"In macro, most of the equations that went into the model seemed to just be assumed. In physics, each equation could be - and presumably had been - tested and verified as holding more-or-less true in the real world. In macro, no one knew if real-world budget constraints really were the things we wrote down. Or the production function. No one knew if this "utility" we assumed people maximized corresponded to what people really maximize in real life. We just assumed a bunch of equations and wrote them down. Then we threw them all together, got some kind of answer or result, and compared the result to some subset of real-world stuff that we had decided we were going to "explain". Often, that comparison was desultory or token, as in the case of "moment matching"."So he begins by discussing one way that math is used by scientists - to represent reality precisely and predict its behavior. He discusses two other uses of math that he thinks are common in economics: signaling that you're smart and obscurantism. I can't disagree that these things go on, although I think it's very hard to get away with it. Most celebrated economists really do have something valuable to say - a contribution - and could not win their acclaim just by doing hard math. Some of the most celebrated papers have fairly accessible math, after all.
I come at this from a different perspective and would add one more way to use math to the list that I think characterizes most math in economics.
In high school, I was "good, not great" at math. I was in the advanced classes, but not the most advanced classes (probably due to where I was tracked early on more than anything else). I had no real interest in doing much of it in college. I knew I wanted to study economics (and very quickly realized I had to do some more math), and had initially planned on double-majoring in government. But in my freshman year, practically randomly, I took a criminology class in the sociology department and was hooked on sociology. I never took any more government after that and I double majored in economics and sociology.
Sociology is very different from economics. There are some great empirical sociologists but sociological theory isn't nearly as formalized as economic theory (there are some important exceptions, for example people dealing with social networks). There are some good reasons for this - what sociologists deal with is harder to quantify and therefore harder to formalize. But that comes with costs. When you take a theoretical outlook in sociology you are juggling lots of different forces and trying to make sense of how they work together and feed back into each other. That's very hard, and the quality of the theory is obviously going to suffer because it's hard. That's not a slight on sociologists - that's just the nature of the problem.
Economics is different. We're dealing with trade-offs - comparisons - which are therefore eligible for mathematical representation. We're also often dealing with quantifiable things. That means math has a real shot in economics.
But Noah is right that it's not quite the same as physics. Our microfoundations and really any equation - microfoundation or not - is not as concrete or testable as it is in physics.
For me, that misses the point of the value of math in economics. Of course it would be nice if we could get more concrete in the way that Noah is suggesting physics is concrete. There are some obvious advantages associated with that. But the big advantage of math in economics is that all those complex interactions and variables that sociological theory is juggling can be much more carefully laid out and related to each other. That is not a guarantee that we are hitting on laws of the universe the way physics is (although good Kuhnians have to take that bit of language about "laws" with a grain of salt anyway). It is, however, a path to a much more coherent and useful way of doing social theory than probably any other social science offers.
This, I think, is very clear if you're coming to economics from another social science. It is less clear if you come from the natural sciences and try to cut and paste your experiences from the natural sciences.
So for me, it's primarily not about signaling or obscurantism. It's about making our ideas clear. That helps to test and arbitrate between different ideas. We're not getting at any kind of precise laws yet. I'm guessing we never will. It's just not the nature of the beast. We are learning a lot about how the world works, though.