Somehow I ended up on Conservopedia reading a debate about whether it is possible that the exponent of r in Newtonian gravity could be slightly different than 2. The context of the debate was open-mindedness, and the original poster (OP) presented an argument that went something like this:

  1. We all know that the formula Newtonian Gravity gives pretty good predictions.
  2. We also know that measurement error exists.
  3. If the exponent of r in the formula were (for example) 2.00000000000000000000001 instead of 2, the predictions would be so similar that any differences would likely not be measurable on the timescale of human civilization.
  4. Therefore it’s possible that the true exponent is really something other than 2.
  5. Therefore if you say that it is not possible, you are closed-minded and falling for the indoctrination of scientists, who are liberals.

Now, point 5 is a bit ridiculous, but otherwise I find this debate fascinating. It reminds me of late-night conversations I would have as a teenager in college, staying up all night in the common area of the dorm building. We didn’t know shit about anything, but we were practicing deliberation, reasoning, and debate. It’s a valuable thing to do to hone one’s thinking skills. So: bravo for that.

Setting aside the jump in reasoning from points 4 to point 5, what interests me about the reasoning shown here is the way it illustrates assumptions about science and what the word “possible” means.

A few people in the discussion pointed out (rightfully) that Newton didn’t calculate the exponent of 2 based on measuring data points and fitting a curve. It was derived based on general assumptions of field theory and geometry. It comes from the fact that the surface area of a 3-dimensional sphere increases by the square of the radius. To ask “are you open-minded enough to think the exponent in Newton’s formulation of gravity should be slightly different from 2” is like asking “do you think the exponent in the formula for surface area of a sphere should be slightly different from 2”: it’s a nonsense question. You may as well ask whether it’s possible that 2+2 is actually 4.000000000000000001, and we only think it’s 4 due to the ambiguities of measurement error.

The OP shot back: “Euclidian geometry does not define gravity, and it’s closed-minded for anyone to imply that it does.”

Is he right?

In my opinion… sort of. (Ignoring the connotations of the term “close-minded”, of course; I don’t want to go down that rabbit hole.)

Taken at face-value, the statement “Euclidean geometry does not define gravity” is plainly true. And if the OP’s question were “Might the entire geometric basis for Newton’s approach to measuring the force of gravity be wrong?” then I don’t think there would have been any debate at all. The answer is plainly: duh, yes, and we have a century of scientific thinking to support that.

But that wasn’t what the OP asked. The way the OP asked the question lead everyone in the debate to interpret the question this way: “If we take the basic assumptions that went in to Newton’s formulation of gravitational force as given, is it possible that the structure of the formula is correct overall but the numerical value of the exponent of r is slightly off and we simply don’t realize it because of measurement error?”

The way that question is asked reflects a basic misunderstanding of where the formula comes from. It also reflects an overly-simplified view of how science works.

Maybe that’s partly the fault of our educational system, too. We spend a lot of time talking about the importance of coming up with hypotheses to explain data. That can easily lead some poor student down the path of thinking that Newton’s gravitational law must have been arrived at by measuring some data points and “fitting a curve”. And everyone knows that when you “fit a curve”, there is margin for error. The flaw in this reasoning is that this formula was obtained through an entirely different kind of scientific reasoning: it was theoretical derivation, not statistical curve-fitting.

Maybe it’s just the former teacher in me, but I’m glad these conversations are happening in the world. It shows that people are at least examining ideas and trying to draw out consequences from their beliefs.

I hope “liberal scientists are indoctrinating us!” dude learned something about how science works as a result of that discussion.