<blockquote>
  <p>What is a reason to be against mathematics in economics?</p>
</blockquote>

<p>The danger that any tool creates: to impose itself on the tool-user, diluting and narrowing its view of the world. It is a matter of Human Psychology why this happens, but it certainly does, and the aphorism "to he who holds a hammer everything looks like a nail" expresses this phenomenon, which has nothing to do with economics specifically.</p>

<p>Mathematics offer a great service to the Economics discipline by providing a crystal-clear path from premises to conclusions. I fear the next time a Keynes with a General Theory book appears -and then we would have to spend decades again deciphering "what the author really meant" by his verbal arguments -and not really agreeing.</p>

<p>The "abuse of mathematics" certainly happens: producers and consumers of economic theory tend to not question/worry/have nightmares about "the premises", to the extend that they should. But once we leave the premises unchallenged, the conclusions become "undeniable truth", since they have been derived in the rigorous mathematical way.</p>

<p>But the ability to challenge the conclusions is always there, if only we take the time to review critically the premises.</p>

<p>Another, more sophisticated way that mathematics may be abused is the belief that the deviation from reality that the premises represent, transfers over to conclusions in a "smooth" manner (call it "the principle of non-accelerating propagation of error"): to consider the trivial example, sure, the assumptions describing a "perfectly competitive" market (the premises) do not hold "exactly" in reality. But, we argue, if they are "close enough" to the structure of a real-world market, then the conclusions we will reach through our model will be "close enough" to the actual outcomes in this market. This belief is not unreasonable, and it is borne by reality in many cases. But this principle of "smooth approximation" does not hold universally. So even if we check the premises and we are satisfied that they are not absurd, it still remains the possibility that the real-world phenomenon we examine is not so well-behaved, it has discontinuities that may make a small deviation lead to very different outcomes in reality. </p>

<p>That's the abstract analysis of the matter. The sociological and historical view would ask "but if a tool, that theoretically can be used in the proper way, has been seen for decades to be used inappropriately and create undesirable consequences, shouldn't we conclude that we must abandon its use?"</p>

<p>...in which instant, we start arguing about <em>the extent</em> of these "undesirable consequences" and whether they overcome any benefits from the use of the tool. In other words, this matter too, dreadfully comes down to a cost-benefit analysis. And we rarely agree on that either.</p>

CC BY-SA 4.0
2018-07-01T15:06:44.743Z
The trouble with this argument is that whatever other stuff we use for economics are also tools. It's not like math is a tool but the other stuff we use are fully legitimate truth-finders blessed with kisses from Jesus Christ. Our views will be inherently "diluting and narrowing", otherwise you are supposing that non-math appraches to economics allow us to see the entire reality as it is.
Red Banana
CC BY-SA 4.0
2018-07-01T15:44:35.233Z
@BillyRubina I am not sure I follow you. Where in my answer is it implied that "other stuff we use" do _not_ constrain us? And where do I imply that we would be better off without math?
Alecos Papadopoulos
CC BY-SA 4.0
2019-10-16T15:55:36.320Z
Regarding "next time a Keynes with a General Theory book appears": Piketty tried to be that next writer. His book was also less mathematical, and the profession immediately poked holes into it, e.g. http://www.econ.yale.edu//smith/piketty1.pdf
FooBar
CC BY-SA 4.0
2018-06-29T10:42:20.997Z
2018-06-29T11:00:49.950Z
2018-06-29T11:00:49.950Z
Alecos Papadopoulos
<p>I would like to point out that <a href="https://economics.stackexchange.com/q/22599/1601">the question</a> is not whether we should have math in economics, but why some people attack mathematical economics. A lot of the recent answers seem to try to answer the first question.</p>
<p>Now then, to cover all bases like a good incumbent in a differentiated product market, I will also post an answer with points that economists have already raised about this question.</p>
<p>Hayek in his Nobel Lecture:<a href="https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1974/hayek-lecture.html" rel="nofollow noreferrer">The Pretence of Knowledge</a> said</p>
<blockquote>
<p>It seems to me that this failure of the economists to guide policy more successfully is closely connected with their propensity to imitate as closely as possible the procedures of the brilliantly successful physical sciences - an attempt which in our field may lead to outright error. It is an approach which has come to be described as the &quot;scientistic&quot; attitude - an attitude which, as I defined it some thirty years ago, &quot;is decidedly unscientific in the true sense of the word, since it involves a mechanical and uncritical application of habits of thought to fields different from those in which they have been formed.&quot;</p>
</blockquote>
<p>Paul Romer coined the term <a href="https://en.wikipedia.org/wiki/Mathiness" rel="nofollow noreferrer">mathiness</a> to describe the issue in his (unrefereed) paper <a href="https://paulromer.net/mathiness/Mathiness.pdf" rel="nofollow noreferrer">Mathiness in the Theory of Economic Growth</a>. He writes</p>
<blockquote>
<p>The market for mathematical theory can survive a few lemon articles
filled with mathiness. Readers will put a small discount on any
article with mathematical symbols, but will still find it worth their
while to work through and verify that the formal arguments are
correct, that the connection between the symbols and the words is
tight, and that the theoretical concepts have implications for
measurement and observation. But after readers have been disappointed
too often by mathiness that wastes their time, they will stop taking
seriously any paper that contains mathematical symbols. In response,
authors will stop doing the hard work that it takes to supply real
mathematical theory. If no one is putting in the work to distinguish
between mathiness and mathematical theory, why not cut a few corners
and take advantage of the slippage that mathiness allows? The market
for mathematical theory will collapse. Only mathiness will be left. It
will be worth little, but cheap to produce, so it might survive as
entertainment.</p>
</blockquote>
<p>He goes on to give specific examples of 'mathiness', including works by high profile economists like Lucas and Piketty.</p>
<p>Tim Harford provides a laymen's summary of Romer's paper in his blogpost <a href="http://timharford.com/2015/06/down-with-mathiness/" rel="nofollow noreferrer">Down with mathiness!</a> In this he writes</p>
<blockquote>
<p>As some academics hide nonsense amid the maths, others will conclude
that there is little reward in taking any of the mathematics
seriously. It is hard work, after all, to understand a formal economic
model. If the model turns out to be more of a party trick than a
good-faith effort to clarify thought, then why bother?</p>
<p>Romer focuses his criticism on a small corner of academic economics,
and professional economists differ over whether his targets truly
deserve such scorn. Regardless, I am convinced that the malaise Romer
and Orwell describe is infecting the way we use statistics in politics
and public life.</p>
<p>There being more statistics around than ever, it has never been easier
to make a statistical claim in service of a political argument.</p>
</blockquote>

CC BY-SA 4.0
2018-06-29T22:07:50.627Z
(+1) for the references, especially Romer's. Setting aside the gossipy issue related to his direct attack on household names like Lucas and Prescott, the most interesting thing here is this concept of "mathiness", which is subtle, because it is not about "garbage premises and then super mathematics" but about something much more subtle but equally critical: mapping verbal concepts to math symbols _without proper justification_. This is much more difficult to detect in a paper, if you are not really experienced.
Alecos Papadopoulos
CC BY-SA 4.0
2019-11-15T00:01:56.430Z
I can't find a precise article, but I could have sworn modern economics violates key mathematical axioms that make the math work in the first place. Would love to have a theoretical mathematician address the problem. Probably, has something to do with the axiom of choice is my guess.
ZeroPhase
CC BY-SA 4.0
2021-06-01T18:03:46.660Z
Robert Solow had a similar opinion: https://economistsview.typepad.com/economistsview/2009/08/solow-dumb-and-dumber-in-macroeconomics.html
Daniel
CC BY-SA 4.0
2018-06-29T12:55:17.900Z
2021-02-21T16:28:08.307Z
2021-02-21T16:28:08.307Z
Giskard
<p>I think there are two important criticisms or limitations.</p>

<p><strong>Limit 1:</strong>  The first, overlapping with what many others have said, is that all mathematic economics are reduced-order models of very highly complex relationships between monumentally complex actors. As Einstein is alleged to have said (approximately) "Insofar as the truths of mathematics relate to mathematics, they are certain. In so far as they relate to the world, they are not certain." 'Does this mathematics apply in this situation?' is always an open question.  Similarly, 'Is there a better mathematics that we haven't yet discovered?'</p>

<p><strong>Limit 2:</strong>  The other problem, and it is bigger for economics than any other field I can think of, is the extent to which the state of the art knowledge of economics changes the economy because it becomes 'common knowledge'.  For example, when you convincingly show that investing in mortgage-backed securities is low-risk compared to the yield, and that home-ownership is a cornerstone of wealth-creation for ordinary people, the economy will pile into those things until the apparent excess value is consumed. This feedback and phase-changeability means that economies are non-ergodic - (apparently NN Taleb makes a lot of this point in Black Swan?)  </p>

<p>Even if economic knowledge were not encoded in policies of economic actors, the changing nature of society and technology will always cause problems under Limit 1.  Neither of these limits argues for excluding mathematics from economics, but they do argue for not excluding non-mathematical considerations (e.g. the political side of political economy) from economics. In practice, this might mean a bit more authority for the judgment of older economists who are wary of, for example, the value of high-speed trading.</p>

CC BY-SA 4.0
2018-06-29T00:34:08.667Z
2018-06-29T00:34:08.667Z
Atcrank
<p>I think that the opposition to mathematics in Economics mainly has to do with the obstacles it poses to <em>indoctrination</em>. </p>

<p>A proposition expressed in terms of a mathematical/logic system is susceptible of objective verification, whence the inconsistencies of a proposition are more visible than where a rigid framework is missing. Moreover, mathematical propositions do not lend themselves to the hyperbole and passionate impetus that fuel a socio-political ideology.</p>

<p>The excerpt cited by @denesp reflects Levinotiz's confusion between the rules of logic and the rules of grammar. Despite the definiteness inherent to Latin grammar and the complexity of expressions it allows, its lack of logical rules and relations of consistency render grammar useless as method of proof.</p>

CC BY-SA 4.0
2018-06-28T19:39:54.617Z
Reminds me of  the words of Roger Beacon: “Neglect of mathematics work injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy.”
EconJohn
CC BY-SA 4.0
2018-06-28T20:04:06.707Z
@EconJohn Exactly, and that leads to a clash of irreconcilable conclusions reached from subjective, unsystematic assessments. Marx's propositions such as "religion is the opiate of the masses" pertain to sociology rather than Economics. Adam Smith's idea of the invisible hand expresses an assumption from which causal arguments can be developed. But the social or subjective origin of an assumption or a perception is not a good reason to exclude a formal, verifiable system of logic for the development of a theory.
Iñaki Viggers
CC BY-SA 4.0
2021-02-20T14:55:35.507Z
Actually I think quite the opposite, mathematics has been used as a great way to indoctrinate generations of people with free-market ideology. The assumptions needed to get to these conclusions e.g. walrasian auctioneer, homothetic preferences, constant returns to scale are absurd but hidden to justify free market agenda.
Papayapap
CC BY-SA 4.0
2021-02-22T00:12:48.510Z
@Papayapap You seem to confuse between assumption and conclusion. *Free market* is most often an *assumption* without which the rest of a model would be pointless and therefore less susceptible of serious or more objective assessment. In turn, interventionists' failure to use mathematics "*as a great way to indoctrinate*" is a clear indicator of the arbitrariness and distortions inherent to their ideology. Mathematics is quite useful for scrutinizing and dismantling the inconsistencies of the rhetoric of interventionism.
Iñaki Viggers
CC BY-SA 4.0
2021-02-22T05:08:59.073Z
@Papayapap I think it would be much more difficult to "indoctrinate" people using math. In math you are forced to disclose all assumptions (which the reader may discard as unrealistic) while in vague qualitative essays even the most well trained reader can fail to identify a hundred assumptions (and without even knowing the assumption, it is hard to see whether the assumptions are reasonable or not). Add to that the fact that qualitative assumptions may mean different things to different people and that it is even difficult to see whether the conclusions even follow from the assumptions or not
Ishan Kashyap Hazarika
CC BY-SA 4.0
2021-02-22T05:12:07.937Z
@Papayapap Adam Smith's qualitative arguments for "free market's efficiency" seemed infallible. It is only when Arrow, Debreau and others mathematically formalised the arguments that we now exactly the know the assumptions under which free markets are efficient and when they are not. This appears, at least personally to me, to be much more honest than verbose arguments. Markets fail when there are externalities, market power, incomplete information, or even in overlapping generations etc.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2021-02-22T07:31:47.703Z
@IshanKashyapHazarika Thanks for citing Arrow and Debreu. Walras was concerned with the question of whether multiple free markets can clear. He concluded that was not possible without an auctioneer setting prices to clear. Arrow and Debreu skipped this fundamental step completely and just assumed that markets would clear. A very bright example of how extensive math can be used to hide important conceptual defficiencies. QED.
Papayapap
CC BY-SA 4.0
2021-02-22T07:32:42.747Z
@IñakiViggers are you an economist? The work done to show that free markets are pareto-efficient is done to show that state intervention is inefficient and markets should be left unregulated. This works starts with absurd assumptions, uses valid mathematical steps, but reaches wrong conclusions. 20th century economists knew that markets need to be regulated to limit market power, set the rules of the game etc. The mathematical dillusions of the later mathematical economists produced only confusion...
Papayapap
CC BY-SA 4.0
2021-02-22T11:27:20.847Z
@Papayapap You did not specify which work in particular you're referring to, let alone which assumptions/conclusions seem absurd/wrong to you. For instance, if by "absurd" you refer to the "auctioneer" being a fiction, you are getting stuck with a pedagogical artifact and missing the abstraction altogether. Likewise, economists adopt the assumption of information symmetry, but that does not mean they preach that such symmetry exists. Regardless, the paper you purportedly have in mind does not invalidate my point that *most often* free market is an assumption.
Iñaki Viggers
CC BY-SA 4.0
2018-06-28T19:32:20.750Z
2018-06-28T19:32:20.750Z
Iñaki Viggers
<h2>"All models are wrong; some are useful."</h2>

<p>The title is really all one needs, but to put a few more words behind it, mathematics is very good at deriving detailed results from very specific premises.  It is very easy to make a mistake in the premises and obscure the consequences with language.</p>

<p>A major issue in macroeconomics is that every policy decision must be self-referential.  It's very easy to accidentally assume that some small actor won't change their decision making slightly in an unexpected way that makes the whole thing fall apart.  It's also very easy to make the mathematics look airtight.</p>

<p>In more microeconomic situations, you have assumptions about how the world will function.  This is most easily seen by developing an AI which can make a killing when fed historical data, but which fails utterly in the real market.</p>

CC BY-SA 4.0
2018-06-30T23:18:18.053Z
For those who don't know, the header is quote by British statistician George Box. One of my all-time favorite quotes!
sam
CC BY-SA 4.0
2018-06-30T23:36:58.560Z
@Sam Good point.  I've put quotes on the header to make it more obvious that it's a quote.  I'm a programmer by trade, so I live and die by that quote!
Cort Ammon
CC BY-SA 4.0
2018-06-30T20:37:32.037Z
2018-06-30T23:36:23.607Z
2018-06-30T23:36:23.607Z
Cort Ammon
<p>In my experience the most important reason is that economics has political implications and that creates a huge moral hazard to use complex incomprehensible math to arrive to politically desirable conclusions.</p>

<p>Unlike in natural sciences, economic models can hardly be verified empirically and require tons of assumptions. Add a thick layer of math on top and you can support pretty much anything. In fact, anything beyond linear regression hardly improves predictive power in practice.</p>

<p>Seasoned economists see through this. Some are in on it (hey, it is very profitable!) and some are pretty unhappy about all this abuse of math, which is unethical from a scientific point of view. But I guess many are both. At the end of the day, we all have bills to pay and families to feed. Nevertheless, we are still scientists. So there is a lot of cognitive dissonance and strong feelings going on.</p>

CC BY-SA 4.0
2018-07-01T13:23:20.380Z
I think most physics models also require a ton of assumptions. It is their empirical verification that is better. Perhaps the system they study can be more frequently decomposed into smaller independent parts.
Giskard
CC BY-SA 4.0
2018-12-10T07:45:48.960Z
Economic models not only can but get constantly empirically verified. Why do people make strong claims about subjects they clearly don't know?

Just see what people are publishing in the frontier journals: https://academic.oup.com/qje/issue. Most, if not all papers published in these good journals empirically verify a theoretical hypothesis or conclusion from a model.
Pedro Cavalcante
CC BY-SA 4.0
2018-12-10T09:25:54.597Z
@PedroCavalcanteOliveira man, [QJE is #1](https://ideas.repec.org/top/top.journals.simple.html). There are thousands of economics journals below it that will publish things of much lower rigor, if any rigor at all, and politicians use those just as well to push the policies of their choice. Guess how many bother to replicate and test any of it? That would require funding. From the same politicians, that is, or an NGO with its own agenda. That's why when I see things that are supercomplex for the sake of a bit higher accuracy but take tons of time and resources to test, I get a bit critical.
Arthur Tarasov
CC BY-SA 4.0
2018-12-10T22:30:52.397Z
You can't look at the worst outlets of a field and claim that there's a problem with it because they're bad. If that's reasonable, then literally all sciences are in big trouble. And bringing this generic argument about politicians basing themselves on bad journals isn't any good. Who are these politicians? Where and when did this happen? Can we blame economics as a field on it? Your claim was that "economic models can hardly be verified empirically", which is clearly wrong. Most of the papers published in any respectable journal ate empiric. That should be your standard.
Pedro Cavalcante
CC BY-SA 4.0
2018-12-11T06:23:44.183Z
@PedroCavalcanteOliveira My point is that many people like it simple when there is moral hazard involved. A good standard for verifying something is an experiment with all the variables controlled. It is a very hard thing to do in social sciences. Not saying we shouldn't push the math forward, just don't build skyscrapers on the sand.
Arthur Tarasov
CC BY-SA 4.0
2018-12-12T18:06:27.843Z
Laboratorial experiment is hard/impossible to do in economics but we have other techniques. RCTs, quasi-experiments, natural experiments... There's an entire field of econometrics dedicated to causal inference in non-laboratorial settings. This has been out there and used in economics for 25-30 years. [Angrist and Krueger, 1990](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.6971&rep=rep1&type=pdf) is a great example, as is [Angrist, 1989](http://harris.princeton.edu/pubs/pdfs/251.pdf)  I feel like you don't know much about what you're complaining about.
Pedro Cavalcante
CC BY-SA 4.0
2018-12-12T18:06:56.267Z
[Here`s a good reading](https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.24.2.3), describing the immense progress we've made as a profession towards credibility.
Pedro Cavalcante
CC BY-SA 4.0
2018-07-01T04:26:45.190Z
2018-12-08T20:59:55.723Z
2018-12-08T20:59:55.723Z
Arthur Tarasov
<p>The problem with math as used in modern economics is that the math is often used to describe models of human behavior. Modeling human behavior, whether with math or otherwise, is incredibly difficult especially over long time scales, if our goal is to make the model match reality. So it's not really that there's a problem with using math per se, but mathematical models of human behavior are by their very nature bound to fail in multitudinous ways, so that the detailed economic models built by economists do not match reality and don't have clear practical utility.</p>

<p>Economics must move away from modeling human behavior and move towards modeling institutions, governments, companies, etc. and the dynamics involving these agents. Mathematical models will be more useful here because the entities I described above have both fewer clearly defined parameters of existence, and their interactions with other human-composite entities are more range restricted than those involving human beings themselves.</p>

<p>Moving away from behavioral economics will restore legitimacy to economic science because a focus on institutions will yield more accurate models and therefore greater predictive and explanatory power.</p>

CC BY-SA 4.0
2018-06-28T22:17:27.547Z
Do you have any reason to think modeling institutions will be any simpler than modeling human behavior? Especially over the longer time scales you note are troublesome?
ako
CC BY-SA 4.0
2018-06-28T22:54:01.000Z
Of course I do, that's why I said it. 

The reasons are that the dimensions of institutional behavior and interaction are much less than that of human behavior, and more importantly, the behavior of actual institutions is much more visible to us than that of people.
credo56
CC BY-SA 4.0
2018-06-29T08:52:34.603Z
Who do you think runs institutions if not humans?
BB King
CC BY-SA 4.0
2018-06-29T08:55:28.397Z
Hi: I just want to add that Nerlove started the attempt to model human behavior in the form of modelling expectations by coming up with adaptive expectations. later on, partial adjustment models were another attempt to do this. then, later on the whole rational expectations revolution went even further in the attempt. How well the RE models work is a different issue but there are definitely mathematical-econometric modelling efforts to model human behavior  through the mechanism of modelling agent's expectations..
mark leeds
CC BY-SA 4.0
2018-06-29T11:21:11.617Z
@credo56 even though I **upvoted** your post, for showing that maths is ineffective with explaining behaviour, I disagree that economics must become more narrow. I think subjects need to be cross-curricular. Personally, I am interested in psychology, and I like the perspective economics has on behaviour. I agree that maths can't describe behaviour to a T, but I think it's fine if maths is left out of behavioural economics (instead, it can focus on understanding irrationality).
ahorn
CC BY-SA 4.0
2018-06-29T14:26:50.657Z
I'd like to add that trying to construct economic models based on human behavior rather than on institutions and companies is like trying to construct an ecological model based on the behavior of individual animals rather than that of whole populations and the interactions of those populations in aggregate. Another analogy: when studying the human body as a whole, we are less concerned with the behavior of individual atoms and molecules and more with the dynamics of entire organs and systems in the body, such as the endocrine system.
credo56
CC BY-SA 4.0
2018-06-29T14:54:24.277Z
@credo56. I see what you are saying but, if you make the assumption that, in aggregate, individual animals tend to behave in some consistent-logical way, you might be okay. and this is pretty much what RE does.
mark leeds
CC BY-SA 4.0
2018-06-29T15:02:23.103Z
@markleeds that is true, but the point still stands that the economy is largely the domain of banks, corporations and governments. Yes populations of consumers and savers (individuals) are part as well, but they are only a part. I am not saying to disregard individuals or populations of individuals, but that a full account of the economy will have to consider institutions as first class citizens along with populations.
credo56
CC BY-SA 4.0
2018-06-29T15:46:41.170Z
@BBKing It is unnecessary to consider humans because we can talk about a company's profit motive, or the function of a regulatory agency in an economy, or a government's tendency to borrow in order to fund a deficit all without considering the humans that are the building blocks of those institutions. If we want to study the internal dynamics of a single company we can and should study its employees, but if we are concerned with sectors of an economy or the economy as a whole, humans (as individuals) add unnecessary complexity.
credo56
CC BY-SA 4.0
2018-06-29T17:40:52.440Z
@Credo56: Excellent points and I agree. I think Muth was thinking more along your lines when he came up with RE but then, over time, it evolved into more of an "individualistic-agent" proposition. Not absolutely sure but my best guess.
mark leeds
CC BY-SA 4.0
2018-06-30T05:26:58.750Z
@Credo56: One last question to you since it sounds like you are very into economics. It's a particularly interesting discussion for me because I have no formal education in economics (background is statistics ) and I kind of learned RE on my own since it's a lot of econometrics.  Here's my question:  If math in economics is not so useful, then why did Sargent and Hansen both get Nobel Prizes ?? Someone must have their work was important and useful I would think. Thanks for insights.
mark leeds
CC BY-SA 4.0
2018-07-02T15:18:43.850Z
@markleeds Math is just a language. I am not saying that math in economics is not useful. I am saying that the language of mathematics is being used to describe models which are less helpful in understanding the economy than models which would place more focus on institutions and corporate entities. Just because a mathematical model is internally consistent does not mean it is helpful or useful.
credo56
CC BY-SA 4.0
2018-07-03T00:52:55.273Z
@Credo56. gotcha. agreed
mark leeds
CC BY-SA 4.0
2018-06-28T21:00:14.113Z
2018-06-28T21:00:14.113Z
credo56
<p>Most economics questions have three parts:</p>

<ol>
<li>Why does a phenomenon occur?  This lets the user understand the answer, understand whether the question is relevant, and understand what factors would change the answer to the next part.</li>
<li>How much of the phenomenon is likely to occur?  This lets the user make decisions based on the answer, and compare the importance of various phenomena.</li>
<li>Under what conditions does a different phenomenon replace this phenomenon?</li>
</ol>

<p>An answer that does not address all three sub-questions is incomplete.  It is likely to either be misunderstood, or be misleading.</p>

<p>Math is needed to get an approximate answer to the second sub-question:  How much?  A person with a good understanding of the math can simplify the math to provide insight to the first and third sub-questions:  Why, and with what limits?</p>

<p>For example, Cobb-Douglas production functions (and mathematically similar utility functions) use math that most non-economists do not understand.  The essential features of these functions can be boiled down to "price elasticities" of supply and demand.  These are terms that most non-economists do not understand, but they can be turned into examples that most people do understand.  For example, such functions for global oil production and demand during the 1980s could be simplified to "In the short run, if OPEC cuts its production by 1 percent of the total world production, then the price of oil will go up by 7 percent."</p>

<p>Unfortunately, many economists use math badly:</p>

<ul>
<li><p>Instead of using the math to generate (and verify) a simplified explanation, some economists go through the details of a complicated "mathematical demonstration".  In the end, the reader has to trust that the economist made the right assumptions, and often only as an answer to "how much", not "why" nor "with what limits".</p></li>
<li><p>Some economists are not careful to explain the uncertainties inherent in their math.</p></li>
<li><p>Some economists use symbols ignorantly.  I once had the displeasure of listening to a lecture by a well-paid, soon-to-be-famous economist.  He had many charts about things like long-term trends of the price of power, which were on a log-log scale.  The x-axis was labelled as log(dollars), and the y-axis was labelled as log(kW).  But his units were actually ln(dollars) and ln(kW).  When politely asked later about it, he did not understand that this was a problem!  (If he had actually wanted to be understood, he would have labelled the y-axis as W, kW, MW, GW, <em>et cetera</em>, and used similar labels for the x-axis.)</p></li>
</ul>

CC BY-SA 4.0
2018-07-02T05:51:53.057Z
I suspect (but don't actually know) that the convention whether log is short for log$_e$ or log$_{10}$ is language dependent.
Giskard
CC BY-SA 4.0
2018-07-02T05:58:44.613Z
@denesp -- The lecture was in American English.  Both the lecturer and I are Americans, and have been part of nearby universities.
Jasper
CC BY-SA 4.0
2018-07-02T04:08:38.950Z
2018-07-02T04:08:38.950Z
Jasper
<p>Mathematics is just language that can be used to provide clear, accurate statements. It should not be seen as an obstacle—rather, it should flow naturally alongside the other language with which it is written (e.g. English). I don't believe that maths is inherently &quot;rigorous&quot; or &quot;authoritative&quot;, as other answers have mentioned, because the reader <em>should</em> be critical enough to spot errors. However, I recognise the limitation here: either because of a limit in human cognition, because people don't put the effort in to study mathematics, or because of a fear of maths, some people <em>aren't good at maths</em>. I think that is where this problem stems from, <strong>but</strong> I don't believe that poor aptitude at maths is a good enough argument for why we shouldn't use it.</p>
<p>Excluding maths from economics is akin to saying that that maths should be kept separate from other subjects.</p>
<p>On the other hand, reading through the answers reminds me of Paul Romer's paper <a href="https://ccl.yale.edu/sites/default/files/files/The%20Trouble%20with%20Macroeconomics.pdf" rel="nofollow noreferrer"><em>The Trouble With Macroeconomics</em></a>. He criticizes (with a good example) that incorrect assumptions that are made for a mathematical deduction can easily be obfuscated. Section 5.3 reads:</p>
<blockquote>
<p>In practice, what math does is let macroeconomists locate the Facts With Unknown Truth Value farther away from the discussion of identification. The Keynesians tended to say &quot;Assume P is true. Then the model is identified.&quot; Relying on a micro-foundation lets an author can say, &quot;Assume A, assume B, ... blah blah blah .... And so we have proven that P is true. Then the model is identified.&quot;</p>
</blockquote>
<p>with the &quot;blah blah blah&quot; making it harder to detect incorrect assumptions.</p>
<p>As <a href="https://economics.stackexchange.com/questions/22599/criticism-of-math-in-economics#comment37184_22600">Wildcard said</a>, the average person may well end up skimming over the maths, in blind faith that it's correct, for lack of effort of checking it themselves.</p>
<p>In closing, sure, economics needs a sociological, psychological or political setting, but mathematics helps to study ideal situations. We can't create complete models of humans or institutions, but economics would be very empty if we didn't study ideal situations. Maths belongs in economics—perhaps those who want it taken out have not satisfied their interest in social sciences enough by studying alternative social science subjects.</p>

CC BY-SA 4.0
2018-06-29T12:40:02.403Z
Romer's [Mathiness](https://en.wikipedia.org/wiki/Mathiness) is indeed lurking in several of the answers.
Giskard
CC BY-SA 4.0
2018-06-29T12:14:15.450Z
2022-03-19T22:43:23.640Z
2022-03-19T22:43:23.640Z
ahorn
<p>In addition, to the excellent previous replies from my experience the biggest downside is the narrowing down the way you think about the economy and life to a way that is mathematically tractable. Some aspects of decision-making and interaction are rather easy to make sense of mathematically (e.g. people on average buy more for a lower price so you can write a downward sloping demand curve) others are highly complex, context-dependent, and unstable (e.g. how advertising or looking what my neighbours have affects my consumption decisions). Unfortunately, in trying to acquire rigour, economists chose to focus only on the first set of questions / explanations because they can be stated neatly using the common mathematical tools they learnt, i.e. optimization. The same with the equilibrium, there is a brilliant passage in MWG graduate Micro textbook on why Economics focuses on equilibrium, they say something like &quot;because in general economists are good at describing what happens in equilibrium and not so good what happens in between&quot;... I find that statement really laughable and sad.</p>
<p><strong>Edit</strong>: I found out somebody much more clever made this argument in a much more compelling way <a href="https://pubs.aeaweb.org/doi/pdfplus/10.1257/jel.20191573" rel="nofollow noreferrer">Akerlof, 2020</a> ;)</p>

CC BY-SA 4.0
2021-02-22T07:35:53.480Z
If you downvote, comment at least
Papayapap
CC BY-SA 4.0
2021-02-20T15:22:56.070Z
2021-06-06T10:16:21.997Z
2021-06-06T10:16:21.997Z
Papayapap
<p>Clearly, mathematics could never cover the full richness of the human experience.</p>

<blockquote>
  <p>…In that Empire, the Art of Cartography attained such Perfection that
  the map of a single Province occupied the entirety of a City, and the
  map of the Empire, the entirety of a Province. In time, those
  Unconscionable Maps no longer satisfied, and the Cartographers Guilds
  struck a Map of the Empire whose size was that of the Empire, and
  which coincided point for point with it. The following Generations,
  who were not so fond of the Study of Cartography as their Forebears
  had been, saw that that vast Map was Useless, and not without some
  Pitilessness was it, that they delivered it up to the Inclemencies of
  Sun and Winters. In the Deserts of the West, still today, there are
  Tattered Ruins of that Map, inhabited by Animals and Beggars; in all
  the Land there is no other Relic of the Disciplines of Geography.</p>
</blockquote>

<p>Jorge Luis Borges, <a href="https://en.wikipedia.org/wiki/On_Exactitude_in_Science" rel="noreferrer">On Exactitude in Science</a></p>

CC BY-SA 4.0
2018-06-29T12:59:36.550Z
I like the image, but this seems to be against modeling in general, not mathematical modeling in economics.
Giskard
CC BY-SA 4.0
2018-06-29T13:39:40.077Z
@debesp The first sentence is undeniably true and the Borges-quote gives the appropriate context.
Michael Greinecker
CC BY-SA 4.0
2018-06-29T17:52:54.447Z
And why should we care about the "full richness of human experience"? It has already happened, let's do something else.
Alecos Papadopoulos
CC BY-SA 4.0
2018-06-29T18:05:50.700Z
@AlecosPapadopoulos The story kinda answers your question.
Michael Greinecker
CC BY-SA 4.0
2018-06-29T11:58:54.780Z
2018-06-29T11:58:54.780Z
Michael Greinecker
<p>To start with, it can be noted that the rise in mathiness in economics is substantially related to increased data processing power, whether in support of theoretical demonstration or empirical application. It is not itself an objective.</p>

<p>Regarding the specific question of why heightened mathiness may be criticized:</p>

<p>1) Economics originates from moral philosophy. There are those who believe that debates involving who gets what, and on what terms, is related to moral philosophy. Mathematical tools can help to express moral concepts or present argumentation about which approach might better serve some moral end.</p>

<p>2) a) Complex math can enable theoretical presentation which is mathematically satisfactory for expressing a theory, but mathematical complexity should not be perceived as demonstration of quality in and of itself, and b) mathematical complexity doesn't necessarily mean that empirical applications will be any better. The risk may be that in order to impress other economists, unnecessarily and/or incorrectly complex math is used to express and/or develop a theory.</p>

<p>I think being open minded in this context would be supported by a belief that diverse economists question the value of heightened mathiness, or that diverse economists view heightened mathiness as a tool (which brings risks, in particular of false overconfidence in results) and not an objective in and of itself. </p>

<p>It can also be noted that one of the main contributions of Marx, aside from proto-macro theory, is extensive development of the idea that technology affects production conditions. And, that production conditions affect the way that all of us live. You don't have to be communist to think that this piece of knowledge is a) useful, and b) not necessarily well served by mathematical demonstration even if some very mathy empirical applications may present results which are very relevant for practical policy considerations. </p>

<p>For most cases, such views should not be perceived as 'anti-math', per se, but rather critical of excessive reliance on (or overconfidence in) mathematical demonstration and/or math-heavy empirical applications as a tool. These can be supplemented by socio-political and/or moral argumentation or reasoning, or if outside of the scope of the work it can at least be explicitly recognized that such considerations are relevant.</p>

CC BY-SA 4.0
2018-06-29T15:47:01.267Z
2018-06-29T15:47:01.267Z
nathanwww
<p>Economics is a social science, not an empirical or laboratory one. It is the study of human behavior in response to competing demands in an environment of scarcity. Human behavior cannot be predicted with mathematical precision — the only way to do it is to make large numbers of gratuitous and unsupportable assumptions about what people will do under a given set of circumstances. </p>

<p>Mathematical economists do not study people. Instead, they study what Nobel laureate Richard Thaler calls “Econs”... perfectly-knowledgeable, perfectly-intelligent, perfectly-logical, perfectly-sophisticated, perfectly-intentioned, perfectly-identical automatons who live and work in an environment of perfect competition; as opposed to humans, who are none of those things and who live on Planet Earth.</p>

<p>It’s not that math is bad — it lets us easily communicate complex ideas clearly and precisely. But we need to remember that the predictions rendered by mathematical economics, very often, will not hold in real life. We need to understand (and promote that understanding in those who look to the economics community for guidance and advice) that math only gets you so far — to make good policy, you need to understand what flawed, fallible, semi-unique, stressed, busy, selfish, sometimes stupid, imperfect human <strong>people</strong> will do. And mathematics cannot tell you that.</p>

CC BY-SA 4.0
2018-06-28T19:38:35.830Z
But most of Thaler's models, which try to include some aspects of human psychology are based in math. Is he then a fraud, or is this a misrepresentation of what he says?
Giskard
CC BY-SA 4.0
2018-06-28T19:40:41.703Z
Not a fraud — like I said, math isn’t bad. It’s a useful tool for communicating complex ideas. It’s just that it only goes so far. You can’t plug values into a formula and accurately predict human behavior with x=y precision. You *can* use math to analyze *past* human behavior though.
Dave
CC BY-SA 4.0
2018-06-28T19:41:42.967Z
Most economists would not claim that that is what they are doing, so this does not seem to directly answer the question. These are models, often simplified to the extreme, to capture one aspect of behavior.
Giskard
CC BY-SA 4.0
2018-06-28T19:42:16.113Z
I think it addresses the question.
Dave
CC BY-SA 4.0
2018-06-28T19:45:33.863Z
@denesp, your own quote would seem to back up what I’m saying.           >“Worst of all, [the ubiquity of mathematical theory in economics] imbues economic theory with an unearned empirical authority”
Dave
CC BY-SA 4.0
2018-06-28T19:45:49.687Z
@Dave Assumptions akin to being rational, perfectly-identical, and so forth are necessary idealizations to develop a manageable body of knowledge. The differences among agents are addressed at advanced stages through the incorporation of probabilistic tools, "error modeling", and other resources. One ultimate point of mathematics is not that agents are necessarily rational or perfectly intelligent, but *how* arbitrage opportunities arise when agents act irrationally.
Iñaki Viggers
CC BY-SA 4.0
2018-06-28T19:54:47.070Z
@IñakiViggers Sure. But how many average joes can understand that stuff? How many even exceptional people, policymakers, and such even? Most people see the Econ-101 supply and demand curves with the empirical-looking equilibrium point and take it as an objective truth.
Dave
CC BY-SA 4.0
2018-06-28T20:10:30.570Z
@Dave Very few, but that's a different issue. Apropos of the analogies about Latin, the suppression of mathematics in Economics is akin to getting rid of literature written in Latin merely because very few average joes & policymakers can read that language.
Iñaki Viggers
CC BY-SA 4.0
2018-06-28T20:20:26.630Z
That’s my point — not that mathematical economics is bad, but that we need to do a better job making people understand that *accurate* mathematical economics is not novice-level stuff. What you learn in Econ 101 is not irrefutable cosmic wisdom (which is how too many people treat it). We need to suppress mathematics in introductory and intermediate economics to make room for the social economcs which will give the majority of people who will never study graduate or doctoral economics a better understanding of how markets actually work.
Dave
CC BY-SA 4.0
2018-06-28T21:06:35.380Z
@Dave But the OP's question is not about pedagogy of Economics; it goes further into its development and systematization. Various aspects of *social* economics are already covered in sociology or political "science". Beyond that, I doubt that one could truly understand, for instance, "why arbitrage opportunities hardly ever occur in *actual* markets" without first learning how those opportunities are detected/exploited. Knowledge of mathematics is essential for the latter. Proficiency in Economics is requisite when serious discussions and informed design of economic policy/theory are intended.
Iñaki Viggers
CC BY-SA 4.0
2018-06-28T21:33:10.017Z
Weather can't be predicted with mathematical precision either, but meteorologists must know quite a bit of mathematics to do their jobs.
Monty Harder
CC BY-SA 4.0
2018-06-28T23:45:32.343Z
@Dave: The rules taught in Econ 101 aren't always sufficient to make perfect predictions.  On the other hand, when someone proposes a policy whose claimed expected results contradict Econ 101 predictions and doesn't address such contradiction, odds are good that the actual results will match Econ 101 predictions more closely than they match the policy promoter's claims.
supercat
CC BY-SA 4.0
2018-06-29T12:07:18.720Z
No, no, no, no.  Literally nothing in the list of "perfectly-knowledgeable, perfectly-intelligent, perfectly-logical, perfectly-sophisticated, perfectly-intentioned, perfectly-identical automatons who live and work in an environment of perfect competition" describes the extend of mathematical economics.
Michael Greinecker
CC BY-SA 4.0
2018-06-29T14:51:24.920Z
@MichaelGreinecker How would you describe what mathematical economics seeks to do? What assumptions or givens does it rely on, if not the ones I’ve laid out?
Dave
CC BY-SA 4.0
2018-06-29T15:13:36.317Z
@Dave Mathematical economists mostly study the consequences of different assumptions. As such, there are no assumptions made by all of them all the time. But every advanced undergraduate should have seen models of imperfect competition, models in which not all agents are the same, and models of imperfect information. To be blunt: You clearly have no idea what you are talking about.
Michael Greinecker
CC BY-SA 4.0
2018-06-29T15:52:16.527Z
@MichaelGreinecker That’s not very charitable. True, I’m still working on my undergrad, but the state of economics in the American political zeitgeist (I would contend) is very much as I’ve described. Whatever it is that economists know, economic policy is most often made by those who are not economists (nor who possess even a bachelors in economics) , and many of whom are ignorant of or content to omit anything beyond the simplistic 101-level model; when they even bother to use economics as more than a buzzword. In short, the math belongs farther on. We need more logic up front.
Dave
CC BY-SA 4.0
2018-06-29T16:41:01.363Z
Well, I think your answer was not very charitable to mathematical economists, a group I belong to. What you now wrote in your comment has very little to do with mathematical economics.
Michael Greinecker
CC BY-SA 4.0
2018-06-29T20:09:31.350Z
@MichaelGreinecker Perhaps it's worth offering a bit of a middle ground. To Michael's point, he's definitely correct- no economist I can think of conducts research that would be recognizable as economics from a 101 class perspective. The whole point of new projects is to relax previously restrictive settings, and see what new dynamics can be explained. That said, Dave's correct too, that often the framework employed by economists is restrictive itself. For example, game theorists like us often make assumptions about strategy spaces- whether they're convex sets, known by all parties, etc....
AndrewC
CC BY-SA 4.0
2018-06-29T20:13:53.600Z
And often research is performed by relaxing these sorts of assumptions- after all, you need certain types of assumptions (such as those regarding strategies, payoffs, etc.) to allow the mathematics to make sense, and be useful. However, it may well be that those seemingly benign mathematical statements used by the field have substantial implications for modeling. A great example is some of Ariel Rubenstein's most recent work in progress. By relaxing the very definition of "strategy" into something that's mathematically much harder to work with, he finds quite interesting results- that how...
AndrewC
CC BY-SA 4.0
2018-06-29T20:17:06.917Z
individuals perceive strategy spaces, and the dimensionality of their reasoning, is instructive for how successful they'll be when playing non-trivially modeled (from the players' perspective) games. Mathematically, this is a very inconvenient prospect, since it makes historically relied upon FPTs useless (among other things, of course). Only by a giant in the field getting away from the "overly math-y mindset" that we often use could this potentially intriguing result be found.
AndrewC
CC BY-SA 4.0
2018-12-10T07:51:50.940Z
Oh boy I just love this "I found out about behavioral economics yesterday and rationality is a myth" attitude. [Behavioral Economics has some of the more mathy/complex models out there](https://www.nber.org/papers/w22954). Why? Well, because modelling people with complete, transitive and reflexive preferences is easier than modelling people with weirder preferences, thus requires simpler math.
Pedro Cavalcante
CC BY-SA 4.0
2018-06-28T19:29:59.733Z
2018-06-28T19:35:17.673Z
2018-06-28T19:35:17.673Z
Dave
<p>I think there are two legitimate sources of complaint.  For the first, I will give you the anti-poem that I wrote in complaint against both economists and poets.  A poem, of course, packs meaning and emotion into pregnant words and phrases.  An anti-poem removes all feeling and sterilizes the words so that they are clear.  The fact that most English speaking humans cannot read this assures economists of continued employment.  You cannot say that economists are not bright.</p>

<hr>

<h2><strong>Live Long and Prosper-An Anti-Poem</strong></h2>

<p><em>May you be denoted as $k\in{I},I\in\mathbb{N}$, such that $I=1\dots{i}\dots{k}\dots{Z}$</em></p>

<p><em>where $Z$ denotes the most recently born human.</em></p>

<p><em>$\exists$ a fuzzy set $Y=\{y^i:\text{Human Mortality Expectations}\mapsto{y^i},\forall{i\in{I}}\},$</em></p>

<p><em>may $y^k\in\Omega,\Omega\in{Y}$ and $\Omega$ is denoted as "long"</em></p>

<p><em>and may $U(c)$, where c is the matrix of goods and services across your lifetime</em></p>

<p><em>$U$ is a function of $c$, where preferences are well-defined and $U$ is qualitative satisfaction,</em></p>

<p><em>be maximized $\forall{t}$, $t$ denoting time, subject to</em></p>

<p><em>$w^k=f'_t(L_t),$ where $f$ is your production function across time</em></p>

<p><em>and $L$ is the time vector of your amount of work,</em></p>

<p><em>and further subject to $w^i_tL^i_t+s^i_{t-1}=P_t^{'}c_t^i+s^i_t,\forall{i}$</em></p>

<p><em>where $P$ is the vector of prices and $s$ is a measure of personal savings across time.</em></p>

<p><em>May $\dot{f}\gg{0}.$</em></p>

<p><em>Let $W$ be the set $W=\{w^i_t:\forall{i,t}\text{ ranked ordinally}\}$</em></p>

<p><em>Let $Q$ be the fuzzy subset of $W$ such that $Q$ is denoted "high".</em></p>

<p><em>Let $w_t^k\in{Q},\forall{t}$</em></p>

<hr>

<p>The second is mentioned above, which is the misuse of math and statistical methods.  I would both agree and disagree with the critics on this.  I believe that most economists are not aware of how fragile some statistical methods can be.  To provide an example, I did a seminar for the students in the math club as to how your probability axioms can completely determine the interpretation of an experiment.</p>

<p>I proved using real data that newborn babies will float out of their cribs unless nurses swaddle them.  Indeed, using two different axiomatizations of probability, I had babies clearly floating away and obviously sleeping soundly and securely in their cribs.  It wasn't the data that determined the result; it was axioms in use.</p>

<p>Now any statistician would clearly point out that I was abusing the method, except that I was abusing the method in a manner that is normal in the sciences.  I didn't actually break any rules, I just followed a set of rules to their logical conclusion in a way that people do not consider because babies don't float.  You can get significance under one set of rules and no effect at all under another.  Economics is especially sensitive to this type of problem.</p>

<p>I do belive that there is an error of thought in the Austrian school and maybe the Marxist about the use of statistics in economics that I believe is based on a statistical illusion.  I am hoping to publish a paper on a serious math problem in econometrics that nobody has seemed to notice before and I think it is related to the illusion.</p>

<p><a href="https://i.sstatic.net/tFQkK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tFQkK.png" alt="Sampling Distribution of Bayesian MAP versus Fisher&#39;s MLE"></a></p>

<p>This image is the sampling distribution of Edgeworth's Maximum Likelihood estimator under Fisher's interpretation (blue) versus the sampling distribution of the Bayesian maximum a posteriori estimator (red) with a flat prior.  It comes from a simulation of 1000 trials each with 10,000 observations, so they should converge.  The true value is approximately .99986.  Since the MLE is also the OLS estimator in the case, it is also Pearson and Neyman's MVUE.</p>

<p>Note how relatively inaccurate the Frequency based estimator is compared to the Bayesian.  Indeed, the relative efficiency of $\hat{\beta}$ under the two methods is 20:1.  Although Leonard Jimmie Savage was certainly alive when the Austrian school left statistical methods behind, the computational ability to use them didn't exist.  The first element of the illusion is inaccuracy.</p>

<p>The second part can better be seen with a kernel density estimate of the same graph.  <a href="https://i.sstatic.net/QoTFO.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QoTFO.png" alt="kernel of sample distribution"></a></p>

<p>In the region of the true value, there are almost no examples of the maximum likelihood estimator being observed, while the Bayesian maximum a posteriori estimator closely covers .999863.  In fact, the average of the Bayesian estimators is .99987 whereas the frequency based solution is .9990.  Remember this is with 10,000,000 data points overall.  </p>

<p>Frequency based estimators are averaged over the sample space.  The missing implication is that it is unbiased, on average, over the entire space, but possibly biased for any specific value of $\theta$.  You also see this with the binomial distribution.  The effect is even greater on the intercept.</p>

<p><a href="https://i.sstatic.net/Wfxdy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Wfxdy.png" alt="alpha"></a></p>

<p>The red is the histogram of Frequentist estimates of the itercept, whose true value is zero, while the Bayesian is the spike in blue.  The impact of these effects are worsened with small sample sizes because the large samples pull the estimator to the true value.</p>

<p>I think the Austrians were seeing results that were inaccurate and didn't always make logical sense.  When you add data mining into the mix, I think they were rejecting the practice.</p>

<p>The reason I believe the Austrians are incorrect is that their most serious objections are solved by Leonard Jimmie Savage's personalistic statistics.  Savages <em>Foundations of Statistics</em> fully covers their objections, but I think the split had effectively already happened and so the two have never really met up.</p>

<p>Bayesian methods are generative methods while Frequency methods are sampling based methods.  While there are circumstances where it may be inefficient or less powerful, if a second moment exists in the data, then the t-test is always a valid test for hypotheses regarding the location of the population mean.  You do not need to know how the data was created in the first place.  You need not care.  You only need to know that the central limit theorem holds.</p>

<p>Conversely, Bayesian methods depend entirely on how the data came into existence in the first place.  For example, imagine you were watching English style auctions for a particular type of furniture.  The high bids would follow a Gumbel distribution.  The Bayesian solution for inference regarding the center of location would not use a t-test, but rather the joint posterior density of each of those observations with the Gumbel distribution as the likelihood function.</p>

<p>The Bayesian idea of a parameter is broader than the Frequentist and can accomodate completely subjective constructions.  As an example, Ben Roethlisberger of the Pittsburgh Steelers could be considered a parameter.  He would also have parameters associated with him such as pass completion rates, but he could have a unique configuration and he would be a parameter in a sense similar to Frequentist model comparison methods.  He might be thought of as a model.</p>

<p>The complexity rejection isn't valid under Savage's methodology and indeed cannot be.  If there were no regularities in human behavior, it would be impossible to cross a street or take a test.  Food would never be delivered.  It may be the case, however, that "orthodox" statistical methods can give pathological results that have pushed some groups of economists away.</p>

CC BY-SA 4.0
2018-07-03T19:23:48.980Z
This is interesting but what was the data and what was being estimated. You say "edgeworth's MLE" but MLE under what distributional assumption of what data ? I may have missed a previous post. Thanks for clarification..
mark leeds
CC BY-SA 4.0
2018-07-03T20:34:40.617Z
The data is from a set of simulations from a time series that is stationary AR(1) with normal shocks.
Dave Harris
CC BY-SA 4.0
2018-07-03T20:43:46.383Z
In that case, you've got a VERY, VERY, VERY,  VERY near unit root process which is going to cause the classical statistical assumptions to fail. So, it sounds more like an assumption problem rather than a problem with classical statistics.  As you're probably aware, unit root processes lead to dickey fuller type distributions rather than t-distrbutions. My best guess is that that's what's going on there. Still, an interesting example. Thanks.
mark leeds
CC BY-SA 4.0
2018-07-03T20:44:48.390Z
That is what started the investigation.  I am looking at nearly and just barely explosive roots.
Dave Harris
CC BY-SA 4.0
2018-07-03T20:48:29.993Z
There is a Bayesian solution for both less than and greater than unit root processes.  The elaborate Frequentist solutions are completely unnecessary.  Being non-stationary is a headache, but only in the sense that the predictions are weaker, not in the calculation sense.
Dave Harris
CC BY-SA 4.0
2018-07-03T20:52:39.953Z
@markleeds if you think about it almost all real economic series are near unit root.  Look at firm survival which is just barely stationary, or low daily growth rates.
Dave Harris
CC BY-SA 4.0
2018-07-03T20:53:40.107Z
@markleeds also, it does not violate any distributional assumptions.  Anything less than one is still t-distributed, it just takes a long time to converge.
Dave Harris
CC BY-SA 4.0
2018-07-04T01:35:40.637Z
Hi Dave: I don't want to have a long discussion about this ( you can have last word. I promise )  but you're entering dangerous territory when simulating a process like that which is close to 1.0 because some propportion of the trials of the simulation will generate data with a unit root so you end up estimating a mixture. If you want to do a fair comparison, you're better off taking the first difference of the simulated data, ,estimating the mean and then adding 1.0. That should give better results. My point is that your estimation process is assuming something that often won't be true.
mark leeds
CC BY-SA 4.0
2018-07-04T05:02:02.953Z
There is a nice, detailed discussion of bayesian ( and some details about classical ) unit root econometrics in "bayesian inference in dynamic econometric models" by bauwens, lubrano and richard. all the best.
mark leeds
CC BY-SA 4.0
2018-07-06T04:54:47.373Z
any time. I bet there are areas where bayesian approaches do generate better results than classical stat. I basically  just wanted to give a heads up about the "dark jungle" called the unit root world. I dealt with it a lot in a previous life.
mark leeds
CC BY-SA 4.0
2018-07-03T16:51:39.413Z
2018-07-03T17:16:58.543Z
2018-07-03T17:16:58.543Z
Dave Harris
<p>This is not so much of an answer as more of a note motivated predominantly by the <em>softness</em> of the question.</p>

<p>It can be the case that the statement</p>

<blockquote>
  <p>"[...] <em>the study of work by classical economists (like Adam Smith, Karl Marx,
  and David Ricardo) is still relevant</em>"</p>
</blockquote>

<p>(insert qualifications) is true <em>irrespective</em> of the truth value of the assertion </p>

<blockquote>
  <p>"[...] <em>the practice of how mainstream economics uses mathematics is
  abusive and is an attempt to fool the masses regarding the "science"
  economists practice.</em>".</p>
</blockquote>

<p>My point is that the relevance of the <em>classics</em> is not necessarily related to the relevance (or lack thereof) of using mathematics in economics. </p>

<p>Obviously, private communications are opaque to anyone not present and as I were not present in the private communications that instigated this question it is not possible to comment on the specific arguments that lend (or detract) support to the math relevance thesis; </p>

<p>I think that there is some renewed interest in the history of economics as a discipline and economic historians are trying to investigate the various paths that economic theory has followed in modern times; I won't use references as I am not an economic historian but I think it is relatively easy for anyone to find material on such issues.</p>

<p>My personal understanding of the subject is that the success of the war effort during WWII attributed (rightly or wrongly, that is debatable) a certain amount of credibility into tools and approaches used in operations research and related fields; obviously these fields were more mathematical in spirit. </p>

<p>With the advent of the Cold War and the political and ideological issues that ensued it was only natural to expect that tools that had proven themselves useful in the recent past (math, op research) would be used again to stave of the <em>red scare</em>. Add to this mix the arms race of Cold War and the subsequent major and minor breakthroughs in <em>hard</em> sciences related to the nuclear effort etc.</p>

<p>It is not difficult to imagine why the agony of the "free world" to emerge victorious from the cold war painted the <em>tools</em> it had invested so much into with favorable colors. </p>

<p>Now, there comes an inversion in this scheme where the tools that had been proven useful once are subsequently used almost ceremonially to impart use value into the body of knowledge that accumulated around their use. That is not to say that the mathematics were 'wrong' or 'too abstract' or 'irrelevant'. But it is the case that at some point the tool-case became more important than the actual problems it could solve.</p>

<p>And this is equivalent to hybris.</p>

<p>On a final note, damning or glorifying economics for its use of mathematics seems misplaced so long as the body of knowledge under the heading 'economics' fails to produce positive results for the society at large. </p>

<p>Resources have competing uses and economists know that very well. </p>

<hr>

<h2>update 1</h2>

<p>this is an update about math and classical econs (as it was too long for a comment)</p>

<p>The classical econs couldn't have used calculus as Leibnitz and Newton invented it during the mid- and late-1600's and it was formalized by mathematicians 100-150 years later into something recognizable; I know Marx did some fondling with infinite calculus by never used it as a proper tool; similarly, the use of linear algebra and systems of linear equations were predominantly <em>popularized</em> by the triumph of Dantzig's simplex algo. The point is that IMO the classic econs did not have that stock of knowledge available to them. </p>

<p>Furthermore, Political economy was to a large extent a discursive enterprise that was meant to <em>convince</em> the hegemon about the proper path to prosperity (whatever that meant to them at that time).
Consider eg the Physiocrats. Quesnay's (a contemporary of A. Smith) Tableau was to a large extent a description of flows that needed little effort to be translated into a linear system of inputs and outputs. It wasn't, because </p>

<p>1.a. his formal education was in medicine (he was trained as a physician)</p>

<p>1.b. the tools to do so were invented by Leontieff in the 60's</p>

<ol start="2">
<li>he and his disciples had all the legitimacy they needed (Turgot, a disciple of Quesnay, eventually become finance Minister)</li>
</ol>

<p>The point I'm trying to make is that lack of mathematical rigor in classical econs does not necessarily mean that they are irrelevant.</p>

CC BY-SA 4.0
2018-07-02T04:40:09.727Z
A major distinction between the "classical economists" and later economists is that the classical economists used neither calculus nor large systems of linear equations to derive their results.  The great classical economists did include a few simple mathematical examples.
Jasper
CC BY-SA 4.0
2018-07-01T17:12:39.683Z
2018-07-02T09:08:42.360Z
2018-07-02T09:08:42.360Z
jojobaba
<blockquote>
  <p>"<em>What is a reason to be against mathematics in economics?</em>"</p>
</blockquote>

<p>IMO, <strong>if you frame your whole economic thinking in mathematic terms</strong> (or too much of it)<strong>, your thought process might become less flexible and innovative</strong>. Mathematically formalizing economic theories can be an arduous task:</p>

<ul>
<li>Some postulates might require extreme care when translating them into mathematical language. This has an opportunity cost in terms of time and intellectual energy which will not be spent on more "productive" tasks (e.g. exploring new, radical ideas to longstanding problems);</li>
<li>Mathematics requires a rigor that is simply absent when a new idea is emerging: you might not be able to formulate mathematically something you are barely starting to understand.</li>
</ul>

<p>As a consequence, your economic thinking might end up "hijacked" by a set of assumptions that enable you to mathematically formalize your theory/model, but which constrains the range of novel economic ideas you can formulate.</p>

CC BY-SA 4.0
2018-07-03T16:15:07.523Z
To address both the points here- Peer reviewed journals deal with both of these issues. when ideas presented they have to go through a process where they are critically reviewed, if they cant stand up to scrutiny (or the author cant handle the criticism) then why publish it?
EconJohn
CC BY-SA 4.0
2018-07-03T17:26:45.183Z
@EconJohn "standing up to scrutiny" involves a fair degree of subjectivity: when L. Bachelier presented his thesis applying Brownian Motion to model stocks, reception was mixed as the Jury deemed it wasn't fully rigorous. Nonetheless, his work has subsequently been tremendously influential in the theory of Finance. Original work might deviate from a profession's prevailing standard (e.g. rigorous mathematical formalization) which do not necessarily invalidate its relevance. So some people might be again an _excessive_ use of mathematics in economics because of that.
Daneel Olivaw
CC BY-SA 4.0
2018-07-03T17:31:04.633Z
Why the down vote, by the way?
Daneel Olivaw
CC BY-SA 4.0
2021-02-20T15:24:52.490Z
This is the most important drawback in the whole discussion. Pity it gets downvoted
Papayapap
CC BY-SA 4.0
2018-07-03T13:18:49.373Z
2018-07-03T15:43:58.950Z
2018-07-03T15:43:58.950Z
Daneel Olivaw
<p>Many of the criticism comes from the recent financial crisis. Economists failed to predict the crisis, beside the super sophisticated models. Many then said that economics is wrong because this super complex models cannot capture essential elements of life and behavior and society. </p>

<p>So part of the movement against mathematics is just in response to the evidence. For many, the sicence is often a failure. </p>

CC BY-SA 4.0
2018-12-09T17:13:50.977Z
2018-12-09T17:13:50.977Z
chatGPT
<p>At the end of the day, economics and its offshoots (ie business, management, marketing etc) are all social sciences. These areas of inquiry are concerned with specific facades of human behavior as individuals or groups. While quantitative methods are very useful in categorizing and generalizing these behaviors, the behavior itself is highly personal and individualistic. For example, you and I could go into the same supermarket, at the same time, buy the same items and leave. This behavior when analyzed quantitatively, will arrive at an average of our behavior and its root causes, it will however completely miss the individual behaviors. By defining a non-existent third behavior (the average) it will model our behaviors, but will not reflect the true nature of the behaviors it is trying to explain. But when you have a large enough sample for a given population the quantitative analysis just about becomes good enough to draw generalizations, but the generalization being drawn usually is not representative of a given individual. </p>

CC BY-SA 4.0
2018-06-30T03:39:46.170Z
2018-06-30T03:39:46.170Z
Sinan Yilmaz
<blockquote>
  <p>What is a reason to be against mathematics in economics?</p>
</blockquote>

<p>I don't think there is a blanket reason to be against math anymore than there is a blanket reason to be against case studies. It is almost a matter of epistemology. What are the knowledge claims made, with what methods, and with what evidence? Some sorts of questions are very well suited for a quantitative treatment: Like, what is the effect of increased accessibility on housing prices? Or, given a number of variables on cost and household demographics, which mode of transport is a household likely to take to work? There are models that are well suited for finding patterns in that type of questions where the domain is fairly specific, and they may work reasonably well even absent a strong theory underlying observed patterns.</p>

<p>Conversely, a number of questions are of a different nature entirely, related to bigger historical shifts. The rise and fall of the labor movement in the US, say, or why did some cities see a revival when others didn't? Such questions are probably better answered by a different approach than using models (this doesn't mean that there can't be useful quantitative components of asking those questions).</p>

<p>Ultimately, I think it has more to do with the sorts of questions different researchers are interested in rather than a wholesale rejection of a practical approach.</p>

CC BY-SA 4.0
2018-06-29T19:34:43.060Z
2018-06-29T19:34:43.060Z
ako
<p>Beyond the quantitative aspects, there are also qualitative factors which do not lend themselves to numeric treatment. My background is electrical engineering, which quite correctly employs quantitative methods extensively. Although investing isn't economics, there is a relationship. As much as possible, I've been trying to read and implement the information and wisdom imparted by Benjamin Graham and his colleague David Dodd. Graham himself was the instructor, and later the employer, of Warren Buffett. Graham felt that when anything more than the 4 basic arithmetic operations were dragged into the model, description, or analysis, then somebody was trying to "sell you a bill of goods". Graham himself was very mathematically adept, and knew calculus and differential equations much better than most students and instructors. So, use of advanced mathematics in some ways acts to obscure, rather than elucidate, matters pertaining to "proper" investment practice. Buffett is still very much alive. Graham himself and most of his employees or students are all long gone, but they all seemed to have died rich. Look through his books "Security Analysis" and "The Intelligent Investor" and you won't find a derivative, integral, ODE, or PDE.</p>

CC BY-SA 4.0
2018-07-02T17:02:41.247Z
You may enjoy reading about the company [Long-Term Capital Management](https://en.wikipedia.org/wiki/Long-Term_Capital_Management).
Giskard
CC BY-SA 4.0
2018-07-03T15:30:27.700Z
@denesb: The LTCM disaster was based on a set of assumptions and confidence about the way people have. It had zero to do with math per-se  but still an interesting read for those interested. OTOH, If you're making a statement about math not  always being applicable in finance, I agree.
mark leeds
CC BY-SA 4.0
2018-07-03T16:55:18.663Z
Actually, Graham was an economist and indeed presented an alternative monetary regime at the Bretton Woods conference.  Just to be fair to Graham, he might, in fact, use those tools today.  The Graham-Dodd method actually lends itself to both statistical and economic model constructions.
Dave Harris
CC BY-SA 4.0
2018-07-03T19:25:42.763Z
should be "people behave" not people have. Not sure how to correct it.
mark leeds
CC BY-SA 4.0
2018-07-02T16:20:10.990Z
2018-07-02T16:20:10.990Z
IchabodKunkleberry
<p>I find that the essay "<a href="https://aeon.co/essays/how-economists-rode-maths-to-become-our-era-s-astrologers" rel="noreferrer">The New Astrology</a>" by Alan Jay Levinovitz (an assistant professor of philosophy and religion, <strong>not</strong> an economist) makes some good points.</p>

<blockquote>
  <p>...the ubiquity of mathematical theory in economics also has serious downsides: it creates a high barrier to entry for those who want to participate in the professional dialogue, and makes checking someone’s work excessively laborious. Worst of all, it imbues economic theory with unearned empirical authority.</p>
  
  <p>‘I’ve come to the position that there should be a stronger bias
  against the use of math,’ <a href="https://en.wikipedia.org/wiki/Paul_Romer" rel="noreferrer">Romer</a> explained to me. ‘If somebody came and
  said: “Look, I have this Earth-changing insight about economics, but
  the only way I can express it is by making use of the quirks of the
  Latin language”, we’d say go to hell, unless they could convince us it
  was really essential. The burden of proof is on them.’</p>
</blockquote>

<p>The essay also makes a (more or less adequate&mdash;which, I leave up to you) comparison with astrology in ancient China to show that excellent math can be used to prop up ridiculous science and grant status for its practitioners.</p>

CC BY-SA 4.0
2018-06-28T22:48:17.230Z
The "_unearned empirical authority_" sounds really weird.  I mean, math's just a precise language that's easy to perform logical operations with.  Putting something into mathematical terms shouldn't be taken as endowing empirical authority anymore than translating a statement to Latin should.  _Barba crescit caput nescit_.
Nat
CC BY-SA 4.0
2018-06-29T01:50:05.023Z
The Latin point doesn't seem much of an argument to me, bordering on straw-man. Latin clearly has nothing to do with economics whereas maths is clearly related. It's straw-man because the reader thinks "well yes, it is completely unreasonable to rely on the quirks of the Latin language to express an economic insight", but that has no relevance at all to whether or not it is reasonable to rely on maths. "It creates a high barrier to entry for those who want to participate in the professional dialogue" *on its own* isn't really much justification either. Many fields have a high barrier to entry.
JBentley
CC BY-SA 4.0
2018-06-29T02:11:22.737Z
Math and logical systems in general conform to ["_garbage in, garbage out_"](https://en.wikipedia.org/wiki/Garbage_in,_garbage_out); so if someone uses mathematical logic on garbage assumptions, then they'll get garbage results.  But isn't that obvious?  (Not being rhetorical - I'm actually asking if this isn't obvious.  Because if it's not, then I could understand why folks could feel misled by seeing garbage expressed in mathematical terms.)
Nat
CC BY-SA 4.0
2018-06-29T02:23:36.073Z
@Nat It is obvious, but technical garbage is more difficult to identify. This comment could be the core of a nice answer IMO.
Giskard
CC BY-SA 4.0
2018-06-29T06:19:07.927Z
@Nat, for those who don't understand mathematics in the first place, there is nothing obvious about what math does or does not prove.  If the math checks out, and the experts agree that there are no errors in the math, then to a layperson the only thing that's "obvious" is that the mathematicians know what they're talking about.  Even if that's actually false.
Wildcard
CC BY-SA 4.0
2018-06-29T11:11:25.843Z
@Wildcard my thoughts about this whole thing is that people encountering maths in economics are obliged to be good at maths. Sure, someone who isn't good at maths can study the non-mathematical aspects of economics, but I think they have an _obligation_ to be good at maths in order to explore the subject in its entirety.
ahorn
CC BY-SA 4.0
2018-06-29T12:27:25.817Z
@ahorn There's a difference between being good at maths in general, and being able to follow an argument (and check the proposed proofs) in any arbitrary mathematical framework out there.
sgf
CC BY-SA 4.0
2018-06-30T17:18:23.770Z
@Nat: I'll take a contrarian position here: *It is just as obvious as the author wants it to be.*  If all of the relevant assumptions are stated up front, then it's entirely obvious when some assumptions (e.g. "All humans are perfectly rational creatures" or similar) are questionable.  But if an author instead buries these assumptions in e.g. the methodology section (so that anyone just reading the abstract misses them), then a [valid but unsound](https://math.stackexchange.com/q/281208/198741) argument may appear to be supported by the math when in fact it is wrong.
Kevin
CC BY-SA 4.0
2018-07-01T10:14:00.230Z
That attitude makes me pity the math-illiterate. Properly written, a page of math can replace about 20 pages of rambling imprecise words, and is hence much _easier_ to check. That is basically the whole point of mathematical language. This guy is like a blind person trying to ban diagrams.
knzhou
CC BY-SA 4.0
2018-07-01T19:09:43.087Z
@knzhou I am not sure who you are referring to as "this guy", but in case it is Paul Romer, I assure you, he knows quite a bit of math. In case you are referring to Levinovitz, I don't think he is math-illiterate either, at least I could not gather this information from his essay.
Giskard
CC BY-SA 4.0
2018-07-01T20:06:04.673Z
@Nat to those who do not know mathematics or latin, they do endow unearned empirical authority. See, for example, https://www.wsj.com/articles/SB10001424127887323374504578219873933502726. Ubiquity of math may not be a problem for economics from a strictly internal viewpoint, since practitioners know some math, but it seems reasonable that it makes it more difficult for non-economists who are not fluent in math to know which economists to listen to.
forget this
CC BY-SA 4.0
2018-12-09T02:25:42.707Z
This is non-sense. We sometimes need to convey ideas more precisely than is possible without using mathematics. Fields progress. People opining for days gone where arguments counted as serious research does not mean that the field has moved in a bad direction. And this is an appeal from authority. I would guess there is someone equally influential willing to argue that introducing precision into economics has been incredibly useful.
123
CC BY-SA 4.0
2018-12-09T07:10:53.397Z
@123 [I would like to point out that the question is not whether we should have math in economics, but why some people attack mathematical economics.](https://economics.stackexchange.com/a/22618/1601)
Giskard
CC BY-SA 4.0
2018-12-10T07:40:21.900Z
This barrier to entry argument just doesn't cut it. Physics too has a high barrier to entry and I don't see people claiming we should abandon the Maxwell Equations. People could describe everything in physics just with english, but it's harder and a lot less transparent. The same happens in economics.
Pedro Cavalcante
CC BY-SA 4.0
2018-12-10T18:02:59.140Z
@PedroCavalcanteOliveira I am not saying we should abandon mathematics, as I pointed out in the comment directly above yours.
Giskard
CC BY-SA 4.0
2018-12-10T18:35:10.813Z
@denesp -- the point I am making refers specifically to the quote. I think it is nonsense. My opinion is independent of the question.
123
CC BY-SA 4.0
2018-12-10T20:47:47.930Z
@123 That is fine, you are entitled to your opinion. Please consider posting it under the article, or on Facebook, and not under my answer, which addresses the question.
Giskard
CC BY-SA 4.0
2021-02-20T14:52:10.210Z
+1 great comparison with astrology. Economics is very much like ancient astrology
Papayapap
CC BY-SA 4.0
2021-05-31T12:51:47.303Z
@knzhou If you have something that is obvious without math, you can state it without math. If you do need the math because it shows something counterintuitive, you should at least note that the math is required because it shows something counterintuitive. But most of the time, you are trying to propose a policy, "doing X makes the world better", and if you cannot show that it is better without using extensive math, maybe it is not so clear after all! Of course, if you are trying to make quantitative predictions (*how much* better) then math is essential.
Criticize SE actions means ban
CC BY-SA 4.0
2021-05-31T16:16:06.840Z
@user253751 Yup, I was naive when I wrote that comment. Now that I'm the ripe old age of 25 I see the wisdom in this -- math gives the researcher too many tricky degrees of freedom to get whatever conclusion they want.
knzhou
CC BY-SA 4.0
2018-06-28T18:30:30.020Z
2018-06-29T14:24:56.590Z
2018-06-29T14:24:56.590Z
Giskard
<p>I've been reading and speaking to a number of educated economists and economics PhDs who are against the use of intense mathematics and mathematical proof in economic theory. Specifically I've been speaking to those of Marxist and heterodox persuasion and reading their work in an attempt to become more open minded.</p>

<p>They emphasize that the study of work by classical economists (like Adam Smith, Karl Marx, and David Ricardo) is still relevant and that the practice of how mainstream economics uses mathematics is abusive and is an attempt to fool the masses regarding the "science" economists practice.</p>

<p>I have difficulty understanding this argument. What is a reason to be against mathematics in economics?</p>

<p><strong>Note:</strong> I'm pretty mainstream and like how economics is taught and structured. I'm not anti math-in-economics, I just want to know why this is an argument.</p>

CC BY-SA 4.0
2018-06-28T18:24:28.013Z
I am not sure what you mean by " how economics is taught and structured". My guess is that this is very diverse and can differ greatly depending on where and when you study economics.
Giskard
CC BY-SA 4.0
2018-06-28T19:11:04.950Z
How about a less sensational title?
Michael Greinecker
CC BY-SA 4.0
2018-06-28T19:14:41.853Z
@MichaelGreinecker I'm taking suggestions.
EconJohn
CC BY-SA 4.0
2018-06-28T19:16:17.733Z
"Criticism of Math in Economics" or "Criticism of the Use of Math in Economics", maybe.
Michael Greinecker
CC BY-SA 4.0
2018-06-28T19:16:40.960Z
How about something like [Mathiness in Economic Theory](https://paulromer.net/wp-content/uploads/2015/05/Mathiness.pdf)?
Giskard
CC BY-SA 4.0
2018-06-28T20:31:43.090Z
Are you talking about the critique of economists for using complex algebraic formulations that assume perfect rationality and don't resemble any real world decisions being made; or is  this the critique of overly convoluted and misused statistical tools that mask the uncertainty of empirical research and make economics look more like hard science than it actually is?
lazarusL
CC BY-SA 4.0
2018-06-28T20:41:59.607Z
@lazarusL both guess. Honestly trying to get it because i'm too mainstream according to some of my peers.
EconJohn
CC BY-SA 4.0
2018-06-29T07:49:17.593Z
One possible reason is that basic maths prove that an infinite growth isn't possible in a finite world.
Eric Duminil
CC BY-SA 4.0
2018-07-02T11:43:34.643Z
I think the criticism is that some economics models make false and inaccurate assumptions. Fancy math is not useful if it builds on faulty assumptions. Mathematically modeling the economy is much more difficult than modeling the physical world, where models have been very successful. (I'm not commenting on the validity of this criticism because I don't know.)
littleO
CC BY-SA 4.0
2018-12-09T02:21:22.483Z
Math is a very precise language that allows us to convey nuanced ideas. Economics is made better, in many cases, by the use of mathematics. These people are probably heterodox for a good reason. And if they are Marxists...welll...they have self-identified as useless.
123
CC BY-SA 4.0
2018-12-10T07:27:27.190Z
What even is "perfect rationality"? Most models assume complete, transitive and reflexive - what we call *rational* - preferences, but that's not a big deal. There's ton of papers on incomplete preferences and an entire field dedicated to studying bounded rationality. 

I find it very funny when an economist says something like that. Apparently it's perfectly acceptable to say people choose bundles of goods based on preferences and budget restrictions but god forbid writing the same idea as a constrained optimization problem.
Pedro Cavalcante
CC BY-SA 4.0
2018-12-10T07:29:59.443Z
People are to this day arguing about what Marx, Keynes and Ricardo "really meant". No one is talking about what Kenneth Arrow really meant, because he wrote in a clear, precise language: math. 

In all honesty, econ majors/grad students who just can't get that whatever logical statement which can be written in plain english can also be written as a mathematical statement - and writing it that way is better because now you prove theorems and do comparative statics in a more transparent way - are bound to be poor professionals and/or irrelevant researchers.
Pedro Cavalcante
CC BY-SA 4.0
2018-12-10T07:37:30.120Z
Oh and just by the way, infinite growth is perfectly possible in a finite world because not all growth in output comes from increasing input consumption. You can conclude that by yourself with some [basic math too](https://en.wikipedia.org/wiki/Solow%E2%80%93Swan_model). Growth can also derive from increased productivity - what we call "input-saving technical change". I see people repeat this absurdly wrong affirmation over and over again and it's to economists what anti-vaccination is to health professionals.
Pedro Cavalcante
CC BY-SA 4.0
2021-02-20T15:10:36.933Z
@PedroCavalcante, complete and transitive preferences are not a big deal? There have tons of experiments showing these assumptions do not hold, yet they continue being used.
Papayapap
CC BY-SA 4.0
2021-02-20T16:07:35.010Z
I think the argument in the question can be stayed like this for clarity: Assumption 1) common people feel that something is scientific if it is expressed mathematically, even if it is not actually scientific, assumption 2) economists want to portray economics as scientific even though it is not, final claim: economists use math and succeed in convincing the common people that economics is scientific even though it is not. Assumption 1 may be true, but I don't think it is economists' fault, assumption 2 is simply an accusation of mal-intent, the conclusion rests on the assumptions.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2021-02-20T16:10:34.447Z
On a different note, any rigorous idea, true or not can be expressed mathematically. It is not that only mainstream economists use maths. I have seen Marxist mathematical models as well, look at the Mahalanobis model for instance, Mahalanobis of the Mahalanobis distance fame in Statistics.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2021-02-20T16:14:31.240Z
@IshanKashyapHazarika In my answer I made exactly the opposite point. Some concepts are much more difficult if not impossible to capture using simple math.
Papayapap
CC BY-SA 4.0
2021-02-20T16:44:36.210Z
@Papayapap Interesting, I will read your answer, haven't read all answers yet. Thanks.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2021-02-22T00:25:00.243Z
Yeah, it's always possible to create conditions in which certain assumptions over preferences do not hold. It's valid and important to understand where, when and how they might fail, so important in fact there's a whole field of research dedicated to better understanding these questions. Most qualified, professional behavioral economists, in spite of knowing in great detail just how irrational people are, aren't willing to throw mainstream economics down the drain, tho. They extend it to account for these shortcomings, it's a thing called scientific endeavor.
Pedro Cavalcante
<mathematical-economics><soft-question>
Criticism of Math in Economics
EconJohn
# Criticism of Math in Economics

(Asked by: username_0 on 2018-06-28)

I've been reading and speaking to a number of educated economists and economics PhDs who are against the use of intense mathematics and mathematical proof in economic theory. Specifically I've been speaking to those of Marxist and heterodox persuasion and reading their work in an attempt to become more open minded.

They emphasize that the study of work by classical economists (like Adam Smith, Karl Marx, and David Ricardo) is still relevant and that the practice of how mainstream economics uses mathematics is abusive and is an attempt to fool the masses regarding the "science" economists practice.

I have difficulty understanding this argument. What is a reason to be against mathematics in economics?

**Note:** I'm pretty mainstream and like how economics is taught and structured. I'm not anti math-in-economics, I just want to know why this is an argument.

**Tags:** mathematical-economics, soft-question

> **Comments:**
> **username_1**: I am not sure what you mean by " how economics is taught and structured". My guess is that this is very diverse and can differ greatly depending on where and when you study economics.  
> **username_2**: How about a less sensational title?  
> **username_0**: @username_2 I'm taking suggestions.  
> **username_2**: "Criticism of Math in Economics" or "Criticism of the Use of Math in Economics", maybe.  
> **username_1**: How about something like [Mathiness in Economic Theory](https://paulromer.net/wp-content/uploads/2015/05/Mathiness.pdf)?  
> **username_3**: Are you talking about the critique of economists for using complex algebraic formulations that assume perfect rationality and don't resemble any real world decisions being made; or is  this the critique of overly convoluted and misused statistical tools that mask the uncertainty of empirical research and make economics look more like hard science than it actually is?  
> **username_0**: @username_3 both guess. Honestly trying to get it because i'm too mainstream according to some of my peers.  
> **username_4**: One possible reason is that basic maths prove that an infinite growth isn't possible in a finite world.  
> **username_5**: I think the criticism is that some economics models make false and inaccurate assumptions. Fancy math is not useful if it builds on faulty assumptions. Mathematically modeling the economy is much more difficult than modeling the physical world, where models have been very successful. (I'm not commenting on the validity of this criticism because I don't know.)  
> **username_6**: Math is a very precise language that allows us to convey nuanced ideas. Economics is made better, in many cases, by the use of mathematics. These people are probably heterodox for a good reason. And if they are Marxists...welll...they have self-identified as useless.  
> **username_7**: What even is "perfect rationality"? Most models assume complete, transitive and reflexive - what we call *rational* - preferences, but that's not a big deal. There's ton of papers on incomplete preferences and an entire field dedicated to studying bounded rationality.   I find it very funny when an economist says something like that. Apparently it's perfectly acceptable to say people choose bundles of goods based on preferences and budget restrictions but god forbid writing the same idea as a constrained optimization problem.  
> **username_7**: People are to this day arguing about what Marx, Keynes and Ricardo "really meant". No one is talking about what Kenneth Arrow really meant, because he wrote in a clear, precise language: math.   In all honesty, econ majors/grad students who just can't get that whatever logical statement which can be written in plain english can also be written as a mathematical statement - and writing it that way is better because now you prove theorems and do comparative statics in a more transparent way - are bound to be poor professionals and/or irrelevant researchers.  
> **username_7**: Oh and just by the way, infinite growth is perfectly possible in a finite world because not all growth in output comes from increasing input consumption. You can conclude that by yourself with some [basic math too](https://en.wikipedia.org/wiki/Solow%E2%80%93Swan_model). Growth can also derive from increased productivity - what we call "input-saving technical change". I see people repeat this absurdly wrong affirmation over and over again and it's to economists what anti-vaccination is to health professionals.  
> **username_8**: @username_7, complete and transitive preferences are not a big deal? There have tons of experiments showing these assumptions do not hold, yet they continue being used.  
> **username_9**: I think the argument in the question can be stayed like this for clarity: Assumption 1) common people feel that something is scientific if it is expressed mathematically, even if it is not actually scientific, assumption 2) economists want to portray economics as scientific even though it is not, final claim: economists use math and succeed in convincing the common people that economics is scientific even though it is not. Assumption 1 may be true, but I don't think it is economists' fault, assumption 2 is simply an accusation of mal-intent, the conclusion rests on the assumptions.  
> **username_9**: On a different note, any rigorous idea, true or not can be expressed mathematically. It is not that only mainstream economists use maths. I have seen Marxist mathematical models as well, look at the Mahalanobis model for instance, Mahalanobis of the Mahalanobis distance fame in Statistics.  
> **username_8**: @username_9 In my answer I made exactly the opposite point. Some concepts are much more difficult if not impossible to capture using simple math.  
> **username_9**: @username_8 Interesting, I will read your answer, haven't read all answers yet. Thanks.  
> **username_7**: Yeah, it's always possible to create conditions in which certain assumptions over preferences do not hold. It's valid and important to understand where, when and how they might fail, so important in fact there's a whole field of research dedicated to better understanding these questions. Most qualified, professional behavioral economists, in spite of knowing in great detail just how irrational people are, aren't willing to throw mainstream economics down the drain, tho. They extend it to account for these shortcomings, it's a thing called scientific endeavor.  

---

## 21 answers

### Answer by username_1 on 2018-06-28. Score: 44 | Accepted answer

I find that the essay "[The New Astrology](https://aeon.co/essays/how-economists-rode-maths-to-become-our-era-s-astrologers)" by Alan Jay Levinovitz (an assistant professor of philosophy and religion, **not** an economist) makes some good points.

> ...the ubiquity of mathematical theory in economics also has serious downsides: it creates a high barrier to entry for those who want to participate in the professional dialogue, and makes checking someone’s work excessively laborious. Worst of all, it imbues economic theory with unearned empirical authority.
>
> ‘I’ve come to the position that there should be a stronger bias
> against the use of math,’ [Romer](https://en.wikipedia.org/wiki/Paul_Romer) explained to me. ‘If somebody came and
> said: “Look, I have this Earth-changing insight about economics, but
> the only way I can express it is by making use of the quirks of the
> Latin language”, we’d say go to hell, unless they could convince us it
> was really essential. The burden of proof is on them.’

The essay also makes a (more or less adequate—which, I leave up to you) comparison with astrology in ancient China to show that excellent math can be used to prop up ridiculous science and grant status for its practitioners.

> **Comments:**
> **username_37**: The "_unearned empirical authority_" sounds really weird.  I mean, math's just a precise language that's easy to perform logical operations with.  Putting something into mathematical terms shouldn't be taken as endowing empirical authority anymore than translating a statement to Latin should.  _Barba crescit caput nescit_.  
> **username_38**: The Latin point doesn't seem much of an argument to me, bordering on straw-man. Latin clearly has nothing to do with economics whereas maths is clearly related. It's straw-man because the reader thinks "well yes, it is completely unreasonable to rely on the quirks of the Latin language to express an economic insight", but that has no relevance at all to whether or not it is reasonable to rely on maths. "It creates a high barrier to entry for those who want to participate in the professional dialogue" *on its own* isn't really much justification either. Many fields have a high barrier to entry.  
> **username_37**: Math and logical systems in general conform to ["_garbage in, garbage out_"](https://en.wikipedia.org/wiki/Garbage_in,_garbage_out); so if someone uses mathematical logic on garbage assumptions, then they'll get garbage results.  But isn't that obvious?  (Not being rhetorical - I'm actually asking if this isn't obvious.  Because if it's not, then I could understand why folks could feel misled by seeing garbage expressed in mathematical terms.)  
> **username_1**: @username_37 It is obvious, but technical garbage is more difficult to identify. This comment could be the core of a nice answer IMO.  
> **username_39**: @username_37, for those who don't understand mathematics in the first place, there is nothing obvious about what math does or does not prove.  If the math checks out, and the experts agree that there are no errors in the math, then to a layperson the only thing that's "obvious" is that the mathematicians know what they're talking about.  Even if that's actually false.  
> **username_24**: @username_39 my thoughts about this whole thing is that people encountering maths in economics are obliged to be good at maths. Sure, someone who isn't good at maths can study the non-mathematical aspects of economics, but I think they have an _obligation_ to be good at maths in order to explore the subject in its entirety.  
> **username_40**: @username_24 There's a difference between being good at maths in general, and being able to follow an argument (and check the proposed proofs) in any arbitrary mathematical framework out there.  
> **username_41**: @username_37: I'll take a contrarian position here: *It is just as obvious as the author wants it to be.*  If all of the relevant assumptions are stated up front, then it's entirely obvious when some assumptions (e.g. "All humans are perfectly rational creatures" or similar) are questionable.  But if an author instead buries these assumptions in e.g. the methodology section (so that anyone just reading the abstract misses them), then a [valid but unsound](https://math.stackexchange.com/q/281208/198741) argument may appear to be supported by the math when in fact it is wrong.  
> **username_42**: That attitude makes me pity the math-illiterate. Properly written, a page of math can replace about 20 pages of rambling imprecise words, and is hence much _easier_ to check. That is basically the whole point of mathematical language. This guy is like a blind person trying to ban diagrams.  
> **username_1**: @username_42 I am not sure who you are referring to as "this guy", but in case it is Paul Romer, I assure you, he knows quite a bit of math. In case you are referring to Levinovitz, I don't think he is math-illiterate either, at least I could not gather this information from his essay.  
> **username_43**: @username_37 to those who do not know mathematics or latin, they do endow unearned empirical authority. See, for example, https://www.wsj.com/articles/SB10001424127887323374504578219873933502726. Ubiquity of math may not be a problem for economics from a strictly internal viewpoint, since practitioners know some math, but it seems reasonable that it makes it more difficult for non-economists who are not fluent in math to know which economists to listen to.  
> **username_6**: This is non-sense. We sometimes need to convey ideas more precisely than is possible without using mathematics. Fields progress. People opining for days gone where arguments counted as serious research does not mean that the field has moved in a bad direction. And this is an appeal from authority. I would guess there is someone equally influential willing to argue that introducing precision into economics has been incredibly useful.  
> **username_1**: @username_6 [I would like to point out that the question is not whether we should have math in economics, but why some people attack mathematical economics.](https://economics.stackexchange.com/a/22618/1601)  
> **username_7**: This barrier to entry argument just doesn't cut it. Physics too has a high barrier to entry and I don't see people claiming we should abandon the Maxwell Equations. People could describe everything in physics just with english, but it's username_28 and a lot less transparent. The same happens in economics.  
> **username_1**: @PedroCavalcanteOliveira I am not saying we should abandon mathematics, as I pointed out in the comment directly above yours.  
> **username_6**: @denesp -- the point I am making refers specifically to the quote. I think it is nonsense. My opinion is independent of the question.  
> **username_1**: @username_6 That is fine, you are entitled to your opinion. Please consider posting it under the article, or on Facebook, and not under my answer, which addresses the question.  
> **username_8**: +1 great comparison with astrology. Economics is very much like ancient astrology  
> **username_44**: @username_42 If you have something that is obvious without math, you can state it without math. If you do need the math because it shows something counterintuitive, you should at least note that the math is required because it shows something counterintuitive. But most of the time, you are trying to propose a policy, "doing X makes the world better", and if you cannot show that it is better without using extensive math, maybe it is not so clear after all! Of course, if you are trying to make quantitative predictions (*how much* better) then math is essential.  
> **username_42**: @user253751 Yup, I was naive when I wrote that comment. Now that I'm the ripe old age of 25 I see the wisdom in this -- math gives the researcher too many tricky degrees of freedom to get whatever conclusion they want.  

---

### Answer by username_10 on 2018-06-29. Score: 39

> What is a reason to be against mathematics in economics?

The danger that any tool creates: to impose itself on the tool-user, diluting and narrowing its view of the world. It is a matter of Human Psychology why this happens, but it certainly does, and the aphorism "to he who holds a hammer everything looks like a nail" expresses this phenomenon, which has nothing to do with economics specifically.

Mathematics offer a great service to the Economics discipline by providing a crystal-clear path from premises to conclusions. I fear the next time a Keynes with a General Theory book appears -and then we would have to spend decades again deciphering "what the author really meant" by his verbal arguments -and not really agreeing.

The "abuse of mathematics" certainly happens: producers and consumers of economic theory tend to not question/worry/have nightmares about "the premises", to the extend that they should. But once we leave the premises unchallenged, the conclusions become "undeniable truth", since they have been derived in the rigorous mathematical way.

But the ability to challenge the conclusions is always there, if only we take the time to review critically the premises.

Another, more sophisticated way that mathematics may be abused is the belief that the deviation from reality that the premises represent, transfers over to conclusions in a "smooth" manner (call it "the principle of non-accelerating propagation of error"): to consider the trivial example, sure, the assumptions describing a "perfectly competitive" market (the premises) do not hold "exactly" in reality. But, we argue, if they are "close enough" to the structure of a real-world market, then the conclusions we will reach through our model will be "close enough" to the actual outcomes in this market. This belief is not unreasonable, and it is borne by reality in many cases. But this principle of "smooth approximation" does not hold universally. So even if we check the premises and we are satisfied that they are not absurd, it still remains the possibility that the real-world phenomenon we examine is not so well-behaved, it has discontinuities that may make a small deviation lead to very different outcomes in reality.

That's the abstract analysis of the matter. The sociological and historical view would ask "but if a tool, that theoretically can be used in the proper way, has been seen for decades to be used inappropriately and create undesirable consequences, shouldn't we conclude that we must abandon its use?"

...in which instant, we start arguing about *the extent* of these "undesirable consequences" and whether they overcome any benefits from the use of the tool. In other words, this matter too, dreadfully comes down to a cost-benefit analysis. And we rarely agree on that either.

> **Comments:**
> **username_11**: The trouble with this argument is that whatever other stuff we use for economics are also tools. It's not like math is a tool but the other stuff we use are fully legitimate truth-finders blessed with kisses from Jesus Christ. Our views will be inherently "diluting and narrowing", otherwise you are supposing that non-math appraches to economics allow us to see the entire reality as it is.  
> **username_10**: @BillyRubina I am not sure I follow you. Where in my answer is it implied that "other stuff we use" do _not_ constrain us? And where do I imply that we would be better off without math?  
> **username_12**: Regarding "next time a Keynes with a General Theory book appears": Piketty tried to be that next writer. His book was also less mathematical, and the profession immediately poked holes into it, e.g. http://www.econ.yale.edu//smith/piketty1.pdf  

---

### Answer by username_1 on 2018-06-29. Score: 29

I would like to point out that [the question](https://economics.stackexchange.com/q/22599/1601) is not whether we should have math in economics, but why some people attack mathematical economics. A lot of the recent answers seem to try to answer the first question.

Now then, to cover all bases like a good incumbent in a differentiated product market, I will also post an answer with points that economists have already raised about this question.

Hayek in his Nobel Lecture:[The Pretence of Knowledge](https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1974/hayek-lecture.html) said

> It seems to me that this failure of the economists to guide policy more successfully is closely connected with their propensity to imitate as closely as possible the procedures of the brilliantly successful physical sciences - an attempt which in our field may lead to outright error. It is an approach which has come to be described as the "scientistic" attitude - an attitude which, as I defined it some thirty years ago, "is decidedly unscientific in the true sense of the word, since it involves a mechanical and uncritical application of habits of thought to fields different from those in which they have been formed."

Paul Romer coined the term [mathiness](https://en.wikipedia.org/wiki/Mathiness) to describe the issue in his (unrefereed) paper [Mathiness in the Theory of Economic Growth](https://paulromer.net/mathiness/Mathiness.pdf). He writes

> The market for mathematical theory can survive a few lemon articles
> filled with mathiness. Readers will put a small discount on any
> article with mathematical symbols, but will still find it worth their
> while to work through and verify that the formal arguments are
> correct, that the connection between the symbols and the words is
> tight, and that the theoretical concepts have implications for
> measurement and observation. But after readers have been disappointed
> too often by mathiness that wastes their time, they will stop taking
> seriously any paper that contains mathematical symbols. In response,
> authors will stop doing the hard work that it takes to supply real
> mathematical theory. If no one is putting in the work to distinguish
> between mathiness and mathematical theory, why not cut a few corners
> and take advantage of the slippage that mathiness allows? The market
> for mathematical theory will collapse. Only mathiness will be left. It
> will be worth little, but cheap to produce, so it might survive as
> entertainment.

He goes on to give specific examples of 'mathiness', including works by high profile economists like Lucas and Piketty.

Tim Harford provides a laymen's summary of Romer's paper in his blogpost [Down with mathiness!](http://timharford.com/2015/06/down-with-mathiness/) In this he writes

> As some academics hide nonsense amid the maths, others will conclude
> that there is little reward in taking any of the mathematics
> seriously. It is hard work, after all, to understand a formal economic
> model. If the model turns out to be more of a party trick than a
> good-faith effort to clarify thought, then why bother?
>
> Romer focuses his criticism on a small corner of academic economics,
> and professional economists differ over whether his targets truly
> deserve such scorn. Regardless, I am convinced that the malaise Romer
> and Orwell describe is infecting the way we use statistics in politics
> and public life.
>
> There being more statistics around than ever, it has never been easier
> to make a statistical claim in service of a political argument.

> **Comments:**
> **username_10**: (+1) for the references, especially Romer's. Setting aside the gossipy issue related to his direct attack on household names like Lucas and Prescott, the most interesting thing here is this concept of "mathiness", which is subtle, because it is not about "garbage premises and then super mathematics" but about something much more subtle but equally critical: mapping verbal concepts to math symbols _without proper justification_. This is much more difficult to detect in a paper, if you are not really experienced.  
> **username_13**: I can't find a precise article, but I could have sworn modern economics violates key mathematical axioms that make the math work in the first place. Would love to have a theoretical mathematician address the problem. Probably, has something to do with the axiom of choice is my guess.  
> **username_14**: Robert Solow had a similar opinion: https://economistsview.typepad.com/economistsview/2009/08/solow-dumb-and-dumber-in-macroeconomics.html  

---

### Answer by username_15 on 2018-06-29. Score: 22

I think there are two important criticisms or limitations.

**Limit 1:** The first, overlapping with what many others have said, is that all mathematic economics are reduced-order models of very highly complex relationships between monumentally complex actors. As Einstein is alleged to have said (approximately) "Insofar as the truths of mathematics relate to mathematics, they are certain. In so far as they relate to the world, they are not certain." 'Does this mathematics apply in this situation?' is always an open question. Similarly, 'Is there a better mathematics that we haven't yet discovered?'

**Limit 2:** The other problem, and it is bigger for economics than any other field I can think of, is the extent to which the state of the art knowledge of economics changes the economy because it becomes 'common knowledge'. For example, when you convincingly show that investing in mortgage-backed securities is low-risk compared to the yield, and that home-ownership is a cornerstone of wealth-creation for ordinary people, the economy will pile into those things until the apparent excess value is consumed. This feedback and phase-changeability means that economies are non-ergodic - (apparently NN Taleb makes a lot of this point in Black Swan?)

Even if economic knowledge were not encoded in policies of economic actors, the changing nature of society and technology will always cause problems under Limit 1. Neither of these limits argues for excluding mathematics from economics, but they do argue for not excluding non-mathematical considerations (e.g. the political side of political economy) from economics. In practice, this might mean a bit more authority for the judgment of older economists who are wary of, for example, the value of high-speed trading.


---

### Answer by username_16 on 2018-06-28. Score: 15

I think that the opposition to mathematics in Economics mainly has to do with the obstacles it poses to *indoctrination*.

A proposition expressed in terms of a mathematical/logic system is susceptible of objective verification, whence the inconsistencies of a proposition are more visible than where a rigid framework is missing. Moreover, mathematical propositions do not lend themselves to the hyperbole and passionate impetus that fuel a socio-political ideology.

The excerpt cited by @denesp reflects Levinotiz's confusion between the rules of logic and the rules of grammar. Despite the definiteness inherent to Latin grammar and the complexity of expressions it allows, its lack of logical rules and relations of consistency render grammar useless as method of proof.

> **Comments:**
> **username_0**: Reminds me of  the words of Roger Beacon: “Neglect of mathematics work injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy.”  
> **username_16**: @username_0 Exactly, and that leads to a clash of irreconcilable conclusions reached from subjective, unsystematic assessments. Marx's propositions such as "religion is the opiate of the masses" pertain to sociology rather than Economics. Adam Smith's idea of the invisible hand expresses an assumption from which causal arguments can be developed. But the social or subjective origin of an assumption or a perception is not a good reason to exclude a formal, verifiable system of logic for the development of a theory.  
> **username_8**: Actually I think quite the opposite, mathematics has been used as a great way to indoctrinate generations of people with free-market ideology. The assumptions needed to get to these conclusions e.g. walrasian auctioneer, homothetic preferences, constant returns to scale are absurd but hidden to justify free market agenda.  
> **username_16**: @username_8 You seem to confuse between assumption and conclusion. *Free market* is most often an *assumption* without which the rest of a model would be pointless and therefore less susceptible of serious or more objective assessment. In turn, interventionists' failure to use mathematics "*as a great way to indoctrinate*" is a clear indicator of the arbitrariness and distortions inherent to their ideology. Mathematics is quite useful for scrutinizing and dismantling the inconsistencies of the rhetoric of interventionism.  
> **username_9**: @username_8 I think it would be much more difficult to "indoctrinate" people using math. In math you are forced to disclose all assumptions (which the reader may discard as unrealistic) while in vague qualitative essays even the most well trained reader can fail to identify a hundred assumptions (and without even knowing the assumption, it is hard to see whether the assumptions are reasonable or not). Add to that the fact that qualitative assumptions may mean different things to different people and that it is even difficult to see whether the conclusions even follow from the assumptions or not  
> **username_9**: @username_8 Adam Smith's qualitative arguments for "free market's efficiency" seemed infallible. It is only when Arrow, Debreau and others mathematically formalised the arguments that we now exactly the know the assumptions under which free markets are efficient and when they are not. This appears, at least personally to me, to be much more honest than verbose arguments. Markets fail when there are externalities, market power, incomplete information, or even in overlapping generations etc.  
> **username_8**: @username_9 Thanks for citing Arrow and Debreu. Walras was concerned with the question of whether multiple free markets can clear. He concluded that was not possible without an auctioneer setting prices to clear. Arrow and Debreu skipped this fundamental step completely and just assumed that markets would clear. A very bright example of how extensive math can be used to hide important conceptual defficiencies. QED.  
> **username_8**: @username_16 are you an economist? The work done to show that free markets are pareto-efficient is done to show that state intervention is inefficient and markets should be left unregulated. This works starts with absurd assumptions, uses valid mathematical steps, but reaches wrong conclusions. 20th century economists knew that markets need to be regulated to limit market power, set the rules of the game etc. The mathematical dillusions of the later mathematical economists produced only confusion...  
> **username_16**: @username_8 You did not specify which work in particular you're referring to, let alone which assumptions/conclusions seem absurd/wrong to you. For instance, if by "absurd" you refer to the "auctioneer" being a fiction, you are getting stuck with a pedagogical artifact and missing the abstraction altogether. Likewise, economists adopt the assumption of information symmetry, but that does not mean they preach that such symmetry exists. Regardless, the paper you purportedly have in mind does not invalidate my point that *most often* free market is an assumption.  

---

### Answer by username_17 on 2018-06-30. Score: 13

## "All models are wrong; some are useful."

The title is really all one needs, but to put a few more words behind it, mathematics is very good at deriving detailed results from very specific premises. It is very easy to make a mistake in the premises and obscure the consequences with language.

A major issue in macroeconomics is that every policy decision must be self-referential. It's very easy to accidentally assume that some small actor won't change their decision making slightly in an unexpected way that makes the whole thing fall apart. It's also very easy to make the mathematics look airtight.

In more microeconomic situations, you have assumptions about how the world will function. This is most easily seen by developing an AI which can make a killing when fed historical data, but which fails utterly in the real market.

> **Comments:**
> **username_18**: For those who don't know, the header is quote by British statistician George Box. One of my all-time favorite quotes!  
> **username_17**: @Sam Good point.  I've put quotes on the header to make it more obvious that it's a quote.  I'm a programmer by trade, so I live and die by that quote!  

---

### Answer by username_19 on 2018-07-01. Score: 5

In my experience the most important reason is that economics has political implications and that creates a huge moral hazard to use complex incomprehensible math to arrive to politically desirable conclusions.

Unlike in natural sciences, economic models can hardly be verified empirically and require tons of assumptions. Add a thick layer of math on top and you can support pretty much anything. In fact, anything beyond linear regression hardly improves predictive power in practice.

Seasoned economists see through this. Some are in on it (hey, it is very profitable!) and some are pretty unhappy about all this abuse of math, which is unethical from a scientific point of view. But I guess many are both. At the end of the day, we all have bills to pay and families to feed. Nevertheless, we are still scientists. So there is a lot of cognitive dissonance and strong feelings going on.

> **Comments:**
> **username_1**: I think most physics models also require a ton of assumptions. It is their empirical verification that is better. Perhaps the system they study can be more frequently decomposed into smaller independent parts.  
> **username_7**: Economic models not only can but get constantly empirically verified. Why do people make strong claims about subjects they clearly don't know?  Just see what people are publishing in the frontier journals: https://academic.oup.com/qje/issue. Most, if not all papers published in these good journals empirically verify a theoretical hypothesis or conclusion from a model.  
> **username_19**: @PedroCavalcanteOliveira man, [QJE is #1](https://ideas.repec.org/top/top.journals.simple.html). There are thousands of economics journals below it that will publish things of much lower rigor, if any rigor at all, and politicians use those just as well to push the policies of their choice. Guess how many bother to replicate and test any of it? That would require funding. From the same politicians, that is, or an NGO with its own agenda. That's why when I see things that are supercomplex for the sake of a bit higher accuracy but take tons of time and resources to test, I get a bit critical.  
> **username_7**: You can't look at the worst outlets of a field and claim that there's a problem with it because they're bad. If that's reasonable, then literally all sciences are in big trouble. And bringing this generic argument about politicians basing themselves on bad journals isn't any good. Who are these politicians? Where and when did this happen? Can we blame economics as a field on it? Your claim was that "economic models can hardly be verified empirically", which is clearly wrong. Most of the papers published in any respectable journal ate empiric. That should be your standard.  
> **username_19**: @PedroCavalcanteOliveira My point is that many people like it simple when there is moral hazard involved. A good standard for verifying something is an experiment with all the variables controlled. It is a very hard thing to do in social sciences. Not saying we shouldn't push the math forward, just don't build skyscrapers on the sand.  
> **username_7**: Laboratorial experiment is hard/impossible to do in economics but we have other techniques. RCTs, quasi-experiments, natural experiments... There's an entire field of econometrics dedicated to causal inference in non-laboratorial settings. This has been out there and used in economics for 25-30 years. [Angrist and Krueger, 1990](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.6971&rep=rep1&type=pdf) is a great example, as is [Angrist, 1989](http://username_31.princeton.edu/pubs/pdfs/251.pdf)  I feel like you don't know much about what you're complaining about.  
> **username_7**: [Here`s a good reading](https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.24.2.3), describing the immense progress we've made as a profession towards credibility.  

---

### Answer by username_2 on 2018-06-29. Score: 5

Clearly, mathematics could never cover the full richness of the human experience.

> …In that Empire, the Art of Cartography attained such Perfection that
> the map of a single Province occupied the entirety of a City, and the
> map of the Empire, the entirety of a Province. In time, those
> Unconscionable Maps no longer satisfied, and the Cartographers Guilds
> struck a Map of the Empire whose size was that of the Empire, and
> which coincided point for point with it. The following Generations,
> who were not so fond of the Study of Cartography as their Forebears
> had been, saw that that vast Map was Useless, and not without some
> Pitilessness was it, that they delivered it up to the Inclemencies of
> Sun and Winters. In the Deserts of the West, still today, there are
> Tattered Ruins of that Map, inhabited by Animals and Beggars; in all
> the Land there is no other Relic of the Disciplines of Geography.

Jorge Luis Borges, [On Exactitude in Science](https://en.wikipedia.org/wiki/On_Exactitude_in_Science)

> **Comments:**
> **username_1**: I like the image, but this seems to be against modeling in general, not mathematical modeling in economics.  
> **username_2**: @debesp The first sentence is undeniably true and the Borges-quote gives the appropriate context.  
> **username_10**: And why should we care about the "full richness of human experience"? It has already happened, let's do something else.  
> **username_2**: @username_10 The story kinda answers your question.  

---

### Answer by username_20 on 2018-06-28. Score: 4

The problem with math as used in modern economics is that the math is often used to describe models of human behavior. Modeling human behavior, whether with math or otherwise, is incredibly difficult especially over long time scales, if our goal is to make the model match reality. So it's not really that there's a problem with using math per se, but mathematical models of human behavior are by their very nature bound to fail in multitudinous ways, so that the detailed economic models built by economists do not match reality and don't have clear practical utility.

Economics must move away from modeling human behavior and move towards modeling institutions, governments, companies, etc. and the dynamics involving these agents. Mathematical models will be more useful here because the entities I described above have both fewer clearly defined parameters of existence, and their interactions with other human-composite entities are more range restricted than those involving human beings themselves.

Moving away from behavioral economics will restore legitimacy to economic science because a focus on institutions will yield more accurate models and therefore greater predictive and explanatory power.

> **Comments:**
> **username_21**: Do you have any reason to think modeling institutions will be any simpler than modeling human behavior? Especially over the longer time scales you note are troublesome?  
> **username_20**: Of course I do, that's why I said it.   The reasons are that the dimensions of institutional behavior and interaction are much less than that of human behavior, and more importantly, the behavior of actual institutions is much more visible to us than that of people.  
> **username_22**: Who do you think runs institutions if not humans?  
> **username_23**: Hi: I just want to add that Nerlove started the attempt to model human behavior in the form of modelling expectations by coming up with adaptive expectations. later on, partial adjustment models were another attempt to do this. then, later on the whole rational expectations revolution went even further in the attempt. How well the RE models work is a different issue but there are definitely mathematical-econometric modelling efforts to model human behavior  through the mechanism of modelling agent's expectations..  
> **username_24**: @username_20 even though I **upvoted** your post, for showing that maths is ineffective with explaining behaviour, I disagree that economics must become more narrow. I think subjects need to be cross-curricular. Personally, I am interested in psychology, and I like the perspective economics has on behaviour. I agree that maths can't describe behaviour to a T, but I think it's fine if maths is left out of behavioural economics (instead, it can focus on understanding irrationality).  
> **username_20**: I'd like to add that trying to construct economic models based on human behavior rather than on institutions and companies is like trying to construct an ecological model based on the behavior of individual animals rather than that of whole populations and the interactions of those populations in aggregate. Another analogy: when studying the human body as a whole, we are less concerned with the behavior of individual atoms and molecules and more with the dynamics of entire organs and systems in the body, such as the endocrine system.  
> **username_23**: @username_20. I see what you are saying but, if you make the assumption that, in aggregate, individual animals tend to behave in some consistent-logical way, you might be okay. and this is pretty much what RE does.  
> **username_20**: @username_23 that is true, but the point still stands that the economy is largely the domain of banks, corporations and governments. Yes populations of consumers and savers (individuals) are part as well, but they are only a part. I am not saying to disregard individuals or populations of individuals, but that a full account of the economy will have to consider institutions as first class citizens along with populations.  
> **username_20**: @username_22 It is unnecessary to consider humans because we can talk about a company's profit motive, or the function of a regulatory agency in an economy, or a government's tendency to borrow in order to fund a deficit all without considering the humans that are the building blocks of those institutions. If we want to study the internal dynamics of a single company we can and should study its employees, but if we are concerned with sectors of an economy or the economy as a whole, humans (as individuals) add unnecessary complexity.  
> **username_23**: @username_20: Excellent points and I agree. I think Muth was thinking more along your lines when he came up with RE but then, over time, it evolved into more of an "individualistic-agent" proposition. Not absolutely sure but my best guess.  
> **username_23**: @username_20: One last question to you since it sounds like you are very into economics. It's a particularly interesting discussion for me because I have no formal education in economics (background is statistics ) and I kind of learned RE on my own since it's a lot of econometrics.  Here's my question:  If math in economics is not so useful, then why did Sargent and Hansen both get Nobel Prizes ?? Someone must have their work was important and useful I would think. Thanks for insights.  
> **username_20**: @username_23 Math is just a language. I am not saying that math in economics is not useful. I am saying that the language of mathematics is being used to describe models which are less helpful in understanding the economy than models which would place more focus on institutions and corporate entities. Just because a mathematical model is internally consistent does not mean it is helpful or useful.  
> **username_23**: @username_20. gotcha. agreed  

---

### Answer by username_25 on 2018-07-02. Score: 4

Most economics questions have three parts:

1. Why does a phenomenon occur? This lets the user understand the answer, understand whether the question is relevant, and understand what factors would change the answer to the next part.
2. How much of the phenomenon is likely to occur? This lets the user make decisions based on the answer, and compare the importance of various phenomena.
3. Under what conditions does a different phenomenon replace this phenomenon?

An answer that does not address all three sub-questions is incomplete. It is likely to either be misunderstood, or be misleading.

Math is needed to get an approximate answer to the second sub-question: How much? A person with a good understanding of the math can simplify the math to provide insight to the first and third sub-questions: Why, and with what limits?

For example, Cobb-Douglas production functions (and mathematically similar utility functions) use math that most non-economists do not understand. The essential features of these functions can be boiled down to "price elasticities" of supply and demand. These are terms that most non-economists do not understand, but they can be turned into examples that most people do understand. For example, such functions for global oil production and demand during the 1980s could be simplified to "In the short run, if OPEC cuts its production by 1 percent of the total world production, then the price of oil will go up by 7 percent."

Unfortunately, many economists use math badly:

* Instead of using the math to generate (and verify) a simplified explanation, some economists go through the details of a complicated "mathematical demonstration". In the end, the reader has to trust that the economist made the right assumptions, and often only as an answer to "how much", not "why" nor "with what limits".
* Some economists are not careful to explain the uncertainties inherent in their math.
* Some economists use symbols ignorantly. I once had the displeasure of listening to a lecture by a well-paid, soon-to-be-famous economist. He had many charts about things like long-term trends of the price of power, which were on a log-log scale. The x-axis was labelled as log(dollars), and the y-axis was labelled as log(kW). But his units were actually ln(dollars) and ln(kW). When politely asked later about it, he did not understand that this was a problem! (If he had actually wanted to be understood, he would have labelled the y-axis as W, kW, MW, GW, *et cetera*, and used similar labels for the x-axis.)

> **Comments:**
> **username_1**: I suspect (but don't actually know) that the convention whether log is short for log$_e$ or log$_{10}$ is language dependent.  
> **username_25**: @denesp -- The lecture was in American English.  Both the lecturer and I are Americans, and have been part of nearby universities.  

---

### Answer by username_24 on 2018-06-29. Score: 4

Mathematics is just language that can be used to provide clear, accurate statements. It should not be seen as an obstacle—rather, it should flow naturally alongside the other language with which it is written (e.g. English). I don't believe that maths is inherently "rigorous" or "authoritative", as other answers have mentioned, because the reader *should* be critical enough to spot errors. However, I recognise the limitation here: either because of a limit in human cognition, because people don't put the effort in to study mathematics, or because of a fear of maths, some people *aren't good at maths*. I think that is where this problem stems from, **but** I don't believe that poor aptitude at maths is a good enough argument for why we shouldn't use it.

Excluding maths from economics is akin to saying that that maths should be kept separate from other subjects.

On the other hand, reading through the answers reminds me of Paul Romer's paper [*The Trouble With Macroeconomics*](https://ccl.yale.edu/sites/default/files/files/The%20Trouble%20with%20Macroeconomics.pdf). He criticizes (with a good example) that incorrect assumptions that are made for a mathematical deduction can easily be obfuscated. Section 5.3 reads:

> In practice, what math does is let macroeconomists locate the Facts With Unknown Truth Value farther away from the discussion of identification. The Keynesians tended to say "Assume P is true. Then the model is identified." Relying on a micro-foundation lets an author can say, "Assume A, assume B, ... blah blah blah .... And so we have proven that P is true. Then the model is identified."

with the "blah blah blah" making it username_28 to detect incorrect assumptions.

As [username_39 said](https://economics.stackexchange.com/questions/22599/criticism-of-math-in-economics#comment37184_22600), the average person may well end up skimming over the maths, in blind faith that it's correct, for lack of effort of checking it themselves.

In closing, sure, economics needs a sociological, psychological or political setting, but mathematics helps to study ideal situations. We can't create complete models of humans or institutions, but economics would be very empty if we didn't study ideal situations. Maths belongs in economics—perhaps those who want it taken out have not satisfied their interest in social sciences enough by studying alternative social science subjects.

> **Comments:**
> **username_1**: Romer's [Mathiness](https://en.wikipedia.org/wiki/Mathiness) is indeed lurking in several of the answers.  

---

### Answer by username_8 on 2021-02-20. Score: 4

In addition, to the excellent previous replies from my experience the biggest downside is the narrowing down the way you think about the economy and life to a way that is mathematically tractable. Some aspects of decision-making and interaction are rather easy to make sense of mathematically (e.g. people on average buy more for a lower price so you can write a downward sloping demand curve) others are highly complex, context-dependent, and unstable (e.g. how advertising or looking what my neighbours have affects my consumption decisions). Unfortunately, in trying to acquire rigour, economists chose to focus only on the first set of questions / explanations because they can be stated neatly using the common mathematical tools they learnt, i.e. optimization. The same with the equilibrium, there is a brilliant passage in MWG graduate Micro textbook on why Economics focuses on equilibrium, they say something like "because in general economists are good at describing what happens in equilibrium and not so good what happens in between"... I find that statement really laughable and sad.

**Edit**: I found out somebody much more clever made this argument in a much more compelling way [Akerlof, 2020](https://pubs.aeaweb.org/doi/pdfplus/10.1257/jel.20191573) ;)

> **Comments:**
> **username_8**: If you downvote, comment at least  

---

### Answer by username_26 on 2018-06-29. Score: 3

To start with, it can be noted that the rise in mathiness in economics is substantially related to increased data processing power, whether in support of theoretical demonstration or empirical application. It is not itself an objective.

Regarding the specific question of why heightened mathiness may be criticized:

1) Economics originates from moral philosophy. There are those who believe that debates involving who gets what, and on what terms, is related to moral philosophy. Mathematical tools can help to express moral concepts or present argumentation about which approach might better serve some moral end.

2) a) Complex math can enable theoretical presentation which is mathematically satisfactory for expressing a theory, but mathematical complexity should not be perceived as demonstration of quality in and of itself, and b) mathematical complexity doesn't necessarily mean that empirical applications will be any better. The risk may be that in order to impress other economists, unnecessarily and/or incorrectly complex math is used to express and/or develop a theory.

I think being open minded in this context would be supported by a belief that diverse economists question the value of heightened mathiness, or that diverse economists view heightened mathiness as a tool (which brings risks, in particular of false overconfidence in results) and not an objective in and of itself.

It can also be noted that one of the main contributions of Marx, aside from proto-macro theory, is extensive development of the idea that technology affects production conditions. And, that production conditions affect the way that all of us live. You don't have to be communist to think that this piece of knowledge is a) useful, and b) not necessarily well served by mathematical demonstration even if some very mathy empirical applications may present results which are very relevant for practical policy considerations.

For most cases, such views should not be perceived as 'anti-math', per se, but rather critical of excessive reliance on (or overconfidence in) mathematical demonstration and/or math-heavy empirical applications as a tool. These can be supplemented by socio-political and/or moral argumentation or reasoning, or if outside of the scope of the work it can at least be explicitly recognized that such considerations are relevant.


---

### Answer by username_27 on 2018-06-28. Score: 3

Economics is a social science, not an empirical or laboratory one. It is the study of human behavior in response to competing demands in an environment of scarcity. Human behavior cannot be predicted with mathematical precision — the only way to do it is to make large numbers of gratuitous and unsupportable assumptions about what people will do under a given set of circumstances.

Mathematical economists do not study people. Instead, they study what Nobel laureate Richard Thaler calls “Econs”... perfectly-knowledgeable, perfectly-intelligent, perfectly-logical, perfectly-sophisticated, perfectly-intentioned, perfectly-identical automatons who live and work in an environment of perfect competition; as opposed to humans, who are none of those things and who live on Planet Earth.

It’s not that math is bad — it lets us easily communicate complex ideas clearly and precisely. But we need to remember that the predictions rendered by mathematical economics, very often, will not hold in real life. We need to understand (and promote that understanding in those who look to the economics community for guidance and advice) that math only gets you so far — to make good policy, you need to understand what flawed, fallible, semi-unique, stressed, busy, selfish, sometimes stupid, imperfect human **people** will do. And mathematics cannot tell you that.

> **Comments:**
> **username_1**: But most of Thaler's models, which try to include some aspects of human psychology are based in math. Is he then a fraud, or is this a misrepresentation of what he says?  
> **username_27**: Not a fraud — like I said, math isn’t bad. It’s a useful tool for communicating complex ideas. It’s just that it only goes so far. You can’t plug values into a formula and accurately predict human behavior with x=y precision. You *can* use math to analyze *past* human behavior though.  
> **username_1**: Most economists would not claim that that is what they are doing, so this does not seem to directly answer the question. These are models, often simplified to the extreme, to capture one aspect of behavior.  
> **username_27**: I think it addresses the question.  
> **username_27**: @denesp, your own quote would seem to back up what I’m saying.           >“Worst of all, [the ubiquity of mathematical theory in economics] imbues economic theory with an unearned empirical authority”  
> **username_16**: @username_27 Assumptions akin to being rational, perfectly-identical, and so forth are necessary idealizations to develop a manageable body of knowledge. The differences among agents are addressed at advanced stages through the incorporation of probabilistic tools, "error modeling", and other resources. One ultimate point of mathematics is not that agents are necessarily rational or perfectly intelligent, but *how* arbitrage opportunities arise when agents act irrationally.  
> **username_27**: @username_16 Sure. But how many average joes can understand that stuff? How many even exceptional people, policymakers, and such even? Most people see the Econ-101 supply and demand curves with the empirical-looking equilibrium point and take it as an objective truth.  
> **username_16**: @username_27 Very few, but that's a different issue. Apropos of the analogies about Latin, the suppression of mathematics in Economics is akin to getting rid of literature written in Latin merely because very few average joes & policymakers can read that language.  
> **username_27**: That’s my point — not that mathematical economics is bad, but that we need to do a better job making people understand that *accurate* mathematical economics is not novice-level stuff. What you learn in Econ 101 is not irrefutable cosmic wisdom (which is how too many people treat it). We need to suppress mathematics in introductory and intermediate economics to make room for the social economcs which will give the majority of people who will never study graduate or doctoral economics a better understanding of how markets actually work.  
> **username_16**: @username_27 But the OP's question is not about pedagogy of Economics; it goes further into its development and systematization. Various aspects of *social* economics are already covered in sociology or political "science". Beyond that, I doubt that one could truly understand, for instance, "why arbitrage opportunities hardly ever occur in *actual* markets" without first learning how those opportunities are detected/exploited. Knowledge of mathematics is essential for the latter. Proficiency in Economics is requisite when serious discussions and informed design of economic policy/theory are intended.  
> **username_28**: Weather can't be predicted with mathematical precision either, but meteorologists must know quite a bit of mathematics to do their jobs.  
> **username_29**: @username_27: The rules taught in Econ 101 aren't always sufficient to make perfect predictions.  On the other hand, when someone proposes a policy whose claimed expected results contradict Econ 101 predictions and doesn't address such contradiction, odds are good that the actual results will match Econ 101 predictions more closely than they match the policy promoter's claims.  
> **username_2**: No, no, no, no.  Literally nothing in the list of "perfectly-knowledgeable, perfectly-intelligent, perfectly-logical, perfectly-sophisticated, perfectly-intentioned, perfectly-identical automatons who live and work in an environment of perfect competition" describes the extend of mathematical economics.  
> **username_27**: @username_2 How would you describe what mathematical economics seeks to do? What assumptions or givens does it rely on, if not the ones I’ve laid out?  
> **username_2**: @username_27 Mathematical economists mostly study the consequences of different assumptions. As such, there are no assumptions made by all of them all the time. But every advanced undergraduate should have seen models of imperfect competition, models in which not all agents are the same, and models of imperfect information. To be blunt: You clearly have no idea what you are talking about.  
> **username_27**: @username_2 That’s not very charitable. True, I’m still working on my undergrad, but the state of economics in the American political zeitgeist (I would contend) is very much as I’ve described. Whatever it is that economists know, economic policy is most often made by those who are not economists (nor who possess even a bachelors in economics) , and many of whom are ignorant of or content to omit anything beyond the simplistic 101-level model; when they even bother to use economics as more than a buzzword. In short, the math belongs farther on. We need more logic up front.  
> **username_2**: Well, I think your answer was not very charitable to mathematical economists, a group I belong to. What you now wrote in your comment has very little to do with mathematical economics.  
> **username_30**: @username_2 Perhaps it's worth offering a bit of a middle ground. To username_2's point, he's definitely correct- no economist I can think of conducts research that would be recognizable as economics from a 101 class perspective. The whole point of new projects is to relax previously restrictive settings, and see what new dynamics can be explained. That said, username_27's correct too, that often the framework employed by economists is restrictive itself. For example, game theorists like us often make assumptions about strategy spaces- whether they're convex sets, known by all parties, etc....  
> **username_30**: And often research is performed by relaxing these sorts of assumptions- after all, you need certain types of assumptions (such as those regarding strategies, payoffs, etc.) to allow the mathematics to make sense, and be useful. However, it may well be that those seemingly benign mathematical statements used by the field have substantial implications for modeling. A great example is some of Ariel Rubenstein's most recent work in progress. By relaxing the very definition of "strategy" into something that's mathematically much username_28 to work with, he finds quite interesting results- that how...  
> **username_30**: individuals perceive strategy spaces, and the dimensionality of their reasoning, is instructive for how successful they'll be when playing non-trivially modeled (from the players' perspective) games. Mathematically, this is a very inconvenient prospect, since it makes historically relied upon FPTs useless (among other things, of course). Only by a giant in the field getting away from the "overly math-y mindset" that we often use could this potentially intriguing result be found.  
> **username_7**: Oh boy I just love this "I found out about behavioral economics yesterday and rationality is a myth" attitude. [Behavioral Economics has some of the more mathy/complex models out there](https://www.nber.org/papers/w22954). Why? Well, because modelling people with complete, transitive and reflexive preferences is easier than modelling people with weirder preferences, thus requires simpler math.  

---

### Answer by username_31 on 2018-07-03. Score: 3

I think there are two legitimate sources of complaint. For the first, I will give you the anti-poem that I wrote in complaint against both economists and poets. A poem, of course, packs meaning and emotion into pregnant words and phrases. An anti-poem removes all feeling and sterilizes the words so that they are clear. The fact that most English speaking humans cannot read this assures economists of continued employment. You cannot say that economists are not bright.

---

## **Live Long and Prosper-An Anti-Poem**

*May you be denoted as $k\in{I},I\in\mathbb{N}$, such that $I=1\dots{i}\dots{k}\dots{Z}$*

*where $Z$ denotes the most recently born human.*

*$\exists$ a fuzzy set $Y=\{y^i:\text{Human Mortality Expectations}\mapsto{y^i},\forall{i\in{I}}\},$*

*may $y^k\in\Omega,\Omega\in{Y}$ and $\Omega$ is denoted as "long"*

*and may $U(c)$, where c is the matrix of goods and services across your lifetime*

*$U$ is a function of $c$, where preferences are well-defined and $U$ is qualitative satisfaction,*

*be maximized $\forall{t}$, $t$ denoting time, subject to*

*$w^k=f'\_t(L\_t),$ where $f$ is your production function across time*

*and $L$ is the time vector of your amount of work,*

*and further subject to $w^i\_tL^i\_t+s^i\_{t-1}=P\_t^{'}c\_t^i+s^i\_t,\forall{i}$*

*where $P$ is the vector of prices and $s$ is a measure of personal savings across time.*

*May $\dot{f}\gg{0}.$*

*Let $W$ be the set $W=\{w^i\_t:\forall{i,t}\text{ ranked ordinally}\}$*

*Let $Q$ be the fuzzy subset of $W$ such that $Q$ is denoted "high".*

*Let $w\_t^k\in{Q},\forall{t}$*

---

The second is mentioned above, which is the misuse of math and statistical methods. I would both agree and disagree with the critics on this. I believe that most economists are not aware of how fragile some statistical methods can be. To provide an example, I did a seminar for the students in the math club as to how your probability axioms can completely determine the interpretation of an experiment.

I proved using real data that newborn babies will float out of their cribs unless nurses swaddle them. Indeed, using two different axiomatizations of probability, I had babies clearly floating away and obviously sleeping soundly and securely in their cribs. It wasn't the data that determined the result; it was axioms in use.

Now any statistician would clearly point out that I was abusing the method, except that I was abusing the method in a manner that is normal in the sciences. I didn't actually break any rules, I just followed a set of rules to their logical conclusion in a way that people do not consider because babies don't float. You can get significance under one set of rules and no effect at all under another. Economics is especially sensitive to this type of problem.

I do belive that there is an error of thought in the Austrian school and maybe the Marxist about the use of statistics in economics that I believe is based on a statistical illusion. I am hoping to publish a paper on a serious math problem in econometrics that nobody has seemed to notice before and I think it is related to the illusion.

[![Sampling Distribution of Bayesian MAP versus Fisher's MLE](https://i.sstatic.net/tFQkK.png)](https://i.sstatic.net/tFQkK.png)

This image is the sampling distribution of Edgeworth's Maximum Likelihood estimator under Fisher's interpretation (blue) versus the sampling distribution of the Bayesian maximum a posteriori estimator (red) with a flat prior. It comes from a simulation of 1000 trials each with 10,000 observations, so they should converge. The true value is approximately .99986. Since the MLE is also the OLS estimator in the case, it is also Pearson and Neyman's MVUE.

Note how relatively inaccurate the Frequency based estimator is compared to the Bayesian. Indeed, the relative efficiency of $\hat{\beta}$ under the two methods is 20:1. Although Leonard Jimmie Savage was certainly alive when the Austrian school left statistical methods behind, the computational ability to use them didn't exist. The first element of the illusion is inaccuracy.

The second part can better be seen with a kernel density estimate of the same graph. [![kernel of sample distribution](https://i.sstatic.net/QoTFO.png)](https://i.sstatic.net/QoTFO.png)

In the region of the true value, there are almost no examples of the maximum likelihood estimator being observed, while the Bayesian maximum a posteriori estimator closely covers .999863. In fact, the average of the Bayesian estimators is .99987 whereas the frequency based solution is .9990. Remember this is with 10,000,000 data points overall.

Frequency based estimators are averaged over the sample space. The missing implication is that it is unbiased, on average, over the entire space, but possibly biased for any specific value of $\theta$. You also see this with the binomial distribution. The effect is even greater on the intercept.

[![alpha](https://i.sstatic.net/Wfxdy.png)](https://i.sstatic.net/Wfxdy.png)

The red is the histogram of Frequentist estimates of the itercept, whose true value is zero, while the Bayesian is the spike in blue. The impact of these effects are worsened with small sample sizes because the large samples pull the estimator to the true value.

I think the Austrians were seeing results that were inaccurate and didn't always make logical sense. When you add data mining into the mix, I think they were rejecting the practice.

The reason I believe the Austrians are incorrect is that their most serious objections are solved by Leonard Jimmie Savage's personalistic statistics. Savages *Foundations of Statistics* fully covers their objections, but I think the split had effectively already happened and so the two have never really met up.

Bayesian methods are generative methods while Frequency methods are sampling based methods. While there are circumstances where it may be inefficient or less powerful, if a second moment exists in the data, then the t-test is always a valid test for hypotheses regarding the location of the population mean. You do not need to know how the data was created in the first place. You need not care. You only need to know that the central limit theorem holds.

Conversely, Bayesian methods depend entirely on how the data came into existence in the first place. For example, imagine you were watching English style auctions for a particular type of furniture. The high bids would follow a Gumbel distribution. The Bayesian solution for inference regarding the center of location would not use a t-test, but rather the joint posterior density of each of those observations with the Gumbel distribution as the likelihood function.

The Bayesian idea of a parameter is broader than the Frequentist and can accomodate completely subjective constructions. As an example, Ben Roethlisberger of the Pittsburgh Steelers could be considered a parameter. He would also have parameters associated with him such as pass completion rates, but he could have a unique configuration and he would be a parameter in a sense similar to Frequentist model comparison methods. He might be thought of as a model.

The complexity rejection isn't valid under Savage's methodology and indeed cannot be. If there were no regularities in human behavior, it would be impossible to cross a street or take a test. Food would never be delivered. It may be the case, however, that "orthodox" statistical methods can give pathological results that have pushed some groups of economists away.

> **Comments:**
> **username_23**: This is interesting but what was the data and what was being estimated. You say "edgeworth's MLE" but MLE under what distributional assumption of what data ? I may have missed a previous post. Thanks for clarification..  
> **username_31**: The data is from a set of simulations from a time series that is stationary AR(1) with normal shocks.  
> **username_23**: In that case, you've got a VERY, VERY, VERY,  VERY near unit root process which is going to cause the classical statistical assumptions to fail. So, it sounds more like an assumption problem rather than a problem with classical statistics.  As you're probably aware, unit root processes lead to dickey fuller type distributions rather than t-distrbutions. My best guess is that that's what's going on there. Still, an interesting example. Thanks.  
> **username_31**: That is what started the investigation.  I am looking at nearly and just barely explosive roots.  
> **username_31**: There is a Bayesian solution for both less than and greater than unit root processes.  The elaborate Frequentist solutions are completely unnecessary.  Being non-stationary is a headache, but only in the sense that the predictions are weaker, not in the calculation sense.  
> **username_31**: @username_23 if you think about it almost all real economic series are near unit root.  Look at firm survival which is just barely stationary, or low daily growth rates.  
> **username_31**: @username_23 also, it does not violate any distributional assumptions.  Anything less than one is still t-distributed, it just takes a long time to converge.  
> **username_23**: Hi username_27: I don't want to have a long discussion about this ( you can have last word. I promise )  but you're entering dangerous territory when simulating a process like that which is close to 1.0 because some propportion of the trials of the simulation will generate data with a unit root so you end up estimating a mixture. If you want to do a fair comparison, you're better off taking the first difference of the simulated data, ,estimating the mean and then adding 1.0. That should give better results. My point is that your estimation process is assuming something that often won't be true.  
> **username_23**: There is a nice, detailed discussion of bayesian ( and some details about classical ) unit root econometrics in "bayesian inference in dynamic econometric models" by bauwens, lubrano and richard. all the best.  
> **username_23**: any time. I bet there are areas where bayesian approaches do generate better results than classical stat. I basically  just wanted to give a heads up about the "dark jungle" called the unit root world. I dealt with it a lot in a previous life.  

---

### Answer by username_32 on 2018-07-01. Score: 2

This is not so much of an answer as more of a note motivated predominantly by the *softness* of the question.

It can be the case that the statement

> "[...] *the study of work by classical economists (like Adam Smith, Karl Marx,
> and David Ricardo) is still relevant*"

(insert qualifications) is true *irrespective* of the truth value of the assertion

> "[...] *the practice of how mainstream economics uses mathematics is
> abusive and is an attempt to fool the masses regarding the "science"
> economists practice.*".

My point is that the relevance of the *classics* is not necessarily related to the relevance (or lack thereof) of using mathematics in economics.

Obviously, private communications are opaque to anyone not present and as I were not present in the private communications that instigated this question it is not possible to comment on the specific arguments that lend (or detract) support to the math relevance thesis;

I think that there is some renewed interest in the history of economics as a discipline and economic historians are trying to investigate the various paths that economic theory has followed in modern times; I won't use references as I am not an economic historian but I think it is relatively easy for anyone to find material on such issues.

My personal understanding of the subject is that the success of the war effort during WWII attributed (rightly or wrongly, that is debatable) a certain amount of credibility into tools and approaches used in operations research and related fields; obviously these fields were more mathematical in spirit.

With the advent of the Cold War and the political and ideological issues that ensued it was only natural to expect that tools that had proven themselves useful in the recent past (math, op research) would be used again to stave of the *red scare*. Add to this mix the arms race of Cold War and the subsequent major and minor breakthroughs in *hard* sciences related to the nuclear effort etc.

It is not difficult to imagine why the agony of the "free world" to emerge victorious from the cold war painted the *tools* it had invested so much into with favorable colors.

Now, there comes an inversion in this scheme where the tools that had been proven useful once are subsequently used almost ceremonially to impart use value into the body of knowledge that accumulated around their use. That is not to say that the mathematics were 'wrong' or 'too abstract' or 'irrelevant'. But it is the case that at some point the tool-case became more important than the actual problems it could solve.

And this is equivalent to hybris.

On a final note, damning or glorifying economics for its use of mathematics seems misplaced so long as the body of knowledge under the heading 'economics' fails to produce positive results for the society at large.

Resources have competing uses and economists know that very well.

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## update 1

this is an update about math and classical econs (as it was too long for a comment)

The classical econs couldn't have used calculus as Leibnitz and Newton invented it during the mid- and late-1600's and it was formalized by mathematicians 100-150 years later into something recognizable; I know Marx did some fondling with infinite calculus by never used it as a proper tool; similarly, the use of linear algebra and systems of linear equations were predominantly *popularized* by the triumph of Dantzig's simplex algo. The point is that IMO the classic econs did not have that stock of knowledge available to them.

Furthermore, Political economy was to a large extent a discursive enterprise that was meant to *convince* the hegemon about the proper path to prosperity (whatever that meant to them at that time).
Consider eg the Physiocrats. Quesnay's (a contemporary of A. Smith) Tableau was to a large extent a description of flows that needed little effort to be translated into a linear system of inputs and outputs. It wasn't, because

1.a. his formal education was in medicine (he was trained as a physician)

1.b. the tools to do so were invented by Leontieff in the 60's

2. he and his disciples had all the legitimacy they needed (Turgot, a disciple of Quesnay, eventually become finance Minister)

The point I'm trying to make is that lack of mathematical rigor in classical econs does not necessarily mean that they are irrelevant.

> **Comments:**
> **username_25**: A major distinction between the "classical economists" and later economists is that the classical economists used neither calculus nor large systems of linear equations to derive their results.  The great classical economists did include a few simple mathematical examples.  

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### Answer by username_33 on 2018-07-03. Score: 2

> "*What is a reason to be against mathematics in economics?*"

IMO, **if you frame your whole economic thinking in mathematic terms** (or too much of it)**, your thought process might become less flexible and innovative**. Mathematically formalizing economic theories can be an arduous task:

* Some postulates might require extreme care when translating them into mathematical language. This has an opportunity cost in terms of time and intellectual energy which will not be spent on more "productive" tasks (e.g. exploring new, radical ideas to longstanding problems);
* Mathematics requires a rigor that is simply absent when a new idea is emerging: you might not be able to formulate mathematically something you are barely starting to understand.

As a consequence, your economic thinking might end up "hijacked" by a set of assumptions that enable you to mathematically formalize your theory/model, but which constrains the range of novel economic ideas you can formulate.

> **Comments:**
> **username_0**: To address both the points here- Peer reviewed journals deal with both of these issues. when ideas presented they have to go through a process where they are critically reviewed, if they cant stand up to scrutiny (or the author cant handle the criticism) then why publish it?  
> **username_33**: @username_0 "standing up to scrutiny" involves a fair degree of subjectivity: when L. Bachelier presented his thesis applying Brownian Motion to model stocks, reception was mixed as the Jury deemed it wasn't fully rigorous. Nonetheless, his work has subsequently been tremendously influential in the theory of Finance. Original work might deviate from a profession's prevailing standard (e.g. rigorous mathematical formalization) which do not necessarily invalidate its relevance. So some people might be again an _excessive_ use of mathematics in economics because of that.  
> **username_33**: Why the down vote, by the way?  
> **username_8**: This is the most important drawback in the whole discussion. Pity it gets downvoted  

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### Answer by username_34 on 2018-12-09. Score: 1

Many of the criticism comes from the recent financial crisis. Economists failed to predict the crisis, beside the super sophisticated models. Many then said that economics is wrong because this super complex models cannot capture essential elements of life and behavior and society.

So part of the movement against mathematics is just in response to the evidence. For many, the sicence is often a failure.


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### Answer by username_35 on 2018-06-30. Score: 1

At the end of the day, economics and its offshoots (ie business, management, marketing etc) are all social sciences. These areas of inquiry are concerned with specific facades of human behavior as individuals or groups. While quantitative methods are very useful in categorizing and generalizing these behaviors, the behavior itself is highly personal and individualistic. For example, you and I could go into the same supermarket, at the same time, buy the same items and leave. This behavior when analyzed quantitatively, will arrive at an average of our behavior and its root causes, it will however completely miss the individual behaviors. By defining a non-existent third behavior (the average) it will model our behaviors, but will not reflect the true nature of the behaviors it is trying to explain. But when you have a large enough sample for a given population the quantitative analysis just about becomes good enough to draw generalizations, but the generalization being drawn usually is not representative of a given individual.


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### Answer by username_21 on 2018-06-29. Score: 1

> What is a reason to be against mathematics in economics?

I don't think there is a blanket reason to be against math anymore than there is a blanket reason to be against case studies. It is almost a matter of epistemology. What are the knowledge claims made, with what methods, and with what evidence? Some sorts of questions are very well suited for a quantitative treatment: Like, what is the effect of increased accessibility on housing prices? Or, given a number of variables on cost and household demographics, which mode of transport is a household likely to take to work? There are models that are well suited for finding patterns in that type of questions where the domain is fairly specific, and they may work reasonably well even absent a strong theory underlying observed patterns.

Conversely, a number of questions are of a different nature entirely, related to bigger historical shifts. The rise and fall of the labor movement in the US, say, or why did some cities see a revival when others didn't? Such questions are probably better answered by a different approach than using models (this doesn't mean that there can't be useful quantitative components of asking those questions).

Ultimately, I think it has more to do with the sorts of questions different researchers are interested in rather than a wholesale rejection of a practical approach.


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### Answer by username_36 on 2018-07-02. Score: 1

Beyond the quantitative aspects, there are also qualitative factors which do not lend themselves to numeric treatment. My background is electrical engineering, which quite correctly employs quantitative methods extensively. Although investing isn't economics, there is a relationship. As much as possible, I've been trying to read and implement the information and wisdom imparted by Benjamin Graham and his colleague David Dodd. Graham himself was the instructor, and later the employer, of Warren Buffett. Graham felt that when anything more than the 4 basic arithmetic operations were dragged into the model, description, or analysis, then somebody was trying to "sell you a bill of goods". Graham himself was very mathematically adept, and knew calculus and differential equations much better than most students and instructors. So, use of advanced mathematics in some ways acts to obscure, rather than elucidate, matters pertaining to "proper" investment practice. Buffett is still very much alive. Graham himself and most of his employees or students are all long gone, but they all seemed to have died rich. Look through his books "Security Analysis" and "The Intelligent Investor" and you won't find a derivative, integral, ODE, or PDE.

> **Comments:**
> **username_1**: You may enjoy reading about the company [Long-Term Capital Management](https://en.wikipedia.org/wiki/Long-Term_Capital_Management).  
> **username_23**: @denesb: The LTCM disaster was based on a set of assumptions and confidence about the way people have. It had zero to do with math per-se  but still an interesting read for those interested. OTOH, If you're making a statement about math not  always being applicable in finance, I agree.  
> **username_31**: Actually, Graham was an economist and indeed presented an alternative monetary regime at the Bretton Woods conference.  Just to be fair to Graham, he might, in fact, use those tools today.  The Graham-Dodd method actually lends itself to both statistical and economic model constructions.  
> **username_23**: should be "people behave" not people have. Not sure how to correct it.  

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