Use of mathematics and imprecise definition of terms As a postgraduate student of economics I've been trying to expand my mathematical &quot;toolset&quot;. While doing so I've talked to engineers, physicists and mathematicians, many of which have disdained the use of mathematics in economics. Their arguments vary, but one common theme is summed up by mathematician Michael Edesess' critique:

Economics pretends to be mathematics, but it is not mathematics. There
is a major difference. No mathematician uses a term in a formula, or a
statement of a theorem, unless that term has first been defined with
excruciating precision.
And while economists may think they’ve defined terms like &quot;aggregate
demand&quot; or &quot;economic growth&quot;, they should try reading some real
mathematics to see what a precise definition truly is. The economists,
I think, leave the work of definition to be inferred from the way the
terms are used in the formulas.

I believe I know the precise definition of (quite a few) economic terms, but maybe Edesess is pointing out to some more profound mathematical foundations which I may not be familiar. Could someone expand on his argument and maybe even counter back?
Edesess is attacking what is really just a straw man of economics. I'm not sure he really understands the field. To start, economics is not math. We're not claiming that it is. It's more of an "applied" science. Economists have never claimed that these definitions are precise in the way that mathematics is. These definitions are modeling constructs---they're for applied work. They're use is temporary in a way. The point is to try to convey an idea in a more precise way than just in words---but everyone knows that they're not a precise as we would like and not as precise as they ultimately should be. They're meant to be debated and later refined. But, as all applied scientists know, you've got to start somewhere and sometimes ideas are best conveyed through simpler---if less detailed mean.  

Coming up with better definitions is a huge part of the economic science. Consider these examples. When the Cowles Foundation was founded in 1932 its motto was "Theory and Measurement" (the motto was first adopted in 1952). Measurement is not an easy thing to do. As another example, a lot of Larry Kotlikoff's work has dealt with how a lot of fiscal measures are not economically well-defined concepts.


  Einstein taught us that neither time, nor distance are well-defined physical concepts. Instead, their measurement is relative to our frame of reference – how fast we were traveling in the universe and in what direction. Our physical frame of reference can be viewed as our language or labeling convention. ... Kotlikoff, along with Harvard's Jerry Green, offered a general proof of the proposition that deficits and a number of other conventional fiscal measures are, economically speaking, content-free, concluding that the deficit is simply an arbitrary figment of language in all economic models involving rational agents.


Also, take another example of current interest. Lars Hansen's recent work (winner of the 2013 Economics "Nobel" prize) has focused on the difficulty and ongoing failure to define certain economic concepts, including "bubbles" and systemic risk. See his essay "Challenges in Identifying and Measuring Systemic Risk". I'm a fan of the dictum he relays, attributed to Lord Kelvin,


  I often say that when you can measure something that you are speaking about,
  express it in numbers, you know something about it; but when you cannot
  measure it, when you cannot express it in numbers, your knowledge is of the
  meagre and unsatisfactory kind: it may be the beginning of knowledge, but you
  have scarcely, in your thoughts advanced to the stage of science, whatever the
  matter might be. 


He notes that "an abbreviated version appears on the Social
Science Research building at the University of Chicago." So, yeah, economists (as social scientists) definitely take this seriously.

So, the point is that economists are well aware of the problems in these "definitions." They are a part of ongoing research in the field; sometimes they're ignored if people don't think they're first-order to the problem; etc...