Mathematical economics, is it a good thing or a bad thing? This seems like more of a philosophical question than an economical question, and being trained in economics and mathematics I am going to try and avoid the philosophical implications at this point. One can read many different Austrian economists give differing opinions of why math is an unnecessary facet of economics today, but Mises has a quote that pretty much sums up the underlying belief about mathematical economics amongst Austrian economists today.
"The mathematical method must be rejected not only on account of its barrenness. It is an entirely vicious method, starting from false assumptions and leading to fallacious inferences." - Mises
This quote, and hence disdain for mathematical economics amongst Austrians is very troublesome to me for many reasons, but there is one main reason: Game Theory.
Lets take a look at this quote by Mises and see if it applies to game theory. Should game theory be rejected on account of its barrenness? Well, it is almost absurd to answer this question being that there have been multiple mathematicians who have won the Nobel Prize in economics for work in game theory. However, we all know a Nobel Prize in economics doesn't mean what it used to. Without going into too much detail, game theory tells us why anti-trust laws are irrelevant in a free market because collusion is not an equilibria. It tells us unions are no longer needed to "protect" workers because managers and employees will reach an equilibrium within the range of the employees demands. It tells us why no nuclear weapons were launched during The Cold War. This list is not even close to being exhaustive. Game Theory's results are rich and can only help economists because it is a mathematical field grounded in logic and truth.
One last thing to consider from this quote is the "false assumptions" portion of Mises's claim. There are three assumptions to game theory. The players are rational. This is self explanatory and no need to argue why it is not false. Same goes for the next assumption: intelligence. All this assumption is saying is that if person X employs strategy A, person Y will employ the best response to strategy A. Furthermore, person X knows that person Y is intelligent and will therefore not employ a strategy in the hopes that person Y will play the wrong strategy. The final assumption is that the actual playing of the game won't affect the outcome of the game. This one is a bit more difficult to explain, but in general it is saying that if we decide what moves are the best before the game is played that each of our outcomes won't change based on something unaccounted for in the game. IE, if my opponent employs strategy B and my best response is strategy A, this will actually be my best response. A good argument for this can be found in "Game Theory: Analysis of Conflict" by Roger B. Myerson. This discussion is in chapter 2.
So, are the assumptions of game theory false and do they lead to fallacious inferences? Not as far as I have ever found. It is odd to me that Mises should be against game theory at all given Oskar Morgenstern was an Austrian economist and laid the groundwork for the theoretical foundation of game theory. Thus, if an Austrian economist makes fallacious assumptions in this area of thought, why should we trust their assumptions in other areas?