The Unreasonable Inefficacy of Mathematics in the Natural Sciences

Before talking about Joy Christian’s unsuccessful run at Bell’s Theorem, I want to talk about the use of mathematics in the natural sciences.

I’ve said some nice things about math previously as a tool for keeping our thinking constrained to consistent pathways. Here I’m going to take a kick at it, and in particular the view of math most famously expressed in Wigner’s crypto-Creationist essay, The Unreasonable Efficacy of Mathematics in the Natural Sciences.

Wigner argues, after introducing some romantic twaddle from Russell, that the mathematical description of nature is far better than we have any right to expect. There are two ways of responding to this: one is to claim that “nature is mathematical” in some deeply meaningful way, as if mathematical principles somehow preceded physical ones. The other is to note that the language of numbers was invented by humans precisely to describe physical reality, be it counting sheep or determining the curvature of the Earth, and as such it would be a pretty poor show if we didn’t get some decent mileage from it.

If the latter were the case, you’d expect that our mathematical description of nature to have some pretty rough edges… and guess what? It does.

Most of modern physics, one way or another, is expressed in terms of second order differential equations, which describe values, the rate at which values change with time, or space, or something, and the rate at which the rate of change changes. God apparently wasn’t up to handling third derivatives, thank heaven.

These equations range from the very simple (the heat equation) to the nightmarishly complex (the Navier-Stokes equation, which describes fluid motion in all its nonlinear glory.) No matter what the complexity, the equations in the end are a set of constraints on the values some physical quantity can possess in given circumstances, be it the curvature of space-time around a neutron star or the temperature of my house on a cold winter’s night. A set of values that fulfills the constraints specified by the equation and its boundary conditions is called a solution to the equation.

These days we generally find approximate solutions via numerical methods, but in days of yore we used to spend a lot of time coming up with exact solutions in terms of second-order-smooth functions. Whole families of functions (Bessel functions, elliptic integrals, and so on) were invented just to solve particularly common differential equations.

And here’s the thing: reality is only one way, but more often than we would like to admit our equations have multiple solutions that we have to pick between based on physical–not mathematical–considerations. It’s almost as if our equations, being at root human descriptions are nature, are still subject to the ambiguities inherent in humanity.

The most famous ambiguity involves the direction of time: every wave equation has two solutions, the so-called “advanced” and “retarded” solutions. One travels backward in time, the other forward. This is a feature of all second order wave equations, and yet we only ever see the solution that travels forward in time. There have been some pretty clever attempts to fix this up in electro-dynamics, notably the Feynman-Wheeler “absorber” theory, but even given that not-yet-successful attempt, it still leaves us with all the other, perfectly ordinary, wave equations that describe things like waves traveling down a piece of string forward in time, but not backward in time.

There are many less dramatic cases as well, typically involving singularities: places where the solution or its derivative become infinite. These are mathematically well-behaved, but physically meaningless. I once delighted my fluids prof in undergrad by solving the Navier-Stokes equation in a way that looked perfectly correct, but which was in fact singular at one particular point. He was clever enough to notice and I was not, thus validating his role as teacher and mine as student.

We find therefore, that mathematics is not only a useful tool for describing reality. It is also a useful tool for describing unreality, as any crooked accountant knows.

Curiously, the human brain itself is the same way: it has capacities far in excess of the ones it was evolved for. Specifically human, tool-making, representational intelligence was not evolved to write sonnets or build spacecraft, but it turns out to be fairly good at those tasks, and uncountably many others as well, in just the way mathematics is good for a vast number of things that don’t have to do with the purpose it was invented to fulfill. I don’t know what that means, but it’s a curious thing, it is.

About TJ

Scientist, engineer, inventor, writer, poet, sailor, hiker, canoeist, father.
This entry was posted in epistemology, mechanics, physics, quantum, science. Bookmark the permalink.

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