Sunday, September 22, 2013

Mathematical Truths Do Not Make Untrue Assumptions in Economics True


Tim Johnston, is a Lecturer in Financial Mathematics at Heriot-Watt University in Edinburgh.  (Heriot-Watt was the first UK university to offer degrees in Actuarial Science and Financial Mathematics and is a leading UK research centre in the fields). He posted this article in Magic , Maths and Money” HERE
How economics suffers from de-politicised mathematics”
“Financial economics produced sophisticated mathematical theorems related to pricing and risk management in the derivative markets and  simply by existing as mathematics they were legitimate. There was no room for debate or discussion because mathematics, based on Hilbert’s formal deduction and Bourbaki’s idealised abstractions, and written in obscure notation, was infallible. It doesn't seem to matter that there were discussion and concerns within mathematics, economics accepted the authority of the theorems and their models simply because they were mathematical.
I have not yet come across what I feel is a credible reason why economics has become so enamored with formalist mathematics. Lawson [7, Ch 10] argues it is because mathematics confers authority, but gives no explanation as to why mathematics should have this power. Lawson challenges the power but one senses that he feels mathematics exists independently of human thought, and this mathematics is irrelevant to social phenomena. He does not seem to think that mathematics could be a product of economic intuition, not just physical intuition. Weitntraub [14] offers a narrative of how mathematical ideas crossed over into economics, without giving what I think is a compelling argument as to why mathematical formalism became so significant. Mirowski argues that ‘Cyborg science’, did not spontaneously emerge but was “constructed by a new breed of science managers” [9, p 15] and it was these managers that promoted the mathematisation of economics. While the emergence of ‘Cyborg science’ as a dominant theme of post-war science may well have been constructed, there is something spontaneous in Wiener, Turing and Kolmogorov, the leading twentieth century mathematicians of the US, UK and USSR, all independently having a youthful interest in biology, becoming mathematicians and making contributions in probability and going on to work in computation.
My own belief is that the critical process was the interaction between (particularly American) economists and mathematicians in the Second World War working on problems of Operations Research. At the outbreak of the war in 1939 the vast majority of soldiers and politicians would not have thought mathematicians had much to offer the war effort, the attitude among the military is still often that “war is a human activity that cannot be reduced to mathematical formulae” [12, p 3]. However, operational researchers had laid the foundations for Britain’s survival in the dark days of 1940-1941, Turing and his code-breakers had enabled the allies to keep one step ahead of the Nazis and Allied scientists had ensured that the scarce resources of men and arms were effectively allocated to achieving different objectives. Alan Bullock argues that the blitzkrieg was the only military tactic available to the Nazis, since they had neither the capability nor the capacity to manage more complex operations [2, pp 588–594]. By the end of the war, it could be argued that the war had been won as much through the efforts of awkward engineers as square-jawed commandos and the Supreme Commander of Allied Forces in Europe and Chief of the U.S. Army, General Eisenhower was calling for more scientists to support the military [12, p 64].
It is hardly surprising that in the post-war years economists embraced mathematics. Pre-war generals would have made the same sort of objections to mathematics that economists had. However, after the war the success of Operations Research could be compared to the failure of economists in the lead up to, and in the aftermath of, the Great Depression that had dominated the decade before the war. But possibly more significant than this theory is the fact that many post-war economists had worked alongside mathematicians on military and government policy problems during the war. Samuelson who was instrumental in introducing stochastic calculus had worked in Wiener’s lab at MIT addressing gun-control problems during the war [8, p 63–64].
Personally I feel prominent economists became over awed by the successes of mathematics, through, for example, observing mathematicians’ abilities to transform apparently random sequences of letters into meaningful messages, something that must have seemed magical and resonant to the economic problem of interpreting data. The problem is codes are generated deterministically but the same cannot be said for economic data. I believe it was a synthesis of the post First World War traumas of mathematics and the post-Second World War optimism and confidence of economics that created the explosion of mathematical economics in the 1950s-1960s.
Today mathematical finance is possibly the most abstract branch of applied mathematics, while mathematical physics is complex it is still connected to sensible phenomena and amenable to intuition, and this state seems to be typical of the relationship between mathematics and economics.  The situation is not irrecoverable, but, as I have said before, it requires a much tighter integration of non-mathematical economists and un-economic mathematicians.  I look with envy at my colleagues carrying out research  in biology using the same mathematical technology I use but, as one said recently, their papers do not need to prove a theorem and clear results are admired, not technical brilliance.”
Refs:
[7] T. Lawson. Reorienting Economics. Taylor & Francis, 2012.
[8] D. MacKenzie. An Engine, Not a Camera: How Financial Models Shape Markets. The MIT Press, 2008.
[9] P. Mirowski. Machine dreams: Economic agents as cyborgs. History of Political Economy, 29(1):13–40, 1998.
[10] PCBS. Changing Banking for Good. Technical report, The Parliamentary Commission on Banking Standards, 2013.
[11] Y. Rav. A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15(3):291–320, 2007.
[12] C. R. Schrader. History of Operations Research in the United States Army, Volume I: 1942–1962. U. S. Government Printing Office, 2006.
[13] I. Stewart. Bye–Bye Bourbaki: Paradigm shifts in mathematics. The Mathematical Gazette, 79(486):496–498, 1995.
[14] E. R. Weintraub. How Economics Became a Mathematical Science. Duke University Press, 2002. 
Comment
The above is an extract from a much longer post discussing aspects of the history of formalised theorems in post 19th-century maths and should be consulted to see where Tim Johnston places his interesting ideas about the consequential shadow of maths in economic thinking, or more particularly, in financial economics.
I tend to agree with Tim Johnston’s general slant (noticing that he is on the faculty of Heriot-Watt University from which I retired in 2005 after 33 years in the Department of Finance and Edinburgh Business School; I have not yet met him).
Broadly, I am remain sceptical of the arrogance of mathematical exponents in economics and their related philosophical ideology of ‘Max U’ thinking that dominated economics since the last quarter of the 19th century. Economic theory is not made any truer by maths. It is a basic error to project mathematical treatment as true of the real world.
Maximisation of utility ideas grip mainstream thinking.  If individuals are assumed to behave as if maximising utility theories are true in the real world, when they may not be true outside the assumption that they are, then even if the maths are true, the assumption of ‘MaxU’ may very well remain untrue.
On their own terms the maths of ‘MaxU’ are unassailable; on any terms ‘MaxU’ is not true of the real world.  Moreover, making decisions based upon the truth of predictions that may very well not be true, is dangerous, as the recent global financial crisis illustrates.
(Hat Tip to Mark Thoma HERE)

4 Comments:

Blogger Vitus Capital said...

"I feel prominent economists became over awed by the successes of mathematics"

Maybe I'm dumb. Could very well be.

"successes of mathematics" seems ridiculous. Have there been failures of mathematics?

I get that this is about a perceived over-mathematization of economics. Lots of good stuff written about this; see, eg, Noah Smith (http://is.gd/LdjgrL). But I don't see the actual point of the present post. Is it "I have not yet come across what I feel is a credible reason why economics has become so enamored with formalist mathematics."? This seems pretty obvious; in fact some of the reasons are mentioned above, then, for some reason, discarded.

As said, I'm probably dumb.

1:29 am  
Blogger John Randomness said...

The GIGO (garbage in garbage out) principle applies to math. Invariably in math one has assumptions, and one sees what follows from those assumption. Nevertheless, the conclusions are only as valid as the assumptions.

If y=4x and x=2, then y=8 (using standard multiplication of ordinary numbers) -- period. Nevertheless, we still have both assumptions, y=4x and x=2. The conclusion is only as valid as the assumptions.

In economics, if people use idiotic assumptions about human behavior it doesn't matter how good the math is. Furthermore, I've seen horrible math -- innumeracy -- in certain economics schools, mainly about exponential population growth facing limited resources.

Some things I've seen in economics and math remind me of Richard Feynman's experience with physics in Brazil. (See "Surely You're Joking, Mr. Feynman".)

5:40 am  
Blogger Gavin Kennedy said...

Vitus Capital
Thanks for your comments.
Posts address different readers. Not all agree with them. Some consider a post unnecessary or wrong. Others agree.
I am not sure in your case which of many sub-groups yours falls into.
Maths, like formal logic, has limits when applied to explain human behaviour. Those limits are not as wide as behaviours can be. That's all.
Gavin

6:40 am  
Blogger Gavin Kennedy said...

John Randomness
Agreed.
Applying philosophy to people likewise. Homo economicus rationality and marginal utility is a case in point too.
Learning it is a useful teaching idea in basic economics. Believing it as the real world is not. Histories of human societies are also useful but not sufficient. Humans are a complex species, with histories and a capacity for reasoning and discussing; the original faculties noted by Adam Smith at the beginning of Wealth Of Nations as the basis for exchange behaviour, from which economic behaviour evolved.
Gavin

6:55 am  

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