Crease: The Great Equations
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Robert P. Crease, The Great Equations : Breakthroughs in Science from Pythagoras to Heisenberg. New York : W.W.Norton & Company, 2008. 315 pages; illustrated; with notes and index.
Relax. This is not a mathematical textbook, nor a mathematical explication of the “great equations” of science and mathematics. Rather they are essays that consider the place of these great equations in history: the intellectual milieu that saw them come into existence, their scientific impact and, through that, their effect on culture and their cultural meaning. Yes, equations can have cultural meaning: for better or worse the Pythagorean theorem, Einstein’s mass-energy equations, and Heisenberg’s Uncertainty Principle all have cultural meaning.
There are other less popularly known equations that are still important. Newton’s laws of motion, Newton’s law of universal gravitation, Euler’s equation, the second law of thermodynamics, Maxwell’s equations of electrodynamics, Einstein’s field equation for general relativity, and Schroedinger’s wave equation for quantum mechanics. Even when written down by one person, these equations represent a coming together of scientific currents of their times, and each one has an interesting story to tell.
Crease tells the stories well, and it’s an important task. Science is too often seen as something outside culture, something culturally other. That’s a big mistake, no matter how frequently it’s made.
Simon Schama's book History of Britain, at 1,500 pages, is a solid history of that country, and has been made the basis for a multipart documentary film. Yet the book contains no mention of James Clerk Maxwell, nor any mention of the role that this scientist played in laying the foundation for electrification, light, heat, communication, and the electronics revolution of the twentieth century, in Britain or elsewhere. Schama's book omits any references to the contributions made by British scientists and engineers to transforming Britain and the world.
The neglect of science, indeed, is common in history books—most disturbingly, even in books that profess to care about the masses, and oppressed and underprivileged peoples. Since it was first published in 1980, for instance, A People's History of the United States, by Howard Zinn, has sold over a million copies and become one of the most influential works of history in the U.S. A popular textbook in schools and colleges, it claims to focus on the "hidden episodes of the past when, even if in brief flashes, people showed their ability to resist, to join together, occasionally to win."
However, Zinn's book makes no mention of people resisting, joining together, and winning when it comes to science. It says nothing, for instance, of the struggles to reduce childhood mortality, to increase life expectancy, or to develop systems of mass transportation. There is no mention of Norman Borlaug, who won the 1970 Nobel Peace Prize for leading the "green revolution," and who helped end hunger for millions of people. Another no-show is the microbiologist Maurice Hilleman, whose vaccines saved more lives than were lost in all the wars to which Zinn devotes chapters. [pp. 152—153]
It’s not by any means a mathematical book, but Crease does display each equation at the head of each chapter in mathematical symbols. That’s obviously important to a book about equations, but I think it’s also very good for the general, non-technical reader. One doesn’t need to understand the equations mathematically, but there’s value in simply seeing what they look like. True, Crease sometimes explains to some extent what an equation means, but understanding an equation in mathematical terms is not his goal. Rather, he would like his readers to appreciate each equation and to have some idea what the equation is expressing.
There is even an aspect of aesthetic appreciation involved. That may sound silly, but mathematicians and mathematical physicists will tell you that correct equations have a certain "beauty" to them. Looking at these equations in that way gives some non-verbal, non-expository feeling of what it's like to work with mathematical expressions.
I appreciated Crease's many insights. For instance, I was taught, like many other physics students, that Heisenberg's uncertainty relations told us something about measurements disturbing the measured. This is ingrained in the popular understanding, too ("the observer disturbs the universe"), but it's wrong. The real implication is deeper and much more interesting.
The uncertainty principle is incomplete in a different sense. It is a mathematical relation, and a feature of the statistical interpretation of the wave function in quantum mechanics. It makes no reference to any underlying physical picture; there are no references to waves or particles, nor to physical experiments. It is not obvious what it refers to, except possibly the clicks of a detector. yet it is about gaps in the world itself. These gaps are not epistemological but ontological; having to do not with our knowledge but with the world.
This is strange, but why? It is important to see what the strangeness is not due to. The strangeness of the uncertainty principle is not due to the measurement process disturbing the object measured, which would be a feature of any Newtonian theory involving exchange of particles. Nor is it due to the presence of statistics. Rather, the strangeness of quantum mechanics is that quantum formulations are not "about" a real or ideal object in the conventional sense. [p. 264]
As an excerpt that may seem a bit cryptic, but as part of his larger narrative I felt that it was an unusually clear laying out of the "meaning" of the uncertainty relations.
Each of the chapters, one each to an equation, is followed by an "interlude", a rumination on some topic that came up in the chapter and deserves its own essay. The remarks quoted above about the lack of science in history books came from such an interlude. So does the following about how science works, and how all the bits of science fit together to create a model of something that we call "reality".
Yet another thing we learn from these journeys is about the nature of scientific concepts. It is tempting to think that there is some pre-existing structure embedded in nature that we are only discovering and translating into mathematical language—that equations are descriptions rather than interpretations or creations. But how we translate depends on the journey we have already taken, on our dissatisfactions with it, and on how we responded to those dissatisfactions. We "fall up," to adapt a phrase of George Steiner. It is thus misleading to picture science as proceeding solely by scientists producing new concepts, then testing and revising them. Two things are wrong with this picture. One is that the meaning of one concept depends on the meaning of all the others; a concept is one element of that fishbowl-like world that Newton discovered at the heart of the world we live in, and needs everything else in that fishbowl for its meaning. Testing one concept thinking you are testing it and not everything else is like asking, Is New York to the right or left of Boston? without knowing where you are; without having the rest of the map. And we not only need the rest of the fishbowl, but the rest of our experience of the world as well. A scientific concept that we trust is really a concept plus that experience, and when our experience changes—new practices, new technologies—so does how the concept applies to the world. That's why concepts never stay put, and always change or are being elaborated; a concept that tests right at one time can be inadequate at another. There is no right way to say something that does not include our experience with it. [pp. 268—269]
This is not a large book, but neither is it a quick read. There are ideas and connections within that reward slow reading and some reflection. It can easily be read one chapter at a time. By the end the reader will have traveled an interesting, less trodden path through the history of ideas, with stops at plenty of unusual historic markers along the way.
-- Notes by JNS