## Calculations and measurements were indispensable tools for the world’s first ruling classes, argues Sean Ledwith

Karl Marx’s writings on economics, history and politics are, of course, his greatest legacy to the revolutionary movement, and the cornerstone of the theoretical tradition he pioneered, along with his great collaborator Friedrich Engels. Less well known, however, is that Marx had a lifelong passion for mathematics and that, typically for such a prodigious intellect, wrote a substantial amount about a subject which he regarded as essential to a ‘conception of nature which is dialectical and at the same time materialist.’ Marx’s major works such as the three volumes of *Capital* noticeably feature complex mathematical formulations on a regular basis as he seeks to identify a solid intellectual foundation for the principles of scientific socialism. In a letter to Engels in 1858, Marx wrote: ‘In elaborating the principles of economics I have been so damnably held up by errors in calculation that in despair have applied myself to a rapid revision of algebra. I have never felt at home with arithmetic. But by making a detour via algebra I shall quickly get back into the way of things.’

#### Marx’s maths

Aside from the core texts of Marxism that have become familiar to millions since his death, the founder also studied specific aspects of pure maths that piqued his intellectual curiosity and which he saw as a challenge for a theory which aspired to encompass the full spectrum of human thought. These works form the content what has become known as the *Mathematical Manuscripts* which were unpublished in his lifetime, but eventually surfaced in Russia in the 1930s. Translations into French and English only became available in the 1980s, and there are now versions in multiple languages. Characteristically, Marx was fascinated by how the history of mathematics has been shaped by socio-economic factors which subvert the common reputation of the subject as being one based on a quasi-mystical realm of autonomous thought.

He discusses, for instance, how Isaac Newton’s pioneering work on calculus in the eighteenth century was held back by the great scientist’s religious faith and refusal to embrace materialistic conclusions fully. As Marx puts it: ‘With the help of secret or evident metaphysical presuppositions, which in their turn led to metaphysical and non-mathematical consequence; what happens is a forcible destruction of certain magnitudes, blocking the path of deduction.’ Going back further in time, Marx noted how mathematics in Ancient Egypt ‘determined the height of the pyramids in terms of the length of their shadows itself’, but the restrictions imposed by the priesthood of that society prevented the subject progressing to a higher level of abstraction. He also discusses the work of the famous figures of ancient Greece such as Pythagoras and Archimedes, along with lesser known thinkers such as Diophantus, describing them collectively as ‘our teachers in mathematics’*.*

#### Dirk Struik

We only have access to this part of Marx’s output partly thanks to the work of Dirk Struik, a remarkable Dutch Marxist scholar and the most important successor to the founder in terms of developing a distinctly socialist approach to the history of mathematics. Struik was born in Rotterdam in the last decade of the nineteenth century and carved out a brilliant academic career as the first member of his family to make it to higher education. His brilliance at mathematics and science took him to the University of Gottingen in Germany in the 1920s, which at the time was one of the great intellectual centres of the world, frequented by geniuses such as Einstein, Rutherford and Marie Curie. Struik also encountered Norbert Weiner, the father of cybernetics, who invited him to take up a post at the Massachusetts Institute of Technology in the US. Struik accepted the offer and would remain in post as Professor of Mathematics up to his retirement in 1960.

Growing up in a Europe dominated by the aftermath of the October Revolution, Struik was attracted by the explanatory power of Marxism, joining the Dutch Communist Party in 1919 where he developed a lifelong commitment to the revolutionary cause. In 1947, he gave a lecture on Lenin’s *State and Revolution* to an audience in Boston – unusual to say the least for a US Professor of Mathematics in the Cold War era. One of his students at MIT commented: ‘From the very beginning, personally and professionally, and continuing even now, Professor Struik’s great concern for people in oppressed situations has been the backdrop for all his activities – has informed his life and work, in fact, even when it brought him hard times.’ He also pioneered the area of study that has now become known as ‘ethnomathematics’; that is to say, the recognition of the theoretical insights of non-Western civilisations in the field, especially those in China and India. At one point, Struik observed that the 3Ms had been the great passions of his life: ‘Marriage, Mathematics and Marxism’*.*

#### 3Ms

On a visit to the USSR in 1930s, he came across Marx’s *Mathematical Manuscripts* and was hugely impressed by their erudition and insight. Struik was entrusted with a copy of the manuscripts and would play a key role in their coming to light in the West decades later. The only serious interruption to Struik’s illustrious career came when he was denounced by the House of Un-American Activities Committee led by the infamous Senator Joe McCarthy. His left-wing connections inevitably brought him to the baleful attention of the US Congress at the height of its red-baiting phase, and in 1951 he was suspended from his academic post for supposedly advocating the overthrow of the American government. Struik was forced to testify in front of McCarthy in Washington but defiantly refused to answer over two hundred questions, pleading the Fifth Amendment, like so many other US leftists in the same predicament.

Eventually McCarthy was discredited as corrupt and fraudulent, and Struik was able to resume his academic career in Boston in 1954 where he would live to the age of 106. His most influential publication is *A Concise History of Mathematics*, first issued at the end of World War II and still in print up to the 1980s. This volume remains the essential starting point for anyone interested in a Marxist slant on the subject and has been translated into nearly twenty languages. One of his colleagues at MIT has noted:

‘As an historian of mathematics, he is particularly important because of his great and influential book … beautifully balanced between technicalities and generalities, translated into uncountable languages, most recently Persian. With this book and his historical scholarship, Struik has become the instructor responsible for half the world’s basic knowledge of the history of mathematics.’

#### Maths in Mesopotamia

As a Marxist, Struik in the *Concise History* was intrigued by the extent to which the evolution of mathematical thinking over millennia has been affected by the socio-economic framework of particular social formations. He wrote:

‘we have been able to give a fairly honest description of the main trends in the development of mathematics throughout the ages and of the social and cultural setting in which it took place. He begins with an account of how the first class societies that developed in ancient Mesopotamia, around 5000 BCE mark the beginning of true mathematics: Counting by fingers, that is, counting by fives and tens, came only at a certain stage of social development. Once it was reached, numbers could be expressed with reference to a base.’

The Sumerian civilisation on the banks of the Tigris and Euphrates rivers was the cradle of the neolithic revolution which marks the epochal transition from a hunter-gatherer society to an agriculture-based one which can generate a significant social product to sustain a larger human population. Struik explains how mathematics emerged as a necessary tool for the new ruling classes that would exploit and benefit from this transformation of the means of production:

‘Oriental mathematics originated as a practical science to facilitate computation of the calendar, administration of the harvest, organization of public works, and collection of taxes. The initial emphasis was naturally on practical arithmetic and mensuration. However, a science cultivated for centuries by a special craft, whose task it is not only to apply it but also to instruct its secrets, develops tendencies toward abstraction. Gradually it will come to be studied for its own sake.‘

#### God of Maths

Inevitably in this era of growing class stratification, mathematical competence became one of the characteristics of the elite, who had the leisure time and exemption from manual work to dedicate themselves to the study of numbers. The ultimate outcome of this process was the reign of King Shulgi of Ur who, in 2074 BCE, presided over what modern scholars have labelled as the ‘first mathematical state’. Shulgi issued a directive to the overseers of his kingdom that they must be accountable for the monthly output of each one of their labourers. Any overseer who could not demonstrate the appropriate contribution would have to pay the shortfall himself, and if the overseer died in deficit, that would be payable by his surviving family. Shulgi’s power expanded to the point where he self-declared as a mathematical god and his subjects were expected to sing hymns to his prowess. One of these hymns contained the immodest words of Shulgi himself:

‘When I was young I learned at school the scribal art on the tablets of Sumer and Akkad. Among the high-born no-one could write like me. Where people go for instruction in the scribal art there I mastered completely subtraction, addition, calculation and accounting.’

The link between early mathematics and social context is also apparent in ancient Egypt, where the formidable power of the Pharaonic dynasties over multiple millennia was founded on control of the annual inundation of the Nile. This facilitated the exceptional fertility of the land on either side of the river, allowing the royals and priests to construct their elaborate religious superstructure, replete with Pyramids and temples, of divinely inspired authority. Most of our understanding of Egyptian mathematics comes from the *Rhind Papyrus,* dated to around 1650 BCE and now in the British Museum which was created by a scribe called Ahmos. The document contains numerous arithmetical exercises designed to teach essential calculations, such as gauging the depth and water levels of the Nile. Ahmos’ text also includes instruction on the techniques of working out fractions, which were essential for calculating the division of land, particularly in cases where the inundation had led to disputes between farmers. Struik comments on how the rigidity of the superstructure in these early class societies profoundly affected the trajectory of their development:

‘We know, for instance, that candidates for examination had to display a precisely circumscribed knowledge of the Ten Classics and that this examination was based mainly on the ability to cite texts correctly from memory. The traditional lore was thus transmitted from generation to generation with painful conscientiousness. In such a stagnant cultural atmosphere new discoveries became extraordinary exceptions, and this again guaranteed the invariability of the mathematical tradition. Such a tradition might be transmitted over millennia, only occasionally shaken by great historical catastrophes.’

Since Struik wrote, it has become increasingly clear that there was more technological development in these ancient societies, including Egypt, than used to be understood. They did, however, show a remarkable capacity to recompose themselves with little superstructural change after periodic crises over the millennia.

#### Revolution in the Aegean

The one crucial location where mathematics was able to make the crucial leap as a subject from primarily an empirical tool to become a form of abstract philosophising was, of course, ancient Greece. Great mathematical minds such as Pythagoras, Archimedes and Euclid remain central to the subject today, and represent one of the great intellectual achievements of the whole epoch of Antiquity. Again, Struik’s Marxist perspective enables him to see that these and other great thinkers did not operate in a political vacuum and that their insights were premised on a quite different type of social formation to the ones that preceded it in Mesopotamia and Egypt.

Situated at the crossroads of these and other great Middle Eastern civilisations, Greece was well placed to take advantage of the upsurge of commercial and trading activity that took place in the eastern Mediterranean following the replacement of bronze-based technology by iron around 1000 BCE. The largely mountainous terrain of the country protected it from the envious ambition of powerful monarchies, especially the Persians, and a spirit of curiosity and independence flourished on the hundreds of island-based city states dotted around the Aegean. Struik writes of the social forces that spearheaded this revolutionary era:

‘During the Seventh and Sixth Centuries B.C. this merchant class won ascendancy and had to fight its own battles with the small traders and artisans, the demos. The result was the rise of the Greek polis, the self-governing city state, a new social experiment entirely different from the early city states of Sumer and other Oriental countries … This new social order created a new type of man. The merchant trader had never enjoyed so much independence, but he knew that this independence was a result of a constant and bitter struggle.’

#### Pythagoras and his theorem

It was on one of these islands, Samos in the eastern Aegean, that the shadowy figure of Pythagoras was born around 580 BCE. Struik points out that the famous theorem forever associated with his name (and cursed by countless generations of schoolchildren) was known to be true by Babylonian and Egyptian mathematicians, but only Pythagoras formulated it as a characteristically Greek piece of abstract reasoning: in a right-angle triangle, the squares of the two shorter sides are equal to the square of the longest side. The eternal veracity of this and similar axioms led Pythagoras to the view that mathematics represents a higher plane of existence and that the physical world we inhabit is a mere product of that foundational realm. Unfortunately for Pythagoras and the philosophical school he founded, this quasi-religious faith in the internal consistency of mathematics was plunged into crisis by their discovery of irrational numbers; that is to say, numbers which cannot be written in the form of fractions, most famously π (Pi).Struik speculates that this crisis in mathematics was affected by a parallel trauma in the stability of the Greek political system:

‘This crisis originated in the later period of the Peloponnesian war, ending with the fall of Athens (404 BCE). We may then detect a connection between the crisis in mathematics and that of the social system, since the fall of Athens spelled the doom of the empire of a slave-owning democracy and introduced a new period of aristocratic supremacy— a crisis which was solved in the spirit of the new period.’

For all his fascination with the social origins of mathematics, and his goal of dispelling the subject’s sometimes mystical reputation, Struik never lost sight of its compelling power to inspire awe at the power of the human intellect. One of his students reflected on Struik’s influence: ‘He taught mathematics not as some esoteric mystery but as practical commons sense. And yet at the same time he gave us a glimpse of the sheer beauty of it. It was at this time that I understood the line, ‘Euclid alone has looked on beauty bare’.

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