56
$\begingroup$

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.

$\endgroup$
  • 10
    $\begingroup$ How about a less sensational title? $\endgroup$ – Michael Greinecker Jun 28 '18 at 19:11
  • 3
    $\begingroup$ "Criticism of Math in Economics" or "Criticism of the Use of Math in Economics", maybe. $\endgroup$ – Michael Greinecker Jun 28 '18 at 19:16
  • 6
    $\begingroup$ How about something like Mathiness in Economic Theory? $\endgroup$ – Giskard Jun 28 '18 at 19:16
  • 18
    $\begingroup$ 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? $\endgroup$ – lazarusL Jun 28 '18 at 20:31
  • 3
    $\begingroup$ @lazarusL both guess. Honestly trying to get it because i'm too mainstream according to some of my peers. $\endgroup$ – EconJohn Jun 28 '18 at 20:41

22 Answers 22

36
$\begingroup$

I find that the essay "The New Astrology" 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 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.

$\endgroup$
  • 11
    $\begingroup$ 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. $\endgroup$ – Nat Jun 28 '18 at 22:48
  • 23
    $\begingroup$ 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. $\endgroup$ – JBentley Jun 29 '18 at 1:50
  • 16
    $\begingroup$ Math and logical systems in general conform to "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.) $\endgroup$ – Nat Jun 29 '18 at 2:11
  • 9
    $\begingroup$ @Nat It is obvious, but technical garbage is more difficult to identify. This comment could be the core of a nice answer IMO. $\endgroup$ – Giskard Jun 29 '18 at 2:23
  • 4
    $\begingroup$ @Nat to those who do not know mathematics or latin, they do endow unearned empirical authority. See, for example, 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. $\endgroup$ – Sarah Griffith Jul 1 '18 at 20:06
27
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ 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. $\endgroup$ – Billy Rubina Jul 1 '18 at 15:06
  • 2
    $\begingroup$ @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? $\endgroup$ – Alecos Papadopoulos Jul 1 '18 at 15:44
  • $\begingroup$ 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. econ.yale.edu//smith/piketty1.pdf $\endgroup$ – FooBar Oct 16 at 15:55
21
$\begingroup$

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. 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 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 to describe the issue in his (unrefereed) paper Mathiness in the Theory of Economic Growth. 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! 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.

$\endgroup$
  • 1
    $\begingroup$ (+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. $\endgroup$ – Alecos Papadopoulos Jun 29 '18 at 22:07
  • $\begingroup$ 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. $\endgroup$ – ZeroPhase 2 days ago
  • $\begingroup$ @ZeroPhase As long as you do not accept the axiom of choice a lot of math results cannot be proven.(I hear axiom of determinancy is a reasonable but imperfect substitute.) I don't economics "violates" any axioms. $\endgroup$ – Giskard 2 days ago
16
$\begingroup$

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.

$\endgroup$
14
$\begingroup$

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.

$\endgroup$
  • 4
    $\begingroup$ 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.” $\endgroup$ – EconJohn Jun 28 '18 at 19:39
  • 3
    $\begingroup$ @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. $\endgroup$ – Iñaki Viggers Jun 28 '18 at 20:04
9
$\begingroup$

"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.

$\endgroup$
  • 2
    $\begingroup$ For those who don't know, the header is quote by British statistician George Box. One of my all-time favorite quotes! $\endgroup$ – Sam Jun 30 '18 at 23:18
  • 1
    $\begingroup$ @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! $\endgroup$ – Cort Ammon Jun 30 '18 at 23:36
4
$\begingroup$

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

$\endgroup$
  • 3
    $\begingroup$ I like the image, but this seems to be against modeling in general, not mathematical modeling in economics. $\endgroup$ – Giskard Jun 29 '18 at 12:59
  • $\begingroup$ @debesp The first sentence is undeniably true and the Borges-quote gives the appropriate context. $\endgroup$ – Michael Greinecker Jun 29 '18 at 13:39
  • 2
    $\begingroup$ And why should we care about the "full richness of human experience"? It has already happened, let's do something else. $\endgroup$ – Alecos Papadopoulos Jun 29 '18 at 17:52
  • 1
    $\begingroup$ @AlecosPapadopoulos The story kinda answers your question. $\endgroup$ – Michael Greinecker Jun 29 '18 at 18:05
4
$\begingroup$

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. 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 harder to detect incorrect assumptions.

As Wildcard said, 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.

$\endgroup$
  • 2
    $\begingroup$ Romer's Mathiness is indeed lurking in several of the answers. $\endgroup$ – Giskard Jun 29 '18 at 12:40
4
$\begingroup$
  • Jacob Theodore Schwartz (1962):

The very fact that a theory appears in mathematical form, that, for instance, a theory has provided the occasion for the application of a fixed-point theorem, or of a result about difference equations, somehow makes us more ready to take it seriously.

The above is probably the single most important criticism of the use (or misuse) of math in economics.

As some have noted, Coase (1937, 1960, etc.) for example could not be published today, because his work — profound as it might be — would not be recognized as such, since the most advanced math it contained was elementary-school arithmetic.

Conversely, useless gobbledygook filled with dozens of pages of intimidating-looking math earns you publications and tenure.

  • Ariel Rubinstein (2012, Economic Fables):

unlike philosophers and linguists, we economists behave as if we do not rely solely on our impressions of the world and introspection.

Along the same lines as the previous point — math helps add the veneer or pretence of scientific "rigor". Math helps convince economists (and perhaps a few others) that their work is better and more important than that of political scientists, historians, and, of course, sociologists.

  • Oskar Morgenstern (1950, On the Accuracy of Economic Observations):

Qui numerare incipit errare incipit. [He who begins to count, beings to err.]

There is the mistaken belief that whatever can be quantified, formalized, and "mathematicized" is necessarily better. Research in economics has thus been reduced to "theory" (by which is meant theorem-and-proof) and "empirical" (by which is meant regression analysis).

Any other method of investigation is banished and branded "heterodox". To reuse our earlier example, Coase was an economic theorist of the highest caliber. Yet he would not count as one of today's "theorists" because he failed to dress his ideas up with enough math.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • 4
    $\begingroup$ 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? $\endgroup$ – Giskard Jun 28 '18 at 19:38
  • 6
    $\begingroup$ 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. $\endgroup$ – Giskard Jun 28 '18 at 19:41
  • 7
    $\begingroup$ Weather can't be predicted with mathematical precision either, but meteorologists must know quite a bit of mathematics to do their jobs. $\endgroup$ – Monty Harder Jun 28 '18 at 21:33
  • 5
    $\begingroup$ 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. $\endgroup$ – Michael Greinecker Jun 29 '18 at 12:07
  • 5
    $\begingroup$ @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. $\endgroup$ – Michael Greinecker Jun 29 '18 at 15:13
3
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ 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? $\endgroup$ – ako Jun 28 '18 at 22:17
  • $\begingroup$ 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. $\endgroup$ – credo56 Jun 28 '18 at 22:54
  • 1
    $\begingroup$ Who do you think runs institutions if not humans? $\endgroup$ – BB King Jun 29 '18 at 8:52
  • $\begingroup$ 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.. $\endgroup$ – mark leeds Jun 29 '18 at 8:55
  • $\begingroup$ @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). $\endgroup$ – ahorn Jun 29 '18 at 11:21
3
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.)

$\endgroup$
  • $\begingroup$ I suspect (but don't actually know) that the convention whether log is short for log$_e$ or log$_{10}$ is language dependent. $\endgroup$ – Giskard Jul 2 '18 at 5:51
  • 1
    $\begingroup$ @denesp -- The lecture was in American English. Both the lecturer and I are Americans, and have been part of nearby universities. $\endgroup$ – Jasper Jul 2 '18 at 5:58
3
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Giskard Jul 1 '18 at 13:23
  • $\begingroup$ 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: 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. $\endgroup$ – Pedro Cavalcante Dec 10 '18 at 7:45
  • $\begingroup$ @PedroCavalcanteOliveira man, QJE is #1. 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. $\endgroup$ – Arthur Tarasov Dec 10 '18 at 9:25
  • $\begingroup$ 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. $\endgroup$ – Pedro Cavalcante Dec 10 '18 at 22:30
  • $\begingroup$ @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. $\endgroup$ – Arthur Tarasov Dec 11 '18 at 6:23
2
$\begingroup$

It is not the math but that authors misuse the math language.

Check out this Article (unrelated to the topic). Where are the Definitions? What is the meaning of S, E, the arrow inbetween, and all these other symbols? Someone who has not studied this subject can not know.

Scientific texts have many quality standards, like citing others, but defining math symbols is not a standard. In my opinion that is not good, especially if such publications are read by the public.

It should be a standard in science to define all symbols in public contexts.

I believe this to be the answer to why your colleagues and most other math haters dont like "math" (which, as i already said, is actually not the problem).

The solution can only come from the scientific community.

For websites there is btw a trivial solution, Hover over the above link to see it.

$\endgroup$
  • $\begingroup$ That's so true. I've been teaching myself RE for ~ two years and the RE literature is extremely difficult to understand. They define very little and often assume signs of coefficients which can make things totally confusing. For example, it took me 2 weeks and help from a top professor in economics to understand a statement on page 2 of the paper at the link below. It turned out that it was because alpha was assumed negative but this was not stated anywhere. We had to go back to an earlier paper to figure that out. jstor.org/stable/2526858?seq=1#page_scan_tab_contents $\endgroup$ – mark leeds Jun 30 '18 at 5:44
  • $\begingroup$ if anyone is interested, it's at the bottom of page 2 where he says " a one percent jump in anticipated inflation, blah, blah". why not state that $\alpha$ is assumed negative ? not everyone would necessarily know that I don't think ? $\endgroup$ – mark leeds Jun 30 '18 at 5:56
2
$\begingroup$

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.


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

  1. 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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Jasper Jul 2 '18 at 4:40
1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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

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

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

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.

$\endgroup$
  • $\begingroup$ 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.. $\endgroup$ – mark leeds Jul 3 '18 at 19:23
  • $\begingroup$ The data is from a set of simulations from a time series that is stationary AR(1) with normal shocks. $\endgroup$ – Dave Harris Jul 3 '18 at 20:34
  • $\begingroup$ 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. $\endgroup$ – mark leeds Jul 3 '18 at 20:43
  • $\begingroup$ That is what started the investigation. I am looking at nearly and just barely explosive roots. $\endgroup$ – Dave Harris Jul 3 '18 at 20:44
  • $\begingroup$ 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. $\endgroup$ – Dave Harris Jul 3 '18 at 20:48
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ You may enjoy reading about the company Long-Term Capital Management. $\endgroup$ – Giskard Jul 2 '18 at 17:02
  • $\begingroup$ @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. $\endgroup$ – mark leeds Jul 3 '18 at 15:30
  • $\begingroup$ 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. $\endgroup$ – Dave Harris Jul 3 '18 at 16:55
  • $\begingroup$ should be "people behave" not people have. Not sure how to correct it. $\endgroup$ – mark leeds Jul 3 '18 at 19:25
0
$\begingroup$

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.

$\endgroup$
-1
$\begingroup$

"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.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – EconJohn Jul 3 '18 at 16:15
  • $\begingroup$ @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. $\endgroup$ – Daneel Olivaw Jul 3 '18 at 17:26
  • $\begingroup$ Why the down vote, by the way? $\endgroup$ – Daneel Olivaw Jul 3 '18 at 17:31

protected by EconJohn Jul 6 '18 at 23:28

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.