Column: On the origin of equations

For this month’s issue of Physics Today, I wrote about a new experiment using molecules as a probe of fundamental forces . As I was researching the topic, I was amazed to learn how little is known about the force of gravity at short range.

In my story, I mentioned that for all any experiment has been able to determine, Isaac Newton’s law of gravity—given by the equation F = GMm/r², where F is the gravitational force between two masses M and m, r is the distance between them, and G is a constant—could be wrong by up to 21 orders of magnitude at the nanometer scale. For distances of tens of nanometers up to microns, the experimental upper bounds are somewhat better but still large; for subnanometer scales, they’re considerably worse. This video sums up the situation well.

If you think about it, it makes sense that the force of gravity between microscopic objects would be extremely difficult to measure. But I’d never thought about it. (And I’ve even written before about searches for non-Newtonian gravitational effects.) My introductory physics classes in high school and college, which I think were typical, all seemed to teach the formula for the force of gravity—like the formulae for everything else—as an incontrovertible truth, without anything interesting to say about where it came from or what its limitations might be.

Miller’s Diary

Physics Today editor Johanna Miller reflects on the latest Search & Discovery section of the magazine, the editorial process, and life in general.

To check my memory, I cracked open the copy of Paul Tipler’s College Physics that sits on my office bookshelf. It’s not the book I learned from, but it’s from roughly the same era. Its discussion of gravitation does mention Newton’s boldness in proposing that the orbits of heavenly bodies might be governed by the same laws of physics that cause apples to fall here on Earth. But then, a couple of paragraphs later, comes a statement with no supporting evidence: “We can use the known value of G to compute the gravitational attraction between two ordinary objects.” This is followed by an example calculation of the gravitational force between two 1 kg masses 10 cm apart. Decimeter-scale gravity, it turns out, has been well tested, beginning with Henry Cavendish’s torsion-balance experiments a century after Newton. Tipler discusses Cavendish’s work in a later section. But it’s presented as a measurement of G, not as a verification of the inverse-square relationship; that part is never questioned.

The focus on formulae and equations extends across a lot of introductory science and mathematics. The area of a circle is given by A = πr², just as F = GMm/r². Plug in some numbers, chug out the answer, and don’t think too much about what any of it truly means. That approach, apart from being a wholly joyless experience, obscures some important differences between scientific and mathematical thinking and the complementary ways they help us make sense of the world.

In my 10th-grade geometry class, as we were getting ready for our first official lessons on mathematical proofs, I remember the teacher explaining to us (poorly) the difference between inductive and deductive reasoning. She made inductive reasoning—extrapolation from patterns—sound like an entirely inferior, even useless, way of thinking, because you can never be sure that a pattern you observe today will continue to hold tomorrow. Deductive reasoning, she said, was the way to go, because the conclusions follow with certainty from the premises.

When limited to the realm of mathematics, what she described isn’t too far wrong. Looking for patterns among specific cases can be helpful in formulating new theorems to prove, but it’s the proof, not the pattern, that makes the theorem valid. Last month my colleague Heather Hill wrote about a curious sequence of integrals that deviates from its apparent pattern by a tiny amount, less than 10⁻¹⁰. From the premises of Euclidean geometry, on the other hand, you can deduce that the area of any circle, no matter how big or how small, is exactly πr². And π has been calculated to more decimal places than you’ll ever need .

(To be fair, one of the axioms needed to derive A = πr² is the parallel postulate , which has long troubled mathematicians with the question of what it means for mathematics to be “true” in the real world. But that’s another story for another time.)

In science, though, inductive reasoning is indispensable. You can reason deductively with Newton’s law of gravity as a premise, but the law itself is based on observations of gravity in action among planets, the Moon, torsion balances, and so forth. Its validity and its predictive value extend only as far as it’s been tested—which, for the nanoscale, is not far at all. Even at larger scales, our confidence in both the inverse-square relationship and the value of G is limited by measurement precision .

If my geometry teacher had her way, we’d be forced to conclude that, although we can be reliably informed about circles, we know nothing whatsoever about gravity. Obviously that’s not true, and no one seriously thinks it is. (Even the crackpots who write to us purporting to have overturned the theory of general relativity would concede, I’m sure, that apples fall when you drop them.) But on more politically contentious topics, the fact that science lacks a mathematical level of certainty in its conclusions—as, indeed, it always does—is much more easily misinterpreted.

I don’t mean to disparage the people who teach (or write textbooks for) math and science classes at the high school and introductory college levels. Students’ understanding has to start somewhere, and perhaps it’s not reasonable to expect them to grasp all the subtleties of the foundations of those fields right from the start. But if students aren’t paying close attention, they can end up with a distorted view of how science and mathematics work together to generate knowledge about the world.

And even if they are paying attention, they can miss out on the ways the limitations of that knowledge occasionally surprise us.

Thumbnail image credit: Linda Williams (public domain)