The Future of Everything

December 14, 2021

Quantum economics – the story so far

This piece gives a brief summary of my work to date (2016-2021) in quantum economics.

The idea that the financial system could best be represented as a quantum system came to me (dawned on me? evolved?) while working on The Evolution of Money (Columbia University Press, 2016). “Money objects bind the virtual to the real, and abstract number to the fuzzy idea of value, in a way similar to the particle/wave duality in quantum physics,” I offered. “Money serves as a means to quantify value, in the sense of reducing it to a mathematical quantity – but as in quantum measurement, the process is approximate.” Price is best seen as an emergent feature of the financial system. I summarised this theory in two papers for the journal Economic Thought: “A Quantum Theory of Money and Value” and “A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle“.

While I had some background in quantum physics – I studied the topic in undergraduate university, taught a course on mathematical physics one year at UCL, and encountered quantum phenomena first-hand while working on the design of particle accelerators in my early career – my aim in the book (co-authored with Roman Chlupaty) was not to impose quantum ideas onto the economy. My primary research interest was in computational biology and forecasting and I had not touched quantum mechanics in many years. The dual real/virtual nature of money just had an obvious similarity to the dual nature of quantum entities, and in fact I was surprised that I appeared to have been the first to make this connection in a serious way and come up with a quantum theory of money.

I was aware that a number of researchers were working in applying quantum models to cognition and psychology, but it was only after finishing the book that I learned about the area of quantum finance (I also discovered a separate paper on “Quantum economics” by the physicist Asghar Qadir from 1978, which argued that the quantum formalism was well suited to modelling things like economic preferences). The reason I hadn’t come across these works in my research about money was because just like in neoclassical economics there was no discussion of that topic. Nor was there much discussion of quantum phenomena such as entanglement or interference. Instead the emphasis in quantum finance (as this paper notes) was on using quantum techniques to solve classical problems such as the Black-Scholes option-pricing algorithm, or portfolio optimisation.

My motivation was completely different. In books such as Economyths, and The Money Formula (with Paul Wilmott), I had investigated the drawbacks and limitations of these traditional models – so rather than invent more efficient ways of solving them, I wanted to replace them with something more realistic. Money was the the thing which linked finance and psychology, so a quantum theory of money could be a first step in developing a new approach to economics.

I sketched out the basic idea as an Economic Thought paper “Quantum economics” which served as a blueprint for my 2018 book of the same name. It tied together the quantum theory of money, with ideas from quantum finance, quantum cognition, quantum game theory, and the broader field of quantum social science. The ideas were also summarised in a piece for Aeon magazine – which was when I found out why no one had probably bothered to develop a quantum theory of money. The article was not well received, by economists but especially it seemed by physicists, some of whom went out of their way to trash the idea.

I was not new to having my work come under criticism. Indeed, much of my career has focused on pointing out the drawbacks and limitations of mathematical models, which has frequently brought me into conflict with people who don’t see it that way, starting with my D.Phil. thesis on model error in weather forecasting (see Apollo’s Arrow). My book Economyths also drew howls of outrage from some economists. However quantum economics felt different, and seemed to touch on a range of taboos, in particular from physicists who have long resisted the adoption of quantum ideas by other fields. But quantum mathematics is not owned by physicists, it is simply an alternative version of probability which was first used to model subatomic particles, but also can be used to describe phenomena such as uncertainty, entanglement, and interference which affect mental systems including the economy.

While writing the book I developed in parallel an online mathematical appendix which presented some key results from quantum cognition, finance, and game theory (an early version was translated into Russian). Because my aim was to develop a theory of quantum economics, I also started applying quantum methods to some key economic problems, including supply and demand, option pricing, stock market behaviour, and the debt relationship which underlies the creation of money. This online appendix later grew into my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, first published in 2020 and now in its second edition.

For supply and demand, my idea was to model the buyer and seller in terms of a propensity function, which describes a probabilistic propensity to transact as a function of price. A simple choice is to describe the propensity function as a normal distribution. The joint propensity function is the product of the buyer and seller functions. The next step is to use the concept of entropic force to derive an expression for the forces which describes the tendency for each party to move the price closer to their preferred price point. The joint force is just the sum of the forces for the buyer and seller. However there is a contradiction because the probability distribution does not match that produced by an oscillation. To resolve this, we quantize the force to obtain a quantum harmonic oscillator whose ground state matches the joint propensity function. This model, which sounds elaborate but is actually quite minimal in terms of parameters, applies to economic transactions in general, so has numerous applications, including the stock market. The paper “A quantum model of supply and demand” was published in the journal Physica A in 2020.

Typical propensity functions for buyer (to the left), seller (right), and joint (shaded).

The question of how to price options is one of the oldest problems in finance. The modern method dates back to a 1900 thesis by Bachelier and is based on the concept of a random walk. For the quantum version, the logical place to start was with the quantum version of this which is a quantum walk. Instead of assuming that the log price will follow a normal distribution with a standard deviation that grows with the square-root of time, the model has two peaks which speed away from each other linearly in time. It therefore captures the psychological stance of an investor who has a bullish or bearish view on the asset (e.g. price might grow by 10 percent each year), but balances that with the possibility that the opposite might happen in order to obtain a fair price for the option. When coupled with the quantum model of supply and demand, the algorithm can be used to predict option price and volume. “A quantum walk model of financial options” was published in Wilmott magazine in 2021, and the theory was reported on the same year by the Economist in an article “A quantum walk down Wall Street“.

Probability distribution for a quantum walk (solid) versus random walk (dashed).

Finally a main question in quantum economics is the interaction between mind and money which underlies the debt relationship, and also the creation of money objects in the first place. Both of these topics are traditionally neglected in mainstream economics. In quantum economics it is easy to show that the debt relationship can be modelled as a simple circuit with two qubits, representing the debtor and creditor, entangled by a C-NOT gate which represents the loan contract. Interestingly, it turns out that the same circuit can be used to represent the decision-making process within the mind of a single person, where there is an interplay between a subjective context and the final decision. In quantum cognition, this is usually modelled as a two-stage process; however it can also be modelled using two entangled qubits, in which the context and the decision are separated out, as in the debt model. This result was published in a 2021 Frontiers in Artificial Intelligence paper, co-authored with Monireh Houshmand, called “Quantum propensity in economics“. A related paper published in Quantum Reports, that discusses applications including mortgage default, is “The color of money: threshold effects in quantum economics“. 

Two-qubit entanglement circuit for debt contract (A is debtor, B is creditor), or quantum cognition (A is context, B is decision).

For a full list of my research in quantum economics and finance, including links to these and other papers, please see the page Quantum Economics Resources. These findings and others are also presented in my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, and for a general audience in Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World (available 02/2022). The work continues! – if readers are interested in getting involved, please drop me a line here or through LinkedIn.

November 11, 2021

Ten reasons to (not) be quantum

Filed under: Economics, Quantum, Quantum Economics and Finance — Tags: , — David @ 11:23 pm

While the use of quantum models is becoming more popular in the social sciences including economics, it is still the case that when many people, especially those with a training in physics, hear the expressions “quantum economics” or “quantum finance” they immediately reach for some off-the-shelf arguments about why it must be nonsense (or some smelling salts). Here is a compilation of the usual ones, along with responses.

  1. Quantum mechanics was developed for subatomic particles, so it should not be applied to human systems. As one website claimed, “It’s only when you look at the tiniest quantum particles – atoms, electrons, photons and the like – that you see intriguing things like superposition and entanglement.” An article wonders why we “see the common-sense [classical] states but not the imponderable superpositions?”

Response: Bohr’s idea of superposition and complementarity was borrowed from psychology, as when we hold (or ponder) conflicting ideas in our heads at the same time, and the concepts of mental interference or entanglement are not so obscure. Also, many ideas from quantum mechanics such as the Hilbert space were invented independently by mathematicians. And calculus was developed for tracking the motion of celestial bodies but we don’t ban its application to other things.

  1. Quantum is too hard for non-physicists to understand. According to the physicist Sean Carroll, “No theory in the history of science has been more misused and abused by cranks and charlatans – and misunderstood by people struggling in good faith with difficult ideas – than quantum mechanics.”

Response: There is often a confusion between quantum probability, which is a mathematical tool, and quantum physics, which is about subatomic particles. Yes, the physics of subatomic particles is complicated – so are things like classical fluid dynamics. But quantum probability is just the next-simplest type of probability after the classical one. And the misuse of mathematical models which has created the most societal problems is the classical methods used in economics. As a side note, most people involved in quantum economics and quantum finance are physicists or (like me) mathematicians. Which brings us to:

  1. Quantum economics is physics envy, or an attempt to “appropriate the high prestige of physics” as one physicist put it.

Response: Mainstream economics is directly inspired by, and based on, concepts from classical mechanistic science. There is nothing inherently wrong with using the same mathematical tools for different areas, what is strange is when the tools used don’t change or adapt. As John Cleese said: “people like psychologists and biologists have still got physics envy but it’s envy of Newtonian physics and they haven’t really noticed what’s been happening the last 115 years.”

  1. Quantum is flaky, pretentious, pseudoscientific hype or woo. Sample usage: “As a quantum physicist, I’ve developed a reflexive eyeroll upon hearing the word applied to anything outside of physics. It’s used to describe homeopathy, dishwasher detergents and deodorant.” Cue nerd jokes about “quantum healing” or “quantum astrology”. In his description of what he called the Intellectual-Yet-Idiot, Nassim Taleb included anyone who “Has mentioned quantum mechanics at least twice in the past five years in conversations that had nothing to do with physics.”

Response: Quantum is a mathematical toolbox – it might come across as flaky or pretentious for a person to talk about it in the wrong context, but not to use it in their work.

  1. Entanglement is unique to special physical systems which can maintain quantum coherence. One science journalist told me that “Dollars don’t become quantum mechanically entangled. If they did, we’d be building quantum computers out of money.” A paper takes it as given that “one could not possibly idealize traders and investors as quantum objects exhibiting non-classical properties, like superposition or long-distance entanglement.” Physicists often conflate entanglement with Bell tests: one explained that “you can never violate a Bell inequality using systems like dice, dollars, or bank accounts. There is simply no way, and certainly no experiment has ever done so. (Maybe one or two ‘crackpot’ people claim otherwise, but they are not to be trusted.)” Another writes: “It is a common mistake to confuse classical statistical correlations with the correlations implied by quantum entanglement. The difference was made explicit by John Bell. There is no chance that correlations in statistical economics will violate a Bell inequality … unless you fiddle the data.” In other words, using entanglement in an economics model is a sign of either gullibility, or scientific fraud.

Response: In mathematical terms, entanglement is a straightforward property of Hilbert spaces, and we can use it to model social and financial systems, including traders’ decisions or the behaviour of money. The Bell test is not a definition of entanglement, it is a way of teasing out a particular form of entanglement for subatomic particles. It is true that we can’t build quantum computers out of money, but nor can we build classical computers – does that mean money is not classical?

  1. Quantum is too complicated and mathematical – we need simpler models and less math. Variants: The economy cannot be reduced to equations, people are not subatomic particles. May quote Lin Yutang.

Response: The need for simple models is a theme of many of my books, however what counts is things like the number of parameters in a model. Quantum probability is more complicated than classical probability, but it is the simplest way to capture phenomena such as superposition, interference, and entanglement, which characterise many key mental and financial processes (for example, the quantum walk model for pricing options or the two-qubit model for quantum decisions are not complicated). People are not subatomic particles, but nor are they classical particles, which doesn’t stop economists from using classical models, or talking about physics-like forces of supply and demand (they are just assumed to be at equilibrium, so cancel out). And while it is true that human behaviour cannot be reduced to equations of any sort, we use equations all the time to simulate the economy. Again, many of my books, such as Apollo’s Arrow, or Truth or Beauty, have criticised the overreach of mathematical models, but that is a separate issue and applies as much to classical models.

  1. Quantum is a forced analogy or a metaphor. As economist Paul Samuelson once wrote, “There is really nothing more pathetic than to have an economist or a retired engineer try to force analogies between the concepts of physics and the concepts of economics … and when an economist makes reference to a Heisenberg Principle of [quantum] indeterminacy in the social world, at best this must be regarded as a figure of speech or a play on words, rather than a valid application of the relations of quantum mechanics.”

Response: Quantum probability is a mathematical tool, which is not the same as an analogy or metaphor. The purpose of a metaphor is usually to describe something which is abstract and complicated in terms of something that is more concrete, so it would make more sense to go the other way and use human behaviour as a metaphor to help describe subatomic behaviour.

  1. The brain has not been shown to rely directly on quantum processes.

Response: Quantum effects appear to be exploited by biological systems in a number of processes (see quantum biology) but whether they are used in the brain or not makes no difference to economics. The argument is not that the economy inherits quantum properties from subatomic interactions in the brain, but that it can be modelled as a quantum system in its own right. For example, a debt contract can be expressed using a quantum circuit in a way which captures effects such as uncertainty, subjective context, power relationships, and so on.

  1. Markets are not quantum because there is no uncertainty. For example, something like a bank account, or an order book for a stock market, has clearly posted amounts and prices. One person compared her bank account to Schrödinger’s cat: “I am a PhD physicist, so for me the word quantum that gets thrown around is a bit ridiculous … So think about your bank account, it might be empty until you open it, so are you telling me that this is uh quantum finance or quantum economics okay you can have a million in your account or you can have zero we don’t know?”

Response: While it may be true that bank accounts are not like Schrödinger’s cat, I will let The Economist answer that one, from an article called “Schrödinger’s markets” in the print edition: “on a closer look finance bears a striking resemblance to the quantum world. A beam of light might seem continuous, but is in fact a stream of discrete packets of energy called photons. Cash flows come in similarly distinct chunks. Like the position of a particle, the true price of an asset is unknowable without making a measurement – a transaction – that in turn changes it. In both fields uncertainty, or risk, is best understood not as a peripheral source of error, but as the fundamental feature of the system.”

As computer scientist Scott Aaronson notes, quantum methods are adapted to handle “information and probabilities and observables, and how they relate to each other.” Since the financial system seems a pretty good example of information, probabilities, and observables (in this case through transactions) it seems like a suitable approach. Much of the confusion comes down to the fact that quantum economics is not quantum physics applied to the economy, but rather quantum mathematics applied to the economy (see figure below). Physicists often struggle with this because they tend to mistake their elegant models for reality, however quantum mathematics should be viewed as a mathematical toolbox that can be applied to either physical or social systems where appropriate.

The idea of quantum economics is not that physics can be directly applied to social behaviour as shown here …
… but instead that quantum mathematics can be applied to both physical and social systems. Figures from: Quantum Economics and Finance: An Applied Mathematics Introduction

Physicists tend to see entanglement for example as a special property of subatomic particles – or “a surprising feature of the world” as one emphasized – and object that the two parties in a loan contract are not entangled in the same way. To understand the entanglement, it is necessary to lift the level of analysis from physical people, to mental constructs – which is entirely appropriate, given that money and value are mental constructs. From the perspective of the debt contract, if the debtor decides to default, then the state of the loan also changes immediately for both parties. And debt contracts are a feature of the world too (even if they are less remarkable or surprising than quarks or whatever).

The above nine reasons for rejecting a quantum approach, which are the ones most commonly produced, are very superficial and are easily dismissed with a little reflection. (Skeptics sometimes prefer to say that they don’t understand or are “not convinced” without giving a specific reason, but my aim is not to convince people of anything, it is to lay out the facts as I see them and let others do their own research and come to their own conclusions.) Also, arguing against these reasons, as I have done above, will in my experience have absolutely no effect. One reason is that getting the quantum approach seems to involve something of an aha moment where it suddenly clicks into place. The other reason though is that they are not the real reason. So why is it that no one even tried to apply quantum methods to the economy until about a century after they were invented? This points to:

    10. Quantum economics touches on a range of taboo topics.

For the full picture read Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World. Finally, given the numerous reasons to not take a quantum approach, I should point out that there also many reasons why the opposite is true, and the economy is amenable to a quantum treatment! In particular, quantum is the best framework for expressing in mathematical terms the complex interactions between mind and money. To see why, the best place to start is again with the books, or see this brief summary. For a mathematical treatment, see Quantum Economics and Finance: An Applied Mathematics Introduction.

[Update] My work in economics has seen me called a number of things including a conspiracy theorist, and the intellectual equivalent of a climate-change denier. More recently an academic physicist read this piece and wrote, in a now-deleted tweet, that I was a charlatan who was ducking and weaving in order to avoid any criticism. I replied that he may have read the post, but he hadn’t understood it. He said “I judge you are not a crank. I judge you are a charlatan.” Then he thought about it (references to names redacted):

Any physicist worth their salt should agree with him that the only test is whether quantum math proves useful in modelling and prediction.

May 17, 2021

Shifting the Economyths

Filed under: Economics, Quantum, Talks — David @ 3:29 pm

This is the text for my contribution to the online conference Beyond the False Dichotomy: Shifting the Narrative

I wrote Economyths a little over ten years ago, and in part the book was my response to the financial crisis. The thesis was that mainstream economics is based on a number of what I called economyths which were defined as beliefs or stories that have shaped economic thought. There were ten in total but to give you an idea here I will just mention four:

One said that the economy is made up of independent individuals. This is the idea that people are like the classical picture of a self-contained atom, and there is “no such thing as society” as Thatcher said.

Another is the idea that the economy is stable and self-correcting. This is the main story of economics from Adam Smith’s invisible hand, to modern equilibrium models.

A third is the idea that the economy is rational and efficient. The assumption that people make rational decisions to optimise utility is connected to the ideas of stability and efficiency. Corporations are defined to be an incarnation of rational economic man.

And then consistent with these is the next economyth which is that the economy is balanced and symmetrical. The issues of power and distribution are viewed as “soft” and peripheral to the subject, and therefore tend to be ignored. As one economist said in an interview, “economists are not good at what’s fair, right?”

Economists have always insisted that their theories are far more complex than these economyths would suggest, but the reality is that these assumptions have been extremely influential and in particular form the basis of the mathematical models used to simulate the economy and make policy recommendations.

In 2017 I did a revised version of the book which included an additional economyth, which was the idea that the economy boils down to barter. This is the myth which in many ways justifies the others, because it means that money isn’t important. Adam Smith for example focused on the “real” economy of labour and commodities and saw money as a kind of veil or a distraction. Economists since then have treated money as just a metric or an inert medium of exchange, and ignored its confounding properties – its dual real/virtual nature, its ability to entangle people through debt, its inherent instability, its tendency to cluster and create inequality, and its psychoactive effects on the human mind.

The drawbacks of omitting things like money and banks from the model became evident after the financial crisis of 2007, when it turned out – as one central banker explained ten years later – that “In the prevalent macro models, the financial sector was absent, considered to have a remote effect on the real economic activity.” And today, the continuing problems of financial instability, social inequality, and environmental degradation can all be largely traced to the money system, which is unstable, unfair, and is reliant on continuous growth to pay off debt.

One reason I called these ideas economyths is because, like myths, their legacy goes back a long way. We could start with post-war economists like Milton Friedman; or go back further to the neoclassical economists who first tried to establish economics as a kind of social physics in the late nineteenth century; or to Adam Smith, who didn’t try to quantify economics but was inspired by Newton. But I would argue that we can actually go back much further. My reason for saying this is that the ten economyths were based directly on a list of opposites, divided into good and evil, from the philosopher Pythagoras, who believed that the universe was based on number, and who lived at a time when coin money was changing the world of commerce. As he said, “number is all” and money is a way of putting numbers on the world.

In order to understand the narrative appeal of this model, it is interesting to compare it with another model which was very influential for a long time, and also reflected the Pythagorean ideals of symmetry and mathematical elegance, namely the Greek model of the cosmos. This model incorporated two main assumptions. The first was that the celestial bodies moved in circles, which were considered the most perfect and symmetrical of forms. The other assumption was that the circles were centered on the Earth. In Aristotle’s version, the planets and stars were thought to be encased in crystalline spheres which rotated around us at different speeds. The fact that planets did not follow perfect circles around the Earth, but sometimes tended to loop back on themselves, was handled by adding epicycles – circles around circles.

This geocentric model was complemented by a theory of physics. According to Aristotle, all matter consisted of the five elements Earth, Water, Air, Fire, and Ether which was reserved for the heavens. His theory was less a theory of motion, than a theory of stability: each element sought its own level, following the same order with Earth on the bottom and Ether on top, and in fact would do so instantaneously in a vacuum. Aristotle deduced from this that a vacuum could not exist: nature abhors a vacuum.

The Greek model lasted for well over a thousand years – it was adopted by the Church, and was not finally overturned until the Renaissance. How did it manage to last for such a long time? And what lessons are there for economics?

One reason for its durability was that it could make accurate predictions of important events such as eclipses. Another reason, though, is related to aesthetics and the fact that, as Aristotle put it, man is a political animal. There was a strong parallel between the perceived order of the cosmos and the order of society. Greek society was structured as a well-ordered hierarchy, with slaves at the base, followed in ascending order by ex-slaves, foreigners, artisans, and finally the land-owning, non-working upper class. These men alone could be citizens, and oversaw everything from above, like the stars in the firmament (women did not take part in political life and took their social class from their male partner). A model of the universe which suggested that everything has its natural place in a beautiful, geometrically-governed cosmic scheme therefore supported the status quo. For this reason it would certainly have appealed to the male leisure class that ruled ancient Athens, and later to the Catholic Church.

The first cracks in the model appeared in 1543, when Copernicus proposed that the Earth might go around the Sun, rather than vice versa. European astronomers began to observe comets which passed between the planets, so if Aristotle’s crystalline spheres had actually existed, they would have broken through them. Finally, in the late seventeenth century, Isaac Newton derived his three laws of motion and the law of gravity. The static circles of classical geometry were replaced with dynamical equations, which had a different but equally powerful aesthetic appeal. Sometimes it takes a model to defeat a model.

So what does this have to do with economics? Well, there is an obvious parallel between the Greek circle model, and our modern model of the economy, because it too pictures the world as rational, stable, ordered, and efficient, and therefore favours the elite. Indeed mainstream economics, in its obsession with rationality and efficiency, sometimes sounds like the PR wing of the financial sector. The model can’t make predictions of things like crises, or indeed the economy in general, but it goes a step further by predicting that it can’t predict, as per efficient market theory. The main difference is that while Aristotle thought that a vacuum could not exist, because otherwise things would find their equilibrium immediately, efficient market theory assumes that prices do reach equilibrium instantaneously – so a vacuum does exist, and it is the market.

Both the Greek circle model and the economics model also show the strength of a mathematical model. Even something like the financial crisis only made a dent. As Paul Krugman wrote in 2018, “Neither the financial crisis nor the Great Recession that followed required a rethinking of basic ideas.” So like the Greek model, the orthodox model of the economy is incredibly resilient.

I think if there was a specific point where this narrative became compelling in economics, it was when economists changed their theory of value, while leaving the rest of the theory mostly the same. The switch was announced 150 years ago by William Stanley Jevons, in the second paragraph of his 1871 Theory of Political Economy, where he wrote that “Repeated reflection and inquiry have led me to the somewhat novel opinion that value depends entirely upon utility.” Here utility was a kind of energy-like quality which roughly equated to happiness. Classical economists such as Smith had followed a labour theory of value which acknowledged the role of power. Utility created a new narrative which flipped this on its head. Economics was now about pleasure and good times.

Of course utility couldn’t be measured directly. Another approach though was to simply assume that utility is reflected by price. Or as Jevons put it: “just as we measure gravity by its effects in the motion of a pendulum, so we may estimate the equality or inequality of feelings by the decisions of the human mind. The will is our pendulum, and its oscillations are minutely registered in the price lists of the markets.”

A side-effect of this emphasis on subjective utility, ironically, was that by reducing value to a number, subjective things like emotion or social power or dignity or ethics were usurped by the theory and thus stripped of all weight. As Tomáš Sedláček wrote for example in his 2011 book The Economics of Good and Evil, “The issue of good and evil was dominant in classical debates, yet today it is almost heretical to even talk about it.” Another thing which also of course didn’t fit into this rational utility approach – it doesn’t compute – was money, with its economyth-defying properties of entanglement, instability, and irrationality.

But still, why is it that more than a decade after the crisis economics still remains rooted in the past? Why is the orthodox narrative so resistant to change?

One reason again is that as with Aristotelian physics, the economyths form part of a connected structure. The story they tell is consistent and self-reinforcing – markets are efficient, stable, rational – and you can’t change one part without changing them all.

Instead, what happens is that economists attempt to fold in new ideas from other areas such as complexity or behavioural psychology, without changing the structure too much. Behavioural economics for example has in my view won acceptance exactly because it can be incorporated in this way, and viewed as an epicycle that can be wheeled out for particular situations. Economists also try to add in so-called “frictions” to their equilibrium models, but assume the equilibrium exists in the first place. Paul Krugman again: “We start with rational behaviour and market equilibrium as a baseline, and try to get economic dysfunction by tweaking that baseline at the edges.” A few more epicycles and it will be perfect, the thinking goes.

The story also offers a powerful restoration narrative, because it says that crises and upsets are caused by external events, and economic forces bring the economy back to this imagined equilibrium. It therefore taps into the human desire for order which is important in politics but also in fiction. And the idea that markets are rational and efficient also justifies the powerful position of the financial sector and the wealthy, while at the same time distracting from the workings of power and the role of money.

To change the narrative it isn’t enough to modify details of the model, instead we need to go back as the neoclassicals did to the fundamental idea of value and its relation with price – in other words, the question of how much something is worth, its numerical cost. There are a number of ways of going about this, but I would argue that a good place to start is with the math.

This might seem counter-intuitive. After all, it is something of a cliché to say that economics is too mathematical. As one recent book put it, “today mainstream economics follows a path of great mathematical rigor that . . . does not make much room for other accounts of economic life.” However rigor isn’t useful if you are using the wrong kind of mathematics in the first place. And while Keynes wrote that “practical men … are usually the slaves of some defunct economist” we could also say that those economists are themselves slave to beliefs about number and value that are inherited in large part from a mathematical model.

Orthodox theory for example is based on the core idea that markets are stable and price is a measure of inherent value. It is best represented by the X-shaped figure of supply and demand. This plots supply as a line which increases with price, and demand as a line which decreases with price. The point where they match in an X represents the stable equilibrium where the market clears. This diagram appears in all introductory economics textbooks, but it is also there in the mathematical models used to simulate the economy.

An odd feature of the graph is that price appears on the vertical axis, and is assumed to be determined uniquely and passively by unknown forces of supply and demand which are in balance, so cancel out. While neoclassical economics is often described as Newtonian, it assumes equilibrium and has no real concept of mass or dynamics or force. The problem though is we never observe supply or demand independently, we only observe transactions. Like the crystalline spheres, these are imaginary constructs. Indeed the whole idea of representing a complex dynamic system as the stable intersection of two lines is very strange and is not done in other areas such as biology, where the only things that are at equilibrium are dead.

To make the diagram more scientific, a first step is to flip it around so that price is the independent variable on the horizontal axis. This seems a trivial change but is actually key to the whole story. Instead of utility, we can then plot the propensity curves for the buyer and seller, which represent the probability of transactions as a function of price. The fact that price is somewhat arbitrary and decisions are subject to effects such as context means these curves are inherently probabilistic. The X-shaped lines of supply and demand from the classical diagram are therefore replaced by probabilistic waves which only collapse down to a particular price during a transaction. This model therefore acknowledges that value is a soft and fuzzy concept, while price is a measure which is subject to intrinsic uncertainty.

The next step is to acknowledge that people are not separate atoms who only communicate by bouncing off one another, instead they are entangled beings who talk to each other, and share things like social norms. An example is the prisoner’s dilemma game, another staple from the textbooks: in classical theory everyone defects and rats out on the other person, but experimental results show people choose to collaborate between a third and a half of the time, which suggests a high degree of entanglement.

We also need to address the fact that people make decisions based on a mix of objective and subjective factors which are entangled in the mind and may interfere with one another. And above all we need to include the dynamics of money, which behaves more like a kind of information than a classical physical object.

The correct mathematical framework for this theory has already been developed by a group of radical thinkers over a century ago. Unfortunately this mathematics has until recently been reserved for the esoteric area of subatomic particles. I’m talking of course about the quantum formalism. For our purposes this refers, not to quantum physics, but to a kind of logic and probability which allows for effects such as interference, entanglement, and the idea of a measurement procedure. The word quantum is from the Latin for “how much” which applies naturally to the economy, where prices are measured through transactions. The point is not that there is a direct map between subatomic particles and humans, but that we can use similar mathematical tools to model each, which is a subtle but important distinction.

While quantum ideas have been around for a while, they are only now starting to reach critical mass in the social sciences. This new quantum narrative is under construction, in economics and finance but also in other areas such as psychology and political science. For example the Carnegie Foundation is funding a series of quantum bootcamps for social scientists at Ohio University starting this summer. There is a new anthology coming up this year from Oxford University Press on Quantizing International Relations. Danah Zohar has been bringing quantum principles into management theory for some time. The area which is seeing the most rapid adoption of quantum ideas is finance, because of quantum computing. As far as I know you can’t take a university course in quantum finance, but you can get a job in it right now in financial centers such as Paris, New York, Toronto, and so on. The development of classical computers in the post-war era changed the way we model and think about systems including the economy, and quantum computers – which have entanglement built in – are now doing the same thing.

What counts for the purposes of today’s discussion though is not the math, but the story told by the math. To summarise, the core narrative of mainstream economics is that people behave like classical atoms: hard, independent, stable. The economy can therefore be modelled as an equilibrium system. The main message of quantum economics is that people are entangled: with their own subjective feelings, with other people, with what they read in the news, and above all through the money system. The economy is a complex, dynamic, living system which can be modelled using a mix of techniques, such as ones from complexity science or systems dynamics, so long as they respect the indeterministic and entangling nature of both mind and money; and particularly the ability of the money system to scale up cognitive and financial entanglements to the societal level.

So as a one word description of the new narrative I would choose quantum or maybe entanglement. If you don’t want to do quantum mathematics, which is understandable, it doesn’t matter because what counts is the idea that the economy is best seen as a complex system which is entangled through a mix of financial and social effects. In practical terms this means that all the so-called “soft” ideas such as subjectivity, emotion, social dynamics, power, money, value, fairness and ethics that have been exiled from economics are now back in play. Debtors and creditors are entangled, shareholders and stakeholders are entangled, and we are all entangled with the climate system. Obviously the quantum approach doesn’t have all the answers, but its built-in emphasis on uncertainty can be liberating, and encourages a pluralistic response. Perhaps it is the model which teaches us to sometimes at least let go of models, because they can’t capture enough of the complex reality. And my hope is that the quantum approach and the idea of entanglement resonates with some of the other ideas and narratives discussed today.

Again, the idea that a new narrative should begin with our system of logic and probability may seem strange but history shows that mathematical models have great influence. And sometimes, as mentioned, it takes a model to beat a model. A new narrative which marks the next evolutionary step of capitalism is going to need a new mathematical framework, if only to better define its language, and help to do an audit on what ideas one may inherited, perhaps unconsciously, from the classical model. It is ironic that we live in an age characterised by volatility, uncertainty, complexity, and ambiguity but our economic theory assumes a deterministic state of placid equilibrium. It is therefore well overdue for an upgrade.

Further reading: quantum economics resources

June 16, 2020

Quantum gateway

Filed under: Economics, Quantum — David @ 2:08 pm

Prior to my July 1 online talk at the CQF Institute on quantum economics and finance, I spoke with Daniel Tudball from Wilmott Magazine about how quantum computing can serve as a gateway into the subject – the interview is here.

May 16, 2020

Quantum Economics and Finance – new book out now

Filed under: Books, Economics, Quantum — David @ 9:35 pm

Quantum Economics and Finance: An Applied Mathematics Introduction


The word “quantum” is from the Latin for “how much” and the new book Quantum Economics and Finance shows how it applies to the world of economic transactions. Written in clear and accessible language, the book covers the essential mathematics behind topics including quantum cognition, option pricing, and quantum game theory, and delves into the nuts and bolts of quantum mechanics, the principles of quantum economic modelling, and the basics of quantum computer logic. On the way the reader will learn how quantum interference can be used to model cognitive dissonance, how a quantum walk goes further than a random walk, and how financial entanglement explains the rate of mortgage default. It is aimed at anyone who wants to understand the quantum ideas working their way into economics and finance, without getting drowned in wave equations.

As interest in quantum computing grows, many companies from established banks to startups are looking at ways to perform financial simulations using quantum algorithms. But what if we should be using quantum models anyway – because the monetary system has quantum properties of its own, and because they work?



May 10, 2020

The quantum coin trick

Filed under: Economics, Quantum, Talks — Tags: — David @ 3:17 pm

Upcoming free talk at the CQF Institute on July 1:

Quantum Economics and Finance: The Quantum Coin Trick

Quantum economics and finance uses quantum mathematics to model phenomena including cognition, financial transactions, and the dynamics of money and credit. This talk takes a particular route into the subject, through a discussion of the quantum coin. Unlike a classical coin toss, which can be either heads or tails, a quantum coin can be – like Schrödinger’s cat – in a superposition of states. This gives it intriguing properties which can be used to simulate everything from the prisoner’s dilemma, to the credit relationship, to the pricing of options. The talk is based on material from the book Quantum Economics and Finance: An Applied Mathematics Introduction.

Slide 1



May 26, 2019

Quantum entanglement, and the strange case of the missing defaulters

Filed under: Economics, Quantum — David @ 6:11 pm

Related imageAccording to the field of quantum cognition, a decision to act is best expressed as a quantum process, where entangled ideas and feelings combine and interfere in the mind to produce a complex, context-dependent response. While the quantum approach has proved successful at modelling many aspects of human behaviour, it is less clear how relevant this is to the economy. This paper argues that the financial system is characterised by three kinds of entanglement: at the individual level between concepts, at the social level with other people, and at the financial level through the use of credit. These entanglements combine in such a way that cognitive processes at the individual level scale up to affect the economy as a whole, in a manner which is best modelled using quantum techniques. The approach is illustrated by making a retroactive “postdiction” about the prevalence of strategic mortgage default during the financial crisis, and a prediction for future such crises.

Read the discussion paper here.

May 1, 2019

A quantum model of supply and demand

Filed under: Economics, Quantum — David @ 10:25 pm

Here is the abstract for a new paper to be published in Physica A: Statistical Mechanics and its Applications. A draft is available at SSRN.

One of the most iconic and influential graphics in economics is the figure showing supply and demand as two lines sloping in opposite directions, with the point at which they intersect representing the equilibrium price which perfectly balances supply and demand. The figure, which dates back to the nineteenth century, can be seen as a graphical representation of Adam Smith’s invisible hand, which is said to guide prices to their optimal level, and features in nearly every introductory textbook. However this figure suffers from a number of basic drawbacks. One is that it doesn’t express a dynamical view of market forces, so it isn’t clear how prices converge on an equilibrium. Another is that it views supply and demand as deterministic, when in fact they are intrinsically uncertain in nature. This paper addresses these issues by using a quantum framework to model supply and demand as, not a cross, but a probabilistic wave, with an associated entropic force. The approach is used to derive from first principles a technique for modeling asset price changes using a quantum harmonic oscillator, that has been previously used and empirically tested in quantum finance. The method is demonstrated for a simple system, and applications in other areas of economics are discussed.


Read the full paper here.

March 29, 2019

Talk for Quantizing IR panel at ISA 2019

Filed under: Economics, Physics, Quantum, Talks — Tags: — David @ 6:49 pm

This is an edited version of my contribution to the panel discussion “Quantizing IR I: Physicists, Meet Social Theorists!” at the International Studies Association conference in Toronto on March 29. The session was chaired by James Der Derian (University of Sydney) and the other participants were Alexander Wendt (Ohio State University), Shohini Ghose (Wilfrid Laurier University/Perimeter Institute), Kathryn Schaffer (School of the Art Institute of Chicago), Michael Schnabel (University of Chicago), and Genco Guralp (San Diego State University).

One nice thing about quantum is that it looks different to people who come at it from different backgrounds and take different paths. My background is in applied math, and my own interest in applying quantum methods to social questions came about several years ago when I was researching a book on the history of money. And I think money serves as a particularly simple and illustrative example of a quantum social phenomenon, so I will give a quick description of that before getting on to more general points.

The word “quantum” is Latin for “how much” and the money system is a way of answering that question – or “quanto costa” in Italian which makes the quantum connection more clear – which when you think about it is not an obvious thing to do since value is a quality not a quantity. Because of this fundamental incompatibility at its core, and because it is related to the transfer of information rather than of classical objects, the money system turns out to have its own quantum properties including indeterminacy, duality, entanglement, and so on.

The most trivial of these is that money moves discontinuously, in sudden quantum leaps. Schrödinger once said “If we have to go on with these damned quantum jumps, then I’m sorry that I ever got involved” but with money the same thing happens every time you tap your card at a store. There isn’t a little needle that shows the money draining out, it just jumps.

Money is fundamentally dualistic, presenting both as a physical object, like a coin, or a virtual object, like a credit transaction, while still retaining the properties of each. A bitcoin for example is a virtual currency but it exists on a real hard drive.

The money system is indeterminate: the price of something like a home is fundamentally uncertain and is only settled at the time of purchase. Transactions therefore act as a measurement process on value.

Money is entangling: here entanglement means indeterminate but coupled at the same time, an example being the state of a loan between two parties. We can model each participant as being in a quantum superposition of cognitive states. If the borrower defaults that acts as a measurement which collapses the state of the loan, even if the other party only finds out later. Of course this is a simpler version of entanglement than the kind seen in physics (there is only one axis, namely default or no default), but an advantage is that you don’t need sophisticated statistical experiments to tease it out.

So money does not behave classically which is one reason it has traditionally played a small or even non-existent role in economic models, which as many commentators have noted is one reason the financial crisis wasn’t predicted. Conventional models didn’t include a banking sector, let alone the financial entanglements represented by a quadrillion dollars worth of derivatives. The fact that money was left out of the picture seems a remarkable omission given its obvious importance not just to economics, but to everything from marital relations to international relations, but fits with the classical view that money is just an inert medium of exchange.

It was only later, through the work of people like Alexander Wendt, that I connected this with the broader areas of quantum cognition and quantum social science, which of course add many completely new dimensions. One way to think of money is as a kind of prosthetic that extends the dualistic properties of quantum mind, as we mentally collapse value down to price.

I agree with the idea that quantum processes are likely to play a role in human cognition. In itself though I don’t think that this will show or prove that society is quantum or that we are best seen as wave functions, because living systems can’t be reduced to their components. Quantum processes are believed to play a role in avian navigation, but this doesn’t really change the way we think about birds. At a trivial level, we are quantum because the universe is quantum, but what counts is the emergent behaviour. The question from my perspective is whether social systems can be usefully modelled as quantum.

The quantum methodology allows scientists to model things like indeterminacy, interference between incompatible concepts, and entanglement, all of which characterise human relations. Bohr’s theory of complementarity for example was inspired by psychology, and our ability to hold two incompatible ideas in the mind at the same time. This position is a little different from the physicalist argument, because it says that something like money is quantum not because it inherits these properties from subatomic particles, but because it is a quantum system in its own right, and I would argue this holds for other social institutions as well.

In this sense I’m not sure that economics needs a model of consciousness, but the problem is that it already seems to have one, which is that people are lifeless automata. This approach is epitomised by the old and, as economist Julie Nelson points out, rather gendered concept of rational economic man, who makes decisions to optimise his own utility based on preferences that are fixed and known. It would be a great improvement to shift to the connected, fluid, and indeterminate idea of quantum economic person.

It might still sound here that quantum is being used as just a metaphor. But the idea of a metaphor is to explain something complex in terms of something that is concrete and immediate, as in “all the world’s a stage”. A wave function is many things but it is not concrete or immediate. For one thing it involves imaginary numbers, and we have no idea what wave function collapse means, with many different interpretations, which is pretty humbling as Kathryn Schaffer noted.

Instead it makes more sense to go the other way, take human experience at face value, and use it as a metaphor to understand the physical world. One reason we don’t do that obviously is because quantum was applied to physics first.

We also need to distinguish between the system and the model of the system. Quantum models are not reality, even if they appear to give exact results in calculations. Instead they are tools adapted from mathematical techniques.

The original quantum idea came from someone, Max Planck, trying to fit a mathematical model to some strange looking data, for black-body radiation. One of the main tools in quantum mechanics is Hilbert space which is a generalisation of normal Euclidean space. This was developed in the early twentieth century and later used by physicists to formalize quantum mechanics. In general, quantizing a system is a fairly clunky mathematical procedure which converts classical equations into quantum ones.
These mathematical methods were adopted in physics not because anyone liked them but because they worked. Social scientists are permitted to do the same thing (models can be applied to different systems and at different scales).

For example if someone decides to quantize some aspect of the economy, the hope is that the resulting model captures the essence of the underlying system, addresses shortcomings in traditional models, is parsimonious in terms of parameters, and can make useful predictions. Such decisions are the prerogative of the modeller and should be made based on sound principles of mathematical modelling.

Of course there has been a lot of pushback from both physicists and economists to this use of the quantum approach, which is all to the good, but I can address a couple of points that often come up. One common objection is that quantum processes don’t scale up (Bohr’s correspondence principle), so what happens at the micro level doesn’t affect us at the macro level because it all washes out. But quantum processes do scale up, especially through the use of technology. In physics we have the atom bomb (which doesn’t wash out), or a laser pointer for that matter, in society we have the financial system which can be viewed again as a kind of quantum social technology that can be used for good or ill.

Another objection is that quantum social science is the ultimate example of “physics envy” – and there is some danger of that. But as someone pointed at another one of these events, that horse already left the barn. An example is the efficient market hypothesis, which is a central theory of economics and finance. The idea that market prices follow a random walk, and the emphasis on probability, was directly inspired by quantum theory, and was developed in part by the many nuclear physicists who switched to finance after the war. However this was a sanitised version of quantum that picked up on stochasticity but omitted its other features; and the theory was widely misused to justify the financial instruments which played a key role in the crisis.

While there is no shortage of model abuse in economics, physics envy is not the main problem. Instead it is institutional pressures that encourage the use of models that look good based on the aesthetic criteria of mechanistic science but have so many parameters and moving parts that they can give any answer you want. This is what economist Paul Romer called in a paper “the trouble with economics”. And in fact you see exactly the same issue in physics – Romer’s paper was named after Lee Smolin’s book The Trouble With Physics.

All this raises a couple of questions. One is that if something like the money system can be described as quantum, then does that tell us something useful about how we should interpret physics, for example the role of information (no idea). Perhaps more relevant from a sociological perspective at least is, why has it taken a century for these ideas and methods to feed into the social sciences.

Of course there is the worry that quantum ideas can be dangerous nonsense when applied outside physics. As physicist Sean Carroll wrote in 2016, “No theory in the history of science has been more misused and abused by cranks and charlatans”. However I would argue that the theory misused to the greatest effect in social science is the idea that we are like classical machines: inert automata, slave to the mechanisms of cause and effect. This has done far more damage than anything like “quantum healing”. Why is there so much controversy about social scientists possibly abusing quantum models, when they have been abusing mechanistic models for years?

One reason is that quantum ideas represented a challenge to our traditional scientific world view, and in some ways we still haven’t got used to them. Einstein for example said quantum reality reminded him of “the system of delusions of an exceedingly intelligent paranoiac, concocted of incoherent elements of thoughts”. Yet these concepts such as duality, indeterminacy, and entanglement seem quite reasonable when applied to our own thought patterns. And I think Einstein’s comments would apply quite well to the social world of finance.

As the Torontonian Marshall McLuhan wrote in 1992: “I do not think that philosophers in general have yet come to terms with this declaration from quantum physics: the days of the Universe as Mechanism are over”. So it is exciting that 25 years later that is starting to change.

January 1, 2019

Quantum social science – reality or metaphor?

Filed under: Economics, Quantum — David @ 8:46 pm

Quantum social science exploits ideas and methods from quantum physics in order to model and understand social behaviour. For example, quantum cognition models human decisions as the collapse of a kind of mental wave function to a particular state, in a process akin to the wave function collapse in physics. But is this social wave function an actual physical thing, or just a metaphor?

Most researchers in quantum cognition adopt the stance that quantum techniques just offer a more flexible toolbox for analysing things like interference between incompatible concepts, or social entanglement between people, while distancing themselves from the idea that the brain is actually based on quantum processes. This is obviously prudent from a strategic perspective, since – while quantum biology has revealed a role for quantum effects in things like avian migration or photosynthesis – they have yet to be detected in the brain. And any assertion that humans are actually quantum entities tend to be met with extreme skepticism (even though comparing them with mechanistic entities, as is customary in the social sciences, gets a pass). It also reflects the “shut up and calculate” approach that is common in physics. In this view, quantum physics is therefore just a metaphor for human behaviour.

A few social scientists however do point out that, just because we haven’t yet detected quantum effects in the brain, doesn’t mean they aren’t there; that it seems reasonable that, if bird brains exploit quantum effects to get around, our own brains might make use of them too; and that areas such as quantum cognition provide circumstantial evidence for the radical notion that we are what Alexander Wendt calls “walking wave functions.” In other words, quantum social science is based not on a metaphor, but on physical reality.

Now, one might think that this question can only be settled by physical proof. Either experiments will eventually show that our brains are quantum, or they won’t. However, as with all things quantum, I think the real answer is more complex.

To start with, if quantum physics is being used as a metaphor, it isn’t a very good one.The usual purpose of a metaphor is to explain something that is difficult or abstract in terms of something that is more simple and concrete. When Shakespeare had an actor read “All the world’s a stage” in As You Like It, he was comparing the vastly complex world to a wooden platform on which the actor was actually standing. In quantum physics, we might think of a wave function as real because it can be expressed using mathematical equations, at least for the most simplified of situations. But no one has actually seen or felt an electron’s wave function (for one thing, it involves imaginary numbers). And one of the major drawbacks of the Copenhagen interpretation is that there is no explanation for how a wave function collapses during a measurement.

It would therefore make more sense to explain the quantum world using human behaviour as a metaphor, then the other way round (it is easier to relate to the experience of a state of uncertainty collapsing to a particular decision, than it is to an electron’s wave function collapsing to a particular eigenvector). But we don’t do that because quantum physics was discovered first. And this raises another question, which is why – given its similarities with human behaviour – quantum physics is usually described as being somehow alien.

Einstein for example said quantum reality reminded him of “the system of delusions of an exceedingly intelligent paranoiac, concocted of incoherent elements of thoughts.” Physicist Steven Weinberg said in an interview that “quantum mechanics, although not inconsistent, has a number of features we find repulsive … What I don’t like about quantum mechanics is that it’s a formalism for calculating probabilities that human beings get when they make certain interventions in nature that we call experiments. And a theory should not refer to human beings in its postulates.” Yet concepts such as duality, indeterminacy, and entanglement seem quite reasonable when applied to our own thought patterns. And consciousness is of course one thing that all human beings have direct personal experience of, no physics course required.

So it doesn’t seem right to say that quantum physics is a metaphor for human behaviour, given that we know less about the former than the latter. But another problem with the metaphor vs physical reality question is that physical proof of quantum processes in the brain would not directly show that our mental processes are best understood as quantum. In the end, everything in the physical world, including our brains, is based on quantum reality at the level of subatomic particles – so in a trivial way we are quantum creatures. But a common argument directed against quantum social science is Bohr’s principle of correspondence, which states that these effects wash out at large scales. So even if quantum effects were shown to play a role in the brain, this wouldn’t in itself indicate that social behaviour is a quantum phenomenon.

Now, while the correspondence argument makes sense for many phenomena, it ignores the fact that quantum effects do scale up all the time, because we design them to. Quantum technologies include everything from lasers to semiconductors to atom bombs. If we can learn to exploit quantum effects to build devices, why shouldn’t eons of evolution accomplish the same thing? Futhermore, quantum behaviour can also appear at large scales in things like phonons – sound waves in crystals or metal bars which appear as discrete quasi-particles and have their own quantum properties. So demonstrating that the brain is quantum would not prove that social behaviour is quantum. And conversely, proving that the brain is based on mechanistic interactions wouldn’t in itself prove that social interactions are not best modelled as quantum phenomena.

This is seen clearly in quantum economics, where money has its own dualistic properties because it merges the incompatible concepts of number and value, and prices are best seen as an emergent property of the money system. As physicist Robert Laughlin notes, “physical law is a rule of collective behavior, it is a consequence of more primitive rules of behavior underneath (although it need not have been), and it gives one predictive power over a limited range of circumstances. Outside this range, it becomes irrelevant.” Quantum behaviour at the level of the money system is not the same as quantum physics at the subatomic level; so while one can make a convincing argument that the brain is probably based on quantum processes, and evidence that this is the case would certainly change the conversation around quantum effects in the social sciences, it isn’t necessary or even apposite in economics to try and draw a direct connection between the two (proof that neurons are quantum wouldn’t prove that dollars are quantum). Instead, each should be handled on its own terms.

One answer to the question of interpretation, then, is to say that we can usefully model society as if it were a quantum system; while at the same time remembering that any mathematical model is only a sketch of the real thing. This might seem like a kind of intellectual dodge – the social sciences version of “shut up and calculate” – but in fact it is the standard practice in mathematical modeling: we model reality as if it obeyed our quantum rules too, even though we know the theory has limitations, which is why physicists continue to work on new ones. And when Max Planck first proposed the idea of the quantum, he didn’t do it in order to make a profound point about the ontology of the universe, he did it because it worked.

In this view, rather than quantum physics being a metaphor for human behaviour, it is more accurate to say that quantum models are a kind of metaphor for both physics and society. And the fact that these have something in common might be telling us something interesting about the nature of both.


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