Social science isn't "experimentable" as biology or physics... And if you try hard to force it you will end up with what's called social engineering instead...
W. Brian Arthur is an economist I like. Complexity theory is also very exciting for me. I do not have any experience in those, but they and other things inspired how I perceive the world.
I probably did not deliver what I think clearly. The link above might be helpful to deliver what I mean.
This isn't really true. When Human Action was written there already was a fair bit of literature on behavioural psychology and other fields, but Mises made a conscious choice to not base his theories on them. In the first part of Human Action Mises does bring this point up but makes a clear distinction between investigations into human behaviour that are rooted in the natural and empirical sciences (i.e. biology) and his own flavour which is rooted in a Kantian-esque introspection. This concept is referred to in Mises' work as well as in the works of his successors as methodological dualism, and is a core tenet of Austrian social analysis.
That said, no one would judge you for doing math. However, the important pieces (axioms) don't require it.
Lack of math doesn't mean anything bad. See Frédéric Bastiat and Gustave de Molinari. Both are amazing and no math required :)
The trouble with praxeology is not the lack of math, though; it is the notion that results need not be validated empirically because they're based on logic. However, your assumptions (i.e. axioms) need not hold in nature and any number of mistakes could had been made in the reasoning process. This is why being based on logic does not make a model magically free of errors.
Problem is, the real world is messy. Some aspects of the world are susceptible to logical reductionism, but not all of it. Enough of it is arbitrary that you very quickly fall off the rails, even when it's not obvious. A theory can often seem superficially more consistent than it really is simply out of coincidence, insufficient precision, or insufficient predictive power. In the real world the proof is in the application.