Jonathan Finegold-Catalan discusses a quote supplied by Robert Murphy. Unless I'm mistaken, Murphy's original intent with that post was merely to highlight the occasionally hilarious use of absolutist language in Mises' writing style. So we should not take Catalan's point as any kind of quibble with Murphy, so much as a tangential exposition.
Anyway, Catalan's post is worth reading and thinking over. I'm not going to excerpt it because it's reasonably short, and I don't think I can describe his point in a more concise way than he did. So go read it. The general question is to what extent Mises objected to the use of "empiricism" (understood here to mean the statistical analysis of historical economic data) in economics. Catalan reasons that if Mises objected to its use - and it is commonly believed that he did - then Mises was wrong to do it wholesale. In short, there are important applications for quantitative analysis in the field of economics.
I seem to recall (and, indeed, here it is) a Mises Daily article covering with academic precision the difference between how Mises saw "history" and how he saw "economics," i.e. theory. And, if memory serves me right, there were elaborations on this throughout the Austrian blogosphere from (I think) Peter Boettke and Steve Horwitz. But don't hold me to that; this is just what I seem to remember.
Anyway, under Salerno's paradigm outlined in that Mises Daily article, Mises felt that "empiricism" was a legitimate question for economic history, while all questions of economics were a matter of pure theory, and thus mathematical models would be of very little use in that arena. This strikes me as being a well-balanced position, and I have no reason to object to that.
I'm not one of these guys who "knew Mises" or "knew the guy who studied under Mises," so I will defer to what those folks say about how Mises felt about empiricism.
What I will say is this: My reading of Mises is always quite a bit more moderate than the typical Austrian economist's reading. I interpret the Murphy/Catalan quote to mean simply that any conflict between the empirical facts and the economic theory needs to be resolved, either by revising theory to make it more exact, or by refining the quality of one's empirical model. In the worst-case scenario, maybe we could say that the empirical model is too much of an approximation to yield a perfect 100 R-square.
I mean, call me stupid, but I have never gotten the impression from any of the Mises I have read that he opposed empiricism so much as felt that descriptive theory is what mattered most. As you say, I think most people already accept that idea, but from what I learned in my old History of Thought course 10 years ago, I had the impression that socialist economists and Soviet planners were moving toward a "mathematical economics" viewpoint where the economy could be planned and assessed via matrix algebra, and this was the kind of thing Mises objected to. Again, in this day and age, who doesn't?
I will add that society has become far more mathematical since the days of Ludwig von Mises. Even my parents grew up in an age when few adults had any reason to know calculus. In contrast, I myself use it almost every day. More to the point, though, society's ubiquitous conversations about growth rates and rates of change of growth rates and derivative finance exchanges and so on and so forth reflects society's growing comfort with calculus and post-calculus mathematical concepts as modes of thought.
To put it another way, every child learns what a mathematical vector is in grade school. Nobody ever sits down and does the kind of math that they teach kids about vectors, though. Instead, we get into physics courses in high school - or advanced mathematics courses in college - and come to understand that a vector is a way of mathematically expressing position, direction, and force in a concise and usable way.
Now, physical sciences can directly apply calculations to vectors. Social sciences - and ordinary people in general - can still use vectors as a sort of "theoretical tool." You can start to see things in your life as "vectors" of sorts. Your career path could be thought of as a vector. Your level of anger in an argument. Or whatever.
The point is that one of the benefits of understanding complex mathematical concepts is that they provide logical insights that we wouldn't otherwise see. What's important, of course, is being able to apply those concepts to our lives theoretically in a meaningful way. In that sense, theory always trumps pure math when we're seeking to understand the world. But it's not a one-way thing. Math influences how we think, and thinking influences how we use math. That's because math is in many ways the formalized version of how humans think, anyway. Wherever a good idea comes from is a good source of knowledge. Those who rule out math as a source of insight demonstrate a lack of appreciation for what mathematics actually is.