Mathematician solves algebra’s oldest problem

Andrew Paul
Thu, May 1, 2025 



Polynomials were first conceived by the Babylonians around 1800 BCE.

Most people’s experiences with polynomial equations don’t extend much further than high school algebra and the quadratic formula. Still, these numeric puzzles remain a foundational component of everything from calculating planetary orbits to computer programming. Although solving lower order polynomials—where the x in an equation is raised up to the fourth power—is often a simple task, things get complicated once you start seeing powers of five or greater. For centuries, mathematicians accepted this as simply an inherent challenge to their work, but not Norman Wildberger. According to his new approach detailed in The American Mathematical Monthly, there’s a much more elegant approach to high order polynomials—all you need to do is get rid of pesky notions like irrational numbers.

Babylonians first conceived of two-degree polynomials around 1800 BCE, but it took until the 16th century for mathematicians to evolve the concept to incorporate three- and four-degree variables using root numbers, also known as radicals. Polynomials remained there for another two centuries, with larger examples stumping experts until in 1832. That year, French mathematician Évariste Galois finally illustrated why this was such a problem—the underlying mathematical symmetry in the established methods for lower-order polynomials simply became too complicated for degree five or higher. For Galois, this meant there just wasn’t a general formula available for them.

Norman Wildberger’s mathematical work rejects concepts like irrational numbers.

Mathematicians have since developed approximate solutions, but they require integrating concepts like irrational numbers into the classical formula.

To calculate such an irrational number, “you would need an infinite amount of work and a hard drive larger than the universe,” explained Wildberger, a mathematician at the University of New South Wales Sydney in Australia.

This infinite number of possibilities is the fundamental issue, according to Wildberger. The solution? Toss out the entire concept.

“[I don’t] believe in irrational numbers,” he said.

Instead, his approach relies on mathematical functions like adding, multiplying, and squaring. Wildberger recently approached this challenge by turning to specific polynomial variants called “power series,” which possess infinite terms within the powers of x. To test it out, he and computer scientist Dean Rubine used “a famous cubic equation used by Wallis in the 17th century to demonstrate Newton’s method.”

You don’t need to try wrapping your head around all that, however. Just trust Wildberger when he said the solution “worked beautifully.”

The same goes for Catalan numbers, a famous sequence of numbers that describes the number of ways to dissect any given polygon. These also appear in the natural world in areas like biology, where they are employed to analyze possible folding patterns of RNA molecules.

“The Catalan numbers are understood to be intimately connected with the quadratic equation,” explained Wildberger. “Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers.”

Outside of headspinning concepts on paper, Wildberger believes the new approach to higher power polynomials could soon result in computer programs capable of solving equations without the need for radicals. It may also help improve algorithms across a variety of fields.

“This is a dramatic revision of a basic chapter in algebra,” argued Wildberger.

Luckily, none of this will be your next pop quiz.

The first driverless semis have started running regular longhaul routes

Alexandra Skores, CNN
Thu, May 1, 2025 



Interior of one of Aurora's driverless trucks. - Courtesy Aurora


Driverless trucks are officially running their first regular long-haul routes, making roundtrips between Dallas and Houston.

On Thursday, autonomous trucking firm Aurora announced it launched commercial service in Texas under its first customers, Uber Freight and Hirschbach Motor Lines, which delivers time- and temperature-sensitive freight. Both companies conducted test runs with Aurora, including safety drivers to monitor the self-driving technology dubbed “Aurora Driver.” Aurora’s new commercial service will no longer have safety drivers.

“We founded Aurora to deliver the benefits of self-driving technology safely, quickly, and broadly, said Chris Urmson, CEO and co-founder of Aurora, in a release on Thursday. “Now, we are the first company to successfully and safely operate a commercial driverless trucking service on public roads.”



The trucks are equipped with computers and sensors that can see the length of over four football fields. In four years of practice hauls the trucks’ technology has delivered over 10,000 customer loads. As of Thursday, the company’s self-driving tech has completed over 1,200 miles without a human in the truck.

Aurora is starting with a single self-driving truck and plans to add more by the end of 2025.

One of Aurora's trucks on the road. - Courtesy Aurora

Self-driving technology continued to garner attention after over a decade of hype, especially from auto companies like Tesla, GM and others that have poured billions into the tech. Companies in the market of autonomous trucking or driving, tend to use states like Texas and California as their testing grounds for the technology.

California-based Gatik does short-haul deliveries for Fortune 500 retailers like Walmart. Another California tech firm, Kodiak Robotics, delivers freight daily for customers across the South but with safety drivers. Waymo, a subsidiary of Google parent company Alphabet, had an autonomous trucking arm but dismantled it in 2023 to focus on its self-driving ride-hailing services.



However, consumers and transportation officials have raised alarms on the safety record of autonomous vehicles. Aurora released its own safety report this year detailing how its technology works.

Unions that represent truck drivers are usually opposed to the driverless technology because of the threat of job loss and concerns over safety.

 

Turbulence in the Global Economy

Vijay Prashad | 25 Apr 2025

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The IMF has predicted “uncertainty” in the global economy, especially in the wake of Trump’s tariff war, yet it seems to ignore the pathways towards stability and growth seen in the East.

IMF briefing. Photo: IMF News

On April 22, 2025, the International Monetary Fund (IMF) released its annual World Economic Outlook report, which has a quiet but decisive subtitle: A Critical Juncture and Policy Shifts. The report, once more a hefty piece of work by the IMF economists and their associates, rushes to make sense of the tariffs first threatened by US President Donald Trump and then delayed, and then, as if the flurry was not sufficient, retained and increased against China. The IMF attempts to make the case that through 2024, “global growth was stable” and that the current downgrade to global growth is largely a factor of the Trump tariff “uncertainty” and “unpredictability”.

The IMF releases this report during the annual week of meetings of the Fund and the World Bank. At the start of the meetings, IMF Director Kristalina Georgieva reflected on the situation in the global economy and claimed that the turbulence is largely due to “an erosion of trust”. No longer, she said, do countries trust each other as they once did, nor do they trust the international system. Apart from the reversal of the tariffs, the IMF says that what needs to be built once more is trust in international economic affairs.

Whispers in the corners of the IMF and World Bank meetings are all about the irrationality of the Trump administration, and – in particular – the unpredictability of Trump’s own statements. With the head of the North Atlantic Treaty Organisation (NATO) beside him, Trump told a press conference in the White House that Canada is not a real country but would be an excellent state within the United States. That sort of comment creates fodder for the hushed giggles at the edges of this meeting, when otherwise serious men and women in business suits maintain a rictus look of concern for high affairs of state.

Errors in judgment

One of the significant errors in the assessment by the IMF is that everything in the Western economies had begun to look stable last year. While it is true that the threat of the tariffs and then the anti-China tariffs themselves created “a major negative shock to growth”, it is not true that growth rates had been expected to reach new heights this year.

Growth in the US has been significantly below its historic trend since the 2007-08 subprime mortgage crisis-induced financial crisis – indeed, US growth since then has been cumulatively lower than in the Great Depression. In the 17 years after the Great Crash of 1929, US annual GDP growth averaged 3.7%, whereas in the 17 years since the International Financial Crisis, US annual growth averaged only 2.0%.

In October 2024, the IMF projected that the United States would grow at 2.2%, and since reduced its forecast to 1.8%. Meanwhile, in October last year, it suggested a rate of 4.5% for China and 6.5% for India, far higher than for either the US projected rate or the advanced economy projections (1.8%). The Trump tariffs certainly compounded the problems for the US, but they are not the cause of the problem. Sluggish growth has been the situation for almost two decades.

On that sluggishness, the IMF’s new World Economic Outlook is remarkably bland. It suggests that the “core macroeconomic policy challenge” of the United States is its federal government debt. This debt, which is 36.2 trillion, is 124% of GDP. Ten global north countries are in the twenty countries with the highest debt-to-GDP ratios: Japan (266%), Greece (193%), Italy (151%), United States (124%), Portugal (122%), Spain (117%), France (112%), Belgium (111%), Canada (109%), and the United Kingdom (105%). Cutting the deficit might make good macroeconomic sense, but it does not by itself propose a way back to growth for the United States. Lower social welfare spending will further deplete private consumption. And Trump’s dream of revitalizing US manufacturing is not going to work merely through a reduced federal government deficit without a massive, massive release of resources for industrialization. Without an attack on living standards, this could only come from measures such as a reduction in excessive US military expenditure or reform of the country’s grotesquely inefficient private health system. These are policies Trump will not adopt.

In fact, the IMF gives notoriously poor advice to the Chinese government. It suggests that China should emulate the United States rather than the other way around. China, the IMF says, should “boost chronically low private consumption” and “dial back industrial policies and pervasive state involvement in industry”. In other words, abandon its long-term growth profile and become like the slow-growing United States!

In November 2024, the IMF released an interesting paper by its economists (Dirk Muir, Natalija Novta, and Anne Oeking) called “China’s Path to Sustainable and Balanced Growth”. The paper and the World Economic Outlook together make the case that China’s strong economic performance comes from its COVID stimulus, its high exports, a high domestic savings rate to finance public infrastructure, its banking system that directs liquidity to small and medium-sized enterprises to generate productive activity rather than property speculation, and an emphasis on high-quality productive forces. This is a fairly good summary of the structure of Chinese growth over the last period. But it is totally counter to the suggestions that the IMF then gives to China: which is to liquidate everything that allowed it to stave off the long term sluggishness of the advanced industrial countries (including to pressure the renminbi to appreciate, as the US would like so that its trade imbalance can be rectified by a foreign exchange shift rather than by greater productivity in the US itself).

The IMF is right. There is great uncertainty ahead. But there is also certainty in its own reports and in its charts. High domestic savings and better sovereignty of resources (including the financial system), alongside canalization of these finances to the productive sector (for infrastructure and industrialization), produce more stability in the long run than an excessive reliance on private financial markets and the whims of the billionaire class. But the IMF does not close its new report with that news. It prefers to look out of the window and see the storms in the Western skies rather than the calm in the East.

Vijay Prashad is an Indian historian, editor, and journalist. He is a writing fellow and chief correspondent at Globetrotter. He is an editor of LeftWord Books and the director of Tricontinental: Institute for Social Research. He has written more than 20 books, including The Darker Nations and The Poorer Nations. His latest books are On Cuba: Reflections on 70 Years of Revolution and Struggle (with Noam Chomsky), Struggle Makes Us Human: Learning from Movements for Socialism, and (also with Noam Chomsky) The Withdrawal: Iraq, Libya, Afghanistan, and the Fragility of US Power.

This article was produced by Globetrotter. 

Courtesy: peoples Dispatch

 

Systemic Agrarian Crisis & Changing Contours of Farm Workers

Vikram Singh | 26 Apr 2025

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The increasing number of agricultural workers and decreasing agricultural work has created new kinds of crises in rural India.

Representational Image. Image Courtesy: Flickr

Manik Ghonshetwad, an agricultural worker from Ambulga village in Mukhed taluka of Nanded district, Maharashtra, works seasonally on farmlands during specific months of the year.  He is employed in June and July for cultivation of soybean or cotton, in August and September for weeding (either manually or with herbicides), and in December and January for harvesting. However, this work is not continuous; he is engaged only for a few days in each of these months.

The limited availability of working days in agriculture in his village and neighbouring areas is insufficient to sustain his family’s basic needs. For the rest of the year, he takes up various forms of daily wage labour, as a headload worker, potter, and construction labourer --essentially any job available.

Even during the cultivation and harvesting seasons, Ghonshetwad supplements his income by taking up non-farm manual work whenever possible. He is also a regular worker under MGNREGA (the 100-day rural job guarantee scheme), though his employment depends on the discretion of government and panchayat officials. Additionally, his family's livestock contributes to their income.

Maroti Shivram Mishkire, from the same village, undertakes similar work but also works as a driver to supplement his earnings. Furthermore, he rears pigs, which provide an additional source of income for his household.

The two examples of multi-occupational profiles of agricultural workers, as encountered during a field survey in this village, illustrate the changing nature of work and the rural workforce in India. These highlight how agricultural workers confront shrinking employment opportunities in agriculture. Multiple factors contribute to this shift, including the indiscriminate use of labour-displacing technology and the growing number of workers reliant on agriculture.

We have a situation of an ever-increasing reserve army of workers, who have no gainful employment in the countryside or assured employment in the urban centres to which they migrate. The deep agrarian crisis in India remains the fundamental reason for this situation.

Read Also: The Modi Years: Rural Wages Saw Further Squeeze

Agriculture is one of the largest sectors providing work to the rural population in India. The share of the agricultural sector in employment has been steadily increasing, rising from 44.1% in 2017-18 to 46.1% in 2023-24. As per 2011 Census, India’s total workforce was 48.17 crore. Of this 72% is from a rural background and more than half i.e. 54.6% or 26.3 crore workers are engaged in agriculture.

In our country, the agricultural workforce primarily consists of cultivators and agricultural labourers. A fundamental distinction between these groups, as delineated by the Census of 2011, lies in land ownership arrangements, specifically characterised by the presence or absence of the ‘ownership rights’, ‘right of lease’ or ‘contract on land’. While a cultivator owns land or has a lease or a contract to operate on it, an agricultural labourer does not, as he or she works on land owned by others in return for wages paid in cash or kind.

Additionally, an important section of poor farmers with small, uneconomical holdings are also forced to labour out on others’ farms or take up other menial jobs. Clearly, there is a blurring of the line as to who is a farmer, a tenant farmer or an agricultural worker.

Agricultural labourers constitute an important segment of the rural proletariat engaged in agricultural production. They are the most downtrodden and marginalised class in rural India. Mostly, they are deprived of all resources and most of them are landless, lack access to means of production and rely solely on their labour for sustenance.

Their socio-economic status renders them vulnerable to exploitation, as they are compelled to accept wages dictated by the landlords, mostly below the minimum wages. Moreover, many lack houses, further exacerbating their precarious conditions.

They are not only economically exploited but also endure social marginalisation, confronting pervasive discrimination and violence. Women agricultural workers face gender deprivation, social oppression and receive far lower wages than men for the same kind of work.

Systematic oppression obstructs the inclusion of the agricultural workers in broader development, entrenching cycles of poverty and inequity. Historically marginalised by illiteracy, their children still face poor education, perpetuating limited awareness and hindering progress. Lack of organisation and unionisation hinder collective bargaining, political mobilisation and resultant lack of exposure to political struggles as well as politicisation.

Agricultural workers were a major part of bonded labour in India. Bonded labour represents the most extreme form of social and economic oppression. The bondage arises from indebtedness and debt burden and is passed from one generation to another. It was known by different names in different parts of the country, such as Hali in Gujarat, Kamia in Bihar, Harwaha in Madhya Pradesh, Gothi in Andhra Pradesh, Jeetha in Karnataka and so on.

Many migrant labourers in the Green Revolution areas were forced to submit themselves due to their helpless situation, to various restrictions imposed by the land owners, virtually reducing them to bonded labourers. In the feudal rural structure, the landless families still depend on landlords and rich farmers for their livelihood and are forced to work on their dictates. Agricultural workers may be legally freed from bonded labour, but in reality, a large number are bound by contracts at meagre wages to escape the clutches of unemployment.

After the country’s Independence, the number of agricultural workers has grown rapidly, especially after the implementation of neoliberal economic policies over the past three decades. The overall number of agricultural workforce in the four decades from 1960 to 2001 consisted of more cultivators than labourers.

However, in Census 2011, for the first time in Indian history, this trend was turned around. The Census revealed that the share of cultivators was less than half (about 45%) and that of agricultural labourers was close to 55% in the total agricultural workforce. In real terms, the total number of cultivators, both main and marginal, is 11,86,69,264, whereas agricultural workers are 14,43,29,833. Back in 1961, there were about 33 labourers for every 100 cultivators; in 2011, there were now about 121 labourers for every 100 cultivators.

There are various reasons for the increase in number of farm workers, but the important reason is the pauperisation of farmers, especially after the implementation of neo-liberal economic policies. With the implementation of these policies, State support for agriculture has reduced, resulting in an exorbitant increase in input costs and no guarantee of minimum support price (MSP) or procurement for agricultural produce.

Hence the whole agriculture production process has become more uncertain. Earlier, weather was the only uncertainty for farmers. They were afraid of drought and rain, but now market uncertainties are harsher than those of nature. When profit is the only guiding force, the lives of agricultural workers hardly stand anywhere.

This persistent agrarian crisis is displacing small and marginal farmers and forcing them to leave agriculture, as it fails to support the livelihood of their families. Studies based on NSSO (National Sample Survey Office) data reveal that lakhs of small and marginal farmers are forced to sell their land, leave cultivation and join the ranks of manual workers.

Not only farmers, small artisans, too, lost their work and are forced to work as agricultural labourers or take up menial work.

The fact that the number of agricultural workers is now more than the cultivators means that dependence on wage labour is more than that on land. Along with the increased number of agricultural workers, employment opportunities in agriculture are not growing. At the same time, the indiscriminate use of labour displacing technology has further reduced the working days on farms. The overall unemployment in rural India has increased.

The increasing number of agricultural workers and decreasing agricultural work has created new kinds of crises in rural India. Unremunerative agriculture, characterised by low returns from farming activities, remains the primary factor compelling agricultural workers and small farmers to migrate in search of improved livelihoods.

Read Also: ‘Modi’s Guarantee’: The Grim Reality of Rural & Agri Workers

The steep decline and crisis in the agriculture sector has caused the youth to migrate for their livelihood. Under these conditions, growing numbers of workers are forced to search employment in non-agricultural sectors. The economic crises and the burgeoning unemployment in cities are creating an increasingly complex situation.

This situation is forcing manual workers in rural India to various kinds of work for survival.  Most agricultural workers are not restricted to specific fields throughout the year. Almost all rural manual workers participate, to different degrees, in agricultural work. They are doing multiple tasks at different times, ranging from MGNREGA work to brick kiln work and from agricultural labour to industrial work in the nearby small towns.

Hired workers in agriculture and in non-agricultural tasks no longer constitute two distinct sections of rural workers. Wage-workers in agriculture work in a wide range of non-farm tasks as well, including as migrant workers in urban areas. These workers nevertheless retain a partial agricultural and rural character, and are, in important respects, distinct from the urban proletariat. It means that there is a growth of rural workers in big numbers who are working in different sectors, but they are still linked to agriculture.

The majority of agricultural and rural workers have been earning wages below the recommended minimum levels. Wages for major agricultural and non-agricultural occupations declined at an annual rate of 3% between 2013-14 and 2018-19.

Amid low incomes and high unemployment rates, the lives of agricultural workers greatly depend on welfare schemes and public sector institutions of the social welfare state. However, the very concept of such a welfare state is opposed by both foreign and domestic capital. With the exception of Kerala’s Left Democratic Front-led state government, public sectors, such as healthcare and education, are undergoing rapid privatisation. Most social welfare pensions are being transitioned into targeted schemes, leaving large segments of rural India excluded. Even the right to work under MGNREGA is being deliberately weakened.

The crisis has reached such severity that agricultural workers are increasingly driven to suicide, reflecting the dire conditions they endure. According to the National Crime Records Bureau report, a total of 40,685 farm labourers have taken their own lives since 2014.

When the work profile and living conditions of agricultural workers is changing, a new class of rural affluent people has emerged in villages. This class has emerged specifically in the past three-and-a-half decades of implementation of neoliberal economic policies. They have been beneficiaries of neo-liberal policies. They are also typically the first to take advantage of possibilities for higher education and modern organised sector jobs.

Apart from acquiring prime agricultural land, they have also invested capital in non-farm businesses. Various studies indicate that many landlords and big capitalist farmers are also involved in lucrative business activities. These include moneylending, operating grain mills and dairies, trading and speculating on crops (like food grains, horticultural, and forestry products), real estate, construction, cinema halls, petrol stations, hotels, transport services, leasing farm machinery, and running private educational institutions.

Education, in particular, has emerged both as a revenue stream and a means to consolidate social influence. This also generates income through financial investments. Beyond economic activities, these families actively seek influence in State power structures, such as local governance bodies (e.g., panchayati raj), state and national legislatures, law enforcement, and the legal sector.

A defining trait of this class is its political dominance in rural and semi-urban areas, exercised through alliances with bourgeois parties. Serving as a pillar of the ruling-class political machinery in villages, this group wields significant control over state institutions. It shapes the implementation of rural welfare schemes—even those targeting marginalised groups—often diverting benefits toward its own interests. This includes programmes like MGNREGA, which they heavily influence. Additionally, powerful contractors, leveraging authoritarian practices tied to their feudal heritage, aggressively maximise profits through exploitative means.

It is also our organisational experience that workers in rural India often shift between agricultural and non-agricultural roles depending on the season. Despite this fluidity, their livelihoods remain rooted in agriculture, with most originating from landless labourer households. Serious efforts are required to unite and organise workers with diverse profiles. It is important task to organise, unite all these workers and wage struggle against the ruling class in rural India.

Crucially, workers’ struggles in rural areas extend beyond workplace demands. They face systemic issues, such as landlessness, stagnant wages, exclusion from welfare schemes, gender and caste-based oppression. This underscores the need to unite rural workers, including migrant workers, both at source and destination of migration, as their common class identity, as the rural proletariat transcends the diversity of their temporary roles. 

A united and organised rural proletariat can build powerful struggles against the ruling nexus of the rural rich and Hindutva forces, paving the way for a qualitative shift in the balance of power in rural India.

The writer is Joint Secretary, All India Agriculture Workers Union. The views are personal.