For Complex Concepts, a Model of Simplicity
Most physicists and mathematicians are grown-up men and women, so it’s surprising to learn how much time they spend playing with toy models.
These models are not exactly the kinds of toy cars or Erector sets or dolls that boys and girls play with. But they do share many characteristics.
A model of a car is simpler than a real car, for example. It has fewer moving parts. It doesn’t require gas. It’s smaller, easier to get a handle on.
In the same way, scientists’ toy models help them grasp situations too complex to understand directly.
A toy model or theory, according to MIT physicist Alan Guth, is a theory “which is known to be too simple to describe reality, but which is nonetheless useful for theorists to study, because it incorporates some important features of reality.”
Cosmologists like Guth obviously need such theories. The kinds of phenomena they study--the origin of the universe in Guth’s case--are fraught with unknowns and dependent on dozens of variables. It would be impossible to make progress trying to deal with all of them at once. So scientists selectively trim pieces out of the puzzle until they whittle it down to a manageable size.
What they have, then, is a “toy model.” Of course these aren’t models in the familiar, everyday sense. They are mathematical descriptions that alter reality to make the problem simpler.
Surprisingly, perhaps, the use of models isn’t relegated to complex systems like universes. Discovering the truth about even everyday things can require creating an artificially simplified view.
Take the well-known fact that all objects fall at the same rate. In a vacuum, even a feather and a bowling ball fall at the same rate. But since we don’t live in a vacuum, this simple relationship between mass and acceleration due to gravity is almost impossible to see directly. If you drop a feather and a bowling ball in Earth’s atmosphere, the ball will drop, but the feather will float on air.
To see the true relationship between mass and acceleration, Galileo used a model that subtracted the distorting influence of air.
In the same way, Einstein discovered his first, or “special,” theory of relativity by getting rid of gravity. As Caltech physicist Kip Thorne describes it, Einstein “idealized our universe as one in which there is no gravity at all. Extreme idealizations like this one are central to progress in physics; one throws away, conceptually, aspects of the universe that are difficult to deal with, and only after gaining intellectual control over the remaining, easier aspects does one return to the harder ones.”
Years later, Einstein eventually figured out how to fold gravity back in, and created his “general” theory of relativity.
Sometimes, as with Einstein, scientists create models that eliminate a force, such as gravity. At other times, they’ll get rid of a dimension or two, or simply sweep most of the known particles in nature under the rug. Any model can work so long as it distills the problem down to its essence.
The problem is: How do you know that you haven’t thrown the essence out too? Or made the model so simple that it doesn’t describe anything at all?
That’s always a worry. Harvard biologist Richard Lewontin recently complained about simplified models of human behavior based on data collected from fruit flies. “Always the same story,” he said. “You take a simple organism because it is simpler to study, but you factor out everything that’s interesting in the process.”
Still, models aren’t meant to actually solve problems. They’re meant to help scientists get a fresh perspective on complex problems. Models, said Stanford mathematician Persi Diaconis, “can be useful in changing our mind-set.”
Indeed, scientists will frequently try and abandon many different models before settling on one that seems fruitful to explore. There are many ways to model a universe or a climate system, just as there are many ways to model a person or a car.
Ultimately, even the best toy models--like supermodels--embody untenable idealizations. They are constructed from ideas in our minds more than from concrete pieces of reality.
As the late physicist Sir Arthur Eddington put it, building such models is not like building a house out of bricks and mortar. “It is like building a constellation out of stars,” he wrote.
The stars are scattered more or less randomly in the sky. Humans, contemplating the bright points of light, grouped them into images of dippers and dragons or maidens. But different humans could just as easily group the stars into constellations of cell phones or Hollywood celebrities.
There is a lesson here, says Eddington, for those who would build toy models of the universe: “The things which we might have built but did not are there just as much as those we did build.”
