My predictions for 2011 went up yesterday. Podcast and chat on Thursday.
Somehow I forgot to review Michael Guillen’s Five Equations that Changed the World
, a very strong look at five equations and the scientists who developed them that’s explained with very little math at all. Guillen’s target is the lay reader, a term which, since I haven’t taken a physics class since 1990, would include me.
The five equations aren’t hard to guess – they are, in the order in which Guillen presents them, Newton’s Universal Law of Gravity, Daniel Bernoulli’s Law of Hydrodynamic Pressure, Faraday’s Law of Electromagnetic Induction, the Second Law of Thermodynamics (discovered by Rudolf Clausius), and, of course, E = mc
* So, does a scientist discover an equation, develop it, or something else? He doesn’t invent it, certainly; these are, as far as we know, immutable laws of our universe. I thought about using “unearth” to describe this process, but it seems to mundane, especially for Clausius’ and Einstein’s contributions. I’m open to suggestions here.
Newton’s and Einstein’s stories are rather well-known, I think, so I would say the most interesting sections of the book were the three that those two bookended. Clausius’ story was probably the least familiar to me, as I probably couldn’t have named him if asked. And what made his story interesting was how many other discoveries or developments had to happen along the way for him to be able to articulate his equation – including the invention of the thermometer, the creation of the calorie as both a unit and as a theory for the source of energy, and the life’s work of Julius Robert Mayer, a Bavarian doctor who first expostulated that all the energy in the universe had to add up to the energy that existed at the universe’s start (that is, the First Law of Thermodynamics), only to find himself rejected and ostracized by both the scientific and religious establishments of the time.
The final section of the book, on how Einstein’s theory of relativity led to the development of nuclear weapons, is a bit poignant as Einstein lived to see the destruction and regretted his role in encouraging President Roosevelt to order the development of the bomb. (I would imagine Einstein realized, however, that since the Germans would have eventually developed it themselves, the Manhattan Project was as much as a defensive move as an offensive one, even though it became an offensive weapon when we figured it out first.) Slightly less interesting, to me at least, was the extent of the family squabbles in the section on Bernoulli, where a pattern of fathers becoming jealous of talented sons tended to repeat itself in a way that would probably land them on Maury Povich today.
If you like the sound of this book but want something mathier, check out my review of Prime Obsession, a book about the development of the Riemann Hypothesis, perhaps the leading unsolved problem in mathematics today.
I would have included keplers laws of interplaetary motion as well or Maxwell’s equations. Either way, most people don’t understand that modern physics was built on the ability of a few to “invent” both the mathematics and equations that describe dynamic proceesses. The most important of these being the invention of calculus by newton and leibnitz. Without calculus, we could not describe even the most basic of processes. Enjoyed your take on the book, if you like ” mathy” ones, try the golden ratio or the story of phi. Thanks
Have you ever read “A Brief History of Time” or “A Briefer History of Time,” both by Stephen Hawking? They’re along the same lines as this book – explaining the universe to the masses, although I found both to be quite reliant on the reader’s ability to focus on the problem at hand while keep the overlaying problem close by. I plan on reading this book – this review piqued my curiosity about the middle three equations. Thanks.
In the first chapter his book, The Big Questions, Steven Landsburg lays out the case that the laws of arithmetic are immutable, eternal, and even exist outside the scope of time. He does this in a non-technical way, just walks through the logic in small bites. Landburg has convinced me that ‘discovery’ is definitely the way to describe this process of advancement, no matter how meaningful a discovery it is.
I like ‘liberation.’ On the surface the Universe seems random and hard to predict. When an equation is found that is not only mathematically sound, but also furthers our understanding of the Universe, I see that as liberating some order from the chaos.
Another thought, I almost wrote that you liberate truth when you find a new equation. Naturally, I am led to police detectives who are trying to uncover the truth. Once the detective proves some man committed the crime in some manner he hasn’t ‘invented’ the crime, or the motive, or the man. Rather, he has discovered the truth. In my analogy the Universe is the crime and the equation is the criminal, the motive, and the method. So maybe we shouldn’t have any reservations saying one ‘discovers’ and equation.
I’d use “developed”. “Discovered” carries a connotation of suddenness which overlooks the painstaking work that goes into a new theory. And that’s what a scientific equation is: a theory. It’s a best guess given the data available at the time, and as new data arrive, the equation is either confirmed, tweaked, or discarded.
We do a disservice to scientific equations by calling them “Laws”. For example, Newton’s Law of Universal Gravitation fit well the data available to Newton at the time he developed it. Since then, however, advances in the measurement technology led to new data that refuted Newton’s Law. For over a hundred years, physicists advanced one new theory or another in an attempt to make Newton’s Law agree with the new data, but Einstein’s General Relativity is the only such theory to continue to withstand scrutiny. Perhaps someday we’ll discover newe data that casts doubt on Einstein, too, at which point someone will have to develop yet another theory of how gravitation really works.
In fact we see this in baseball too! Think of the “theory” of defensive metrics. For a long time, many analysts looked to Fielding Percentage to determine the quality of a players’ defense. As new and better data collection (and processing) technology has come, we’ve largely abandoned Fielding Perrcentage in favor of UZR, etc. Perhaps someday someone will develop a truely satisfying “Theory of Defense” that allows us to understand and predict a players’ defensive prowess (including catchers!) with greater accuracy.
The best book I ever read on the development of scientific theories is Thomas Kuhn’s “The Structure of Scientific Revolutions”. Perfectly accessible to the intelligent lay reader, not at all mathy, but profoundly enlightening and thought-provoking.
@Jake Russ: I haven’t read Landsburg, but to offer a counter-perspective: in their book “Where Mathematics Comes From”, George Lakoff and Rafael Núñez propose the idea that mathematics as we know it is a purely human invention. After all, humans are the only species we’ve ever discovered that does mathematics. We have absolutely no evidence of a non-human mathematics, and we have no reason to believe that what we call math isn’t unique to our specific brains, rather than universally true. Who knows whether or not hypothetical alien races do math as we know it? Who knows whether they would conceive of integers or infinity? Who knows what they would call “logic”? To call math “universal” is a profoundly anthropomorphic viewpoint.
Addendum to my previous comment: I should have said, “‘Discover’ carries a connotation of suddenness… and of ‘inevitable truth’ which scientific theories do not actually possess.”
Great review, Keith! For a slightly different spin, you might enjoy The Ten Most Beautiful Experiments, which focuses not on the great theories, but on the practical side of things. It’s definitely on the brief side, but short science books are a good thing.
@Alex Basson. The relationship between energy, mass, and the speed of light described by Einstein’s equation existed before any human was aware of it. To say Einstein ‘developed’ E=mc2 suggests that he created that relationship. No, he merely found out that the relationship was there. Look in your thesaurus, develop!=(find==discover).
P.S. To say that math equations are not universal truths is just wrong. Just to get ahead of the naysayers, I am not going to argue this point, I am busy at work being a physicist.
Ooh, I also have to plug Michael Brooks’s Thirteen Things That Don’t Make Sense. Numeric science titles FTW!
@Obo: Your comment suggests you may be misunderstanding an important but subtle distinction between physical phenomena and scientific theory. A scientific theory isn’t a physical phenomenon; rather it’s a description of a phenomenon which makes testable predictions. The success of a scientific theory is predicated upon its ability both to satisfy existing data and to make accurate predictions about future phenomena.
Yes, of course, the relationship between energy, mass, and the speed of light existed before Einstein. But Einstein’s *equation* didn’t—he had to develop it based on all the data he had available to him. Understand: the equation isn’t a relationship. It’s a *description* of a relationship. “Ceci n’est pas une pipe.”
Like any good scientific theory, Einstein’s equation makes testable predictions about physical phenomena that, to this point, have all been confirmed, and this gives us confidence in future predictions based on the equation. In other words, Einstein’s description of the relationship between energy and mass is superior to, say, my description that energy equals mass. My description fails to satisfy the data, while Einstein’s has, so far, held up to such scrutiny.
But if a time should come to pass—as in fact happened to Newton’s Law of Universal Gravitation—that new data reveals the equation to make faulty predictions, then we will be forced to abandon it in search of a superior theory. Someone will have to develop (not “discover”) this theory. This kind of thing has happened frequently throughout the history of science. Universally accepted scientific theories must be, and often are, abandoned due to contradictory evidence.
I mean, consider the alternative. Do we say that Newton “discovered” rather than developed his Law of Universal Gravitation? But if so, what does that mean when we later find out that he was wrong? Can one “discover” a false theory?
Put yourself in the shoes of a physicist two hundred years ago; you’ve made a successful career using Newton’s Law of Universal Gravitation, and you accept it as fact. But now new telescopes and measuring devices have been developed that allow you to track the path of the planets with unprecedented accuracy. So astronomers the world over re-plot the paths of Mercury and Venus, and behold! they don’t match what Newton’s Law of Universal Gravitation says they should. At first, you doubt the new measurements; they must be error-prone. After all, Newton’s Law is fact! But more and more people come up with the same data time and time again, and the data consistently disagrees with Newton.
What do you say in such a case? “But Newton’s Law tells us what the relationship between gravitational force, mass, and distance IS! It is fact! The planets are doing it wrong!” But of course that’s absurd. Newton’s Law ISN’T the relationship between gravity and mass; it’s a *description* of that relationship, one that Newton worked hard to develop, and it’s a pretty decent one up to a point. After that point, though, it fails to satisfy the data and ultimately we must abandon it in favor of some new theory that more closely satisfies the data and makes accurate predictions.
Who’s to say such a thing won’t happen to Einstein’s equation in the face of new, as-yet undiscovered data? And if that should happen, some new physicist will have to develop a new theory that satisfies the data better than Einstein’s equation can. If new data reveals “E = m c^2” to be false, will we still say that Einstein “discovered” it?
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As for math equations not being universal, you are correct that your assertion to the contrary is not, in fact, an argument. The fact remains that all mathematics of which we are aware is human mathematics.
Here’s the key point: As far as we are aware, the math we know is the only math that does exist and has ever existed. And from that, it’s not a huge jump to say that therefore the math we know is the only math that *can ever* exist, that it is “universal”. But that jump is unfounded. It’s based on the premise that because it’s the only math *of which we are aware*, it’s the only math that can be.
A mathematical analogy: For over two thousand years, mathematicians thought that Euclidian Geometry was the only geometry that could be. It wasn’t until a very deep examination of the Parallel Postulate revealed Hyperbolic Geometry did we discover we were wrong. But we *were* wrong; there *are* non-Euclidian geometries (in our human mathematical system). We were wrong all along and just didn’t know it. We assumed we were correct because we had no evidence to the contrary.
Point being: in the absence of contradictory evidence, it is tempting to think one’s assumptions are correct. We assume that math is universal because we have no evidence to the contrary. But we are also the only species, to our knowledge, that does any math at all. When it comes to math-doing species, humans are the *only data point*. To conclude, therefore, that human math is universal is to make a sweeping generalization based on a single datum. Readers of this blog understand the fallacy of such generalizations.
@Alex Of course mathematics is a human construct. I wasn’t precise when I said ‘math equations’ are universal truths. I was referring to those equations referenced in my first paragraph that are mathematically sound and illuminate our understanding of the universe such as your favorite law, F=ma, as well as many others like Maxwell’s equations, Poisson’s equation, and of course E=mc2.
As for your example about F=ma, that is still a fundamental truth of the universe. Abandoned?! We crashed a satellite into a freaking satellite (an aside: read about Deep Impact, truly an amazing human accomplishment) using F=ma. We did discover that a more restrictive premise is required, however. Specifically, F=ma is true for things larger than the atomic scale and things much slower than the speed of light.
We crashed a satellite into a freaking asteroid*
You asked, “So, does a scientist discover an equation, develop it, or something else?”
That’s a challenging one.
Translate could work. The “immutable laws of our universe” are languages that only a relatively small number of mathematicians understand. They translate the concepts into equations for the world.
Translate lead me to transliterate. So could we also use transnumerate?
Would transnumerate work in narrative non-fiction?
For a fun read, with more math (assuming you can stand the early 20th century bigotry – note the title):
Men of Mathematics by E.T. Bell.
For a good history of mathematics with even more math: Kline’s Mathematical Thought from Ancient to Modern Times.
For an actual introduction to mathematical thinking, Polya’s How to Solve it is a classic, but a bit stiff in style.
They are “pop” books, but I found had more meat.
I would have added Planck’s constant (E=hv). It is such a simple concept and they way it was identified, at least to my understanding, was through simple experimentation and regression analysis. Thus providing the lesson that well-crafted experiments often do not require fancy analysis.
Another equation would have been the exponential growth model (y=y0e^rt).
But neither probably make a readable story.
As to the question of scientific discovery, I think part of the process is understanding the problem. Specifically, the scientist obtains data or insight and is able to understand the problem better or differently than other scientists. I think this is true mostly for theoretical problems but can also be attributed to experimental problems.
For those interested, the PBS documentary “Naturally Obsessed” provides a wonderful introduction to the making of an independent scientist. It sure portrays my life in graduate school.