the principle of bivalence (every statement is either true or false), also known as the law of the excluded middle, the basis of the syllogism.

    Devised by the Ancient Greeks, the syllogism is 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 two dominoes can only be adjoined where they have identical patterns of dots, so two propositions can only be joined where their terms are the same. 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". The truth of that new proposition is guaranteed by the fact that there is no mutually exclusive alternative to the second proposition; that is, there is 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. The primary example of knowledge obtained by deduction from axioms is Euclid's plane geometry. Starting with a set of self-evident axioms, Euclid showed how to deduce theorems about figures that can be drawn by a person using only a compass and a straightedge. It is a purely mental construct that can be represented in drawings. What Maxwell did was similar, except that he did not begin with self-evident axioms. Ampere's law certainly is not self-evident: it had to be inferred from observations and experiments. Maxwell's method, then, was a kind of hybrid between the empirical-inductive method and the axiomatic-deductive method.

    Can we go beyond the 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 (though I must confess to being haunted by the knowledge that Aristotle actually reasoned much as I have done, laying out several of my own arguments, though he did it in order to deny the existence of the vacuum and thus did not carry the arguments to my conclusions). 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. The theory of Special Relativity was originally obtained 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.

    The technique that Einstein used in his deductions is what 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 the essentials were clear 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 (somewhat in the manner of Faraday's aids to the imagination). In the course of this treatise I have been showing you such fantasies and explaining how they lead to a valid description of the relation between space and time. Following Einstein's lead, I have worked out the rules governing the relations between distance and duration through the use of certain caricatures of familiar situations to aid your imagination: in particular, I asked you to imagine the operations of a railroad in a world in which the speed of light is merely one hundred miles per hour. As improbable as it seems, such fantasies do, indeed, yield valid results.

    From those simple little imaginary experiments we have deduced an ever-widening array of laws and found that, so far as we can tell, they mimic the actual laws of physics perfectly. And in those results we may glimpse faintly Humanity's destiny. I want to banish the black despair from our culture and show that we are not mere accidental scum floating upon Reality, but that we have made ourselves the proper heirs of this Creation. We have transcended evolution and drunk deeply from Mimir's well, so if we can take the next step and deduce the laws of Nature from simple, self-evident axioms, then we are truly little less than a deity in our collective self.

    But then we must confront G÷del's theorem. The Map of Physics is a formal system because it includes the rules of arithmetic. Such a system can be complete or consistent but not both, according to Kurt G÷del; if we declare that our system is consistent, that it contains no contradictions, then it must be incomplete; that is, according to G÷del, it must contain at least one proposition that cannot be proven rationally. Do we thus create a Rationalist physics only to find it subverted by its own rationalism?

Epilogue

    Steven Weinberg, one of the physicists who helped create the Standard Model by showing how electromagnetism and the weak force can be related to each other, once noted that the more the Universe becomes comprehensible the more it seems pointless. In that observation some see the seed of a profound nihilism, a belief that nothing matters because the Universe itself is meaningless.

    We are indeed lost in a vast, cold, indifferent Cosmos. We do seem to be pretty much adrift in a trackless void, so I can't find a basis for disagreeing with Dr. Weinberg. Nonetheless, I don't agree that nihilism is the proper response to the pointlessness of the Universe.

    Our situation reminds me of the Parable of the Talents. The talents in the parable were merely flattish pieces of silver that had patterns stamped onto them; in themselves, they were inherently meaningless. Nonetheless, people chose to give them meaning in order to use them as tools and markers of value. The value was not in the talents themselves, but in what the servants chose to do with them. In the same way the point of the Universe is not in the fact that it exists, but in what we choose to do with it: the Universe is our talent.

    As our scientific knowledge becomes ever more esoteric our misconceptions of it become more subtle in their ability to mislead us. When he devised the theory of General Relativity, Einstein noticed that the equations implied an expansion or contraction of space itself, so he inserted a "cosmological constant" to make the equations describe a steady-state universe. When Edwin Hubble announced a few years later his discovery that the Universe appears to be expanding, Einstein repudiated the cosmological constant as his greatest blunder. Similar misconceptions continue to vex and bedevil scientists.

    Astrophysicist David Spergel provides a good example of what I mean here. In an article on the Microwave Anisotropy Probe he offers only two views of the Universe. The infinite universe he dismisses because it is strange in that every possible arrangement of matter and energy would be repeated an infinite number of times. The alternative, which he hopes to explore via the MAP satellite, is a finite universe in which light emitted early in its history has actually circumnavigated space and come back to its point of origin, possibly more than once. His study group will be examining data from the satellite for patterns that show a repetition that would come from objects seen more than once, as in a hall of mirrors, by virtue of light having gone around the Universe.

    Misconceptions of our current knowledge mislead us in formulating the questions aimed at expanding our knowledge. Empirical science is founded upon the concept of using Reality checks in the form of experiments and I expect, as a consequence, that the results from the MAP satellite will reflect the true shape of the Universe and thereby correct the misconceptions. Such correction of misconceptions is the chief virtue of empirical-inductive science.

    Rationalist science lacks that advantage. I have only succeeded in creating my Map of Physics (insofar as I have been able to do so) by cheating, by using the knowledge gained from the standard empirical-inductive scientific method to guide my construction of an axiomatic-deductive alternative science. I compare my Rationalist work with the solving of a jigsaw puzzle by someone who looks at the picture on the lid of the box. That's fine for creating a Rationalist parallel to current scientific knowledge: in essence I am using the natural corrective of empirical science to correct errors in my Rationalist science. But if I am to extend the Map of Physics into realms of knowledge not yet probed empirically, I will need a new means to correct misconceptions, lest I fail as Descartes did.

    I must have some method of so clarifying thought that I can be certain that I will not be misled by misconceptions, such as "faster than light". Now I have, indeed, come full circle. I have returned to what the Greeks intended natural philosophy to be; that is, I have now proclaimed that our study of Reality is aimed at nothing more nor less than the refinement of the human mind. That, ultimately, gives the Universe the meaning that we crave.

References

Folger, Tim, "The Magnificent Mission", DISCOVER, Volume 21, Number 5, May 2000, Pages 44 - 51.

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