Let me begin by saying that 'interpretivism' — not a term I've used before, but will here for the sake of the discussion — does not reject applications of the natural sciences in the absolute sense implied here. It is quite willing to use procedures such as statistical analysis when and where they are helpful. The problem is that the methods of the natural sciences aren't always that helpful; they are not sophisticated enough to address the kinds of problems that a society provides.
The natural sciences are powerful because they are able make a number of useful assumptions:
- Non-identity: different items of a given 'type' can be swapped indifferently, because they share precisely the same properties
- two object of the same mass will fall the same way in a vacuum; two samples of a given compound will react chemically in the same way; etc.
- Continuity: every object of a given category will retain the same properties and tendencies across different times and places
- a drop of water will behave exactly the same in any place and any time, so long as the conditions for maintaining a drop of water exist.
- Conservation: a system remains in energy balance unless disturbed by an outside force
- principles of inertia, thermodynamic equilibrium, etc.
These assumptions allow natural scientists to reduce dimensions of analysis and simplify equations until the equations are actually solvable. For instance, if we want to know how a thrown object behaves, we don't have to think about whether it's a baseball, bowling ball, or bag of duck feathers. We don't have to worry about whether we're in Africa, Asia, or Europe; we don't need to remember whether it's Thursday (in case things work differently on Thursdays). We never have a problem with objects suddenly changing course of their own volition. All we need to know is mass, initial force, and angle of trajectory, and we're good to go. And no, obviously it's never perfect, but any inconsistency can be attributed to measurement errors (which we can improve on) or unanticipated forces (which we can anticipate and account for).
Natural science works because it can make simplifications and approximations that work very well for most purposes. They excel where they can take a hard problem and make it (comparatively) easy.
These assumptions don't really apply to the social world. They don't even apply to all of the natural world, to be frank — think turbulence, non-linear dynamics, quantum entanglement — but in the social world it's often a blatant error to make them. I mean, could you and I switch places without any impact on the social world around us? Do we behave exactly the same now as we did when we were six, or as our stone-age ancestors did? Do we simply bounce around like molecules in a solution, neither adding to nor detracting from the world around us? Maybe... But those are really challenging assumptions to make about the social, human world.
And note, this hasn't even gotten to the question of value which has no easy correlate in the natural sciences. I mean, the natural sciences can manipulate the physical properties of certain materials to produce nuclear fission, but there is no natural sciences distinction between the fission that occurs in a power plant and the fission that occurs in a bomb. There's a human distinction between the two — one heats and lights my house and the other reduces my house to radioactive rubble — but there's no 'scientific' reason to build or not to build the bomb. How would we begin to analyze nuclear proliferation politics using the methods of the natural sciences? What statistical analysis tells us how many and who should be incinerated?
