To Math or Not to Math is not the question

Peter Radford's message (stated previously by Paul Krugman, Lars P Syll, and countless others) about the economics profession is something I read again and again on blogs and in articles: economics/economists have become entranced with mathematical rigor and have forgotten models and equations are supposed to say something about reality and actual human beings.

I hear this message so much sometimes I  think "but nobody really thinks like that".  Math is a tool you can use to simplify your theory or organize your thoughts.  (My old professor used to say Joan Robinson's verbal reasoning is like take a train while Blanchard's mathematical reasoning is like taking a plane- both get you where you need to go).  No one would intentionally throw in more complex math just to make their model more rigorous.  No one would add microfoundations they didn't believe in just because it was "the way to do it" or "the way to get published".  No one would pile shaky assumptions into their model just because that would mean using some profound theorem or doing some crazy programming.  Would they?

I've been watching some financial theory lectures by John Geanakoplos on Yale's Open Courses and so far they've been less about finance and more about the place of math and methodology in economics.

I've only watched about four of the videos and I haven't been biased by discovering his economic pedigree so my judgment may be a little premature but John really makes me not want to blame math for all of econ's woes. 

I don't quite buy the "economics as an objective science" line. (Feminist economists have long argued economists want to make economics more like physics because you can't argue with scientific laws.)

But I do buy the notion math is one of many ways of looking at something and if we are serious about developing theories about the economy that are rooted in real phenomena, math should be a part of the way forward.

Just as Robert Solow argued capitalism OR socialism was a false dichotomy, I think perhaps we can think about math in this way too. 

How can we utilize math as a means to explain, predict, and simplify the way the economy actually works?

How can we use math to compare our theories to what we observe in the world?

Can math help incorporate the idea of a society, the implications of history, the idea of uncertainty into our theories about the economy?