## There is a sound reason for the growth of statistical theory.

from **Gerald Holtham**

Econometrics like more casual empiricism can be done well or badly, intelligently or stupidly, dogmatically or with an open mind. But are these gentlemen saying that statistical analysis can never reveal anything in economics that is not obvious to simple observation? Evidently that is untrue. What is revealed is never a “law” and will obviously be contingent in space and time. That follows from the nature of society and economic data. Statistical analysis nevertheless has an indispensable place in social studies whether economics, psychology, medicine or sociology.

Econometrica may be boring and contain articles of no evident application but there is a sound reason for the growth of statistical theory. Physicists do not have to contend with lousy data. They can set up a controlled experiment and repeat it to generate as much data as they need. If the data are well conditioned simple statistics is good enough. Economists do not have that luxury. Economic data contains measurement errors, multicollinearity, erratic distributions and a host of other issues. Econometric theory is concerned with how to extract information as efficiently as possible from a sparse and noisy data set. Economists use more fancy statistical methods than physicists not out of perversity but because their data is lousier. Statistical sophistication cannot make up for lousy data and you cannot extract information that isn’t there but that is no reason for using simple statistics that you know are inappropriate, ignoring problem of which you are aware.

Gerald. How can single or multiple regression solve the problem of lousy data for mean, median, and mode? Just for argument’s sake.

There’s been a bit of debate about mathematics on quite a few threads. That debate is relevant here too. It seems to me (though I might be wrong) that mathematics proponents are somehow claiming the primacy of mathematics over philosophy and also mathematics indispensability for all science; in other words that mathematics is the first, final, only or greatest arbiter of truth claims.

In other words, I am, just a little, accusing mathematicians of mathematical epistemic snobbery. In turn, I might rightfully be accused of (amateur) philosophical snobbery. Religionists and ideologues might be accused of dogmatic or axiomatic snobbery, respectively, and so on.

Of interest I think is this article;

https://www.academia.edu/15391638/On_Mathematical_and_Religious_Belief_and_On_Epistemic_Snobbery

There’s a lot to unpack in the above article and it goes to the heart of some of the ontological arguments we are having on this blog site (with me perhaps “provoking” at least half of them as some kind of self-appointed, ontological provocateur). I admit I am saying to people: “Show me your ontology. Prove it, or at least support it with some epistemic claims so as to meet the requirements of justification, warrant, rationality, and probability.”

I am saying that when one attacks an ontology, on this blog the ontology of conventional economics (which attack I wholly agree with), then one must front with a new, supportable and consistent ontology. I also consider that the economically heterodox will not coalesce into an effective intellectual opposition to orthodox economics until a new and persuasive ontology for political economy unites a wide part of the heterodox field. Of course if that occured and it came to dominate, it would be a new orthodoxy.

A footnote in the article above links one to:

https://global.oup.com/academic/product/science-without-numbers-9780198777915?cc=au&lang=en&#

This is a book (and author) of which I was unaware and now hope to read. Overall, I consider new ontological arguments to be central, implicit and critical in critiquing any existing ontology. How can it be otherwise?

Ikonoclast, What a good link! According to https://silviajonas.com/ this seems to deserve a riposte from a mathematician’s viewpoint. I’d like to comment on it on my blog. Unfortunately the text you point to gets garbled when I try to quote it. Have you managed to download a straightforward text?

(On a quick read, I think her critique of mathematicians is quite reasonable from a social scientist’s standpoint, and in particular it is interesting to see how she makes use of relevant results in mathematical logic. But (to adopt Keynes’ terminology) is it fair to lump category theory in with pseudo-mathematics?)

I have the paper (which downloaded as a Word document which I converted to PDF). Have you been able to download it Dave? If not I can make it available.

Ikonclast. On your “mathematics proponents are somehow claiming the primacy of mathematics”, I agree that many (not just economists) claim the primacy of pseudo-mathematics. I also tend to favour the ‘primacy’ of mathematics but only in the sense that mathemtics is the logic of mathematical structures (e.g., numbers, graphs) and I prefer things that can be said logically to things that can’t to things that are contra logic.

What we may be able to agree on is that mathematicians are human and so anything that they say should be critiqued. But I tend to think that whatever logic survives such critique is more dependable than common sense, empirical reasoning etc. What logic do you commend?

(Wishing I could quote Jonas at this point!)

Beautifully said Dave! Bravo, you inspire me to further my studies in mathematics for the sheer joy of it.

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Dave’s comment above I find to be a kindred spirit. Indeed, some I have observed make a fetish of mathematics, failing to recognize its limitations thereby breaking mathematical sense and descending into a form of pseudo-science more akin to mathematical masturbation than the true use of mathematics in support of true science. Scientism is alive and well and it takes a discerning mind able to apply the acid test of sound philosophy to discern the difference between the proper use of mathematics and its misuse in the service of some unconscious and/or conscious philosophical loyalty to mechanistic materialism and philosophical reductionism. In economics, as Söderbaum[1] notes, this equates to “monetary reductionism” at the expense of our socio-political and environmental health. There are domains where methodological reductionism and mathematics are the proper tools for understanding the problem at hand; and so too there are domains where dogmatic insistence upon

reductionismand mathematical modeling are inappropriately applied based upona prioribeliefs (worldviews, held consciously or unconsciously) of philosophical reductionism lead down dead end paths eventually to be proven false. History is replete with such examples..

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Consider that within the domain of physics a scientist was able to creatively use the power of mathematical thinking to predict the existence of the neutrino before it was ever empirically proven to exist. And that is one of many examples. In the domain of matter-energy, the material universe, where mathematics is the language of science, and appropriately so, makes uncanny predictions before observations confirm their reality.

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Mathematics can also be misused, abused, and placed into the role and service of dogmatically held critical assumptions with false appeals to mathematization being the

sine qua nonof science and the “scientific method.” A fine example of mathematical hubris leading to erroneous and dogmatic claims can be found in the history of plate tectonics. The physicists insisted that the continents couldn’t move; mathematics proved that they couldn’t. Wegener (and a few others) insisted they could and indeed disparate lines of evidence showed they did. The problem was that the mathematics didn’t account for radioactive elements deep within the earth causing convection currents. Such dogmatism delayed the theory of plate tectonics from advancing for decades. Mathematical hubris won the day simply because it was impressive, despite the large volume of non-mathematical but historical evidence that to an unbiased observer clearly validated the hypothesis that the continents move..

In the domain of economics there are those who believe that they can mathematically model an entire world economy disregarding human agency (socio-political interest groups, lobbyists, business interests, Cock Brothers, etc.) and its ability to manipulate and distort economic outcomes contrary to neat theories; this is a dead end in my view, as there it is a pipe dream of those who dream the dream of a “social mathematics.” Game theory doesn’t cut it. Complexity is in the ballpark but still fails to address, as far as I can see, the element of human agency and human creativity as individuals embedded in social context. But it along with information theory and systems biology have promise in providing metaphorical modeling tools for understanding better how to organize our economic lives. In the end though, it is morality and ethics that must be the arbiters of how we relate to each other in a complex socio-economic context. Ultimately, it is spiritual, and comes down to the Golden Rule: treat others as you would have others treat you in the highest philosophical and religious meaning humans can achieve.

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There are a multiplicity of ways of approaching truth through discernment of facts, meanings, and values. Both philosophy and religion are constrained by scientific fact if they are to be intelligent and progressive and able to be corrected. Science, at its best, purifies philosophy of erroneous presuppositions (e.g., time and space are absolute) and religion of superstition. But this doesn’t invalidate either philosophy or religion, and the road runs both ways. Science can give rise to new philosophical insights, and philosophical insights (e.g., Einstein’s thought experiments) can lead to new scientific discoveries. True religion (opposed to institutional religions of authority) are the supreme desire to discover for oneself experiential truth, beauty, and goodness; experiential recognition of facts, meanings, and values; and to synthesize and coordinate these in the love and service of one’s fellows in practical and meaningful ways.

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~ ~ ~

[1] Peter Söderbaum, Economics, ideological orientation and democracy for sustainable development. WEA, 2018, p. 13.

This seems mostly in line with my own thinking, although coming from a different place. But this struck me:

“A fine example of mathematical hubris leading to erroneous and dogmatic claims can be found in the history of plate tectonics. The physicists insisted that the continents couldn’t move; mathematics proved that they couldn’t. Wegener (and a few others) insisted they could and indeed disparate lines of evidence showed they did.”

My own reading is that Keynes and Smuts were co-subversives, and Smuts was a supporter of the theory of continental drift. Certainly, the idea that mathematics ‘proved’ anything about the real world was only ever a delusion. Surely the role of mathematics was to highlight the previously ‘hidden’ assumptions in the physics of the day, leading to the reform of science that Smuts reported on in 1929 ( https://djmarsay.wordpress.com/science/complexity/smuts/smuts-scientific-world-picture-of-today/ ).

It seems to me that all references to ‘mathematical hubris’ might be replaced with ‘pseudo-mathematical hubris’. This is not to claim that the mathematics of 1929 was all ‘true’ or that mathematicians are not liable to human failings. Its just that wherever there is a valid criticism of mathematics (as there was of Euclidean Geometry) ‘proper’ mathematicians have always risen to the challenge, and by 1970 it

seemedto have reached a sustainable conclusion. So, for now, my ‘working hypothesis’ is that non-pseudo mathematics is good enough and useful. But how do we differentiate?Dave, you are a gold mine ;-) I always enjoy your comments, you always raise thought provoking questions, and your blog is a wealthy of material.

What are the origins of science and mathematics, and how do they fit together, if they do at all?

Chemistry, metallurgy, and the materials sciences in general originated in the knowledge created by ancient miners, smiths, and potters. Physics is a special case. Its origins are in the work of ancient farmers, hunters, and priests. Physics since the 19th century denies this heritage. Claiming instead origins in the theories of the intellectual elites of 17th, 18th, and 19th century Europe. Physics also warps science in the 20th and 21st centuries. Hunter-Gatherer Sapiens invented a process of creative problem-solving in which hypotheses are continuously tested against evidence, rejecting those which do not stand up and replacing them with better hypotheses. They invented science, that is, long before any physicist drew breath. 20th and 21st century physicists are often arrogant to the point of absurdity. Particularly, theoretical physicists. Frequently belittling such disciplines as botany and paleontology by likening their intellectual content to stamp collecting. Clearly implying that physics is “more scientific” than are less theory-driven disciplines—a reflection, once again, of the ancient prejudice that proclaims intellectual labor more honorable than manual labor. But what physicists do is not typical of what most other scientists do. The methodologies of biology, anthropology, ecology, psychology, and sociology have very little in common with the abstractions of theoretical physics, and yet the general ideology of modern science places physics on a pedestal as the model science which all other sciences should strive to emulate. The “imperialism of physics” was part the creation of American governmental policy. Because of their role in developing the atomic bomb, a few “aristocrats of physics” emerged in the post-World War II period as the primary spokespersons for American science. Think Albert Einstein and Edward Teller. While their views on science were somewhat similar, their views on society and social concerns were opposites. This idealization of physics has consequences. First, it demands science and scientists focus on theory. The more complex, abstract, and mathematical the better. Second, it reinforces the idea that science must be “value-free,” especially with respect to social problems. In physics, objectivity is equated with neutrality; scientists (all scientists) are expected to be neutral and dispassionate about the subject of their inquiry. Neutrality may be workable for physicists, but in sciences closer to social concerns—such as medicine, anthropology, psychology, sociology, and political economy—the appeal to neutrality supports the status quo, which is supported by racist, sexist, or elitist assumptions of which the scientists themselves are often unaware. In short, it’s bad science and bad policy to base any science on the aberrant science physics, and particularly the social sciences. Including economics.

The social sciences must reflect the concerns they were created to address. The history, ongoing welfare, and future life of the species Sapiens. Some have dealt with this need better than others. Since the end of World War II, however, all, save history and anthropology have become too “physics-like.” Too concerned with theories and value-free (objective) research. Too little concerned with observation, case study, experiment, ethnography, and archival exploration about the species their disciplines were created to reveal and assist. But theory-focused science is a more direct threat to Sapiens. In the ongoing ‘science wars,’ the science traditionalists portray science as “pure theory” in order to place it beyond criticism. That view of science is frequently an adjunct to reactionary political views because it supposedly offers a source of unchallengeable authority, like religion, and so serves as a support for authoritarianism. A little trick that economists mastered 75 years ago, and Donald Trump is attempting as I write.

Mathematics is too often the tool invented to dazzle us into submission. Mathematics owes its existence and a great deal of its development to surveyors, merchants, clerk-accountants, and mechanics of many millennia. Mathematics’ origins are thus humble and everyday. Mathematics was not created by theoreticians of ivory-towers. Mathematics emerged from the work of surveyors, mapmakers, instrument makers, navigators, and mechanics whose activities represented the leading edge, in the 15th and 16th centuries, of innovation in practical mathematics. Later, the astronomers of Mesoamerica and Mesopotamia created codices and cuneiform tablets showing the use of written record-keeping and mathematics, in the creation of accurate and precise calendars, building on and refining the fundamental knowledge inherited from the sky-watchers of many millennia past. All developed from the routine economic activities of farmers, artisans, and traders. Though the activities of farmers, artisans, and traders created both literacy and numeracy, as political power centralized and bureaucracies took command of commerce in ancient empires, writing and reckoning came under the control of educated elites—astronomer-priests and high-ranking court scribes. “The appearance of writing,” Claude Lévi-Strauss declared, was always and everywhere linked to “the establishment of hierarchical societies, consisting of masters and slaves, and where one part of the population is made to work for the other part.” Seems economists today want to continue this control, but with a democratic hue.

Ken, almost all physicists and mathematicians are human, social and ’employees’ or at least reliant on others for their daily bread. This has consequences. Mainstream physics has assumptions that Lars, I and others don’t accept when applied to more ‘complex’ domains, and it is a real problem when people want to read methods across from physics to social systems (unless they are exceptionally thoughtful). I regard mathematics as logic applied to mathematical structures, and so if some mathematics text contained some unwarranted assumption or agenda, such as you describe, then I would look to reform the text. For me, the question is what will survive such reform? I tend to think mathematical logic and its ‘proper’ applications, such as category theory, which seem to me consistent with your views. The laterntive seems to me to be that nothing survives: a ghastly thought!

@Dave – I was once entranced by Mathematics, having done my BS in Math from MIT and Ph.D. level courses at Stanford. But eventually I learnt that, in the social sciences, NOTHING does survive of it — it was a ghastly thought when I first thought it – all those years of studying elegant and powerful theorems gone to waste. I am now more aligned with CK Raju on Cultural Foundations of Mathematics — though this book is not easily available for reading. Giving up on a hopeless enterprises, trying to fit mathematical models to human behavior, does lead to clarity and insight on alternatives

Dave, “almost” all physicists and mathematicians are human. Do you know something about extraterrestrials the rest of us don’t? All kidding aside, what most distinguishes our species is imagination. Physicists present us the most extreme and unusual imaginings, sort of like Mark Twain’s liars. There is a role for people like this in most every society. They push the edges of cultural creation. But physicists sometimes present two other sides that can at times become a concern. First, they too often believe they can identify how the universe works with certainty. Sorry, certainty is simply not available to humans. In fact, our whole existence is built around uncertainty. As the ancient Greek hero, Achilles said, the gods admire humans for this uncertainty. It provides an exhilaration to living that the gods cannot have. Second, physicists have great facility with mathematics. With the construction of mathematical logics, and the linking of mathematic algorithms with one another. Such facility soon leads to favoring mathematics as the “language of the universe.” Into which they then want to indoctrinate the rest of us. It’s not an accident that often physicists are labeled the “high priests of science,” since physics has become a religion as well as a science. Look to how the death of Stephen Hawking was treated in the press. Like the death of Pope John Paul or minister Billy Graham.

Mathematics has shown itself a useful and flexible cultural construct. Without is our earliest ancestors could not have found their way to hunting sites year-to-year, the ancient Greeks and Persians could not have built ships or sailed beyond sight of shore, or Americans have walked on the Moon. Everyone should be educated in mathematics. But mathematics can reach no further than the humans who invent it. But what humans invent with mathematics can take us places we’ve never been before. But ultimately, it’s humans’ imagination that gets us there. Humans imagined mathematics into existence. We should not allow our attachment to mathematics to squeeze us out of new imaginings, even if non-mathematical.

Ken, Your long post at August 5, 2019 at 1:27 pm contains a lot of information. I agree with

you that “The methodologies of biology, anthropology, ecology, psychology, and sociology have very little in common with the abstractions of theoretical physics,” but your story about “The “imperialism of physics” and “the creation of American governmental policy” sounds very much like conspiracy theory. It seems better to argue differences of the methods of biology, anthropology, ecology, psychology, and sociology and the method that economics should and can take. Just like modern physics had a very “peculiar” origin and method very different from chemistry, metallurgy, and the materials sciences, it is logically possible that economics can have different method than other social and human sciences.

In the post as a reply to Dave Marsey at 1:44 am August 6, 2019, Ken pointed that “physicists have great facility with mathematics. With the construction of mathematical logics, and the linking of mathematic algorithms with one another.” This is true but he should also note that this “linking” is not a natural combination. As he knows well, calculus (differential equations in particular) is mathematics invented by Newton, Leibniz and others for the use and applications to physics (kinetic theory of solid matters in particular).

Mathematics is not a fixed box of ready-made tools. As Dave says, mathematics is “logic applied to mathematical structures”. If the object of economics (i.e. economy) is complex and does not permit an easy application of ready-made mathematics, what is lacking may be to create a new mathematics. Of course, this is not an easy task which may risk whole mathematicians’ life to end in vain. As Asad Zaman has failed, there is a risk of producing a large number of failures. But, is it assured that there is another easier way which leads to a good economics? This depends much on the fields that an economist wants to study. If it is the very basis of economic inquiry such as how the large economic system works, it seems that long history since Adam Smith (at least 250 years) has demonstrated that verbal intuitive argument did not produced almost nothing. This is the reason why Arrow and Hahn boasted that their theory on the existence of competitive equilibrium was the “the most important intellectual contribution that economic thought has mad to the general understanding of social process” (Arrow and Hahn 1971, p.1). I am not praising the works of Arrow, Debreu and Hahn (and many others). In order to compete with their theory, we need mathematics.

A typo:

that economic thought has mad to >> that economic thought has made to

(in the citation from Arrow and Hahn 1971, p.1)

Yoshinori Shiozawa, good post. Each science has its own history. Physics, for example, owes more to star gazing, religion, and mathematics worship than any of the other sciences. It’s not the oldest of the sciences and certainly not the most useful. But it is the most self-promoting. As to its history after World War II, that’s established history. So is the effects of American government policy on pushing physics up the science status scale after the War.

As to facility with mathematics and mathematicians, this from Wittgenstein,

“A mathematical proof must be perspicuous” … I want to say: if you have a proof-pattern which cannot be taken in, and by a change in notation you change it into one that can, then you are producing a proof, where there was none before. (Wittgenstein 1978, 143) Ultimately, a proof is a narrative for human consumption, a “procedure that is plain to view” (Wittgenstein 1978, 173), not a superhuman objective structure. For the primary function of a proof is to convince, and logical structure is just a means to that end. Thus, mathematical knowledge is founded on human persuasion and acceptance. The mathematician is an inventor. Wittgenstein’s philosophy of mathematics undermines the often-unacknowledged Platonism which leads mathematicians to believe that the discovery of mathematical propositions and objects is predetermined, once a mathematical system is specified. This view suggests that the conclusions of mathematical proofs, the terms of mathematical sequences, and so on, force themselves on us as soon as the initial conditions are laid down. Wittgenstein refutes this by demonstrating that no such compulsion exists. But beyond this, Wittgenstein’s view is that all such consequences, sequences, and so on, have to be created by mathematicians, and cannot be said to exist until they are created. As he says “the mathematician is not a discoverer: he is an inventor” (Wittgenstein 1978, 111). Even after listing untold millions of terms in a sequence given by a rule, the continuation “has to be invented just as much as any mathematics” (270) The mathematician who constructs a new proof always creates a new concept (or changes the meaning of an existing one).

Of what were Newton and Leibniz, and many others attempting to persuade us about with Calculus? In simple terms, that humans can deal with quantities that are growing, and with rates of growth. With what mathematicians informed us are divided into two classes: constants and variables. Why ought we accept the judgement of mathematicians?

You are correct that mathematics is not a fixed box of ready-made tools. For that reason alone, it would be wrong to believe it is “logic applied to mathematical structures.” Logic and mathematics are both human inventions. And there are no “mathematical structures” or “logical structures,” per se. There are only logic and mathematics used in everyday life, usefully or not, or at worst harmfully. The universe is not mathematical till humans create it as such. Neither is the universe logical till humans create it as such. These are humans’ tools to deal with the things encountered.

Dear Ken Zimmerman

mathematical structure is not confined to geometry. algebra, and calculus. Modern mathematics has discovered/created many varieties. Graph theory, a field of combinatorics, has discovered a wide range of applications. In my international value theory, I discovered that the notion of spanning tree of a bipartite graph composed of set of countries and set of production techniques is a crucial one when we want to define regular value in the interior of the production possibility set. See two pf my papers:

(1) The nature of international competition among firms (with Fujimoto)

https://www.researchgate.net/publication/328098431_The_Nature_of_International_Competition_Among_Firms

(2) A large economic system with minimally rational agents

https://www.researchgate.net/publication/334075195_A_Large_Economic_System_with_Minimally_Rational_Agents

These papers provide a general theory of many-country and many-commodity economy with choice of production techniques. Trade of input goods (or intermediate goods) are assumed to be freely traded across countries. This is a very important condition because global value chains are networks of productions connected by trade between countries.

However, traditional trade theory excluded input trade by assumption only because it becomes extremely difficult when input products are traded. The production cost of a firm F in country A depends in this case not only on wage rate of country A but also countries B1, B2, … , BK from which A imports input good for the production of the firm F.

Curiously enough, in this simple case, the Ricardo economy that excludes input trade has an mathematical structure which I named Subtropical convex geometry. See my paper:

International trade theory and exotic algebra

https://www.researchgate.net/publication/280646264_International_trade_theory_and_exotic_algebra

Subtropical algebra is a set of positive real numbers in which two operations are defined by the formula:

a ⊕ b = max {a, b},

a ⊗b = a ・b.

I named this algebra subtropical algebra, because another algebra of real numbers defined by

the formula:

a ⊕ b = max {a, b},

a ⊗b = a +b,

is called tropical algebra and used as powerful tool to investigate complex algebraic equations of higher order. Curiously, this exotic algebra has an applications in control theory such as discrete event dynamical systems. These mathematics is now called tropical mathematics. You can easily find many papers and books written in this theme.

My conclusion is that mathematical structures need not to be restricted to conventional mathematics. We can find mathematical structures anywhere on any topic. They are ubiquitous. It is our limit of intellectual capacity that prevents us to detect an underlying structure and to find a useful relations in that structure.

Yoshinori, I agree that “mathematical structures” are not limited to just what’s labeled mathematics. But that is not surprising since humans began creating mathematics and embedding it in their cultures over 9,000 years ago. It’s a part of human music, literature, religion, etc. It’s a part of these because humans put it there. Just like they put it into textbooks and “control theory.” Nothing mysterious about it.

Ken, don’t dodge the point. Mathematics remains one of most promiseful tools of investigation for economics. Even if you want to say that there should be more variety of approaches for economics, you should not exclude mathematics for that reason.

Economics is a science that has an object of research. If it is possible and effective, we can use any tool of investigation. Those people who want to exclude mathematics from economics are romanticists who claim that, considering the ontology of economy, they can build a better, right economics. It is free to think like that, but we should not forget whether their philosophical research strategy has produced anything concrete. We should judge by the product, not by methodological arguments.

Yoshinori, I’m not dodging anything. As I’ve said, repeatedly mathematics is a useful tool, but not just for scientists. In my view there is no need, and no useful application in economics for mathematics beyond basic arithmetic and descriptive statistics. Going further provides economics a false veneer of exactness that is simply not possible. Moreover, standard statistical testing’s applicability in economics is limited, at best due to its underlying assumptions. Some non-parametric testing may be useful for economics. Particularly since most data in economics is classified as nominal or ordinal. Such tests as Friedman, Goodman Kruska’s Gamma, Mann-Whitney, Mood’s Median, etc. may be useful for economics. In my view doing any test beyond correlation (non-parametric) is not justified in economics. Finally, I’m interested in building a better economics only if that means revealing more clearly and completely how and why the people involved built the economics they did. Don’t need a lot of fancy numbers to do that. Or deep dives into philosophy.

Please read my post. Yoshinori Shiozawa August 8, 2019 at 3:37 am in

https://rwer.wordpress.com/2019/08/01/non-normal-normality

Our arguments are dispersed in different articles. It is already done. I do not repeat it. see also my comment after the comment above. You are presenting an important point when we discuss research strategy in economics.

Hi,

I’ve been distracted for a while, so excuse me jumping in. I would like to endorse Yoshinori’s points about the general relationship between mathematics and practical subjects, such as economics. The particular mathematics he cites seem to me relevant. But how ‘useful’ are they? Are they ‘fit for purpose’?

We should not trust the opinions of either economically naive mathematicians or mathematically naive economists on this. Rather we seem to at least need real collaborations doing real hard grind in genuine good faith between real ‘experts’ in all relevant disciplines (including mathematics). If they can’t agree, why should we trust them?

Dave, people do not invent economies and economics to count, solve equations, or do any other kind of mathematics. They invent these to provide for their welfare, their needs and wants for survival first and then for comfort and enjoyment. Mathematics can be useful in achieving these but it is not them. No one invents a market, or money, or supermarkets, or auctions for the sake of mathematics. You’re losing sight of what and why humans create economies. And, we certainly don’t need any more “experts” to intervene to reconstruct the work ordinary people have done to create the kinds of economies these people, not experts, need. Economies should always remain in the control of the people who create and use them. I’m not certain what a mathematical economy would look like or how it would operate, And I don’t want to find out.

What you say, Ken, is sometimes true, sometimes false. People in the economy usually ignore economic theory, especially in manufacturing where their intellectual sourcing is with engineers and natural scientists. The great exception, in recent times, has been in the application of financial theories to investor models, which oft have turned out badly because of the black swans. Then we have examples of economic theories, derived from people who mainly started in operations research, having to be bailed out by the taxpayer.

Robert, American exceptionalism is socioeconomic theory. But economists did not create it. It was created by politicians, merchants, working people, etc beginning even before the American Revolution. It was made stronger and more central in American life by the Revolution. And still stronger by the resistance of America in the War of 1812, the Indian Wars, the Civil War, American colonization, and American world powerism after WWII. The major downside of the fundamental notion that America is exceptional and destined to lead and control the world is dealing with failure. For many, long before Trump the only way America or Americans could fail was by sabotage and infiltration of outsiders, non-Americans. This foundational American mythology should not be ignored. It influences the beliefs and actions of many Americans, of all political leanings. Not as much as before WWII but still substantial.

Ken, I don’t wish to challenge what I think are your substantive points, only the way you make them. Something that mathematicians (and others) find difficult in working with social scientists of all kinds is to try assign an appropriate ‘category’ to the various claims.

Too often the mathematicians and social scientist come to an understanding on the categories and hence on what economists call ‘the mathematical model’ without either side seeming to consider the possibility of catgeory errors. Hence (it seems to me) one gets what Keynes called pseudo-mathematics. Much of what you say seems reasonable, but my head spins at your “people do not invent economies and economics to count, solve equations, or do any other kind of mathematics. They invent these to provide for their welfare, their needs and wants for survival first and then for comfort and enjoyment.”

I would rather say that small groups working closely together may ‘invent’ economic theories, whose interpretation co-evolves with aspects of reality, including the intended domain of the theories. For example, Greenspan thought mainstream economics was a kind of fiction whose psychological credibility the benificiaries could maintain. But, Cybernetically, how could they, without an adequate understanding of the impact of their theories?

People may invent what they want, but what will they get? This is not about what economists call ‘mathematical expectation’ but more about the mathematics of ‘expectation’. (IMHO.)

Dave, I assume you’re referring to mathematical categories. When mathematicians work with physical scientists, who assigns the categories? It’s the scientists. Mathematicians have to take what they’re given and through dialogue work out a translation of scientist categories into mathematical categories. This is less of the problem with physical scientists such as physicists, who are deeply embedded in mathematics.

If a social scientist “finds” four instances of behavior X, then after treatment only finds two instances of behavior X, the scientist concludes treatment helps remove two instances of behavior X. Next, according to clinical protocol larger research would be undertaken to determine if the relationship holds in the “larger” population. That would likely use parametric or nonparametric statistics. Behavior X is a scientist category, based on client or patient categories.

But your “head spinning” confuses me. Why would your head spin when recognizing that people invented the economies in which they act. Sure, some had more input than others, and some are almost excluded. But that doesn’t change the basics. In this sense people don’t always get what they want. And certainly as in “Brave New World” experts such as economists, mathematicians, etc. can construct an economy “for the people” and with enough political and police power force them to live “in” that economy. But alternative economies will pop up, as people organize to meet the needs and desires the “professional” economy does not. Even without political or police power economists, etc. can invent, at least on paper any kind of economy they want. But can it ever be functional or meet the needs of people? In in the end it boils down to a conflict over how to set up and operate an economy. With different “theories” and expectations. Dave, don’t assume just because you have a PhD and are a “professional” that the theories and expectations you like, or others like you like are better or will have better impacts than those of merchants, mechanics, or zealots. If your interest is mathematicians making economies then study that. Don’t assume it will work out or work out as the better option.

Robert, It is quite possibly true that failed operational researchers are part of the problem, just as some mathematicians are. But there’s a lot in O.R., such as the notion of ‘wicked problems’, that mainstream economists could do well to pay attention to. (Although I quibble on some of their small print.)

Ken, It seems to me that you are not distinguishing between mathematics as presented by mainstream economists and non pseudo-mathematics. I wish you would.

Dave, I can see only two ways to misuse mathematics. Violate one of its many, many axioms or just lie about a number or equation..Otherwise, it’s just the continued emergence of mathematics.

Ken, What about misinterpreting it? For example, Kolmogorov’s probability theory.

Dave, mathematicians argue so much about axioms it’s sometimes difficult to distinguish an argument from a “misinterpretation.” Kolmogorov’s probability axioms are widely accepted by most mathematicians. Applying them in another question. That’s why axioms are listed in each proof so that, hopefully those reviewing the proof can determine how the axioms are applied. But this is not a misuse of mathematics.

Ken, https://en.wikipedia.org/wiki/Probability_axioms#Simple_example:_coin_toss has an application of Kolmogorov to a coin toss. Keynes would add the assumption that the probability is measureable, and question it. I agree that some mathematicians are in the habit of taking this as an axiom, but my point is that even if true it would have to be an axiom some other subject (e.g., mathematical physics), not mathematics as such.

Mathematics as such is silent on coins, urns, horses, the weather, stock markets, … . It mostly provides tools that others use (or misuse). But I am trying to point out that it also provides some means to critique computational models. (And it seems to me that many of the critiques of mathematics as used in economics are pretty similar to those of mathematicians as such.)

Dave, these are Kolmogorov’s three axioms for computing probability.

First axiom

The probability of an event is a non-negative real number:

F is the event space. It follows that P(E) is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom.

Second axiom

See also: Unitarity (physics)

This is the assumption of unit measure: that the probability that at least one of the elementary events in the entire sample space will occur is 1.

Third axiom

This is the assumption of σ-additivity:

Any countable sequence of disjoint sets (synonymous with mutually exclusive events) E1, E2 … satisfies

Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra. Quasiprobability distributions in general relax the third axiom.

Easy to learn and apply. Unless one has facility with the jargon of mathematics (actually several jargons) and mathematical proof protocols, challenging these is impossible. So, we are taught them, learn them, and apply them. Generally without question. Some social applications researchers have and do challenge both the axioms and the rules derived from them. Including the derivation process. And by so doing have modified them to create a style of mathematics less obtuse and more closely connected to everyday life. Which is, of course how mathematics began before mathematicians took it over.

Ken, The axioms of Euclidean Geometry are also easy to learn and apply, and I gather that generations of students were taught to make the obvious identification between mathematical and physical lines. But so what?

I have some sympathy with your “Unless one has facility with the jargon of mathematics (actually several jargons) and mathematical proof protocols, challenging these is impossible.” I’m guessing you have no idea what I’m talking about when I refer to category theory, so let me switch to https://en.wikipedia.org/wiki/Category_mistake . Mathematics is about ‘the category’ of mathematical structures. If you are not a mathematician then presumably your notion of ‘probability’ is not a priori a mathematical one: it must belong to some different category. (For example, some theory of science, or physics, or statistics.) To borrow some of Lars’ terminology, where is your ‘warrant’ for supposing that Kolmogorov’s axioms apply to your concept? Kolmogorov had no idea what your concept is, and even if here were alive today, he was not a psychologist (or anthroplogist). So even if Kolmogorov made some broad claim such as ‘No-one will ever use the term ‘probability’ except in cases where my axioms apply’ (did he?), this would not be a mathematical claim.

My mathematics teachers taught me mathematics. Axioms were never ‘things that should be assumed’ but things that need to be accepted by relevant subject matter experts (e.g. physicists) in order to apply the mathematics. (E.g., see https://en.wikipedia.org/wiki/Axiom ). If you want to pass a mathematics exam, you need to ‘make the usual assumptions’. But if you are doing logic or science you

shouldbe taught to question them, as Einstein and Eddington did about 100 years ago.If, as you say, some mathematics teachers are teaching a kind of alchemy, then they deserve you scorn.

I’d be interested in your ‘less obtuse’ ‘style of mathematics’, though. Maybe it provides some language that would help us to apprciate each other’s viewpoint?

Dave, according to Lev Vygotskian, Paul Ernest, Jean Piaget, etc. Euclid’s geometry as well as category theory, etc. in mathematics is the result of the empirical lives of people who work with measuring but are not mathematicians. All three say this is the way people learn anything, including mathematics. Mathematics of mathematicians may be about mathematical structures. But mathematics in everyday life is about measuring to do a job, travel, build a ship, climb a mountain, etc. So, in this sense, I agree with you that probability was brought to mathematics. It was not invented by mathematicians. Best I can phrase this is “probability is a human notion about the uncertainty of human life.” It’s included in every culture we know of. However, we’re not discussing what mathematicians do. Yes, they are trained to accept axioms. But this is irrelevant for social scientists. The question we need to answer is, do social scientists accept these axioms in using mathematics? My conclusion is that many say they do, but in practice do not; and that social scientists frequently use mathematics in an effort to “up their standing” in the community of their science.

Less obtuse. How about this.

First, a large share of research in education, social work, psychology, etc. is performed using a multiple methods approach. This is referred to as triangulation social science.

Barrow summarizes the approach in his text “An introduction to philosophy of education.”

Since educational issues are of many different kinds and logical types, it is to be expected that quite different types of research should be brought into play on different occasions. The question therefore is not whether research into teaching should be conducted by means of quantitative measures (on some such grounds as that they are more ‘objective’) or qualitative measures (on some such grounds as that they are more ‘insightful’), but what kind of research can sensibly be utilized to look into this particular aspect of teaching as opposed to that.

Second, the quantitative part of this methodology is any data that is in numerical form such as statistics, percentages, etc. The researcher analyses the data with the help of statistics and hopes the numbers will yield an unbiased result that can be generalized to some larger population. Research in mathematical sciences, such as physics, is also “quantitative” by definition, though this use of the term differs in context. In the social sciences, the term relates to empirical methods originating in both philosophical positivism and the history of statistics, in contrast with qualitative research methods.

Typical education and social science research include:

Survey that concludes that the average patient must wait two hours in the waiting room of a certain doctor before being selected.

An experiment in which group x was given two tablets of aspirin a day and group y was given two tablets of a placebo a day for headache where each participant is randomly assigned to one or other of the groups.

You want to improve student essays, but you don’t believe that teacher feedback is enough. So, you devise an experiment. Your hypothesis: peer workshopping prior to turning in a final draft will improve the quality of the student’s essay. The study is at one school involving three teachers, each teaching two sections of the same course. The treatment in this experiment is peer workshopping. Each of the three teachers will make the same essay assignment to both classes; the treatment group will participate in peer workshopping, while the control group will receive only teacher comments on their drafts. The treatment and no treatment groups are self-selected. Making this a quasi-experiment. After teachers’ grades are recorded, papers are evaluated for a change in sentence complexity, syntactical and grammatical errors, and overall length. Any statistical analysis is done at this time if you choose to do any. Notice here that the researcher has made judgments on what signals improved writing. It is not simply a matter of improved teacher grades, but a matter of what the researcher believes constitutes improved use of the language.

Ken, Excuse my tardiness, but I was having fun!

If I can deal with one of your points, on probability. “[It] was brought to mathematics. It was not invented by mathematicians. Best I can phrase this is “probability is a human notion about the uncertainty of human life.” It’s included in every culture we know of.”

So, we have two distinct concepts of probability: the ‘human notion’ and that of mathematicians such as Kolmogorov. Many economists seem to equate them. But why? Whose responsibility is it to make this correspondence? Why would anyone trust a mathematician to do it? Or a human scientist? How might we properly do it?

I take the critiques of Boole and Keynes seriously, hence my blog.

Ken, on your “we’re not discussing what mathematicians do. Yes, they are trained to accept axioms. But this is irrelevant for social scientists. The question we need to answer is, do social scientists accept these axioms in using mathematics? My conclusion is that many say they do, but in practice do not; and that social scientists frequently use mathematics in an effort to “up their standing” in the community of their science.”

My question is, do the social scientists take responsibility for the mathematics that they use? Do they even realise that there is a potential problem?

What does it say about a community of practice if the flagrant abuse of mathematics (or any other theory) in this way ‘ups their standing’? Why has mathematics been blamed for the failings of financiers and economists? (Not that mathematicians are not at fault, but if fingers are going to be pointed, it helps to point them where there is actual fault.)

Ken, Your ‘less obtuse … how about this?’ notes deserve a fuller response. Briefly, I would distinguish between your ‘quantitative’ methods and what I think of as a (more broadly) mathematical approach. My reading of Keynes, Whitehead, Russell et al is that from their mathematical perspective a narrowly quantitative approach (such as you deem ‘mathematical’) would be absurd. We clearly need to consider problems from broader perspectives. But among those, why not include the mathematical critiques of the likes of Keynes?

To take probability as an example, just because Kolmogorov’s axioms for the quantification of ‘probability’ are inappropriate, why stop there? (I would further argue that sometimes even a naive view of probability has its uses, so why not use a broader view of the subject to give some guidance on the use of the simplistic theory ‘as a tool’?)

Dave, the two don’t appear to be equivalent, when examined in terms of how each is used. Translating between them is difficult, since each translator would approach the work from only one of the bases. What is needed is a translation team that can simultaneously approach from both use bases.

I know of few social scientists who are interested in any form of mathematics except those useful to their craft. They do not actively seek other forms. Although a few social scientists do have backgrounds in non-social scientific mathematics. These have become well known in a somewhat heroic way in their areas of social science. As to problems in their use of mathematics, most social scientists don’t consider mathematics in that broad of a way. But this is improving.

On your final point, I have no objections to the consideration of social scientific questions from a quantitative perspective. The issue is, how do we decide which questions are open to quantification and which are not. In other words, which questions quantification will illuminate and which it will conceal.