Welcome Iain
A good lecture about reasoning in austrian economy can be found here. Just a short remark, if only empirical testing counts as science, we would have to give up math altogether basically, as math is at its core axiomatic 
And I would strongly disagree to say philosophy is not science.
A good starting point might be the reading list found here.
Regarding the critisism of austrian economy being to qualitative and imprecise, because they are not mehtematical enough, one quick argument.
Economic actions are simply action of individuals. How an individual act at a given time in a given environment can not be expressed in mathematical terms. Without offering a proof here, each individual strives to achieve his goals. One does so by choosing means he employs to get there. What action is taken depends on the preferences a person has at the time of the action. The end with the highest preference that seems attainable with means thought to be at hand will be choosen. In fact people do no only have one goal at a time. So you have a list of preferences of wich you always choose the one you regard highest. You show that by acting accordingly. All other preferences are lower on the list. Yet, they are simply rankings. This means they are ordinal. You can not use ordinal numbers to calculate anything, this can only be done with cardinals. If the basis of economy is human action and the action is based on picking the one with the highest preference which, at any time, are a list of rankings,e.g. ordinal, you can not possibly build any mathematical system on that no matter how complicated the equations are you use.
If you do not accept that 1 and 1 is two than you can not have any useful discussion about calculus at all.
That does not mean austrian economics doesn’t use math. It just means any math based on preferences humans have counterdicts itself right from the start.
Have a lot of fun researching