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The way of the abstract

The realm of theoretical physics is teeming with abstract and beautiful concepts, and the process of bringing them into existence, and then explaining them, demands profound creativity according to Giovanni Vignale, author of The Beautiful Invisible: Creativity, imagination, and theoretical physics. In the excerpt below Vignale discusses the beginnings of theoretical physics and the abstract.

Physics, most of us would agree, is the basic science of nature. Its purpose is to discover the laws of the natural world. Do such laws exist? Well, the success of physics at identifying some of them proves, in retrospect, that they do exist. Or, at least, it proves that there are Laws of Physics, which we can safely assume to be Laws of Nature.

Granted, it may be difficult to discern this lofty purpose when all one hears in an introductory course is about flying projectiles and swinging pendulums, strings under tension and beams in equilibrium. But at the beginning of the enterprise there were some truly fundamental questions such as: the nature of matter, the character of the forces that bind it together, the origin of order, the fate of the universe. For centuries humankind had been puzzling over these questions, coming up with metaphysical and fantastic answers. And it stumbled, and it stumbled, until one day—and here I quote the great Austrian writer and ironist, Robert Musil:

. . . it did what every sensible child does after trying to walk too soon; it sat down on the ground, contacting the earth with a most dependable if not very noble part of its anatomy, in short, that part on which one sits. The amazing thing is that the earth showed itself uncommonly receptive, and ever since that moment of contact has allowed men to entice inventions, conveniences, and discoveries out of it in quantities bordering on the miraculous.

This was the beginning of physics and, actually, of all science: an orgy of matter-of-factness after centuries of theology. Careful and systematic observation of reality, coupled with quantitative analysis of data and an egregious indifference to theories that could not be tested by experiment became the hallmark of every serious investigation into the nature of things.

But even as they were busy observing and experimenting, the pioneers of physics did not fail to notice a peculiar feature of their discipline. Namely, they realized that the laws of nature were best expressed in an abstract mathematical language—a language of triangles and circles and limits—which, at first sight, stood almost at odds with the touted matter-of-factness of experimental science. As time went by, it became clear that mathematics was much more than a computational tool: it had a life of its own. Things could be discovered by mathematics. John Adams and, independently, Urbain Le Ferrier, using Newton’s theory of gravity, computed the orbit of Uranus and found that it deviated from the observed one. Rather than giving up, they did another calculation showing that the orbit of Uranus could be explained if there were another planet pulling on Uranus according to Newton’s law of gravity. Such a planet had never been seen, but Adams and Le Ferrier told the astronomers where to look for it. And, lo and behold, the planet—Neptune—was there, waiting to be discovered. That was in 1846.

Even this great achievement pales in comparison with things that happened later. In the 1860s, James Clerk Maxwell trusted mathematics—and not just the results of a calculation, but the abstract structure of a set of equations—to predict the existence of electromagnetic waves. And electromagnetic waves (of which visible light is an example) were controllably produced in the lab shortly afterwards.

In the 1870s Ludwig Boltzmann undertook the task of finding out, by mathematical analysis, how a hypothetical world made of atoms would behave. Nobody had seen an atom, and very few believed seriously in what, at the time, must have looked like a very artificial concept. With the help of a revolutionary mathematical approach in which probability was the main actor, Boltzmann was able to show that his artificial world behaved pretty much like the real world. At least, the behaviour of gases was the same!

These three examples illustrate three different ways of practising the strange kind of science known today as theoretical physics. They are like three different literary genres, such as essay, poem, and novel. In the first, one applies a general theory, summarized in a set of mathematical equations, to the solution of a concrete problem. In the second, one plays with the mathematics to find new equations that are more satisfactory from an intellectual, aesthetical, or practical point of view. Finally in the third way—the Boltzmann way—one constructs an artificial world with building blocks that obey the laws of a previously established theory. Then one tries to find out whether the behaviour of this artificial world matches the behaviour of the real one.

By the early twentieth century theoretical physics had become a well—established science within the science. The two great triumphs of that period—relativity and quantum mechanics—spawned a host of revolutionary concepts such as ‘antimatter’ and ‘black holes’, which were discovered in later experiments and have since become staples of popular scientific literature. No other science, as far as I know, can boast a comparable record of successful predictions. And yet, at the beginning of the twenty-first century theoretical physics stands aloof in the middle of a culture that is deeply suspicious of abstract thought. Indeed, to many people a theory is the exact opposite of a science. The very word ‘theory’ suggests a loss of contact with reality, which in turn evokes a lack of vital strength, of feeling, of generosity—in short of all the virtues that are most prized in a human being.

I must concede that the prejudice against the abstract is not entirely unjustified. All around us we see abstract concepts, laws, and classifications prevail upon basic considerations of humanity. Governments and individuals have been known to commit the worst crimes in the name of abstract ideals. Yet I believe that the main problem in those cases is not the faith in the abstract, but a specific degeneration of the abstract, for which I suggest we use the term formality.

Giovanni Vignale is Curators’ Professor of Physics at the University of Missouri-Columbia. After graduating from the Scuola Normale Superiore in Pisa in 1979 and gaining his Ph.D. at Northwestern University in 1984, he carried out research at the Max-Planck-Institute for Solid State Research in Stuttgart, Germany, and Oak Ridge National Laboratory. He became a Fellow of the American Physical Society in 1997. Professor Vignale’s main areas of research are many-body theory and density-functional theory. You can read more about OUP’s science books here.

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