Mathematics falls from grace – Part 1

Science, Eugenics, and Statistics

Much of what is taught as higher Mathematics today was developed in the 16th and 17th centuries, as part of a tremendous flowering of natural science that ushered in the rule of the bourgeoisie. Spurred by the expansion of marine navigation and trade, of mining and metallurgy, and of military technology, first Astronomy and then Mechanics – the study of the physical displacement of bodies in space – were the central sciences of the day. The generalisations of Isaac Newton brought these two disciplines together under a single set of natural laws. The mechanical conception of nature these sciences developed proved very fruitful in many fields – for example, in human physiology, by the discovery of circulation of the blood, modelling the body as a hydraulic system, with a pump, valves, and so on. It became the prevailing metaphor for nature in European science for the next three hundred years.

Isaac Newton, physicist and mathematician. Portrait by Godfrey Kneller, 1689

The philosophical grounding of this science was empiricism: it based itself on the empirical evidence of the senses, challenging and defeating medieval scholasticism in one field after another. The empirical scientists were thoroughly materialist in tendency – that is, their starting assumption was that a material world existed outside of human consciousness, and that human knowledge was a reflection of that objective reality. And while relying on the evidence of the senses, they probed beneath the superficial appearances to discover the essential processes involved. Beneath the appearance of the sun rising, moving across the sky and setting lay the deeper truth that the earth was revolving on it axis.

The search for cause and effect in nature was essential to this deeper probing: seeking to explain the processes of nature without recourse to human or superhuman intervention. “We are to admit no more causes of natural things, than such as are both true and sufficient to explain their appearances,” declared Newton.

However, the empirical scientists operated within the framework of the fixed, immutable categories of formal logic: time, space, the separate forces of mechanical force, heat, electrical, etc. Cause and effect were likewise conceived in the simplest terms, singular and mutually exclusive. ‘Force’ became almost a synonym for ‘cause.’

The mechanics of the day was also deeply quantitative in character – measurements and the relationships between quantities were its concern, to the exclusion of other aspects of reality; the ultimate goal was to express these relationships in a mathematical formula.1

Rene Descartes, whose analytic geometry and algebra accelerated physical sciences. Portrait by Frans Hals, 1648

Thus, Mathematics developed rapidly in this period along with the other sciences, as the indispensable science of counting and measurement, the ‘handmaiden’ of Mechanics, Physics and Astronomy, as Engels describes it [Introduction to Dialectics of Nature, p23 ]. Logarithms, developed by John Napier in 1614, facilitated calculation in the very large magnitudes needed for astronomy. The analytic geometry and algebra developed by Descartes permitted the calculation not just of unknown but of variable magnitudes, and thus became a crucial tool of generalisation. With differential calculus Mathematics began to burst the bounds of fixed formal categories and become the mathematics of matter in motion.

John Napier, inventor of logarithms, in 1616

Yet mathematical theory remained distinct from all the other sciences, in that it continued largely in a scholastic framework. While helping to advance the physical sciences and the mechanical worldview, Mathematics itself remained firmly rooted in idealist metaphysics: it claimed to be entirely derived from unprovable ‘first principles’. The very subject matter of the science of counting and measurement seemed to consist of abstractions rather than nature itself. As the philosopher George Berkeley put it, “Number is entirely the creature of the mind.”

George Berkeley, philosopher of subjective idealism, in 1727.

In fact, contrary to Berkeley’s assertion, even mathematical abstractions have their origin in the real world – the decimal number system in the biological fact of humans having ten fingers; the cylinder in the shape of a tree trunk, the cone in the shape of a volcano, the parabola in the path of a stone thrown into the air. But of all the sciences, Mathematics was least affected by the prevailing – though still largely unconscious – philosophical materialism. It was the only science which remained in a state of grace, at God’s right hand.

So it was for several hundred years, and  as far as the Mathematics that proudly bears the label ‘pure’ is concerned, so it remains today. It appears to unfold according to its own internal logic, and nothing else.

Nonetheless, a materialist Mathematics has arisen, still an infant science, but one that is gaining strength all the time (even, to a certain extent, at the expense of the old Mathematics in educational institutions.)2 It states plainly that its point of departure is not axioms of thought, but empirical observations of the real world. Like the other sciences, it deals in messy approximations, hypotheses, and judgment calls, and subjects itself to the rigours of verification rather than ‘proofs’ of its perfection.

Francis Galton, father of modern Statistics, and of eugenics, in 1890s

This materialist Mathematics is called Statistics. Modern Statistics really begins in the late 19th century with Francis Galton’s development of the scatterplot and investigations of bivariate data. Up to that point no more than a descriptive tool, Statistics now became an investigative tool, capable of answering the question: is there a relationship between two phenomena?  Is there a relationship between cigarette-smoking and lung cancer? Between atmospheric temperature and the distribution of animal species? Whole new fields of investigation were opened up by this powerful new tool.

The precursor to this development was the exploration by Gauss and Laplace of what is now known as the Normal Distribution. When the physicist Maxwell demonstrated that probability distributions were not just a mathematical tool but also a fact of nature, the idea then arose of continuous variation within and between the categories of nature (such as organic and inorganic, the different states of matter, species, the chemical elements and so on) and with that, the mathematics of Probability came into its own. Probability up to that point had been the ‘handmaiden’ of gambling and insurance; now, in close connection with Statistics, it extended its range to all of nature, and addressed the question of chance and necessity in nature.


When the fall from grace came, it was not by Mathematics aligning itself with the empirically-based sciences, but in connection with the anti-scientific nostrums of eugenics. It is not a coincidence that the three foremost theoreticians of modern Statistics, Francis Galton, Karl Pearson, and Ronald Fisher, were all ardent advocates of eugenics, who developed statistical theory in an attempt to prove its anti-scientific notions. Statistics today still bears the eugenic birthmark, in the terminology of ‘regression.’ In Galton’s original usage, this referred to a supposed hereditary tendency of ‘regression to mediocrity.’

Galton’s lifelong concern was the ‘inheritance of genius,’ especially his own. Galton was a cousin of Charles Darwin, and was fascinated by the fact that genius seemed to run in families. He founded the Eugenic Records Office, which collected pedigrees of Fellows of the Royal Society.

The eugenic idea, essentially the fear that the human race (or more particularly, the white race) was deteriorating in quality, grew very popular in bourgeois circles in the late 19th century and well into the twentieth, in the UK and Europe, and in the United States. It was generated by fear of the rapidly-multiplying proletariat – viewed from above, the surly, uneducated masses. The proletariat was growing disproportionately, both as the inevitable consequence of the development of industry in the advanced capitalist economies of Europe and North America, and also with the spread of colonial domination of the rest of the world, its consequent mass migrations and proletarianization of the peoples of the world.

Families compete in the “Fitter Family” contest, meant to find the most eugenically perfect family. Topeka, Kansas. 1925.

The ideology of eugenics took the form of a malignant outgrowth of Darwinian ideas: according to the eugenicists, the advances of modern industry (and especially the social conquests of the working class) had blunted the effect of the ‘survival of the fittest’ which conditioned the vitality of species in nature. Therefore harsh measures were needed to rid society of the ‘weaker elements’ by restricting their access to reproduction, while encouraging breeding by better quality people… like the middle-class advocates of eugenics themselves, for instance.  Supported by statistical arguments from, among others, leading statistician Ronald Fisher, forced sterilisation of ‘feeble-minded’ and ‘unfit’ individuals was practised widely in the United States in the 1920s and 1930s. It also became a common practice in Germany – well before the rise of the Hitler regime and its eugenic horrors. Eugenic fears took a particularly strong hold in late-nineteenth and early-twentieth-century Britain, as Britain progressively lost its industrial monopoly: how else could it be explained why no Englishman had invented the aeroplane, except by the ‘decline in the national stock’? In the United States, the imagined degeneration of the national stock through immigration of undesirables was the focus of eugenic insecurities.

Although it arose in close connection with advances in the science of genetics, the eugenic movement was essentially a social and political movement rather than a field of scientific investigation. Its class base was not the bourgeoisie – which was relatively secure in its inherited wealth – but rather the professional middle class, the self-styled meritocracy, whose status and economic security depended on social recognition of their merits. Often this meritocracy was liberal in its political sentiments. Eugenics overlapped with middle-class movements for socialism and women’s equality: Bernard Shaw, Havelock Ellis, H. G. Wells, birth control advocates Margaret Sanger and Marie Stopes, and anarchist Emma Goldman were among its supporters. (Daniel J Kevles, In the Name of Eugenics, pp 85-90).  Karl Pearson, protégé and admirer of Galton, was like Galton a capable mathematician and statistician. Pearson moved on the fringes of these circles of social reformers, becoming an acquaintance of Bernard Shaw, Havelock Ellis, and Eleanor Marx, and founding a Men and Women’s Club to discuss the social and sexual emancipation of women. Shaw invited him to join the Fabian Society.

Fabian socialist Sidney Webb commented on a survey that compared birth rates in rich and poor boroughs of London, “…we are, in London at any rate, multiplying most prolifically from our least wealthy stocks…. [the poor boroughs] happen to include not only those containing the greatest numbers of Irish Roman Catholics, but those in which the great bulk of the Jews are to be found.”

A family afflicted with rickets, photographed by the Eugenics Society to demonstrate that their condition is hereditary and could be controlled through selective breeding. (In fact, rickets can be controlled simply by correcting a deficiency of Vitamin D). Photo was used “to illustrate Professor Karl Pearson’s Cavendish lecture, 1912.”

It is not without irony that Karl Pearson, in pursuit of a scientific grounding for such eugenic bigotry, laid the basis for modern Statistical theory. Among other things, Pearson devised an objective measure of the strength of the relationship between two attributes of a bivariate data set, the correlation coefficient r (for ‘regression’). He thereby opened the road for the further development of biology, (especially ecology) and medicine (especially epidemiology) through statistical evidence, as well as probabilistic models in physics, genetics, and other fields. The geneticist JBS Haldane likened the situation to Columbus, who set out for China but discovered the Americas.

The irony didn’t end there, for Pearson was also a philosopher of epistemology, the theory of knowledge. Pearson was very much of the idealist school. Contrary to the materialists, who hold that the natural world is primary and knowledge is an image in the human mind of that objective reality, idealists believe that at least some ideas are innate or from immaterial sources. The primary theorist of an essentially materialist branch of mathematics was thus a conscious and consistent idealist. In this role he was among the philosophers on the receiving end of a sharp polemic from Vladimir Lenin in the early years of the twentieth century. Pearson’s idealist outlook also had important consequences for Statistical theory, which I will discuss in Part 2.


  1. George Novack in Empiricism and its evolution traces this unconscious bias to the economic thought of the merchant capitalists, (p 40-41). When a farmer goes to buy a tool, they are interested in its concrete, qualitative properties, its use-value – will it do the job? But when a merchant buys a tool in order to re-sell it at a profit, they are interested in a more abstract property: its exchange-value, the property which the tool shares with all other commodities. Since ancient times the merchant class has produced some of humanity’s greatest abstract thinkers. Novack’s Empiricism is one of his most brilliant little books, one of the best I know from any author for explaining the social roots of an ideology.
  2. In this writer’s opinion, pure Mathematics holds a priority in most countries’ high school curricula far out of proportion to its actual usefulness in a world where mechanics is no longer the central science. I believe this can be attributed to the false and archaic belief that the idealist formal reasoning of pure Mathematics is the most rigorous form of scientific thinking. In this respect the status of Mathematics resembles the status once held by Latin, the teaching of which was once quite common – as the supposedly indispensable language of higher learning – but which has almost completely died out in the last fifty years.)



One response to “Mathematics falls from grace – Part 1

  1. Pingback: Physics, Materialism, Causality, Statistics – and Vladimir Lenin | A communist at large·

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