Jason Smith asserts: “The primary purpose of mathematical theory is to provide equations that illustrate relationships between sets of numerical data.”
This is a bit shallow and does not reach the level of Wikipedia: “Mathematics is the study of topics such as quantity (numbers), structure, space, and change. … Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.”#1
Why mathematics is so admirably appropriate to the objects of reality is not fully understood: “I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with what is going on in the original thing.” (Feynman) see also (Wigner) and (Velupillai)
How does this relate to economics? Walrasian economics, too, is axiomatized, the hard core premises are verbally given as follows: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.” (Weintraub)
It should be pretty obvious that the Walrasian axiom set contains THREE NONENTITIES: (i) constrained optimization (HC2), (ii) rational expectations (HC4), (iii) equilibrium (HC5). Every model that contains a nonentity is A PRIORI false. And this is why mathematics does not work in economics and why economics is a failed science or what Feynman famously called a cargo cult science.
In practical terms, it follows immediately: as soon as the word equilibrium/disequilibrium appears in an economic paper or textbook it can be thrown into the waste basket. The same holds for all other nonentities. Note well that this holds also for Jason Smith’s information equilibrium.
The decisive insight for the role of mathematics in economics is: “If it isn’t macro-axiomatized, it isn’t economics.”
The natural math of economics is the elementary math of accounting. This is the formalism to start with and NOT SS-function-DD-function-solution.#2 The problem with economists is that they grab a piece of math from the math department and apply it without deeper understanding.
The cargo cultic methodologist Jason Smith asserts: “A big step in using math to understand the world is when you’ve collected several different empirically successful models into a single paradigm or framework. That’s what Newton did in the seventeenth century. He collected Kepler’s, Galileo’s, and others’ empirical successes into a framework we call Newtonian mechanics.”
And how does Newtonian mechanics start? Yes, with the axioms of motion, see Axiomata Sive Leges Motus at the very beginning of Principia.#3
Physicists got the axioms right and this is why math works, economists messed up the axioms (first and foremost with HC5, i.e. equilibrium), and this is why the representative economist became a scientific laughing stock. The economist-turned-physicist Jason Smith is no exception.#4
#1 Wikipedia Mathematics
#2 Macro for dummies
#3 Wikipedia Newton’s laws of motion
#4 You are fired!
For details of the bigger picture see cross-references Math/Mathiness
You say: “Physics envy again. … Changing the framework in physics doesn’t fundamentally alter the behaviour of the elements studied ― even in quantum physics. It does in social sciences and economics ― because they involve people with brains and emotions that change their mind.”
This is not a case of physics envy but of social science delusion. Economics is a system science, i.e. about the structure/behavior of the economy, and NOT a social science, i.e. about Human Nature/motives/behavior action. This is the subject matter of psychology, sociology, political science, etcetera.
Economists are simply at the wrong party and have not realized it since the founding fathers.
Standard economics is based on behavioral axioms (constrained optimization, rational expectations, equilibrium) and mathematics simply does NOT work with nonentities. The calculation that when three angels and four angels dance on a pinpoint then the total is seven angels is not applied arithmetic but brain-dead crap.
It is well-known among mathematicians, but not among economists, that not all mathematical structures incorporate “certain aspect of empirical reality”, which means, that there is a “... whole crop of monster-structures, entirely without application.” (Bourbaki)
Standard economics, i.e. supply-demand-equilibrium, is such a monster-structure, entirely without application. This is NOT the fault of mathematics but of abysmally incompetent economists. Count Jason Smith and yourself among them.*
* Incompetence — the original sin in economics
You quote: “Newton’s law of gravity hasn’t changed for eons, Derman said, but human behavior in markets changes all the time, wreaking havoc on even the best models made by scientists.”
WOW, what a revelation! Guess what, this has been known to scientists of all ages except, of course, to retarded economists: “The bifurcation of motion into two fundamentally different types, one for natural motions of non-living objects and another for acts of human volition … is obviously related to the issue of free will, and demonstrates the strong tendency of scientists in all ages to exempt human behavior from the natural laws of physics, and to regard motions resulting from human actions as original, in the sense that they need not be attributed to other motions.” (Brown)#1
Economists, in their ignorance, built economics on a set of BEHAVIORAL axioms with utility maximization at the core. This was 140+ years ago and neither has Orthodoxy abandoned this proto-scientific rubbish nor has Heterodoxy come forward with something better. Economics has never risen above the level of folk psychology and folk sociology and second-guessing other people’s expectations.
Of course, there is NO such thing as a behavioral law or regularity or stable functional relationship, never was, never will be. Being axiomatically false, economics has to be abandonment and fully replaced.
There is no use to criticize Walrasianism, Keynesianism, Marxianism, Austrianism. It is ALL proto-scientific sitcom blather. The only question is how to replace these failed approaches as fast as possible.
The first step to a paradigm shift is to understand that economics is not a so-called social science like psychology/sociology and not a natural science like physics but a system science. The economist’s proper task is to look out for objective systemic laws and to empirically verify/falsify them. Science is about the invariants beneath changes on the surface and not a commonsensical description of what happens here and now. There are NO BEHAVIORAL laws but there are SYSTEMIC laws, and they are as objective, certain, mathematically exact, and eternal like physical laws.#2
#1 The existence of economic laws and the nonexistence of behavioral laws
#2 New Economic Thinking: the 10 crucial points
Everybody can climb on a soap box and make an economic proposal ― except an economist. In the political realm, anything goes and no qualification is needed, only an emotionally backed opinion and some rhetorical talent. An economist, on the other hand, is supposed to be a scientist and to know what he is speaking about, that is, to know how the economy works.
Economists who have no scientific knowledge are at the same level with political cranks and these morons thrive when the economy is sluggish or worse: “A sure sign of a crisis is the prevalence of cranks. It is characteristic of a crisis in theory that cranks get a hearing from the public which orthodoxy is failing to satisfy. In the thirties we had Major Douglas, and social credit — it can all be done with a fountain pen — and Warren and Pearson who convinced President Roosevelt that raising the dollar price of gold would raise the price of everything else and bring the slump to an end. The cranks are to be preferred to the orthodox because they see that there is a problem. Nowadays we have plenty of cranks taking up the problems that the economists overlook.” (Joan Robinson)
The difference between a crank and an economist is NOT in the political orientation but in the necessity that the proposal of the economist must have a sound theoretical foundation. That is, the economist must be in the possession of the true theory: “In order to tell the politicians and practitioners something about causes and best means, the economist needs the true theory or else he has not much more to offer than educated common sense or his personal opinion.” (Stigum)
Now, scientific truth is defined by material and formal consistency and the problem of economics is that we know for sure that economics is false: the four main approaches — Walrasianism, Keynesianism, Marxianism, Austrianism — are materially and logically inconsistent.
So, people like Krugman have to be opposed NOT because they are liberal or conservative but because they are scientific fraudsters: they speak in the name of economic science but there is NO such thing as economic science.
How is scientific knowledge established? “Research is in fact a continuous discussion of the consistency of theories: formal consistency insofar as the discussion relates to the logical cohesion of what is asserted in joint theories; material consistency insofar as the agreement of observations with theories is concerned.” (Klant)
And this is the point where math comes in. Math is simply the best available means to establish logical cohesion. And here, in turn, is where the problem of economists with math comes in: as natural born muddleheads and cranks they abhor nothing more than logic. Fact is that economists do not even get the elementary mathematics of accounting right.* Where things become comical is when these folks cover their stupidity by posing as philosophers.
* Macro for dummies
The curriculum at British schools is the business of the British people. Likewise for the US. No economist has to tell these or other countries how to organize their education and what to teach. The business of economists is to figure out how the economy = world economy works. Fact is that economists are failed/fake scientists. They do not even know the elementary mathematics of accounting, which, if anything, is the minimum condition of doing economics.*
Economists are a public nuisance because they have an opinion on everything but knowledge of nothing.
* See also ‘A new curriculum for swampies?’