|Muck and Mystery
Loitering With Intent
|blog - at - crumbtrail.org|
The other day I recommended this site, Gravity and Levity, to some folks as an example of the type of stuff that I read for phun and general edification. The author has a suitably divergent attitude and says:
as I went further into physics, it began to be more than a game. Little by little, all the equations and “rules of the game” started coming together into a coherent perspective on the universe and how it works. A good physics class would leave me with a sense of how some great minds of the past must have looked at the world and its natural laws: Isaac Newton, James Maxwell, Albert Einstein, Erwin Schrodinger, etc. And that was fascinating. These men had some audacious and crazy-sounding ideas. The idea of gravitational attraction, for example, — that every object in the universe pulls on every other object by some mysterious “action at a distance” — is so ridiculous that it took a man with a deep interest in the occult sciences to come up with it. Over years of study my interest in physics gradually but completely shifted from “the game is fun” to “I want to know how to think about what the universe is made of and how it works.” ...We wish to talk to The Man Behind the Curtain.
In physics, every equation is a sentence. Mathematics is just a language: a concise way of writing precise statements that comes along with a complicated grammar. So every equation should be translatable into English to produce an idea. So every important equation in physics must have a corresponding important idea. But the translation process Math <-> English is a difficult one. It’s easy to abandon the English version altogether and just “talk in math.” ...
Translating math into words and pictures can be difficult for intellectual reasons, but talking about those words and pictures can be difficult for social reasons. Physics students, for the most part, are eternally insecure. Physics has sort of a cult of intelligence, where we worship the “heroic” figures of the past, pass on mythologies of their other-worldly brainpower, and glorify their acts of intellectual arrogance. It is no surprise, then, that every student is secretly afraid that s/he isn’t really smart enough to do physics as a career, and that if they talk too much they will reveal their secret. So in our conversations we stick to the math and the precise “formalism” left for us by other smart people, and it is only with great hesitancy and trepidation that we attempt to discuss the ideas behind all the math.
In the first week or two of any freshman physics course, students are exposed to the force of friction. They learn that friction impedes the motion of objects and that it is caused by the microscope interaction of the two surfaces sliding past one another. It all seems quite plausible, even obvious, yet regardless of any high falutin’ modeling, with molecular mountain ranges resisting each other’s passage or running-shoe soles binding to tracks, friction produces heat and hence an increase in entropy. It thus distinguishes past from future. The increase in entropy—the second law of thermodynamics—is the only law of Nature that makes this fundamental distinction. Newton’s laws, those of electrodynamics, relativity … all are reversible: None care whether the universal clock runs forward or backward. If Newton’s laws are at the bottom of everything, then one should be able to derive the second law of thermodynamics from Newtonian mechanics, but this has never been satisfactorily accomplished and the incompatibility of the irreversible second law with the other fundamental theories remains perhaps the greatest paradox in all physics. It is blatantly dropped into the first days of a freshman course and the textbook authors bat not an eyelash. ...If I had to read and speak only maths the conversation wouldn't be very interesting unless I spent a few more years brushing up on old stuff and learning much that is new. I appreciate these fellows who seek to make the math<->english translations not only because they save me some tedious work, but also because they reveal some of the blemishes and gaps in what is too often sold as an essentially complete and correct account. It isn't just that there is more to learn on the frontier, it is that the frontier begins with the first step.
Physicists have long believed that mathematics is the Rosetta Stone for unlocking the secrets of Nature and since a famous 1960 essay by Eugene Wigner entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” the conviction has become an article of faith. It seems to me, though, that the “God is a mathematician” viewpoint is one of selective perception. The great swindle of introductory physics is that every problem has an exact answer. Not only that, students are expected to find it. Such an approach inculcates our charges with an expectation that is, in fact, exactly contrary to the true state of the world. Vanishingly few problems in physics have exact solutions and a physicist’s career is one of finding approximations and hopefully not being too embarrassed by them. ...
As Einstein famously put it, contra Wigner, “As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality.”
The maxim might be taken to heart a few weeks later in a freshman course when instructors introduce their students to Newton’s law of gravity. The famous law works exquisitely well, of course, but a singular strangeness goes unremarked. According to the equation, if two objects become infinitely close to one another, the force of attraction between them becomes infinite. Infinite forces don’t appear in Nature—at least we hope they don’t—and we dismiss this pathology with the observation that real objects have a finite size and their centers never get so close to each other that we need to worry. But the first equation in any freshman electricity and magnetism course is “Coulomb’s law,” which governs the attraction or repulsion of electrical charges and is identical in form to Newton’s law. Now, in modern physics we often tell students that electrons and protons are point particles. In that case, you really do need to worry about infinite forces and it is exactly this difficulty that led to modern field theories, such as quantum electrodynamics. Well Newton himself said, “Hypotheses non fingo”: “Look guys, the equation works, usually.”