When I studied mathematics, many years ago, we began by doing analysis on the real number line. The course proved all the fundamental theorems of calculus on real numbers. In the second year I studied analysis on the complex plane and learned the proofs of the same theorems for complex numbers. The next year I moved on to n-dimensional space and, yes, the same theorems work in n dimensions. In my final year I took an optional course which proved the same theorems on topological space*.
When I went on to study economics I learned all sorts of results in simplified worlds where there were two goods and a budget constraint or a production possibility frontier. I assumed that the same results could be demonstrated for worlds with n goods and n-1 hyperplanes as budget constraints or PPF. Strangely no-one bothered to show that they did.
Thanks to Steve Keen's book, I now know the answer. Serious economists have indeed checked whether the simple models can be generalised. For example, the consumer theory model (the one with two goods where the consumer has a budget constraint) is used to demonstrate the downward slope of a demand curve. It might work when there are more than two goods but it falls apart once there is more than one consumer. Consumer theory only gives the traditional downward sloping demand schedule under conditions which amount to there being only one consumer.
In the first part of Debunking Economics Keen uses the results of economic research to show that most of microeconomic theory has been tested and found wanting. This matters for macroeconomics as the last few decades have been dominated by an approach which insists that macro has microfoundations. In effect, modern macro is built on crumbling foundations.
The second part of the book does the same wrecking job for macroeconomics. The argument here is more complex and demands more from the reader. One difficulty is that macro models are less well known than supply and demand. Few outside the economic profession understand the DSGE models used, nor even the simpler IS-LM model. Nevertheless, it is worth persevering; Keen is writing for a general audience. He points out the failures of conventional macro to deal with the nature of money and credit, the effect of time, and the analysis of disequilibrium. By the end the case for abandoning neoclassical economics is made.
In a final part, Keen sets out the alternatives. This section sits less well with the overall argument and has the effect of making the volume read like two books joined together. Perhaps it is, as the earlier version of the book - published before the crises - did not go into the alternatives in such detail.
I strongly recommend this book to anyone interested in how economics has failed in predicting or dealing with the current depression. It is an angry book which pours scorn on the mainstream economics profession. If you would prefer an less polemical approach, then I would point you towards a book by Marc Lavoie, Introduction to Post-Keynesian Economics.This is a short book with a more academic approach which covers similar ground.
*A topology is defined in such a way that it has all the properties needed to make calculus work, and no more.