Groucho Marxism

Questions and answers on socialism, Marxism, and related topics

  • Why aren’t we investing in renewables?

    The urgency of the task we face in combating climate change should now be clear to everyone. In one sense, implementing the measures required to achieve this is straightforward. The key element required is a rapid conversion to renewable energy across all sectors of the economy. We could begin this immediately as the technology already exists. Indeed, if you listened to our politicians you would be forgiven for thinking this task is already well underway, as they have a habit of exaggerating their achievements when it comes to combating climate change. For example, in 2024, Prime Minister Rishi Sunak (remember him?!) claimed that Britain had the best record in the world for cutting greenhouse gas emissions. But is that really true?

    The short answer is: no. Britain has been effectively ‘exporting’ its emissions, mainly to China, for the past 30 years. If the greenhouse gas emissions from the manufacture of imported products are included – as they really should be – the figures look very different, as many of the most polluting industries have moved offshore from Britain in recent decades. The result is that overseas emissions increased from 200 million tonnes CO2 equivalent in 1990 to 400 million tonnes CO2 equivalent in 2020. All major Western countries have followed a similar strategy in off-shoring emissions. Even taking overseas emissions into account, however, Britain has still reduced its overall greenhouse gas emissions over the past 30 years. So shouldn’t our politicians get some credit for that at least?

    Again, the short answer is: no. The main driver for the reduction in greenhouse gas emissions was the switch from coal-fired to gas-fired power stations. This was done for a political reason – namely, smashing the miners’ unions – rather than an environmental one. And of course natural gas is a fossil fuel so the burning of natural gas still emits greenhouse gases, just not to quite the same extent as coal. Another driver in the reduction of greenhouse gas emissions has been the widespread installation of wind farms, with Britain now deriving around 30% of its power from wind. But this figure is a lot lower than it could be. The equivalent figure for Denmark is 60%, which undercuts the frequently-made claim that there is no point in investing further in wind power due to its intermittency.

    A recent study by Ernst & Young found that Britain has huge potential for future offshore wind energy generation, possibly as much as 1000 TWh per year, equivalent to several times our average energy consumption. And that’s just offshore generation. Other options are onshore wind farms and micro-generation, whereby wind turbines are installed on the roofs of buildings. The problem with the latter option is that it is less efficient than grouping turbines into wind farms because economies of scale are lost. Plus many people would not want to have a big wind turbine sticking out of their roof. But even if we discount micro-generation completely, it is clear that a combination of offshore and onshore wind farms would be more than sufficient to meet our energy needs.

    The fact that many wind farms are already operational shows that there are no major technical barriers to connecting them with the national grid. The intermittency problem can be addressed by developing ways to store large quantities of energy. Fitting each wind turbine with a lithium-based battery would help even out energy availability. There are other possibilities too, such as storing excess energy by hauling rail wagons uphill, then releasing the energy when there is an energy deficit by allowing them to roll back downhill. Another solution is to deploy back-up sources such as hydro-power or green hydrogen (nuclear power is often mentioned as a potential back-up source too, although I argued against that in a previous blog post). AI can also be used to help balance supply and demand.

    One reason often given by the government to explain why more onshore wind farms have not been built is objections from local residents. But this obviously does not apply to offshore wind farms. In short, all the excuses made by politicians for not building more wind farms, or investing in renewable energy more generally, are just that: excuses. These excuses make little sense when renewables provide a cheap, clean source of energy that can easily meet our energy needs. So what’s really going on here? The real reason the government is so reluctant to invest in renewable energy sources is not that they aren’t effective: on the contrary, it is that they are too effective! To understand why this is, we need to understand something fundamental about how capitalism works.

    In order to make money under capitalism, you need to control something that other people need but which is in short supply. This explains why it is so easy for capitalists to make money out of fossil fuels. Capitalist society has been set up in such a way that everyone needs access to these fuels, but they are difficult to get hold of as they are buried deep underground. If a cheap source of energy became freely available it would be much more difficult for capitalists to control the supply and thereby make money out of it. This explains why capitalists and their politician lackeys are so reluctant to invest in renewable energy. Instead, they resort to lying about their achievements in cutting emissions to deflect attention away from their lack of action. We will never solve the climate crisis until we get rid of capitalism.

  • ADHD is not a myth

    Channel 4 courted controversy recently by airing a documentary entitled The Great ADHD Myth?, which centres on the struggles of a stereotypically hyperactive schoolboy named Mason. The show went to great lengths to stress that many people exhibit similar behaviour and questioned the wisdom of diagnosing such people with Attention Deficit Hyperactivity Disorder, or ADHD. The program is presented as being concerned only with uncovering the facts. In reality, however, it sets out to demonstrate that ADHD is not a genuine neurological disorder but is instead an invented social construct. Much of the program is taken up with ‘experiments’ designed to prove this conclusion which was clearly arrived at in advance.

    It has since transpired that the words of one of the experts interviewed were, in her words, “cherry picked… and presented… out of context to make them fit into their narrative”. At one point the presenter, Max Pemberton, even resorts to taking ADHD medication himself. Quite what he was trying to prove with this little stunt is not entirely clear. Despite these obvious defects I suspect that the show’s central thesis would have been music to the ears of many. There seems to be a perception amongst a significant proportion of the population that neurological disorders like ADHD are essentially made up. This view was recently epitomized by ‘comedian’ Ricky Gervais, who went viral for mocking current trends around diagnoses for ADHD.

    What does the science say? A comment in a recent edition of New Scientist helpfully debunks many of the claims put forward in the show. One claim made is that ADHD is a new condition, which if true would suggest that it is in fact a social construct. But documented descriptions of what we would now call ADHD can be found back as far as 1798, and the condition has been medically described many times since then under various different names. Whilst it is true that rates of diagnosis have increased rapidly in recent years, just 1% of the population of England is currently diagnosed with ADHD. Another claim made is that ADHD cannot be a real phenomenon as it cannot be diagnosed using a brain scan. But this is true of most, if not all, psychiatric and neurological conditions.

    Pemberton seeks out a diagnosis for ADHD via an online assessment, presumably to demonstrate how easy it is to be diagnosed (Pemberton does not actually have ADHD). But in reality, ADHD is diagnosed via a lengthy process that includes multiple questionnaires and an interview. Having apparently been diagnosed, Pemberton is then handed a prescription for medication, the implication being that it is far too easy for people to get hold of ADHD medication even if they do not actually have the condition. There is certainly an argument that we need more data on the effects of long-term medication use. But what we do have is evidence that ADHD treatments – drug-based or otherwise – improve quality of life for those suffering with the condition.

    The program in fact demonstrates this by taking Mason off his medication, only to find that his schoolwork and behaviour decline to the point that his parents are forced to put him back on the treatment. This effectively blows a hole right through the show’s central argument (they don’t mention that of course). It seems that in making this program, Channel 4 chose shock tactics over scientifically accurate reporting, presumably because they thought it would boost their ratings. They must have known that the show would strike a cord with a certain section of the population. Why are some people so unwilling to accept the reality ADHD? I think it is because its core traits look like personal choices rather than a medical condition.

    Western capitalist society places a high value on hard work and self-control. When people succeed or fail, others tend to blame only personal effort. But as I have pointed out in previous blog posts, this idea is based on the fallacy that we have free will. In truth, all of our choices and actions are ultimately predetermined by physical processes. Nonetheless, societal emphasis on productivity, individual willpower, and self-sufficiency significantly hinders the acceptance of ADHD as a legitimate disability. Of course everyone gets distracted or feels restless at times. The problem is that sceptics will tend to use their own mild experiences to dismiss the disorder, ignoring the fact that ADHD is the result of hidden differences inside the brain.

    Because people cannot physically see an external wound, they often misinterpret real neurological struggles as bad behaviour. Forgetfulness is judged as disrespect; fidgeting is viewed as rudeness; procrastination is labelled as laziness. We humans are often quick to judge others. The reason for that, I think, stems from our egos, which are always trying to exert their superiority over others’ egos. This makes it easy for us to pass judgement; conversely, displaying compassion and understanding requires overcoming your ego, which takes effort. It also requires giving up pre-existing beliefs, which can feel jarring. Some people clearly find it easier to reject ADHD rather than give up long-held beliefs on fairness and opportunity.

    The strange thing is that many of those who reject ADHD as a real neurological phenomenon would consider themselves rational thinkers. Ricky Gervais, for example, is always going on about the superiority of science over religion, yet when it comes to ADHD his love for science apparently goes out of the window. As I have pointed out in a previous blog post, Gervais is clearly a man with an overly inflated sense of his own intelligence and importance. He is also a man with an unfortunate habit of ‘punching down’ with his comedy. Gervais’ recent statements on ADHD have nothing to do with science or rationality and instead serve as an object lesson on what can happen if you allow your ego to take control.

  • Linear and dynamic programming

    Linear programming, also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships. It is widely used in mathematics and, to a lesser extent, in business, economics, and engineering. More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. The canonical linear programming problem can be expressed as: find y ≥ 0 that maximizes the quantity bTy subject to the constraints Cy ≤ a, where y is an n by 1 vector, b is a known n by 1 vector, C is a known m by n matrix, and a is a known m by 1 vector.

    Every linear programming problem can be converted into a so-called dual problem, which can be expressed as: find x ≥ 0 that minimizes the quantity xTa subject to the constraints xC ≥ b, where x is an m by 1 vector, and a, b, and C are defined as above. The strong duality theorem of linear programming states that if y* is a solution to the original (primal) problem and x* is a solution to the dual problem, then bTy* = x*Ta. This result can be used to prove the well-known minimax theorem from game theory, which states that if C is an m by n matrix then maxp minq pTCq = minq maxp pTCq, where maxp is taken over m by 1 vectors p ≥ 0 satisfying  ∑i p(i) = 1 and minq over n by 1 vectors q ≥ 0 satisfying  ∑j q(j) = 1.

    Dynamic programming, also called dynamic optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model which can be broken down into stages or time steps. The simplest dynamic programming problem can be expressed as: find the sequence (u0,u1,u2,…) which minimizes v(i0) = ∑k g(ik,uk) subject to the constraints ik+1 = f(ik,uk) for each k, where f and g are known functions. The interpretation is that ik represents the state of some system at time step k which is can be controlled using the inputs (u0,u1,u2,…). Let v* be such that v*(i) attains the minimum for each i. Then it can be shown that v* satisfies the so-called Bellman equation, named after the American mathematician Richard Bellman: v*(i) = minu {g(i,u)+ v*[f(i,u)]}.

    This formulation can easily to be extended to stochastic systems, where instead of a transition function f we have a transition probability distribution P. In this extended formulation, P(i,u,j) represents the probability that the next state of the system is j given that the current state is i and the control input is u. The Bellman equation then becomes v*(i) = minu{g(i,u)+ ∑j P(i,u,j)v*(j)}, which can be written in a more compact notation as v* = Tv*, where Tv(i) = minu{g(i,u)+ ∑j P(i,u,j)v(j)}. It is straightforward to show that for all vectors v, if v ≤ Tv then v ≤ v*. It follows that if v is the maximum vector satisfying v ≤ Tv then v = v*. Thus the dynamic programming problem just described can be reformulated as the problem of maximizing v subject to the constraints v ≤ Tv.

    This reformulated problem is not a linear program as the operator T is nonlinear, but we can turn it into a linear program by rewriting v ≤ Tv as v ≤ g(u)+P(u)v for each u, where g(u) is the vector whose ith component is g(i,u) and P(u) is the matrix whose (i,j)th component is P(i,u,j). This may be rewritten as [I-P(u)]v ≤ g(u) for all u, where I is the identity matrix. To convert this into the canonical linear programming form described above, we can define n to be number of possible states, m to be the total number of states multiplied by the total number of possible control inputs, a to be the m by 1 vector whose [i+n(u-1)]th element is g(i,u), b to be an n by 1 vector of 1s, and C to be the m by n matrix whose [i+n(u-1),j]th element is P(i,u,j).

    This demonstrates that a dynamic programming problem can be reformulated as a linear programming problem, thereby providing a link between two seemingly unconnected types of mathematical optimization.

  • AI is coming for your job

    The term ‘AI apocalypse’ is used to describe any of a number of hypothetical futures where artificial intelligence causes catastrophic harm to the human race. These futures can be broadly classified into four scenarios. The first involves mass automation replacing vast portions of the global workforce, leading to severe economic instability. The second involves large-scale automated cyber attacks targeting critical infrastructure like water and hospitals. The third involves the proliferation of autonomous weapons.And the fourth involves super intelligent machines developing goals that do not align with human survival or well-being. In this blog post I will go through the first of these scenarios and assess how worried we should be about it (I will leave the other scenarios for a future blog post).

    AI is not developed enough to take over entire jobs right now, but it is already automating routine tasks and drastically reshaping how people work. Companies are increasingly using AI to handle repetitive tasks, leading to hiring freezes and fewer entry-level openings. This means that young people are being hit hardest by the AI boom and explains why there are so many NEETs: young people aged 16 to 24 who are not in education, employment, or training. The jobs most immediately at risk are customer support, entry-level coding, and administrative roles. Other jobs will be at risk in the medium-term as AI becomes more developed. Conversely, those least at risk are skilled trades, those involving human empathy, and those involving complex strategy.

    In theory, AI should free humans up from routine tasks and allow us to focus on higher-level work. Indeed many commentators argue that this is precisely what will happen. Work published recently by Boston Consulting Group, for example, argues that AI will “reshape more jobs than it replaces”. Dig a bit deeper though and things don’t seem quite so rosy. Despite the optimistic headline, the research suggests that 15% of jobs in the US could be eliminated in the next five years, a pattern that will presumably be replicated across the Western world. Will sufficiently many jobs be created in the meantime to compensate for this decline? It seems unlikely to me. And that’s just in the next 5 years. It’s very difficult to predict what will happen beyond that, other than that even more jobs will be replaced.

    It is clear therefore that in the not-too-distant future, AI will eliminate a significant proportion of jobs that are currently being done by human beings. So what are governments doing to prepare for this? Sadly the short answer is: very little. This is despite the fact that existing safety nets are woefully ill-equipped to cope with what is coming. Traditional unemployment insurance programs often require a specific work history, which can exclude new graduates and entry-level workers most vulnerable to early AI displacement. Unfortunately our politicians are asleep at the wheel and it is left to tech moguls like Bill Gates to sound the alarm. In a recent essay, Gates has warned that there is no plan for the upheaval AI will create.

    So what should governments be doing to prepare? The two obvious solutions are a Universal Basic Income (UBI) and a Job Guarantee (JG). A UBI is a social welfare model where every citizen regularly receives a set, unconditional amount of cash from the government, regardless of their work status or personal wealth. A JG, on the other hand, is an economic policy proposal that aims to create full employment and by having the state promise to hire unemployed workers as an employer of last resort. There are arguments for and against both of these policy options which I won’t go into here (I will leave that to a future blog post too). But either would be better than the alternative: mass unemployment with no safety net. Unfortunately no politicians seem willing to even discuss these proposals at present.

    Thus we find ourselves in the absurd situation where we are on the cusp of developing a technology that could completely free us from drudgery and allow us to focus on higher-level tasks; but instead of feeling excited about this, the overall feeling people have is one of fear and trepidation about the job losses that will result. If there is anything that demonstrates what a perverse system capitalism is this, it is this. Defenders of the status quo would argue that if it wasn’t for capitalism AI would not have been developed in the first place. That may or may not be true (personally I think it would have been developed sooner under socialism). But even if you accept this premise, you have to ask what the point of developing this technology is if it just ends up making peoples’ lives worse.

  • Zionism and racism

    Sacha Baron-Cohen is facing growing outrage over his new Ali G film, with some alleging that the pro-Israel actor and ‘comedian’ has built his career on racist mockery of Muslims and other marginalized groups. Many have taken to social media to accuse Ali G, who in the trailer is seen wearing dreadlocks and a Rastafarian hat, of being an outdated and racist caricature. Anti-racism activists have apparently sent over 37,000 emails to cinemas urging them not to screen the film. Much of the outrage has focussed on concern over a white comedian engaging in a modern form of ‘blackface’: dark makeup worn to mimic and caricature black people, historically used by white entertainers to promote racist stereotypes.

    It is not the first time that a prominent British pro-Israel celebrity has engaged in such behaviour. Back in the 1990s, ‘comedian’ David Baddiel infamously wore blackface to mock Black footballer Jason Lee in the television show Fantasy Football League. Baddiel and his sidekick Frank Skinner repeatedly targeted the Nottingham Forest striker across several episodes of the show, with Baddiel often wearing a pineapple on his head to mimic Lee’s dreadlocks. Lee later stated that the racist caricature made him feel violated and added a severe layer of trauma to the abuse he already faced. Some 25 years after the show aired, Baddiel met Lee in person to make a long overdue apology, admitting that the sketches constituted a targeted bullying campaign.

    Whilst Baddiel’s apology was a positive step, the suspicion among many was that he had only made it in order to help sell his book Jews Don’t Count, which was released around the same time. The book’s core thesis is that progressive anti-racists frequently overlook, downplay, or ignore judaeophobia, treating it as a second-class racism. (I use the term ‘judaeophobia’ rather than the more commonly used ‘antisemitism’ to refer to bigotry against Jews, for reasons outlined in a previous blog post.) In order to put forward this thesis, Baddiel had to position himself as a kind of authority on racism; but it is difficult to do that when you have a history racist bullying. This, I think, is the real reason behind Baddiel’s attempt to clear the air with Lee.

    As disturbing as Baddiel’s racism may have been, it almost pales into insignificance when compared to the bigotry of Baron-Cohen, whose ‘comedy’ invariably involves a privileged, Cambridge-educated man performing caricatures of marginalized groups. First there was Ali G, a caricature of Black street culture. Then came Borat, a fictional and grotesquely exaggerated television reporter from Muslim-majority Kazakhstan. After that came Brüno, a gay fashion designer through which Baron-Cohen demonstrated that he is equally happy mocking LGBTQ+ people. Then finally came the film The Dictator, which stars Baron-Cohen as Admiral General Aladeen, the absurdly brutal ruler of the fictional North African state of Wadiya.

    Baron-Cohen claims to oppose racism, but this makes little sense when his entire shtick involves giving mainstream audiences permission to laugh at characters constructed from the very stereotypes he supposedly opposes. Thankfully it seems that Baron-Cohen is finally facing some push-back – not just because everyone has realized that Ali G is just him doing blackface, but also because of his support for Israel. The trailer for the Ali G film even features an Israeli government talking point, which is that the only state actor currently standing up to Israel – namely, Iran – is ‘dangerous’. (Dangerous to whom?) Baron-Cohen has apparently failed to noticed that the zeitgeist has moved on since Ali G first aired and nobody finds it funny any more. Many are wondering how we ever found it funny in the first place.

    It is notable that both Baron-Cohen and Baddiel are graduates of Habronim Dror, the Zionist youth movement. (Israeli politician Mark Regev is another graduate.) It seems that Zionism and racism invariably go hand-in-hand. Further evidence for this can be seen in the actions of American pro-Israel ‘academic’ Nathan Cofnas, the self-described race realist who was instrumental in the character assassination of the British academic Jason Arday which ultimately led to Arday’s tragic death just a few weeks ago. It is beginning to emerge that, far from being an isolated actor, Cofnas is in fact part of a network of race scientists and eugenicists that has infiltrated top-tier universities. Cambridge University apparently knew about this network as far back as 2021.

    We ought to be extremely concerned about this. Universities play a key role in educating our young people, and it seems they are increasingly becoming centres of racist pseudo-science and propaganda. Or perhaps it has always been this way and we are only just noticing. It is remarkable that in the wake of Arday’s untimely death, many academics have appeared to side with Cofnas, with hundreds signing an open letter defending Cofnas’s ‘academic freedom’ following his suspension by Ghent University. The letter was co-authored by prominent Harvard psychologist Steven Pinker, who claims not to be a Zionist but has defended several Zionist talking points, such as the claim that those who accuse Israel of genocide are committing a blood libel.

    It is no surprise that Zionism and racism go hand-in-hand, as Zionism is a racist, supremacist ideology. I don’t see how it is possible to be a Zionist and not be racist. Of course I am not in any way suggesting that Zionism is the source of all racism in society. Racism has been around a lot longer than Zionism has. But it is certainly a significant source of racism which gets amplified through the words and deeds of a number of prominent pro-Israel public figures. This demonstrates that we will never fully rid ourselves of racism until we defeat Zionism for good.

  • Testing the labour theory of value

    In a recently published article, the Canadian economist Blair Fix has argued that all empirical tests of the labour theory of value, henceforth LTV, are based on a pseudo-scientific approach he refers to as ‘metaphysical alchemy’. This is not the first time Fix has criticized the LTV. I dealt with his previous critique in another blog post, in which I argued that although Fix is correct in pointing out the flaws in other economists’ attempts to empirically validate the LTV, he is incorrect in arguing that this implies the LTV is fundamentally flawed as a concept. Here I will look at Fix’s recently published critique, which focuses on the work of the Pakistani Marxist economist Anwar Shaikh, who was the first to attempt to empirically validate the LTV.

    The thrust of Fix’s argument is that economists of every persuasion misinterpret empirical data because they subconsciously impose their own ideological biases onto it. This results in them effectively seeing things in the data that aren’t really there, and this is what Fix refers to as ‘metaphysical alchemy’. Fix claims that Marxian economists such as Shaikh are guilty of this when they use empirical data to test the LTV. He also argues that national accounts tables used to test the LTV are particularly susceptible to this type of mis-analysis due to the many accounting identities present in the data. These identities create many correlations that can be easily misinterpreted as underlying causal relationships.

    Fix’s definition of ‘metaphysical’ relates to concepts that cannot be directly observed. In the context of the LTV, he argues that ‘socially necessary labour’ is just such a concept, as it is unobservable and only gets ‘revealed’ through prices. The fundamental problem is that the LTV is a theory about commodities, but the data required to test it is not available at commodity level as it is kept private by firms. For this reason the LTV has only ever been tested at sector level. But a correlation between prices and labour time at sector level does not imply such a correlation holds at commodity level, as a sector-level correlation could just be a result of the number of commodities in each sector correlating with itself.

    To make this a bit more rigorous: suppose our economy has m sectors, and for i = 1,2,…,m let qi, pi, and li be the quantity of commodities produced by sector i, average price of commodities produced by sector i, and average labour employed per commodity in sector i, respectively. Empirical tests of the LTV have demonstrated that P = (Pi) is correlated with L = (Li), where Pi = piqi is the total monetary value of output in sector i and Li = liqi is the total labour-time employed in sector i. But to test the LTV properly we need to show that p = (pi) is correlated with l = (li). We cannot infer this from the fact that P is correlated with L, as both P and L are dependent on q = (qi) and could be correlated simply for that reason.

    In theory we can get around this problem by dividing Pi and Li by qi for each i. In practice, however, this qi is not made publically available, so we cannot do this. It is difficult to see how we might get around this problem. However it is important to note that this is a problem with the data, not with the underlying theory. Fix seems to be arguing that the lack of available data with which to test the LTV somehow invalidates the theory itself; but that is obviously not the case. There are many historical examples of scientific theories that took years to be test empirically due to a lack of available data, but did not mean these theories were invalid. In fact Fix provides one such example himself.

    As Fix notes in his introduction, the ancient Greeks proposed that matter is made up of fundamental ‘quanta’, but it was not until two millennia later that scientific advances allowed atoms to be observed, thereby proving them correct. The Greeks’ proposition was a metaphysical one, in Fix’s sense, as it related to a concept that could not be directly observed. But that does not mean that for 2,000 years the idea that matter is made up of quanta was fundamentally invalid; it just meant that it hadn’t been experimentally verified yet. This is the situation we currently find ourselves in with the LTV. At the present time, the LTV is best thought of as hypothesis that has yet to be tested empirically. The task in front of us is to figure out how to do that.

  • Rational quantum mechanics

    In a previous blog post I questioned the existence of irrational numbers. So I was interested to see an article in the most recent edition of New Scientist which suggests that removing irrational numbers from quantum mechanics could eliminate its weirdest features. Irrational numbers are firmly embedded in the standard theory of quantum mechanics. However according to Tim Palmer, a physicist at the University of Oxford, it doesn’t need to be this way. Palmer argues that banishing irrational numbers from quantum mechanics would remove oddities such as entanglement an superposition. If true, this would revolutionize our understanding of quantum mechanics and have a huge impact on quantum computing.

    To be clear, Palmer does not have a problem with irrational numbers per se (unlike me, and unlike Pythagoras, whose followers allegedly drowned his contemporary Hippasus for suggesting √2 is irrational). Rather, he questions the formalism that underpins the standard mathematical exposition of quantum mechanics. To analyse the behaviour of a quantum particle such as an electron, physicists assign it a mathematical object that lives in something called a Hilbert space, named after the German mathematician David Hilbert. In Palmer’s view, not all such objects should be allowed. In particular, Palmer argues that those with length equal to an irrational number should be banished. We may refer to this view of quantum mechanics as ‘rational quantum mechanics’.

    Palmer claims that rational quantum mechanics is consistent with all the experimental findings of quantum mechanics, but does not suffer from the oddities that come with the conventional interpretation. To exemplify this, Palmer focuses on a famous experiment referred to as the Bell test, named after the Irish physicist John Stewart Bell. Suppose that a pair of entangled particles are produced, with one sent to an experimenter called Alice and the other to another faraway experimenter called Bob. Bell determined that Alice’s and Bob’s measurements of the entangled particles will be correlated in a non-local way, somehow influencing each other across large distances.

    Some interpretations of quantum mechanics simply accept this non-locality, whereas others argue that some law of physics must pre-determine the measurements (this view is known as superdeterminism; see my previous blog post on this for an introduction). Rational quantum mechanics apparently cuts through all this by stipulating that some measurements are impossible to make in principle. In Palmer’s view, carefully thinking through which quantum states are theoretically possible can resolve the problem of non-locality, plus all manner of odd quantum scenarios, including Erwin Schrödinger’s famous thought experiment in which a cat is both dead and alive at the same time.

    Rational quantum mechanics also has practical significance for quantum computing (see my previous blog post on quantum computing for an introduction to this). There is currently just a small number of problems that mathematicians have proved can be solved by a quantum computer but not a standard computer. Moreover, no quantum computer currently exists that is powerful enough to solve these problems. In rational quantum mechanics, qubits – the things that are manipulated by a quantum computer – have a fixed information capacity. This suggests that the problems that could theoretically be solved by a quantum computer but not a standard computer could not in fact be solved by a quantum computer after all.

    If a future quantum computer is invented that solves these problems, rational quantum mechanics goes up in smoke. But this is a strength rather than a weakness of the theory, as it not only shows that it is falsifiable but demonstrates exactly how it could be falsified. Another strength of rational quantum mechanics, in my view, is that it is a discrete rather than a continuous theory. I have argued in several previous blog posts that aspects of mathematics which rely on the existence of a continuum are not as rigorous or ‘real’ as those made up out of finite or discrete parts. This is because the continuum relies of the existence of infinite sets, and infinite sets do not exist in the physical universe.

    Palmer is not the first to suggest that reality might be discrete rather than continuous. Several physicist have argued for the existence of a discrete rather than a continuous space-time. However this view is marginal and the majority of physicists still work under the assumption that space-time is continuous. It is well-known that physicists have not been able to find a theory that successfully combines quantum physics and gravity, despite having searched for one now for well over 100 years. My hunch is that the assumption of a continuous space-time is clouding their judgement and hindering their search. Of course the only way to prove that is to develop a discrete theory of quantum gravity.  

    Palmer thinks that gravity ought to play a role in the mathematical space that dictates the reality of quantum objects, and has built this assumption into his theory. However at present this assumption raises more questions than it answers. For rational quantum mechanics to replace standard quantum mechanics it will have to address more than the information capacity of qubits. In practice, given how entrenched the standard theory is, it will probably have to make a breakthrough that the standard theory has so far been unable to make. A theory of quantum gravity would be one such breakthrough and would suggest that we have been thinking about physics all wrong for over a century.

  • Input-output analysis

    An input–output model is a quantitative economic model that represents the inter-dependencies between different sectors of an economy. The idea dates back to the 18th century French economist François Quesnay, who developed a simple input-output model which he referred to as a ‘tableau économique’. Karl Marx’s economic analysis provided another early example involving a set of tables where the economy consisted of two interlinked departments. The Russian economist Alexander Bogdanov has been credited with originating the modern concept of an input-output model in 1921. However it was another Russian economist, Wassily Leontief, who did most to develop the field of input-output analysis; he was awarded the Nobel Prize in Economics for this work in 1973.

    The basic input-output model depicts inter-industry flows within an economy, showing how output from one industrial sector may become an input to another industrial sector. Suppose that we have an economy with m sectors, each of which produces a quantity qi units of single homogenous commodity. Suppose further that Aij units from sector i in order to produce 1 unit from sector j. If Qi is the quantity of goods demanded from sector i then we must have qi = ∑j Aijqj +Qi for each i. This can be written in matrix form as q = Aq+Q, which after re-writing becomes (I-A)q = Q, where I is the identity matrix. If the matrix I-A is invertible then we can write q = (I-A)-1Q, and q can be shown to be positive under a relatively weak assumption on the matrix I-A known as the Hawkins-Simon condition.

    In addition to the quantity model just described, there is also a ‘value’ model that examines the relationship between commodity values and inter-industry flows within an economy. In this model, the value of commodity j can be expressed as vj = ∑i viAij +Vj, where Vj is the value added per unit output of sector j. Thus can be written in matrix form as v = vA+V, which after re-writing becomes v(I-A) = V. Similarly to the quantity model described above, if the matrix I-A is invertible then v = V(I-A)-1, and v can be shown to be positive provided the matrix I-A satisfies the Hawkins-Simon condition. Note that we have vq = vAq+vQ = vAq+Vq, and therefore vQ = Vq. In other words, the total value added is equal to the value of the demand vector Q.

    There is also a third ‘price’ model that examines the relationship between commodity prices and inter-industry flows. In this model, the price of commodity j can be expressed as pj = (1+r)∑i piAij +(1+r)wLj, where r, w, and Lj are the profit rate, wage rate, and labour inputs in sector j, respectively. This can be written in matrix form as p = (1+r)(pA+wL). Rearranging gives p[I-(1+r)A] = (1+r)wL, and similarly to the models described above, if the matrix I-(1+r)A is invertible we can write p = (1+r)wL[I-(1+r)A]-1, which is positive provided the matrix I-(1+r)A satisfies the Hawkins-Simon condition.  In Marxian economics, wages are usually assumed to be set at subsistence level, so w = pb for some subsistence commodity vector b. We then have p = (1+r)p(A+bL), so p is an eigenvector of A+bL with eigenvalue 1/(1+r).

    A well-known result called as the Perron-Frobenius theorem says that under a weak condition on the matrix A+bL, such a p and r exist and are unique. Furthermore, as w = pb, if the subsistence bundle b is uniquely determined, then wages are uniquely determined too. (We can ensure that b is uniquely determined by assuming that if there is more than one subsistence bundle, wages will be determined by the one with the lowest price.) In general there is no simple relationship between these prices and the values v defined above. Moreover, there is no logical reason why the values added in production, V, should be equal to the labour inputs, L; that is, there is no logical reason why the labour theory of value should necessarily hold. This is something that is usually just assumed in Marxian economics.

  • The truth about Jason Arday

    Almost everyone agrees that the suicide of Cambridge professor Jason Arday last week was a terrible tragedy, both for him and his family. There is far less agreement on who should be held responsible for his untimely death. Many have blamed the drive towards greater equality, diversity, and inclusion (ED&I) for propelling Arday to a level above his natural ability. According to this narrative, Arday was a con artist who got to where he was by plagiarising other peoples’ work and exaggerating his achievements; and the universities that employed him turned a blind eye because Arday was black, and they knew that employing him would improve their ED&I statistics. This narrative has been pushed hard by the right-wing media, who claim they were simply doing their job by pointing this out.

    Does this narrative hold any water? Let’s start with the accusations of plagiarism. In 2023, David Harris, an emeritus professor at Plymouth Marjon University, emailed Arday to enquire about similarities between sentences in Arday’s 2018 Social Sciences article and an earlier BMJ Open article by Anjum Memon. Social Sciences published a correction to Arday’s article, with revised citations, later that year. Times Higher Education reporter Jack Grove then investigated Arday’s alleged plagiarism and assembled a 63-page report. Arday responded with a letter from Carter-Ruck, a legal firm specializing in libel, after which the Times Higher Education chose not to publish the story, presumably because they knew it wouldn’t withstand legal scrutiny.

    Having been alerted to allegations of plagiarism in Arday’s PhD thesis in 2025, Liverpool John Moores University – who had awarded him his PhD – established a misconduct panel. The panel concluded that the similarities between Arday’s thesis and an earlier PhD thesis by another academic were the result of “honest and reasonable error”, not deliberate plagiarism, and were “well within the accepted range of compliance with the academic standards of the time”. Nonetheless, allegations of plagiarism became public in July this year through a Substack post by Nathan Cofnas, a postdoctoral researcher in philosophy at Ghent University and self-described ‘race realist’. This post was then picked up and amplified by the mainstream media.

    Cofnas used AI software to demonstrate that Arday had copied and pasted some passages from another academic into his PhD thesis. However it turns out that these passages were all in the introductory chapter, which Arday would not have been examined on when he was awarded his PhD. That explains why the misconduct panel did not find him guilty of plagiarism. Thus the sum total of the plagiarism claims levelled against Arday are based on one 2018 article which has since been corrected, and the introductory chapter of his PhD thesis. I don’t know about you but that hardly screams ‘serial plagiarist’ to me. Listening to the mainstream media talk about this, however, you’d think that Arday had plagiarised at least half of his academic output.

    So much for the plagiarism accusations then. What about the accusations that Arday exaggerated his achievements? Arday apparently made some fantastical-sounding claims: that he was non-verbal until the age of 11; that he was illiterate until he was 18; that he played semi-professional football; that he personally raised “£5.5m for over 80 charities”; and that he ran 30 marathons in 35 days, the last nine marathons with a hairline fracture of the fibula. It turns out however that at least two of these claims – about playing semi-professional football, and running 30 marathons in 35 days – are actually true. I suspect that some of the other claims have been taken out of context and do not faithfully reflect what Arday actually said or meant.

    Even if Arday was a fabulist who exaggerated his non-academic achievements, this should have had no bearing on his meteoric career progression, which would have been based solely on his academic output. And it seems that his academic work was highly valued by other academics in his field. A quick look at Google Scholar shows that Arday’s publications regularly attract hundreds of citations; and his 2018 book Dismantling race in higher education: Racism, whiteness and decolonising the academy, co-authored with Heidi Safia Mirza, has been cited over 600 times. These are citation numbers most academics can only dream of. Compare that with Nathan Cofnas, Arday’s tormentor-in-chief, whose academic output has achieved less than 700 citations combined.

    The image of Arday that has been portrayed in the media – an ED&I hire who was propelled to a level above his natural ability through a combination of plagiarism and falsified claims – does not tally with the facts. On the contrary, Arday was clearly a talented academic who was well-respected by his peers. This is another example of a lie that gets repeated so many times, everyone assumes it must be true. It is not difficult to understand why the right-wing media would want people to believe this lie: it deflects attention away from the fact that they are the ones that need to take some accountability for Arday’s death. The truth is that Jason Arday was a gifted academic whose work shone a light on institutionalized racism in academia and wider society. And the right-wing media hounded him to his death for it.

  • Capitalism can’t solve the climate crisis

    The consequences of climate change are hitting us squarely in the face. Temperatures are soaring, drought conditions are spreading, and wildfires are raging across the UK and other countries around the globe. The world is quite literally burning. Meanwhile millions of workers are forced to live and work in sweltering heat, in the knowledge that extreme heat events are the new normal. Yet those with their hands on the levers of power are doing little to help the situation. Politicians have demonstrated time and again that they serve the interests of the super-rich and will not act to save our futures, despite the fact that green technology that would alleviate the crisis is now readily available and is becoming cheaper all the time.

    In a previous blog post I explained how the United Nations Climate Change Conferences, or COPs, have made no meaningful progress in tackling climate change. Despite the consistent failures of these meetings, capitalist politicians claim they are doing what they can to prevent global warming. So what’s going on here? One issue is that the methods put forward to solve climate change and other environmental problems by capitalists and their academic lackeys, neoclassical economists, are often based on the so-called ‘property rights’ approach. This relies on direct negotiations between parties who own property and results in an efficient solution for these property owners, but not for society as a whole.

    An alternative approach put forward by capitalists is to control emissions through the price system, by introducing eco-taxes. The idea is that increasing the price of polluting resources will create an incentive to use them less and seek substitutes. The task then is to put a monetary value on the cost of global warming. The problem is that it is very difficult to meaningfully measure this cost in monetary terms. It must be estimated by a roundabout route which involves establishing a so-called ‘shadow’ market value, which is related to how much a consumer is willingness to pay to avoid environmental ‘bads’. This can only be properly assessed by considering future generations, whose willingness to pay could vary dramatically. Of course it is impossible to do that in practice.

    Another approach that has been put forward by capitalists involves trading pollution permits. These permits are issued by governments to corporations who can then freely trade them on the market. This is often seen by capitalists as preferable to eco-taxes, which they regard as a dangerously ‘socialist’ idea. But in truth, all the methods put forward by neoliberals to solve the climate crisis – the property rights approach, eco-taxes, and permit trading – suffer from the same fundamental drawback: namely, they don’t work. This is despite the fact that the long-term costs of global warming are generally agreed to far exceed the immediate costs of mitigation. So why do capitalists find it so difficult to come up with a workable solution?

    In the past it was much easier for governments to tackle environmental issues as they were largely restricted within national boundaries. That is manifestly not the case with global warming. Moreover, antagonisms between countries now are greater than they were 100 years ago when there were just a handful of dominant world powers. A useful concept to understand the factors preventing meaningful agreement being reached on global warming is imperialism, as defined by Vladimir Lenin in his 1917 work Imperialism: The Highest Stage of Capitalism. Lenin identified five features that characterise the imperialist era: concentration of capital, merging of finance and industrial capital, export of capital over and above commodities, development of international cartels, and territorial re-division of the world.

    The first four of these features are clearly still relevant today. The concentration of capital has reached levels way beyond what Lenin would have experienced in the early 20th century. The merging of finance and industrial capital has continued apace, with finance capital becoming more and more dominant. We increasingly see the export of capital over and above commodities, with multinational corporations establishing operations throughout the globe. We have also seen the development of international cartels, particularly the Organization of Petroleum Exporting Countries (OPEC). On the other hand, the territorial re-division of the world was forestalled after WWII. This was largely due to the existence of the Soviet Union, which led to greater cooperation between capitalist states in the post-war period.

    This cooperation broke down after the collapse of the Soviet Union in 1991. Today, the multinational companies dominating the world economy fiercely resist anything that will threaten their short-term profits, and look to their ‘home’ countries for assistance in doing this. These home countries are often all too willing to oblige. The fundamental problem is that under capitalism, politicians are effectively installed by these multinational corporations, and are then constrained to act in a such a way so as to not affect these corporations’ bottom line. This inevitably leads to a lack of action when it comes to mitigating the effects of climate change. We will never solve the climate crisis until we get rid of capitalism and replace it with a system based on international cooperation.

  • Indo-European velars after *s

    It has long been recognized that the distribution of Proto-Indo-European (PIE) velars is highly skewed after *s, with plain velars occurring far more frequently in this position than palatovelars and labiovelars. In Pokorny’s 1959 Indogermanisches Etymologisches Wörterbuch (IEW), there 118 entries with a plain velar occurring after *s or mobile *s, as opposed to 13 entries with a palatovelar and 4 with a labiovelar (here I have included entries with a velar after *z, the voiced allophone of *s). Similarly, in Rix et al.’s 2001 Lexikon der Indomanischen Verben (LIV2) and Wodtko et al.’s 2008 Nomina im Indogermanischen Lexikon (NIL), there are 35 entries with plain velar occurring after *s or mobile *s, as opposed to 7 entries with a palatovelar and 3 with a labiovelar.

    How can we account for this distribution? The obvious explanation is that the opposition between the velar series was neutralized after *s in PIE. The problem with this is that we still find all three types of velar occurring in this position. In the case of mobile *s we can posit analogical change to account for this, but that doesn’t work for roots with non-mobile *s. There are 5 entries where a palatovelar occurs after non-mobile *s in IEW: sk’ed- ‘cover’, sk’eH(i)- ‘shimmer’, *sk’er(d)- ‘defecate’, *sk’eu- ‘throw’, and *mosg’hos ‘young’. There are 4 additional entries where a palatovelar occurs in this position in LIV2 and NIL: sk’eH2i- ‘separate’, sk’eH2id- ‘split’, sk’end- ‘cover’, and *tresk’- ‘squeeze’. Let us go through each of these in detail.

    The root sk’ed- ‘cover’ only has reflexes in Indo-Iranian among the satem languages. Whereas the Iranian reflexes suggest an original palatovelar, the Indic reflexes point to a plain velar. This discrepancy can be explained by positing that palatalization was not blocked after *s in Iranian when the velar was followed by a front vowel. This also applies to sk’end- ‘cover’ which is obviously a nasal-infix variant of the same root. The same explanation can be applied to the root *sk’eH2i- ‘separate’ if we assume that the laryngeal was lost in the zero-grade forms sk’H2i- prior to the palatalization in Iranian. It can also be applied to the roots *sk’eH(i)- ‘shimmer’ and *sk’er(d)- ‘defecate’ if we assume that the root was originally *s-mobile and that the Balto-Slavic reflexes are derived from the variant without *s.

    The root *mosg’hos ‘young’ only has reflexes in Armenian among the satem languages, where the palatalized reflex can be explained by positing that *-sgh- yields -z- in Armenian. This rule was established by the Australian linguist Robert Woodhouse in 2014, as was the rule that palatalization was not blocked after *s in Iranian when the velar was followed by a front vowel. The root *tresk’- ‘squeeze’ should be reconstructed with a plain velar it only has unpalatalized reflexes in the satem languages. The same applies the root sk’eH2id- ‘split’. This root is clearly derived from *sk’eH2i- ‘separate’ (see above), which provides further evidence that the latter root should be reconstructed with a plain velar. The root *sk’eu- ‘throw’ is not probative as it only has reflexes in Balto-Slavic.

    Thus none of the entries with a palatovelar after non-mobile *s stand up to scrutiny. What about the examples with a labiovelar in this position? There are no entries where a labiovelar occurs after non-mobile *s in IEW, and just 1 entry where a labiovelar occurs in this position in LIV2 and NIL: *Hosgw– ‘knot’. In his 2014 paper Woodhouse also provides 3 examples of a labiovelar apparently occurring after non-mobile *s: *sgwhH2el- ‘stumble’, *skwelH- ‘split’, and *mosgw– ‘knot’. However *sgwhH2el- ‘stumble’ is reconstructed with a mobile *s in LIV2 to account for the Latin reflex; and  *skwelH- ‘split’ is reconstructed with a plain velar in LIV2 to account for the Hittite reflex. The labiovelars in *Hosgw– ‘knot’ and *mosgw– ‘knot’ are reconstructed based on single words from Gaulish and Old Norse respectively and are therefore far from certain.

    Thus none of the entries with a labiovelar after non-mobile *s stand up to scrutiny either. The obvious conclusion is that the opposition between the velar series was indeed neutralized after *s in PIE.

  • What is to be done?

    Vladimir Lenin was born Vladimir Ulyanov into a upper middle-class family in 1870 in the provincial Russian city of Simbirsk, which has since been renamed Ulaynovsk in his honour. His elder brother was executed by Tsarist forces for revolutionary activity when Lenin was just 17. This event had a huge impact on the young Lenin and from that point on he became a committed revolutionary himself. In 1897 Lenin was exiled to Siberia, after which he fled Russia for Western Europe. He returned to Russia in 1917 and in October that year led the successful revolution which removed the Tsar for good. However leading the revolution took a huge toll on Lenin’s health and he fell seriously ill four years later. His health never recovered and he died in 1924.

    What Is to Be Done? is a book written by Lenin in 1901 and published in 1902. Having just finished reading What Is to Be Done? I thought I would provide a short summary. The book’s central focus is the ideological formation of the proletariat. Lenin first confronts the so-called ‘economist’ trend in Russian socialism that followed the line of the German Marxist Eduard Bernstein. Economism in Marxist usage means reducing the workers’ movement to immediate economic demands – wages, hours, and workplace conditions – and treating broader political struggle for independent class power as secondary, derivative, or unnecessary. Lenin argues forcefully that workers will not develop class consciousness through economism.

    Reflecting on the wave of strikes in late 19th century Russia, Lenin observes that “the history of all countries shows that the working class, exclusively by its own efforts, is able to develop only trade union consciousness”; that is, the conviction that it must combine into unions and petition the government for pro-labour legislation. Lenin refers to this unguided, raw, and natural struggle of the working class as ‘spontaneity’, and argues this alone will never result in the overthrow of capitalism. Instead, Lenin argues, workers need a centralized vanguard party of professional revolutionaries to lead them. According to Lenin, socialist theory and political awareness must be brought to the working class from the outside by educated intellectuals and dedicated activists.

    Lenin argues further that the revolutionary movement requires a tight, highly disciplined organization of professional revolutionaries, and that a centralized newspaper should act as the collective organizer, linking scattered local groups and spreading political ideas. His call for a core of professional revolutionaries stirred controversy and contributed to the Bolshevik-Menshevik split in 1903. Many treated What Is to Be Done? like it was Lenin’s last word on revolutionary organizing, when in fact it was an early formulation by him on how a small group of Russian socialists could begin to build an effective movement. Many also misread Lenin’s views on spontaneity, claiming he was hostile to spontaneous struggles when he was not against these struggles per se but against relying on them exclusively.

    There has been much debate about whether What Is to Be Done? is still relevant. In my view, the three core arguments put forward in the book – the insufficiency of economism and spontaneity, the need for an organized group of professional revolutionaries, and the importance of theory in underpinning the revolutionary movement – remain as applicable today as they were when Lenin wrote about them 125 years ago. On the other hand, there is a lot of detail that is specific to Russia at the turn of the 20th century that is not so relevant now. What Is to Be Done? is written as a manual for underground resistance, which was necessary in Tsarist Russia but not so applicable to the freer societies most of us live in today.

    One criticism often leveled at What Is to Be Done? is that the concept of a centralized vanguard party is fundamentally undemocratic and inevitably leads to dictatorship and totalitarianism. Those putting forward this criticism tend to point specifically to the red terror: the brutal campaign of mass arrests, torture, and executions launched by the Bolsheviks from 1918 to 1922. Enforced by the secret police known as the Cheka, its goal was to crush political opposition and secure communist power. Some high-profile contemporary Marxists, such as Rosa Luxembourg and Karl Kautsky, were highly critical of the Bolsheviks for this approach. There are even some who even argue that the vanguard party concept paved the way for Stalinism and all the horrors that unleashed.

    Proponents of the red terror argue that extreme violence was a compulsory, defensive reaction to existential threats. The Bolsheviks faced civil war against the White Army, foreign invasions, economic collapse, and assassination attempts, including the near-fatal shooting of Lenin in 1918. These proponents claim that crushing the bourgeoisie and counter-revolutionaries was required to prevent the return of Tsarism or capitalist exploitation. There is certainly some truth to this. When assessing the red terror we do need to take into account the fact that there was a brutal war going on at the time. And as the architects of the first ever socialist revolution, the Bolsheviks faced an existential threat from capitalist countries whose leaders were terrified of the revolution spreading beyond Russia’s borders.

    It is notable that the red terror did not begin immediately when the Bolsheviks came to power, but only after the assassination attempt on Lenin, which occurred almost a year later. The Bolsheviks had brought in a lot of progressive policies in the meantime, such an 8-hour workday, free education, a literacy campaign, and land redistribution. Thus rather than seeing the red terror as an inevitable consequence of the vanguard party concept, I think it is better viewed as a defensive manoeuvre instigated in response to the situation the Bolsheviks found themselves in. They may have overstepped the mark in terms of the level of brutality required, but it’s easy to say that when there isn’t a civil war going on and with the benefit of 100 years’ hindsight.

  • Kantian and correlated equilibrium

    Game theory is the study of mathematical models of conflict. Such models generally involve a finite set of players labelled 1,2,…,n; a finite set of actions Ai available to each player i; and a reward or utility function ri for each player i. The reward function maps the set A = ∏j Aj to the real numbers, with the interpretation that ri(a) represents the reward or payoff to player i when the players take the action profile a ∈ A. A mixed strategy pi for player i is a probability distribution over the set Ai, with the interpretation that pi(ai) represents the probability that player i will take action ai ∈ Ai. It is generally assumed that each player will play a mixed strategy. It is also usually assumed that these strategies are statistically independent of each other.

    Statistical independence means that the action taken by each player tells us nothing about the actions the other players will play. This is expressed mathematically by saying that probability the players take the action profile a ∈ A is given by the product p(a) = ∏j pj(aj). Then the expected payoff for player i is given by ∑A ri(a)p(a). A key concept in game theory is the Nash equilibrium, named after the American mathematician John Nash. The mixed strategy profile p is said to be a Nash equilibrium if for all players i and actions ai,bi ∈ Ai, we have ∑A ri(a)p(a) ≥ ∑A ri([a,bi])p(a), where [a,bi] ∈ A is defined by [a,bi]i = bi and [a,bi]j = aj for i ≠ j. In other words, in a Nash equilibrium no player can do better by unilaterally changing their strategy.

    Recently, the American Marxist economist John Roemer has devised a novel solution concept for game theory he refers to as a Kantian equilibrium, named after the 18th century German philosopher Emmanuel Kant. It formalizes Kant’s categorical imperative – acting according to a maxim you would wish to see universalized – into mathematical optimization. Instead of asking “What is my best response if I hold others’ strategies fixed?” (the Nash approach), a player in a Kantian equilibrium asks: “What action would I want everyone to take if everyone were to do the same thing as me?” This solution concept only makes sense when the players’ action sets are all the same, and the players only have two actions available to them, so we can set A1 = A2 = … = An = {0,1}.

    In this model a mixed strategy for player i can be represented as a number qi in the interval [0,1] representing the probability that player i takes action 1, and a mixed strategy profile can then be represented as a vector q = (qj). For a ∈ A let J(a,0) and J(a,1) denote the set of indices j where aj = 0 and aj = 1 respectively. The probability that the players take action profile a ∈ A is then given by the product p(a,q) = ∏J(a,1) qjaj ∏J(a,0) (1-qjaj). The mixed strategy profile represented by  q is said to be a (multiplicative) Kantian equilibrium if for all players i and real numbers c with 0 ≤ cqj ≤ 1 for all players j,  the following inequality holds: ∑A ri(a)p(a,q) ≥ ∑A ri(a)p(a,cq). In other words, no player can do better if all players scale their mixed strategies by the same amount.

    A correlated equilibrium is a game theory solution concept which relaxes the statistical independence assumption and is therefore more general than the well-known Nash equilibrium. If p is a probability distribution over A then p is said to be a correlated equilibrium if for all players i and actions ai,bi ∈ Ai, we have ∑A ri(a)p(a) ≥ ∑A ri([a,bi])p(a), where again [a,bi]i = bi and [a,bi]j = aj for i ≠ j. Thus, a Nash equilibrium is a special case of a correlated equilibrium where the actions taken by the players are statistically independent; that is, where p(a) = ∏j pj(aj). On the other hand, a Kantian equilibrium can be seen as a special case of a correlated equilibrium where A1 = A2 = … = An = {0,1} and the actions taken by the players are linearly dependent. I will leave the demonstration of this to a future blog post.

  • Debunking race science

    In 2023, Jason Arday was appointed a professor of sociology of education at Cambridge, attracting media attention as the youngest black professor in the university’s history. Earlier this year, it emerged that Arday may have plagiarized some of his writings, falsified research, and made false claims in his academic history and autobiography. An intense witch-hunt followed, orchestrated by media outlets such as the Times, the Telegraph, and the Spectator. Last night, Arday was found dead in his London home having apparently committed suicide. It appears he was hounded to death by the right-wing media. This tragic episode tells us a lot about how racism is alive and well in this country. It also tells us that the concept of ‘race science’ has not yet been fully eliminated from our public discourse.

    Race science is based on the belief that humanity divides into biologically distinct groups with inherent behavioural or intellectual characteristics. It emerged during the European Enlightenment and 19th-century colonialism to justify social stratification, slavery, and imperial dominance. Early race science relied on flawed and biased practices like craniometry (measuring skull sizes) and IQ testing to assign intellectual capacities to different groups. It led to the development of the field of eugenics, which advocated for improving the genetic quality of the human population through selective breeding. This inspired harmful social policies and forced sterilizations under the guise of improving human hereditary traits, and ultimately led to the atrocities committed by the Nazis in the mid-20th century.

    These atrocities demonstrated what happens when these ideas are taken to their logical conclusion and led to a significant diminution of interest in race science. It did not go away completely however and was kept alive by a small group of dedicated individuals. One such individual is Charles Murray, who in 1994 co-authored a now infamous book with Richard J. Herrnstein called The Bell Curve. Murray and Herrnstein argued that human intelligence (measured by IQ) is a better predictor of life success than parental socioeconomic status and contributes heavily to group performance disparities. In so doing, they argued that humans can neatly be divided into races, and that these races have measurable disparities in intelligence. These are the core propositions of race science.

    Another such individual is Nathan Cofnas, a self-defined ‘race realist’. Cofnas was sacked by Cambridge University in 2024 and underwent a disciplinary investigation into a blog post about black academics which suggested that in a meritocracy, there would be ‘close to zero’ black professors. He is at the centre of the storm around Arday as it was Cofnas who first alleged plagiarism in Arday’s PhD thesis. Of course, there is nothing wrong with academics highlighting plagiarism; indeed, they have a duty to do so. And it may be that Arday was guilty of plagiarism, although that is not clear at this stage. But there is no doubt that Cofnas went out of his way to find a black professor who he could discredit so as to back up his preexisting beliefs.

    It is tempting to simply dismiss individuals such as Murray and Cofnas as racist cranks – particularly as their ideas have been roundly debunked by the scientific community. However the furore around Arday demonstrates that these ideas have not yet been fully discarded by the wider public. This is despite the fact that the core propositions of race science are embarrassingly easy to debunk. Modern genomic research demonstrates that human genetic variation is continuous rather than categorical, meaning two people of the same perceived race can be genetically more distant than two people from different racial groups. Thus the idea that humans can neatly divided into races does not tally with the scientific evidence. This demonstrates that race is a social (i.e. constructed) rather than a scientific concept.

    The fact that humans cannot be neatly divided into races means that it doesn’t make any sense to argue that different races have measurable disparities in intelligence. But let us suspend disbelief for a moment that humans could be neatly divided in this way. Would it make sense then? The answer is no, for the obvious reason that there can be no single measure of intelligence. There are many types of intelligence: linguistic, logical-mathematical, spatial, bodily-kinesthetic, musical, interpersonal, intrapersonal, naturalistic, existential, creative, practical, and so on. Many of these are difficult or even impossible to measure quantitatively. How would we go about quantifying interpersonal intelligence, for example?

    The idea that these different types of intelligence can be boiled down into a single ‘intelligence quotient’ metric is nonsense. The standard IQ metric really measures just logical-mathematical and spatial intelligence. And it doesn’t even do that particularly well. So even if it was the case that human beings could be neatly divided into races, and it could be shown that there was a difference in IQ between these races, that wouldn’t really tell us anything interesting. It certainly wouldn’t tell us anything about why different people experience different outcomes. The truth is that disparities in health, education, and socioeconomic outcomes are driven by environmental, historical, and systemic inequities rather than innate biological traits. So why can’t scientists convince people of this?

    In a 2025 article published in Evolutionary Human Sciences, evolutionary biologists Kevin Lala and Kalyani Twyman identified five barrier to the effective countering of racist pseudoscience. The first such barrier is a widespread belief in genetic determinism. The second is an overly simplistic conception of heredity that attributes the inheritance of traits, and of their differences, to genes and genetic variation. The third is belief in the naturalistic fallacy: the false idea that traits that evolved are desirable or inevitable. The fourth is the failure of relevant scientific disciplines to take responsibility for teaching the science of racism. And the fifth is the self-promotion of academic fields, many of which are tarnished with a history of racism and eugenics which they are too embarrassed to talk about.

    All of these factors are undoubtedly relevant. But I think a better explanation lies in the fact that a widespread belief in race science among the general public is beneficial to the ruling class, as it makes it that much easier for members of this class to pursue a divide-and-conquer strategy, which in turn makes it that much easier for them to cling to power. The implicit message underlying all the furore around Jason Arday in the mainstream media is that people from ‘lesser’ races should not get ideas above their station. Arday may well have been a fabulist who arguably should not have been made a professor. But he was also a human being who was unfortunate to find himself directly in the firing line of the ruling class’s divide-and-conquer culture war. Perhaps now he will find some peace.