## The misuse of mathematics in economics

from **Lars Syll**

Many American undergraduates in Economics interested in doing a Ph.D. are surprised to learn that the first year of an Econ Ph.D. feels much more like entering a Ph.D. in solving mathematical models by hand than it does with learning economics. Typically, there is very little reading or writing involved, but loads and loads of fast algebra is required. Why is it like this? …

One reason to use math is that it is easy to use math to trick people. Often, if you make your assump-tions in plain English, they will sound ridiculous. But if you couch them in terms of equations, integrals, and matrices, they will appear more sophisticated, and the unrealism of the assumptions may not be obvious, even to people with Ph.D.’s from places like Harvard and Stanford, or to editors at top theory journals such as Econometrica …

Given the importance of signaling in all walks of life, and given the power of math, not just to illuminate and to signal, but also to trick, confuse, and bewilder, it thus makes perfect sense that roughly 99% of the core training in an economics Ph.D. is in fact in math rather than economics.

Indeed.

No, there is nothing wrong with mathematics *per se*.

No, there is nothing wrong with applying mathematics to economics.

Mathematics is one valuable tool among other valuable tools for understanding and explaining things in economics.

What is, however, totally wrong, are the utterly simplistic beliefs that

• “math is the *only* valid tool”

• “math is *always and everywhere* self-evidently applicable”

• “math is all that really counts”

• “if it’s not in math, it’s not really economics”

*• *“almost *everything* can be adequately understood and analyzed with math”

Mainstream economists have always wanted to use their hammer, and so have decided to pretend that the world looks like a nail. Pretending that uncertainty can be reduced to risk and that all activities, relations, processes, and events can be adequately converted to pure numbers, have only contributed to making economics irrelevant and powerless when confronting real-world financial crises and economic havoc.

How do we put an end to this intellectual cataclysm? How do we re-establish credence and trust in economics as a science? Five changes are absolutely decisive.

(1) **Stop pretending that we have exact and rigorous answers on everything**. Because we don’t. We build models and theories and tell people that we can calculate and foresee the future. But we do this based on mathematical and statistical assumptions that often have little or nothing to do with reality. By pretending that there is no really important difference between model and reality we lull people into thinking that we have things under control. We haven’t! This false feeling of security was one of the factors that contributed to the financial crisis of 2008.

(2) **Stop the childish and exaggerated belief in mathematics giving answers to important economic questions**. Mathematics gives exact answers to exact questions. But the relevant and interesting questions we face in the economic realm are rarely of that kind. Questions like “Is 2 + 2 = 4?” are never posed in real economies. Instead of a fundamentally misplaced reliance on abstract mathematical-deductive-axiomatic models having anything of substance to contribute to our knowledge of real economies, it would be far better if we pursued “thicker” models and relevant empirical studies and observations.

(3) **Stop pretending that there are laws in economics**. There are no universal laws in economics. Economies are not like planetary systems or physics labs. The most we can aspire to in real economies is establishing possible tendencies with varying degrees of generalizability.

(4) **Stop treating other social sciences as poor relations.** Economics has long suffered from hubris. A more broad-minded and multifarious science would enrich today’s altogether too autistic economics.

(5) **Stop building models and making forecasts of the future based on totally unreal micro-founded macro models with intertemporally optimizing robot-like representative actors equipped with rational expectations.** This is pure nonsense. We have to build our models on assumptions that are not so blatantly in contradiction to reality. Assuming that people are green and come from Mars is not a good – not even as a ‘successive approximation’ – modeling strategy.

Mainstream economic theory today is still in the story-telling business whereby economic theorists create mathematical make-believe analog models of the target system – usually conceived as the real economic system. This mathematical modeling activity is considered useful and essential. To understand and explain relations between different entities in the real economy the predominant strategy is to build mathematical models and make things happen in these ‘analog-economy models’ rather than engineering things happening in real economies.

Without strong evidence, all kinds of absurd claims and nonsense may pretend to be science. Let us not forget what Paul Romer said in his masterful attack on ‘post-real’ economics a couple of years ago:

Math cannot establish the truth value of a fact. Never has. Never will.

We have to demand more of a justification than rather watered-down versions of ‘anything goes’ when it comes to the main postulates on which mainstream economics is founded. If one proposes ‘efficient markets’ or ‘rational expectations’ one also has to support their underlying assumptions. As a rule, none is given, which makes it rather puzzling how things like ‘efficient markets’ and ‘rational expectations’ have become standard modeling assumptions made in much of modern macroeconomics. The reason for this sad state of ‘modern’ economics is that economists often mistake mathematical beauty for truth.

I am less concerned with the math dominace than the exclusion of historians from the cohort of people involved in real world studies. alfred d chandler jr for example and the business historians in. general

Reblogged this on Calculus of Decay and commented:

EXCELLENT

I agree with everything Prof. Syll says. I would add one thing to his list of what economists should stop doing. This is to start by describing what an economy is and what it’s purpose is. If you say the purpose of the economy is to generate a situation where the most possible wealth ends up in the fewest possible hands, then your economics will be a certain way. If you say the purpose of the economy is to make sure every living person has at least the minimum of everything necessary for a healthy, safe, secure, and productive live, then your economics will probably look slightly different. If you are not willing to say clearly what you think is the purpose of economic activity (“the economy”, taken as a whole) then there is no reason to listen to you, no matter how fancy your mathematics are. It is not possible to prove that a better world means fewer people with tremendous wealth and more people living in poverty. Economists in the audience, please take note.

I always thought math was misued in Economics for the purpose of getting the title of a good science

..but it gets people confused an deceived…

“Many American undergraduates in Quantum Physics interested in doing a Ph.D. are surprised to learn that the first year of a Quantum Physics Ph.D. feels much more like entering a Ph.D. in solving mathematical models by hand than it does with learning quantum physics. Typically, there is very little reading or writing involved, but loads and loads of fast linear algebra is required.”

Is it ironic that physicists who say “there are no laws of physics” (please search for the phrase) probably think that mainstream economics has proved the rational expectations hypothesis, because both those physicists and Lars rely on superficial, popular versions of the others’ subject?

As for the misuse of mathematics in economics, we had a long discussion at the comment page on Lars Syll’s post

Mathematics and economicson May 5, 2018. There are 45 comments (to date).As Syll comments in the above and this posts,

Mathematics is one valuable tool among other valuable tools for understanding and explaining things in economics.However, it seems there is a kind of disrespect or contempt for mathematics in

The Real-World Economics Reviewand this RWER Blog.A typical example is given by the paper by Philip George, The giant blunder at the heart of General Equilibrium Theory,

Real-World Economics ReviewIssue 101, 2022. A part of the paper is reproduced as a post by the Editor in this RWER Blog:The real economy is never in equilibriumon December 5, 2022. Philip George claims thatIt is true that (sets of) prices are not a vector in a strict mathematical sense, but the term “vector” is simply used by convention as an abbreviation of “set of prices” or “relative prices”. This does not affect the validity of Arrow and Debreu’s existence theorem at all. However, George claims this is a giant blunder at the heart of General Equilibrium Theory. This is a pure allegation against General Equilibrium theory based on a simple misunderstanding. But the problem is much worse. This is in fact a big blow to the Real-World Economics movement and to heterodox economics in general. It damages the reputation of the Real-World Economics Review. George’s paper presents to mainstream economists a good example to claim that heterodox economists are persons who have no basic training in mathematics and lack common sense and hence heterodox economics is unworthy of serious examinations.

Hi Lars!

Thank you for your reflections, much of which I agree with. Economists’ belief in mathematical modelling may be as strong as you have described in your text, but my impression is that this belief is articulated today in a way that is much more cautious about criticism of the hardcore version of belief in mathematical modelling. So, mainstream economists typically promote themselves as ‘modern economists’ in the sense of modern empirical methods that rely heavily on ‘empirical facts’, of being free of theoretical school thinking (the latter is considered overcome) and (truly) ‘neutral’. Therefore, I wonder if we should not formulate our criticism of mathematical modelling in another way.

However, there is another problem: Mathematical modelling also dominates the field of heterodox economics. Although some heterodox economists agree with the criticism of mathematics in economics, this agreement often seems to be only lip service: The criticism of mathematics is used to put their own (heterodox) modelling in a nicer light, so that this criticism does not lead to a different pluralistic understanding of the use of scientific methods, but only to a different economic modelling – a heterodox economic modelling culture – … and the associated original critique is then shattered hopelessly through their own (heterodox) practice of mathematical modelling. In the end, my impression is that much of what you rightly criticise about mathematics in mainstream economics probably also applies to a large part of heterodox economics. And that is a rather uncomfortable aspect of it for heterodox economics.

Mr T.

How are you thinking about? Do you admit any difference of mathematics usage in mainstream and heterodox economics? If you don’t, it may imply that you are criticizing the usage of mathematics

per sein economics.My contention is totally different. It is wrong to criticize mathematics usage

per se. We should observe how mathematics is used in this and that economics (or this and that paper). There is good economics with good usage of mathematics. There is practically no good economics with bad usage of mathematics. (But don’t confuse this with the claim that there is good economics with no mathematics. I am not against it.)The problem with Lars Syll’s way of arguments is that he seldom enters to the contents of the economics that we examine. He is almost always questioning the apparent styles (or superficial appearance) of research. If “mathematics is one valuable tool among other valuable tools for understanding and explaining things in economics,” why doesn’t he argue whether mathematics is usefully or creatively applied in this and that research?

Yoshinorishi,

I am not sure you have met my point. My argument is aimed at a slight inconsistency in the heterodox critique.

Anyway, the debate about pluralism in economics shows different arguments against using mathematics in economics: some heterodox economists argue against the dominance of the mainstream’s way of applying mathematics, implying there are other kinds of mathematics (think about neoricardian economics versus neoclassical economics); some other heterodox economists criticize the mainstream economists’ inconsistency with their positivist Popperism-base of mathematical modelling (e.g. they concentrate on the true/ false assumptions); and finally, there are some heterodox economists concentrating on serious arguments that limit the use of mathematical modelling for economics (such as the criticism on deductivism by Tony Lawson).

In addition we should keep in mind that firstly hardly one heterodox economist calls to ban mathematics from economic curricula (there is simply no call for banning mathematics!). Secondly the pluralist movement criticized the dominance of mathematics in economics in general for already about 20 years. They do not call for another mathematics but they demand a fundamental different economics… they want more non-mathematical methods, more history of economic thought, more ethics and more epistemology in economics curricula. Keep in mind that mathematical modelling does not provide any proper insight for research – for instance – on history of economic thought or normativity in economics. At this point, any heterodox appeasement of mathematical modelling or defence of it is a major problem for pluralist economics, it is counterproductive. Thirdly, I guess most of ‘established’ heterodox economic approaches are mathematical modelling approaches. They may differ from mainstream economics in the kind of mathematics used, in some assumptions (e.g. ergodicity vs. non-ergodicity), and – perhaps – the attitude of applying mathematics and mathematical modelling. In the end, heterodox modelling differs from the modelling of mainstream economics, but it remains simply mathematical economics, just as mainstream economists insist on understanding economics as a mathematical discipline. There is a distance from what the movement of pluralist economists originally called for and the dominance of heterodox modelling economics.

As a result, the question for mathematics and the use of mathematics in economics is confronted with a lot of fundamental problems. Of course, there are serious arguments against mathematical modelling in economics, insofar as mathematics offers no help in understanding normative problems in economics or the history of economic thought, or in developing a vision of the economic future that is not necessarily based on markets. (But again, there may be strong arguments restricting the use of mathematical modelling in economics, but no heterodox economist has called for a ban on mathematical modelling in economics).

That gives a reasonable answer to your question: “why doesn’t he argue whether mathematics is usefully or creatively applied in this and that research?” The problem is that you want to apply mathematics (!) and you probably don’t like arguments that restrict the application of mathematics in economics. And that’s why I think you missed Lars’ point. He wasn’t arguing for mathematical modelling (as you would like), his critique implies attention to those marginalised parts of economics that are not necessarily based on mathematical modelling. It is about alternatives to the mathematical modelling culture in economics (heterodox or not).

Mr. T.,

you know very well what is happening in heterodox economics. It seems we can discuss productively.

First, please read my post on December 15, 2022 at 3:50 am in this comment page. Please skip first three paragraphs and read the part after that. How you do think about Philip George’s contention? Do you believe it helps to develop economics further? Or do you believe it is an effective criticism against mainstream economics? (As I have posted several other “Replies”, I do not enter in this question further.)

Second, you raised a good point of argument. Your third paragraph of your post on December 19, 2022 at 10:08 am deserves due attention. The problem is closely related to the intellectual situation of heterodox economics.

To make you grasp my position quickly, let me introduce my theory or research program briefly. In the words you used in the second paragraph of your post, I am a neo-Ricardian or Sraffian. However, it is necessary to note that Sraffians are not monolithic. Normally most Sraffians (or Ricardians) emphasize the importance of long-period analysis (See Kurz and Salvadori for example). I am thinking oppositely. The lack of a suitable framework for short period analyses was the greatest weakness of Sraffian economics. To build a framework that is lacking, I believe mathematics provides us various useful tools. This is not a pure wish. We have produced a new framework based on classical theory of value, but with a new theory of quantity adjustment process. See our book Shiozawa, Morioka, and Taniguchi (2019)

Microfoundations of Evolutionary Economics, Springer Japan. See also Marc Lavoie’s short introduction of our work in Subsection 3.7.4 in his new edition ofPost-Keynesian Economics2022. He admits that “the results achieved by Shiozawa et al. (2019) constitute a great breakthrough.” The main results obtained by Morioka were impossible without a deep analysis in linear algebra.Now, let me return to the problem raised in your third paragraph. I am much more fundamentalist than pluralist, because I believe a new unified economics is necessary and possible that can replace mainstream economics. Actually, there are many strands in heterodox economics. There is no need to oppose those various positions and stances. But I believe it is necessary to provide new foundations for almost all heterodox economics. As it is hinted in the Preface, our theory provides microfoundations not only for evolutionary economics but also for Post Keynesian (PK) economics. Of course, as you know well, there are many different stances among Post Keynesians (PKs) and many PKs are opposed to the very idea of microfoundations. In my understanding such PK economists have neoclassical backgrounds and cannot think that a new microeconomics is possible.

But, it is not my intention to make a war against them. What is most important is to build a new theory. If you agree with me on this point, the next problem is which stances are most productive and promising. Now, we must come back to your second paragraph. You raised three stances toward mathematics:

(1) Group of economists who consider (or hope) that different economics is possible and mathematics are often necessary and useful for building this new economics.

(2) Group of economists who endeavour to criticize the bad usage of mathematics and recommend to abandon mathematics.

(3) Lawson’s group who appeal to ontology to point out that good usage of mathematics is impossible because social economic system is open.

I do not agree with Lawson’s arguments but it is not our issue here. A half century has passed since 1970’s, the time where PK and NK antagonism became clear. Which was the most productive one among above three groups? I believe (1) was relatively more productive than other two. I admit that there are many new economics that do not use mathematics, like for example, Elinor Ostrom. But those people are rather indifferent about criticizing usage of mathematics in economics or more wider social sciences. For example, Elinor Ostrom considers that game theory treats too narrow topics as social sciences, but does not criticize its usage itself. She contends that we have much wider problem to study and analyze about rules (origins of rules and others) than solutions of games with which game theory people are uniquely concerned.

As a non-mathematician and a non-economist I am doubly unqualified to comment on this topic. So of course I will. The non-expert is confronted by thr experts who all disagree. In that situation, the non-expert might well be justified in saying, “I don’t know (the truth) and you don’t know either.” However that might be a little too much Pyrrhonism.

With respect to math what can I say as a non-mathematician? I mean I remember a little math from high school: some algebra for example some geometry and some basic physics equations but I have forgotten just about all of what little calculus I could once do. Will general reasoning suffice for me to take a position on this topic?

It seems to me the problem is the applicability of math to the real world. Leaving aside pure math we are here talking about applied math. To oversimplify, applied math does one thing. It counts real things. It applies a number, a quantity to an aggregation of real things or real stuff. Example, 10 apples. A number and real things.

Then in economics we do the following. We apply numbers to real things. 10 widgets. But we also apply numbers to nominal or formal things. 10 dollars. This is where all our trouble starts. We are trying to manage real systems, specifically the real economy and the real environment, with a hybrid theoretical system which counts real things and equates themn with counts of nominal non-things (dollars or any numeraire).

Real things obey real fundamental Laws, like the Laws of Thermodynamics, or if you don’t like couching if that way, real things obey fundamental, detectable, reliable regularities. Nominal non-things don’t obey these same Laws or regularities, like the Laws of Thermodynamics. Nominal non-things, formal objects, can be created ex nihilo, out of nothing, and destroyed back into nothing. We wave such things into and out of existence by the stroke of a pen or a computer key. I am talking about money.

So if you try to manage or theorize a real system by a melange of quantified real characteristics and of quantified nominal quantities you get a great big mess. You get what we have today: a system (capitalism) destroying the natural world.

The answer, in my opinion, is to demote the importance of money and finance. They will need to exist as chits or permissions to consume but that’s about it. Do all the calculations for real stuff needs with science and judge everything else via ethics, not via money as a default ethic. We currently use money as a proxy (a very poor one) for merit and even need. How much money people have is assumed to be merited and it is then assumed to be a right to selfishly keep and selfishly spend this money on ever-expanding excessive desires paraded as needs. My prescription? Stop managing society and systems with money. Sounds simple, in theory. ;)

How can you demote the importance of money and finance without using the threat of state violence to restrict fundamentally nonviolent behavior, and thus lose moral arguments to political populists who will keep getting elected to repeal the regulations that immorally restrict the liberty of those who just like to trade?

Instead, what if you let finance disconnect itself entirely from real things, by giving everyone a universal, unconditional, non-tax-funded, generous, inflation-indexed basic income to insure them against irrational financial panics, and just let traders knock themselves out distracting themselves from doing actual physical harm by playing in financial markets with fictitious assets?

Ikonoclast

I posted a comment on your long Reply on December 15, 2022 at 9:17 pm to the page

https://rwer.wordpress.com/2022/12/05/the-real-economy-is-never-in-equilibrium/#comments

Because it is still blocked with the reason that it is awaiting moderation, I post the same comment here:

Number 6 is factored as 2 times 3. Since 2 is not a prime number, the fundamental theorem of arithmetic does not hold. Hence, the mathematics based on arithmetics is all wrong.

It is free to post your opinion on how the economics should be (or what the economics should be concerned with). But, expressing such a general opinion on the economics and the issue I raised in “the real economy is never in equilibrium” page are two different things. I wonder why you mix up everything in a single argument.

As for many citations from “Philosophy of Complex System” (Edited by Cliff Hooker), I am glad to see that you have become interested in questions of complexity. It is one of most important questions in economics, because there exists a neglect of complexity at the core of mainstream economics and because an economy is a large complex system. Your post recalled me of the arguments between us of about two years ago:

https://rwer.wordpress.com/2021/02/18/information-take-two/#comment-178487

I have no intention to enter here in the question of complexity. But, let me add a few words on it.

Some mainstream economists talk about complexity, but they think complexity as a question of second approximation. So, they do not think we need fundamental reformulations of economic behaviors including optimization. If we consider about complexity, theory of intentional actions must change. But, to repeat, the issue I raised in the “the real economy is never in equilibrium” is totally irrelevant with this complexity arguments.

I beg pardon of dear readers. I made a mistake in the above post. The following part should be a block quotation within the citation of my post on December 16, 2022 at 1:57 pm:

“Number 6 is factored as 2 times 3. Since 2 is not a prime number, the fundamental theorem of arithmetic does not hold. Hence, the mathematics based on arithmetics is all wrong.”

Is all arithmetics incomplete, meaning that true math statements exist that you cannot prove? How important are those true statements that your model can’t prove?

Also, why doesn’t the Banach-Tarski paradox lead to classical explosion and thus, trivialism?

Dear rms,

Please be aware that I am not claiming that mathematics is all wrong. The illustration I gave here is an example which is logically wrong. 2 is a prime number. (There is no relation with the incompleteness theorem of axiomatic arithmetics, or Gödel’s theorem.)

Nobody can claim something that is deduced from a false contention. I raised this example in order to show that Philip George’s claim that there is “a giant blunder at the heart of General Equilibrium Theory” (RWER #101) is a contention similar to the above illustration.

《There is no relation with the incompleteness theorem of axiomatic arithmetics, or Gödel’s theorem.》

If mathematics is incomplete, how do you know there is not a true mathematical statement that contradicts any statement you prove using mathematics?

If mathematics is inconsistent, can you prove anything, even Philip George’s claims?

(For the record, may I note that my personal opinion is that direction is not necessary to define vectors because qubits have no direction; however, does it all just come down to opinion since mathematics is either inconsistent or incomplete, or both?)

All very abstract. Should the Fed raise interest rates again? What effect will that have? Should it do so by the expedient of paying commercial banks enormous sums of money – its current technique for raising interest rates? What purpose did Quantitative Easing actually serve? Can a country like the United States where the executive power is not in control of fiscal policy have sensible macro-economic policies?

There are lots of real questions in economics, affecting people’s lives. Would it be more productive to discuss topics in economics rather than discussing “economics”? A great deal of academic economics and economic commentary is nonsense. But good practice applied to real problems is a more potent instrument of reform than discussion of methodology conducted at a level of abstraction that exceeds even that of the useless models against which Lars preaches.

Well that all depends if you think the current paradigm is still fundamentally okay or has fundamental problems. I happen to think the current paradigm, capitalism, has fundamental problems. But I admit that me railing against it will do no good at all. Then again, I also admit me taking positions against QE or fiscal policy and practice in the current system will do no good either. I have no power or influence. This is just as the great majority of citizens in our system have no power or influence. Nobody listens to us.

So why participate in debate at all? I admit I ask myself this question a lot more these days. Nobody with a formed ideological, political economy or religious view ever seems to be swayed by debate, or facts, in my experience. So who is the debate for and who it is directed against? Well, it is true that unformed, uneducated minds can be captured. maybe in that time window from age 15 to age 25. So, capture them in education I guess. Which is why education is contested: culture wars, history wars, “woke” wars etc.

At the political level, no matter who we vote for in this system we get a neoliberal (a market fundamentalist) into power; one who took massive campaign donations from capitalists and always does exactly what the dominant capitalists want. Both major parties are almost the same on economic and broad social issues but differ somewhat on the window-dressing aspects of identity issues. That’s about it.

So most of the debate, posting and argey-bargey is about careerism and ego. Since I have no career and as I have said no power or influence in any shape or form, then for me it must about my lack of life achievements and fragile ego. Being philosophically consistent I hope, I think an iconoclast certainly should smash and trash his own image first.

I think we now have an automatic system: everyone is taught to automatically apply its axioms and rules with little question, at least with little question as to the validity of the entire system itself. This automatic system assumes infinite resources, equilibrium theory and has in a general a great very normalcy bias. It cannot envision anything is wrong with it, it cannot envision it is not eternal, it cannot envision any alternative.

The only way such a system can change is from exogenous factors, external shocks, which stress it to its core and shatter it. This is happening right now. The system is breaking, shattering, but normalcy bias cannot see it and will deny it all as long as possible. This will only make the shattering worse.

When a system fails, catastrophically, then people do begin to question again and they can come to believe in different possibilities: for example in forms of socialism, equality and sustainability. It just might happen. Or we might totally collapse into barbarism and perhaps even extinction. The latter looks far more likely now.

Hi Gerald,

please read my reply posted above on December 15, 2022 at 3:50 am. Although it is not related to economic policies, it is closely related to how we do economics. It is true that we should not misuse mathematics in economics, but it is also wrong to misuse wrong mathematics to accuse mainstream economics. If we do, there is no difference with fake news propagators.

Good to see you commenting Gerald. I hope you and your loved ones are well. I think you are right, there are “lots of real questions in economics, affecting people’s lives,” and learning about “good practice applied to real problems” is how reform happens in the world. I have read many good books (including one’s published by WEA). But they are not books published by WEA in general that address such pragmatic real-world problems. Have you read any you recommend lately?

Missed the most important universal failing by economists, i.e. math cannot be applied without first defining the units appropriately. The use of math in terms of $ or € etc., is inapplicable to any reality without first defining those units intelligibly in terms of that reality.