What is the proper role of mathematics in economics?

There is a widespread public misconception about the level of difficulty of mathematics used by economists.

Perhaps the real issue is not the difficulty but the "proper" use of maths in this field.

Theoretical physicist Leo Smolin has lamented how "lesser" physicists are often contaminated by the improper use of maths. To a large extent, this is even worse among economists, especially those from the Chicago school who pay little attention to common-sense propositions. But the most advanced maths used by economists today may baffle even a well-trained physicist.

It is simply not true that much of mathematics for economics could be learned by a competent physicist in two months. In finance, one deals with random processes where the time series of price are not differentiable, but can be integrated.

This requires further study of such subfields as measure theory and Lebesgue integrals. To understand how they work further requires some heavy-duty understanding of Brownian motion, something for which Einstein won an Nobel prize and what is called Ito calculus.

Briefly put, this results in establishing an equivalence between a continuous partial differential equation and an Ito integral. It is not something I could easily master with just two months of intense study.

Even if you understand all that in a deep mathematical way, it still can't account for so-called black swans, improbable but high-impact events like the global financial crisis.

Investment bank charlatans tell their clients financial products are customised to hedge away portfolio risk. That's why Long-Term Capital Management (LTCM) - which had Robert Merton and Myron Scholes on its board and which went belly up after the Asian financial crisis, dared to operate with a 50 to 1 leverage.

Other examples of weak hypotheses abound in macroeconomics with their assumption of the existence of general equilibrium. An economy is a complex system with many components, each of which is an unknown feedback system with its characteristic time. The "equilibrium" in some financial markets might be reached in seconds while that of commodities such as crude oil might take months or years.

Nineteenth century French economists tended to be more mathematical than their English-speaking counterparts. Maths among American economists was unpopular until Paul Samuelson came on the scene in the 1960s.

Most Harvard economists initially rejected his approach at first. Essentially, his use of maths is somewhat similar to an undergraduate study in thermodynamics - more accurately in thermostatics. Nowadays, one finds in introductory economics textbooks the subject of comparative statics in the study of general equilibrium.

Milton Friedman made outstanding contributions to economics, but he also misled with statements that economic models don't have to comply with physical reality.

He famously wrote: "A theory cannot be tested by comparing its "assumptions" directly with "reality". Indeed, there is no meaningful way in which this can be done. Complete "realism" is clearly unattainable, and the question whether a theory is realistic "enough" can be settled only by seeing whether it yields predictions that are good enough for the purpose in hand or are beat predictions by alternative theories.

"The belief that a theory can be tested by the realism of its assumptions independently of the accuracy of its predictions is widespread and the source of much of the perennial criticism of economic theory as unrealistic."

In effect, Friedman gives licence to lesser economists who can get away with ignoring realities with their mathematical formalism.

Alas, even in the hallowed halls of Nobel laureates, they don't require stringent testing of hypotheses. Many prize winners advocate theories that are totally wrong but are still hailed as outstanding achievements. Worse, they fall in love with their own theories and pontificate them like political ideologues.

Mak Wai-lim holds a doctorate in physics and is a former university lecturer in the United States