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On mathematics and economics

from Lars Syll

Studying mathematics and logic is interesting and fun. It sharpens the mind. But economics is not pure mathematics or logic. It’s about society. The real world. Forgetting that, economics becomes nothing but an irrelevant and uninteresting ‘Glasperlenspiel.’ Or as Knut Wicksell put it already a century ago:

Knut-WicksellOne must, of course, beware of expecting from this method more than it can give. Out of the crucible of calculation comes not an atom more truth than was put in. The assumptions being hypothetical, the results obviously cannot claim more than a vey limited validity. The mathematical expression ought to facilitate the argument, clarify the results, and so guard against possible faults of reasoning — that is all.

It is, by the way, evident that the economic aspects must be the determining ones everywhere: economic truth must never be sacrificed to the desire for mathematical elegance.

  1. yoshinorishiozawa
    April 28, 2022 at 6:42 pm

    The mathematical expression facilitates the argument, clarifies the results, and so guards against possible faults of reasoning — that is amazing!

    The economy is complex, complicated, and full of vague and ambiguous facts. Who can understand this complex system that is a world-wide network of productions and transactions? Mathematics gives only a dim light on it but it is important part of economics investigation.

  2. Romar Correa
    April 29, 2022 at 6:42 am

    I would distinguish between mathematical economics and the use of mathematics in economics. The former is the Bourbaki practice of axiom-theorem-proof exemplified in the agenda of general equilibrium theory. The development of that research program as in the ongoing work on the multiplicity of equilibria will naturally take on a pure math character. New mathematical spaces are found to throw light. At the same time, the founding fathers of yesteryear were equally concerned with public economics and the pricing of utilities in their economies. Importantly, they were alive to the assumptions and limitations of their models and warned against their misuse. Who can forget Frank Hahn’s excoriating critique of DSGE?! We have lived down the socialist calculation debate but if multi-level hierarchical system-wide planning was to be entertained tomorrow as an antidote to the anarchy of the market this mathematical economics would be helpful.
    Mathematics is a natural language in many branches of theory that attempt to shed light on society. Avinash Dixit is a theorist whose math passes the reality check every single time. In the evolution of a line of work, the topology may overtake the economics but experienced economists will be able to draw the separating line. The recent work on networks is a case in point. Who will question the value of theory in illuminating the facets of social networks? Yet the profusion of papers in the leading journals is mathematically dense with economics that must be seen with a magnifying glass. Kaivan Munshi stands out as scholar of networks which are institutionally dense. The theory has been internalised in the empirics.
    Wicksell’s cumulative process with the market interest rate chasing the natural interest rate might benefit from mathematical formalism. In particular, the turning point between upward and downward movements could be rigorously specified.

    • Robert Locke
      April 29, 2022 at 9:45 pm

      historians, who are copletely ignored in this blog, usually stress institutions in their work, e.g. Mccloskey, Visible Hand.

      • Meta Capitalism
        May 1, 2022 at 1:17 pm

        [H]istorians, who are completely ignored in this blog, usually stress institutions in their work, e.g. McCloskey, Visible Hand.

        Anthropology and economic history do not help you to understand economics. They serve you as epistemological obstructions. Of course, you will not understand it as you never did. I finish my intervention here. Please continue your eloquent speech. (Shiozawa on economic history & anthropology, RWER, January 6, 2020)

        It seems to me that philosophical arguments are useless for this kind of practical questions. (Shiozawa, RWER, Philosophy useless, 12/28/2019)

        I have no intention to exclude institutions by defending mathematical reasoning among economics. At the actual stage of mathematics it is difficult to analyze most of institutions and organizations by mathematical methods…. What I want to say is not exclude mathematics among one of many necessary tools of consideration and analyses. It must be used where it suits the subjects. (Shiozawa, RWER, Flip-Flopping On history, math, and where they suit their subjects)

        History and institutions matter. Consistency matters.

    • yoshinorishiozawa
      April 30, 2022 at 7:32 am

      Hi, Robert, happy to know that you are going fine. We may be approaching the final phase of COVID-19 pandemic. But disaster comes after disaster. There is no end. We are now in the middle of Putin’s War against Ukraine.

      I have no intention to exclude institutions by defending mathematical reasoning among economics. At the actual stage of mathematics it is difficult to analyze most of institutions and organizations by mathematical methods. Game theory may cover a small part of such a study, but it is not impossible to cover all the aspects. What I want to say is not exclude mathematics among one of many necessary tools of consideration and analyses. It must be used where it suits the subjects.

      Although he does not tell explicitly, mathematics that Lars Syll imagines as typical seems to be econometric models. I wonder if he has even a single experience in which mathematics helped him to understand an underlying logic beneath the confusing surface. We are not omnipotent or omniscient. For a boundedly rational entity as we are, there are many relations the connections of which are often difficult to understand. On a rare occasion, we find such a connection by the aid of mathematics. Syll’s arguments seem to be based on too narrow range of (intellectual) experiences.

      • Meta Capitalism
        April 30, 2022 at 11:37 am

        Although he does not tell explicitly, mathematics that Lars Syll imagines as typical seems to be econometric models. ~ Shiozawa spewing absurdity & nonsense for straw men arguments

        Lars has been both on this blog and in his book more than explicit that his critique is aimed at “econometric models” and the formal statistical/probabilistic methodologies that such models is based upon. Lars could not have been more explicit in the target of his critique, yet nevertheless, Shiozawa disingenuously claims Lars “does not tell explicitly” the subject of his many critiques. This is nothing but a dishonest straw man that reveals Shiozawa is afraid to engage the actual context honestly of Lars arguments. And from a purported academic no less. Shameful in my view.

      • Meta Capitalism
        April 30, 2022 at 1:13 pm

        No, there is nothing wrong with mathematics per se.

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

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

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

        • “math is the only valid tool”

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

        • “math is all that really counts”

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

        • “almost everything can be adequately understood and analyzed with math” (Lars Syll on Proper Use of Mathematics, RWER, February 8, 2018)

        ~ ~ ~

        What I want to say is not exclude mathematics among one of many necessary tools of consideration and analyses. It must be used where it suits the subjects. (Shiozawa, on math, and where they suit their subjects, which is exactly what Lars says above and Shiozawa has read before. So why ignore it and make false straw men arguments?)

        Shiozawa knows full well that Lars is making the same argument he makes himself for he read Lars post above (here), yet rather than honestly engage in the same argument he allows for himself he chooses instead to ignore it and misrepresent Lars views with rhetoric using logical fallacies to pretend he doesn’t know what Lars real substantive arguments are even though he has clearly read them:

        I wonder if he has even a single experience in which mathematics helped him to understand an underlying logic beneath the confusing surface…. Syll’s arguments seem to be based on too narrow range of (intellectual) experiences. ~ Shiozawa spewing absurdity & nonsense for straw men arguments redux

        This self-serving fallacious argument lacks integrity. If Syll is “too narrow” in his argument above which is the essentially the same argument Shiozawa makes what does this say about Shiozawa’s consistency and logic? What I see is a passive aggressive pseudo-intellectual argument of someone who lacks the courage and integrity to engage the actual content of Lars arguments, so instead he engages in malpresentation and straw men arguments and psychological projections to avoid the simple fact that if Lars is right his axiomatic-deductive theory is doomed from the start. Since He cannot muster the degree of intellectual and philosophical sophistication to address Lars arguments Shiozawa turns to passive aggressive ad hominem. That is a sad place for so-called scholar to be.

        Shiozawa claims his theory “has succeeded to explain how a large economy as big as world economy works” (see Shiozawa et. al., 2019, Kindle Locations 1792), yet this claim is proven false by the simple fact his theory is purely an axiomatic-deductive exercise which he calls “pure theory” divorced from the real-world economy in that it “excludes important parts of the modern economy (Shiozawa et. al., 2019, Kindle Locations 1890)” — relevant reality (e.g., finance, agriculture, mining, consumption, institutions, history, etc.) for tractability (, here). True irony can be found in the self-contradictory claims made plain via juxtaposition. On the one hand his theory “has succeeded to explain how a large economy as big as world economy works” and on the other hand excludes important realities because “we have to sacrifice such completeness for the sake of tractability and comprehensiveness (Kindle Locations 1822-1829).” How one excludes important realities and claims “comprehensiveness” while also admitting “The range of validity of the new theory is narrow” is akin to illogical double-speak. Shiozawa explicitly states his theory, comprised of a set of eighteen (plus auxiliary) “complicated postulates (Kindle Locations 1822-1829),” had to “sacrifice such completeness for the sake of tractability” and only models “a small part of the total economic system (Shiozawa et. al., 2019, Kindle Locations 2165)” and “It is not a theory that can explain all aspects of a modern economy (Shiozawa et. al., 2019, Kindle Locations 2165).”

      • Meta Capitalism
        May 1, 2022 at 1:11 pm

        “malpresentation” should be “misrepresentation.” Malpresentation refers to when your baby is in an unusual position as the birth approaches. Ironically, it seems, Shiozawa’s arguments might fit the definition as they often come out in rather strange contortions of illogic (logical fallacies) ignoring Lars clearly stated meanings so he can falsely assert meanings Lars clearly never stated or intended.

    • yoshinorishiozawa
      May 6, 2022 at 7:18 pm

      Romar Correa,

      it may be a good idea to distinguish mathematical economics and use of mathematics. Mathematics (or more exactly speaking, mathematical expression) is simply a tool of formulating and expressing our ideas. Some idea can be well expressed in common language. Some idea is better expressed in mathematics.

      We can observe a similar situation in economic history. We have old style history investigated and written in prose or in common language. We have a field called cliometrics, which uses statistical method (strictly speaking, this is not mathematics per se) for the main method of investigation. I am of the opinion that both methods are possible. We cannot judge by the form: cliometrics or history in prose. Both may produce a deep insight. We should judge each research whether it is good or not, or has a deep meaning or not.

      In the similar way, we should judge each theory or strand of theories whether it is good or not by its contents. To this judgement, if the theory takes mathematical form or not is irrelevant.

      Tony Lawson and his followers often talk that the economy is an open system and it is impossible to apply mathematics to this kind of system. The correctness of this contention depends on the domain of mathematics. If we define mathematics narrowly, it may consists of geometry, algebra and calculus. If we use mathematics in this narrow sense, Lawson’s contention may be right. However, mathematics is developing. New fields of mathematics are emerging. Network theory is one of them. In my case (the new theory of international values), some concepts in bipartite graph theory was essential to the characterization of what I call regular international value.

      Georg Cantor once declared that “The essence of mathematics lies in its freedom.” There is no rigidly fixed mathematics. Classical mathematics is the mathematics that are known at the end of the 19th century. If necessary, we can construct new mathematics, although it would be much more difficult than to develop a theory in prose.

      • Meta Capitalism
        May 6, 2022 at 10:11 pm

        Georg Cantor once declared that “The essence of mathematics lies in its freedom.” (Shiozawa on Georg Cantor)
        The second area we shall examine is set theory and more particularly the revolution that arose through the work of the mathematician Georg Cantor. Cantor’s work arises directly from a new understanding of the actual infinite. Like many radical changes in the world of ideas, Cantor’s work engendered virulent opposition in its time, yet its conclusions are today accepted without question in the world of mathematics. They involve the very nature of mathematical truth and its relationship with reality. (Byers, William. How Mathematicians Think (pp. 146-147). Princeton University Press. Kindle Edition.)

        Byers is a fascinating read and reveals that mathematics is not reducible to logic, but is rooted in genuinely creative thought.

  3. Edward Ross
    May 1, 2022 at 1:08 am

    Here above I see Robert Locke’s concern for the way many economists and academics ignore history. Here i member seeing this quote, “True ignorance is not the absence of knowledge but the refusal to acquire knowledge.” Karl popper. Thus from my perception when academics refuse to consider history they are refusing to acquire knowledge. My point here is that history can point us in a direction that may help to avoid repeating the same mistake repeatedly.
    For example some of my friends who had escaped as children from a Russian camp in Siberia were appalled at the way neither NATO or anyone else stood up to Russia when they were amassing a large militarily force on Ukraine’s border.
    Recently talking to some Hungarian friends whose mothers had made terrific efforts to bring them out of Russian oppression and since passed away. When i was talking to them about the state of affairs’ in THE Ukraine they said they were happy that their mothers had passed away before they saw history repeating it self. My point here is that these people had lived history and need to be included in the conversation. Thus i think we have to listen to the experience of elderly people with many years of experience such as Robert Locke and Iconoclast. Before we think we know it all. Thankyou Robert Locke and Iconoclast and Dave Taylor who we have not herd from for some time.Ted

  4. yoshinorishiozawa
    May 1, 2022 at 4:46 pm

    Ted, I read your comment on the article by David Lane
    What Caused Russia to Invade Ukraine?

    It may show the weakness of geopolitical thinking.

  5. Econoclast
    May 1, 2022 at 11:08 pm

    Well, here’s part of my take on history: I utterly fail to see how any profession (economics, engineering, social work, whatever) can thrive and progress if its members have no knowledge of the history of philosophical thought in their profession. What would such people consider to be the source of their knowledge, much less wisdom?

    I’ve thought about this recently after reconnecting with a professor of the history of economic thought (my teacher a half century ago) before he died. He considered himself an historian, rather than an economist, even though he had been taught by Frank Knight. And, of all the mental data I acquired from two degrees, his work I remember most.

    • Meta Capitalism
      May 2, 2022 at 10:27 am

      Nevertheless, if I may throw out a word of council to beginners, it is: Treasure your exceptions! When there are none, the work gets so dull that no one cares to carry it further. Keep them always uncovered and in sight. Exceptions are like the rough brickwork of a growing building which tells that there is more to come and shows what the next construction it to be.

      — William Bateson (1908)

      To understand evolution we must first understand the historical development of ideas on evolution. But to understand its history, we must first understand evolution. The paradox implies that the study of evolution and of its history must go hand in hand. (Forsdyke 2008: xi)

      (….) For a scientist looking at history there are three overriding double questions: What did he (she) know and when did he know it? What did he think and when did he think it? What did he do and why did he do it? Answers to these questions give us information on process — the process of scientific discovery. If the elements of that process have remained essentially unchanged since Bateson’s day — an assertion assumed and not argued here — than a better understanding of ways this outstanding contributor to scientific progress operated may help us improve the process of discovery as it now operates. (Forsdyke xvii)

      (Cock, Alan G. and Forsdyke Donald R. Treasure Your Exceptions: The Science and Life of William Bateson. New York: Springer; 2008; p. xi; xvii.)

      As the relations between development and evolution were explored in the last decades of the twentieth century, mutations in those genes that regulate development also assumed great significance. Regulatory (homeobox) genes were reported that affected whole sets of genes, and some paleontologists [and developmental geneticists, etc.] have emphasized that mutations in such regulatory genes result in large discontinuous evolutionary changes (a reshuffling of parts) and may be a macromechanism of evolution. But symbiosis–the inheritance of acquired genomes, wholes or parts–was trivialized or ignored by paleontologists. Gould himself regarded the symbiotic origin of mitochondria and chloroplasts as “entering the quirky and incidental side” of evolution. (Sapp 2003: 250-251)

      Most of the great controversies and conceptual oppositions of the nineteenth century are still present at the beginning of the twenty-first century: religion and vitalism versus evolution and materialism, structuralism versus functionalism, reductionism versus holism, gradualism versus saltationism, selectionism versus nonadaptationism, the inheritance of acquired characteristics, and nurture versus nature. What has changed is not so much the nature of the ideas but the evidence supporting them and the intensity of the debates. (Sapp 2003: 267)

      (Sapp, Jan, . Genesis [The Evolution of Biology]. Oxford: Oxford University Press; 2003; pp. 250-251; 267.)

      Prior to taking up a deep dive into the history, philosophy, and theory of economics I spent decades (thirty to be exact) studying the history, philosophy, and theory of evolutionary biology in all its ramifications. I have come to see major themes running through both the history of biology and economics similar to the themes noted by Fullbrook with regards to physics. History, contrary to those who are philosophically naïve claim history doesn’t help us understand economics, or biology, or any other scientific field. Frequently, those who are so philosophically naïve as to dismiss history are also slaves to their own unexamined philosophical presuppositions; intellectual parrots to old dead men’s ideas.

  6. May 2, 2022 at 3:28 pm

    Or, as Sismondi explicitly put it already well over two centuries ago (my translation): “Economics does not rest solely on calculation, the facts are constantly altered by a host of observations of a moral order which are not subject to calculation. If indeed permanent abstractions are to be made of the latter, it would be just as if a mathematician were to chance that suppressing some essential terms in each of his equations [wouldn’t be making a substantial difference in the outcome]”*1.
    He had come to this most unorthodox conclusion after having formulated algebraic equations of lagged aggregatively-obtained income, that were fully in line with those devised by modern Post Keynesians*2; but, for the reason as mentioned, swore off to ever make use of these again as an appropriate tool to interpret an economy’s workings – to the utter astonishment of especially his otherwise quite sympathetic modern interpreters, like Sowell, Hyse, etc; being all steeped of course in a math approach to economics.
    And then there is Marx, who, in ch. 3, sect.10 of his TSV, be it indirectly and quite inadvertently proved that math and economics aren’t compatible. Reasoning from materialistic first principles, as a perceived reality of existence at any given time, he discovered that the vertically integrated economy of his analysis wasn’t able to ongoingly replace itself, thus materially, from charged depreciation allowances; and hence couldn’t possibly exist as such – when factually accounted for, and with profits also being charged as a matter of course. But being unwilling/unable to drop his materialistic point of departure; he acted as though having no choice but to resort to subterfuge, in order to conceal this to him most embarrassing fact*3.

    Marx’s unwittingly arrived-at but ever so crucial discovery, of non-existing material-capital replacement producers however, is equally devastating for all present day conventionally-taught economics’ persuasions. For this would mean that while we know that our economy does exist without any fundamental problems, it can apparently only do so (at least temporarily) in some form of non-material reality at its base. But this could only be factually so, by virtue of being able to coherently suspend any identifiable materiality in real terms (and, consequently with it, invalidate a logical approachability in a mathematical format of identification, as a static reality!); and now instead exclusively identify an economy’s ongoing operation in terms of a non-material unit of account. That is, until having been booked but still being materially-indeterminate values-in-exchange, as produced and distributed, become determined as real – at the point of mutually resolving through retail-level purchases of final output. Whereby only then a generally sought-after positively-valued real materiality, as being inherent of course in exploited natural resources, becomes restored in terms of economy-exogenous values-in-use; a no longer seeking of monetary returns on earlier booked efforts occurs; and all is set for an unencumbered systemic rebirth. The latter however is far from certain to indeed be happening – with the always open option to reinvest in existing “assets” (read: to be resolved claims, or systemic debts; i.e., negative values). Note that Marx’s deterministic-material approach had assumed (capital) values to be positively real at each economic level, thus before these were realized through final demand at the retail level; rather than negatively “real”, as to be passed-on down systemic debts. Hence, in the reality as he perceived it to be, no reproducers of (material) capital were to be found where he expected them to be. But orthodox economics also, although for very different reasons, assumes that same faulty “fact” as being operational too.
    Furthermore, in accordance with the herewith touched-on alternate-economic reasoning*4, our money cannot exist in a material form either*5; so better forget about its attributes as being a material means of exchange and a store of positive value (in equilibrium or not). The common-sense “utility of money”, as being a positively-valued materiality, is thus sourced outside a modern economy as “animal spirits”; and potentially impairs an economy’s efficient operation – through a wide-spread accumulation of unresolvable, though as such not immediately apparent, debt. Its sole redeeming feature, that at least conceptually could yet be understood as adding to an economy’s efficiency, would be limited to an at certain times necessary pruning of economic deadwood. But the accumulation of monetary “wealth” by the so-called 1%, aggregatively more or less the sole ‘net’ savers in our excessively indebted economy, has been most inefficient; and far from something to be endeavoured.

    The crux of the matter however in the context of “On mathematics and economics” is that our economy’s identification, as being a vertically-integrated system of accounts, is rendering all algebraically-depicted points of economic departure, like e.g. Y=C+I (and also of course a subsequent differential analysis in those terms), inappropriate for determining an economy’s generally agreed-to purpose fulfillment; and thus consequently would be wrongheaded – in these being no more than a begging the question of the terms’ real (material) static identities ex ante; meaning: so much for the bulk of Keynes’ writ and therefore its modern offshoot Post Keynesianism as well. The essential question though is: will heterodoxy be able to come to terms with this now exposed alternate economic reality?

    It’s this supposition of a modern economy’s actual state of affairs, as perpetually only on its way to become manifest; but never, as materially, is – that’s the reason why at least in my reasoned opinion there simply can be no role for mathematics in an economics where people matter*6; other than just plain arithmetic, as it’s being utilized in accountancy. And although perhaps some restricted use of “higher” mathematics in ‘what-if’ projections could possibly remain validly applicable yet; it should never be lost from sight that an economy’s definitive identity rolls out of what ought to be the obvious: no returns = no economy. This would mean that our economy is demand-side determined; and that fact, all by itself already, as a shortcut to the above laid-out argument, makes algebra and calculus in economics about as useful in implementation as these tools are in cost accounting.

    *1) “de la Richesse commerciale” 1803: Vol I, p. 104. As noted on p. 71, in my partially translated and annotated critique of Sismondi’s 1827 (2nd ed. of) “Nouveaux Principes…” http://www.vcn.bc.ca/~vertegaa/sismondi.pdf
    *2) Thomas Sowell, “Sismondi: A Neglected Pioneer”. History of Political Economy, Spring 1972.
    *3) http://www.vcn.bc.ca/~vertegaa/Marx_Debunked.pdf
    *4) For more – http://www.vcn.bc.ca/~vertegaa/ontology.pdf
    *5) On the reason why a coherent theory of money still hasn’t been written – http://www.vcn.bc.ca/~vertegaa/money.pdf.
    *6) The paraphrased subtitle of E.F. Schumacher’s iconic and math-devoid economics book “Small is Beautiful”.

  7. Ken Zimmerman
    May 10, 2022 at 3:43 am

    Economic life is the activities through which people produce, circulate and consume things, the ways that people and societies secure their subsistence or provision themselves. Activities by which communities sustain themselves. ‘[T]hings’ is, however, an expansive term. It includes material objects, but also the immaterial: labor, services, knowledge and myth, poems, religion, names and charms, and so on. In different times and places, different ones of these will be important resources in society, and when they are important they come within the purview of economic anthropologists. That is, where some economists have identified economic life in terms of the sorts of mental calculus that people use and the decisions that they make (for example, utility maximization), which stresses the form of thought of the person being studied (theories), most economic anthropologists would identify it in terms of the substance of the activity; even those who attend to the mental calculus are likely to do so in ways that differ from what is found in formal economics (for example, Gudeman 1986; Gudeman and Rivera 1991). This substance includes markets in the conventional sense, whether village markets in the Western Pacific or stock markets in the First World. However, these markets are only a sub-set of economic life, and in accord with their tendency to see the interconnectedness in social life, economic anthropologists tend to situate things like markets or other forms of circulation, or production or consumption, in larger social and cultural frames, in order to see how markets, to continue the example, affect and are affected by other areas of life.

    You might ask, how do top subjects of modern economists such as financialization, stock and commodity markets, hedging, private equity, etc. serve this definition? They do not. So, why do economists today label them as necessary to sustain modern societies? Perhaps they believe societal sustainability is improved when a few hundred society members control so much wealth they alone can determine societal actions in almost every important area of life. Or, when there is widespread and endemic poverty. Or, when over half the society’s members lack affordable healthcare, housing, and education for their children and themselves.

    Humans began inventing ‘economic’ activities in prehistory. The first time this is seen in the historical record is about 5,300 years ago. Affixed to a wall at the British Museum in an unadorned manner, in a plain exhibit on the history of human writing, is a gypsum tablet that is some 5,300 years old (more than 3,000 years older than the Rosetta Stone). This tablet is only several centimeters long on each side and contains lines and dots made in the gypsum. These lines and dots, we now know, represent quantities— likely quantities of grain or some other item that featured in an economic transaction. They are more systematic than the marks evident in the Paleolithic record, as they are not just tallies of quantities. Instead they represent a standardized form of communication in two dimensions. They are the first true written symbols we know of, as each line and dot represents a specific abstract quantity. In other words, the marks in the gypsum are true numerals. Combined with the names of the items involved in the transaction. For historical purposes this can be considered the invention of numerals, writing, and economics. Economists continue to ignore these origins at the peril of their discipline and the disordering of societies today.

    This tablet drives home an important and often overlooked point: human survival is contingent on knowledge stored in the repository of culture, accessed through linguistic means. Daily, we rely on knowledge that is not really our own but can be easily extracted from the minds of others and has, in many instances, been brutally and often randomly acquired over the course of millennia.

    In the course of human history, cultures have died off completely due to degradations of their survival- related know-how or because of the loss of basic material technologies that could not be easily replicated. Such cases directly contravene the popular, some would say mythologized, notion that humans excel simply because we are inherently smarter than other species. It turns out that this idea has little empirical support. While we are obviously smarter than other species and do have a high encephalization quotient (large brains for our body size), in some ways our innate cognition is not as advanced as we once assumed. Many of our distinguishing intellectual attributes are not genetically hardwired but learned in culturally dependent ways. While natural selection has undoubtedly yielded remarkable human brains, what is really most striking about our species is what we have managed to do with those brains through the invention of culture. I want to emphasize this point. Culturally dependent innovations like language initiated a cognitive and behavioral revolution in our species. A set of conceptual tools called “numbers”— words and other symbols for specific quantities—is a key set of linguistically based innovations that has distinguished our species in ways that have been underappreciated. Numbers are human creations that, like cooking, stone tools, and the wheel, transformed the environments in which we live and evolve. While anthropologists and others have long been enamored with highlighting such inventions and their role in changing the script of the human story, the role of numbers has received insufficient attention in the past. The motivation for that inattention is simple: we are only now beginning to appreciate the extent to which the tools called “numbers” have reshaped the human experience.

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