Plasticity, maths and history – an intervention in materialist dialectics

To hear experts repeat that our brains are ‘hard-wired for plasticity’ should lead us to ask whether they listen to what they say. Their oxymoron is one sign of the muddle-headedness that pervades thinking about thinking. Because thoughts are intangible they lend themselves to all manner of Philosophical Idealism, from mind-over-matter cures for cancer to beliefs that only numbers are real. Fear of dementia needs to be matched by alarm at the pandemics of mumbo-jumbo.

Hard-wired: the mischief of metaphor
‘Hard-wired’ is no slip of the tongue. Its status as cliché has roots deep in culture. ‘Hard-wired’ sneaks in a biological determinism, code for class, ethnic and gender prejudices, in short, that the poor, chicks and chinks are ‘born that way’, that is, are innately inferior to propertied, white males. That there is no escape from metaphor is no excuse for perpetuating figures of speech that are harmful. No brain can be hard-wired since no stem cell transmutes into metal. Analogies with fibre, chord or thread are closer to the connections in the brain, though even those terms come nowhere near the patterns through which thoughts and emotions are communicated.[1]

In addition, ‘hard-wired’ is detritus from the attempts to install artificial intelligence (AI) in a computer so that it will replicate human thought. Even as a metaphor, ‘hard-wired’ is old technology. The Mac on which I am composing this sentence has none of the cables, valves or punch cards of the first commercial IBMs, UNIVAC, in 1952; those elements have been replaced by a circuitry of transistors and micro-chips.[2]

The AI experimenters failed for the reason that keeps the brain plastic. Computers do not have bodies to interact with the rest of the world. Hubert Dreyfus gave the example of how we respond to hearing that ‘George Washington was in the Capitol’. We know without being told that ‘so was his left foot’. Because we have a body like Washington’s, we do not need to be told that his left foot was also in the Capitol. The computer cannot know that fact without being told.[3] The result is that engineers have to supply all the minutiae that we absorb from our experience as beings in the world.

During decades of chasing AI, Douglas Lenat had fed in millions of items but still had not built a machine indistinguishable from a human respondent, and thus pass what is known as the Turing Test. Current researchers propose a different test that computers are unable to pass:

‘What’s wrong with this picture?’ Solving that simple problem requires having lots of contextual knowledge, vastly more than can be supplied with algorithms that advanced computers depend on to identify a face or detect credit-card fraud. A room full of IBM supercomputers cannot fathom what makes sense in a scene. The evolution of our visual system, our neurological development during childhood and a lifetime of experience enable us to know instantly whether all the components fit together properly: Do the textures, depths, colours, spatial relations among the parts, and so on, make sense? Does the Chicago skyline, seen at a distance from the approaching highway, resemble a burned tree grove emerging from the mist? Answering such questions … would require countless dedicated software modules that no one could build in anticipation of that particular question.[4]

Similarly, computers cannot pick out one thread of speech when two or more people are talking. We manage ‘the cocktail-party problem’ because we acquire our first language while over-hearing conversations.[5]

As an emetic for the mechanistic materialism of AI and the waffle of mind-over- matter, this discussion is grounded in materialist dialectics with a focus on the ideas and experiences of Marx and Engels. The paper proceeds through three stages:

  1. the plasticity of the adult brain within the mutability of the rest of nature;

  2. the expansion of mathematical abilities as an instance of cultural transmission;

  3. ground-rules for writing a materialist history of mathematics as a realm of social labour.

The aim is to stimulate discussion around the pertinence of materialist dialectics to the accumulation of greater relative knowledge in every domain of human endeavour.

Plasticity as the norm
Although the plasticity of the adult human brain is of special interest to our species, that condition is but an instance of the mutability of the natural world. The acceptance by scientists that regeneration also applies to adult neurons followed 250 years of demolishing fixed categories for the rest of the universe. In 1701, the prominent British naturalist Rev John Ray put the orthodox view in The Wisdom of God Manifested in the Works of the Creation: ‘the Works created by God at first, and by Him conserved to this Day in the same State and Condition in which they were first made’. However, Ray’s researches into plants had revealed a complexity which he could fit into this assertion. Variation happens. Ray, therefore, admitted ‘that God uses for these Effects the subordinate Ministry of some inferior Plastick Nature’.[6] He believed that the Deity had limited the effectiveness of this device to secondary aspects of rigid systems and autarkic species.

Distinct approaches arose regarding the extent of mutability. Some conceded penetrability of the orders of nature but within a static whole; others glimpsed how those orders could transmute in a dynamic system. The following sketch does no more than underline the twists and turns in the acceptance of qualitative change as a universal possibility. This survey will place the recent recognition of the plasticity of the adult brain in the protracted struggle to establish evolution as orthodoxy. Three layers of nature are involved: one, the inorganic world of geology and cosmology; two, the organic in plants and animals; three, whether we are part of the animal kingdom. Telescopes and microscopes were the tools that helped to initiate multiple revolutions in science from the solar to the cellular. Beyond this trio is the transforming of modes of production, also touched on.

Cosmology: Although Copernicus had dislodged the earth from the centre of the universe, Kant’s cosmology was more alarming because he speculated on the death of the sun. In geology, nineteenth-century materialists began by upholding uniformitarianism against the catastrophism associated with a Biblical Deluge. Before the 1960s, this mentality delayed the acceptance of continental drift until plate tectonics was identified as the mechanism.[7]

Biology: ‘Grotesque’ derived from the Pagan depiction of the inter-twinings of the vegetable and the animal, the human and the non-human discovered in Roman Grottoes during the fifteenth century. This blurring was important for Renaissance art but suppressed by the Counter-Reformation.[8] ‘The Great Chain of Being’ in most of its expressions did not allow mutations between the orders of nature though its proponents disputed over how to rank us within a pre-ordained hierarchy.[9] Linnaeus strengthened the classificatory boundaries, despite his emphasis on the sex life of plants, which underpinned discoveries regarding their mutability by Darwin, Hooker and Mendel.[10]

Notwithstanding Kant’s views on solar extinction, he clung to Christian doctrine for botany in ‘The Critique of Teleological Judgement’, asserting that it was

quite certain that we can never get a sufficient knowledge of organised beings and their inner possibility, much less get an explanation of them, by looking merely to mechanical principles of nature. Indeed, so certain is it, that we may confidently assert that it is absurd for men even to entertain any thought of so doing or to hope that maybe another Newton may some day arise, to make intelligible to us even the genesis of but a blade of grass from natural laws that no design has ordered. Such insight we must absolutely deny to mankind. But, then, are we to think that a source of the possibility of organised beings amply sufficient to explain their origin without having recourse to a design, could never be found buried among the secrets even of nature, were we able to penetrate to the principle upon which it specifies its familiar universal laws? This, in turn, would be a presumptuous judgement on our part.[11]

Within eighty years, Kant’s teleology was being declared ‘presumptuous’.

In 1856, William Perkin extracted dyes from coal tars instead of from root madda.[12] Perkin went beyond the artificial selection by pigeon fanciers, with whose achievements Darwin introduced his argument for natural selection in On the Origin of Species, three years later. Over millennia, plants had turned into coal but now human ingenuity yielded ‘mauve’, adding a new dimension to mutability. Darwin dyed pigeons magenta to see what effect it had on their mating success, which was nil.[13]

Although it is never possible to isolate scientific discoveries from social attitudes, cultural consequences tell us nothing about natural or industrial processes. In this case, mass production changed attitudes towards purple as a dress colour. The cost of natural dye had been so great as to confine purple to emperors and the like, voiced in the expression ‘born to the purple’.[14] Perkin put purple within reach of most consumers so that it became associated with colonials, the indigenous, the nouveau riche and women of ill-repute.[15] 

Frederick Engels grasped the significance for all the sciences from what Perkin had wrought through ‘experiment and industry’ in chemistry:

If we are able to prove the correctness of our understanding of a natural process by making it ourselves out of its conditions and making it serve our own purposes into the bargain, then there is an end to the Kantian ungraspable ‘thing-in-itself’. The chemical substances produced in the bodies of plants and animals remained such ‘things-in-themselves’ until organic chemistry began to produce them one after another; whereupon the ‘thing-in-itself’ became a thing for us ...[16]

Indeed, even as Kant wrote about the impossibility of a botanical Newton, the agricultural revolutions of crop rotation, deep ploughing and fertilisers had been undermining of his thing-in-itself. Justus von Liebig’s Chemistry and Its Application to Agriculture and Physiology in 1840 argued for the restoration of minerals to the soil to repair its plunder by capitalists, an authority whom Marx and Engels had read before they met and quoted throughout their lives.[17] Farmers improved seed cultivation, artificially selected their livestock and grafted fruit-trees before the controversies around Mendelian genetics versus natural selection led to the neo-Darwinian synthesis from the late 1920s.[18]

Voicing a dialectical approach to the universe, Engels in 1875-6 drew on his profound knowledge of scientific discoveries since the Renaissance to attack the notions of ‘the absolute immutability of nature’ that had ‘dominated the entire first half of the nineteenth century, and in substance is even now still taught in all schools’.[19] A decade later, he rejoiced at the obliteration of teleology along with its concomitant ‘rigidity and absoluteness’:

And since biology has been pursued in the light of the theory of evolution, in the domain of organic nature one fixed boundary line of classification after another has been swept away. The almost unclassifiable intermediate links are growing daily more numerous; closer investigation throws organisms out of one class into another, and distinguishing characteristics which had become almost articles of faith are losing their absolute validity; we now have mammals that lay eggs … [20]

Engels was thinking of the platypus, long known as the ‘Australian Paradox’. An 1884 report confirmed that the creature was neither wholly reptile nor entirely mammal, a glimpse of the transmutation of species in action.[21]

By the start of the twentieth century, species evolution had been widely accepted by scientists. The realm for resistance was the human brain. Where Descartes had proposed dualism to escape the strictures of Christian doctrines, its revival snuck the teleological back into science during the eclipse of Darwinism.[22] For example, a 1914 Introduction to Lyell’s 1863 The Geological Evidence of the Antiquity of Man rejoiced that

a peculiarly dreary kind of materialism … has been gradually replaced by speculations of a more positive type, such as those of the Mendelian school of biology and the doctrines of Bergson on the philosophical side.

In Creative Evolution (1907), Bergson rejected Darwinism in favour of an elan vital, as part of a Catholic revival in France.[23] The retreat of ‘shame-faced materialists’ before the discoveries of sub-atomic physics and a consequent resurgence of fideism provoked Lenin to hit back with Materialism and Empirio-Criticism in 1908.[24]

A brain-for-itself?
Fifty years on, the man who became the lead researcher into the regrowth of adult brain cells, Salk Institute professor Fred Gage, was being taught to accept ‘as a dogma’ that ‘once we had reached maturity the brain was fixed and immutable’. The scientific consensus in the 1960s disparaged contrary evidence from Joseph Altman regarding rats. In the early 1990s, the generation of new cells became okay for some birds (‘bird brains’) and for apes, but not for Homo sapiens. Fearing peer ridicule, Gage and his team withheld their results until 1998 when they announced a double discovery. Not only does regeneration persist throughout life, but it is stimulated by interaction with environments, cultural and physical:

Neurogenesis is not static, it doesn’t just keep occurring but it is affected quite dramatically by experience so that enrichment, having greater experiences and more complex experience, the number of new neurons that are born and survive is dramatically increased by virtue of environmental enrichment. And physical activity alone, basically repetitive walking for a period of time each day over a period of 30 days, can result in up to a doubling or tripling in some cases of the number of cells that are dividing and adding into the circuitry.

Activity and learning are complementary, the first making new stem cells, the second prolonging their survival.[25] Sodoku needs physical exercise to stimulate the production of Serotonin as neurotransmitter.

Hence, the brain does not change itself as Norman Doidge pretends in the title of his best-seller, a fancy he refutes in an Appendix on ‘The Culturally Modified Brain’.[26] The brain is not changed by acting on itself any more than Yuri Geller’s brain, by itself, can bend a spoon. All that that organ will do if left to itself is to rot. A living brain is changed through its engagements with the rest of its body and with the outside world. As Marxist bio-chemist, Steven Rose puts it:

… systems do not exist in the brain in abstract; they are called into play by actions, and are as transient and dynamic as the actions themselves.

That is why ‘procedural memory (remembering how to do something) is the most robust’.[27]

In 2002, Gage introduced a series of papers surveying the demolition of the dogma that the adult nervous system could not generate new neurons:

One reason why neuronal replication in the adult was considered unlikely was the complexity of most neurons; with their highly branched dendrites and polysynaptic axonal combinations, they were considered to be terminally differentiated and unable to re-enter the cell cycle and divide, strictly from a mechanistic view.

The dominant metaphor of brain function as a computer with fixed circuits had blocked research into how new cells could ‘functionally integrate into the brain without disrupting existing circuits’. Gage realises that any extension of his team’s discoveries to more sections of the brain will require ‘a reconceptualisation of neural plasticity that incorporates structural plasticity’.[28]

None of this research would have surprised Engels who had championed mutability all along the line. Nor would he have been surprised at how long it had taken to add the adult brain to the catalogue of natural objects accepted as mutable. The privileging of the human brain was the last gasp of a special creation for our kind, with Darwinism admissible for every species but our own.[29] Darwin had accommodated this prejudice by delaying his endorsement of the transmutation of species through natural selection to humankind until The Descent of Man in 1871.[30]

That was not the end of it. Evolution was acceptable for the human body and even the brain but not for the mind, also known as the soul. Many of those who abandoned faith in an afterlife with the Almighty clung to the power of their own intellects, as Engels noted in ‘The Part Played by Labour in the Transition from Ape to Man’:

All merit for the swift advance of civilisation was ascribed to the mind, to the development and activity of the brain. Men became accustomed to explain their actions as arising out of thoughts instead of their needs (which in any case are reflected and perceived in the mind); and so in the course of time there emerged that idealistic world outlook which, especially since the fall of the world of antiquity, has dominated men’s minds. It still rules them to such a degree that even the most materialistic natural scientists of the Darwinian school are still unable to form any clear idea of the origins of man, because under this ideological influence they do not recognise the part that has been played therein by labour.[31]

By ‘labour’, Engels intends all human activities, physical and mental, which he understands to be inseparable.

Behind this privileging of the brain lurks an ache to reinsert the soul by making the mind distinct from and superior to the brain. The modern version of this argument began in the 1640s when Descartes got out from under the repression of the Church by proposing a line between using mathematics to investigate extensions in space (bodies) but not thoughts, which were unquantifiable.[32] He gave the body, including the brain  and the pineal gland, to scientists and left the ‘soul’ to the Inquisitors.[33]  Several commentators suspect that this ‘dualism’ was a maneouvre to avoid his being sent to the stake as a libertine as much as it was an ontological or epistemological conviction.

In physiology, the anatomist Thomas Willis (1621-75) led the way in recognising the brain as the site of the intellect. He thought of the human being as a ‘double-soul’d animal’, with a material mortal soul for sensations and an immortal soul for intellect and morality. Like many contemporary commentators, Willis misunderstood Descartes’ claim that other animals did not possess rational souls into a belief that they therefore also lacked sensitive souls.[34] Descartes’ description of other animals as automata is now misinterpreted because of the modern sense of an automata as an unthinking machine when in the 1640s the term meant any process that did not require outside drivers.[35]

Experiments on other animals contributed to the advances made by Willis and Descartes but it was not then possible to practice vivisection on fellow human beings. A soldier with a head wound from the American Civil War allowed surgeons to observe something of his brain’s functioning. By the 1890s, Santiago Ramon y Cajal and Camillo Golgi independently identified the synapse, but concluded that cellular regeneration was impossible.[36] Much of what we know about brain function has come in the past twenty years from non-invasive procedures. A rush to extrapolate from that ill-digested data to the mind and to consciousness is a fount of speculative philosophising.

Gage recalled the fear that ‘if the brain were changing too much all the time we may lose that sense of identity or may not be able to retain memories’.[37] In studying the links between identity and memory, Steven Rose tracked processes of interaction among billions of cells and their trillions of connections to confirm that memories are retained only by being remade, and thus altered ever so slightly each time they are recalled. He rejected a notion that each memory had a neuron of its own: one molecule, one memory. Rose endorsed a variant of plasticity when he preferred ‘patterns’ to ‘pathways’, In addition, he challenged eh fixedness of memory:

Indeed there is good evidence that the act of recall, retrieval, evokes a further biochemical cascade, analogous to, through not identical with, that occurring during initial learning. The act of recall remakes a memory, so that the next time one remembers it one is not remembering the initial event by the remade memory from the last time it was involved.

… What they absolutely are not is ‘stored’ in the brain in the way that computer stores a file. Biological memories are living meaning, not dead information.

However, Rose insists that all explanations remain provisional because, despite an avalanche of information about the operation of the brain, researchers still know less than 5 percent of what there is to be known about consciousness.[38]

The notion of personality as a fixture contrasts with Marx’s insight when he criticises Feuerbach for resolving ‘the essence of religion into the essence of man’. Not so, says Marx: ‘But the essence of man is no abstraction indwelling in each separate individual. In its reality it is the ensemble of social relations’.[39] In opposition to bourgeois apologists who treat the individual, isolated outside history, as a concrete instance of humanness, Marx considers that ‘idiot’ to be the true abstraction.

To the extent that our species now has ‘a’ nature, it derives from interactions of the physiological with the socio-cultural. We become what we do, as Marx illustrated with the instance of first-language acquisition:

Man is a Zoon polikikon [social animal] in the most literal sense: he is not only a social animal, but an animal that can be individualised only within society. Production by a solitary individual outside society – a rare event, which might occur when a civilised person who has already absorbed the dynamic social forces is accidently cast into the wilderness – is just as preposterous as the development of speech without individuals who live together and talk to one another. It is unnecessary to dwell upon this point further.[40]

Marx’s account of labour as a collective and natural practice demolished a related prop to individualism – the Robinsonade.[41] To avoid the transitory nature of capitalism, its apologists explained its development by extrapolating the global expansion of capital from an isolated individual like the fictional Robinson Crusoe.[42] However, Marx was naïve to think that ‘this inanity’ of a self-forming individual had been disposed of. It persists for reasons that he indicated in his critique of political economy.

Labour is social in the sense of being shared but it is not confined to human society, for, as he reiterated, ‘labour … is only the manifestation of a force of nature’.[43] Because we are discussing natural sciences, it is important to be clear about what Marx and Engels meant when they called their analysis scientific. They were in no doubt that absolute truths existed for natural phenomena despite the inevitably of our relative knowledge of them. Those absolute truths included the qualitative changes in the geo-physical universe and life forms discussed above. Nothing that human beings thought could alter those actualities.[44]

Different criteria apply to human society and to our understanding of it. Marx’s critique of political economy demonstrated that both the capitalist mode of production and ideas about it were historical, that is, transitory:

But this definite, specific, historical form of social labour which is exemplified in capitalist production is proclaimed by these economists as the general, eternal form, as a natural phenomenon, and these relations of production as the absolutely (not historically) necessary, natural and reasonable relations of social labour.[45]

Marx lambasted the ideology that capital was natural and eternal, a fallacy by which ‘the whole of history can be brought under a single great natural law’, though he had to admit that here was a ‘very impressive method – for swaggering, sham-scientific, bombastic ignorance and intellectual laziness’.[46] Hence, the mass of writing against Marxism as a science has missed the point of what Marx and Engels understood by science in regard to human affairs.[47] Their approach opened the way to a science of history and political economy because they recognised the gulf between the laws of nature and those of social relationships.

During the nineteenth century, two strands of anthropology challenged the ideology of eternal social categories. The first step was to look from gods to men, as Feuerbach had done when he saw that we make gods in our own image and likeness.[48] The second step came with the systematic reporting of pre-class societies, as in the work of Lewis Henry Morgan and on Australian Aborigines.[49] Marx and Engels welcomed these developments, with the latter writing Origins of the family, property and the state to sketch qualitative changes in human affairs. Marvin Harris contends that this materialist outlook was not in Morgan but resulted from ‘skillful editing’ by Engels who supplied Marx with

a suitably materialist Morgan. But it is Engels and not Morgan who presents the first clear-cut periodisation of prehistory, based on the mode of production … The modifications which he introduced were on the whole quite sound and imported to Morgan’s scheme a logical coherence that it lacked in the original.[50]

What matters is the acceptance of the transitory nature of social organisations - not who said what first.

A larger controversy than the transitory nature of modes of production is whether hominisation has altered the relations between our species and the rest of nature. The need for materialists to refute faith in a special creation distracted many of them from the unique position that humankind had reached through evolution. The fact that our species articulates an explanation of its origins proves that a factor had been added to how Homo sapiens interacts with nature. The co-discoverer of natural selection, Alfred Russell Wallace, advocated investigating this uniqueness. His hypothesis was dismissed by equating it with the discredited notion of acquired characteristics[51] and because of his attachment to Spiritualism following the death of his eight-year old daughter.[52]

The break-out to which Wallace drew attention had been attained through natural selection, even if he suspected otherwise. The flowerings that this peculiarity made possible no longer depend on that mechanism. By 1963, Julian Huxley accepted that human culture had so altered our relations with our environment that biologists should consider natural selection to be inadequate:

Man is also unique in having markedly reduced the impact of natural selection on the survival of individuals by artificial means, such as medical care and sanitation.

The human situation is so different from the biological that it may prove best to abandon the attempt to apply concepts like natural selection to modern human affairs. All the evolutionary differentials now operating, whether in survival or in reproduction, have their roots in the special psychosocial character of human evolution. It would seem best to accept the fact that a novel form of selection, psychosocial selection, or more simply social selection, is now operating …[53]

Huxley’s highlighting the importance of culture is not a claim for an alternative physical device for the transmutation of species. He had denounced any notion of inheriting acquired characteristics associated with Soviet Mincurism and Lysenko.[54]

Our brains allow us to circumvent or short-circuit some of the barriers that biology and physics place on our evolution. We can, as Engels cautioned,

by no means rule over nature like a conqueror over a foreign people, like someone standing outside nature – but … all our mastery of it consists in the fact that we have the advantage over all other creatures of being able to learn its laws and apply them correctly.[55]

For instance, the huge size of our brains relative to body mass (Dubois’s encephalisation quotient) imposes limits on the speed of connections. Through evolution, specialisation of zones in the brain has helped to get around some of those difficulties. Even if we waited for natural selection to bring about further improvements, how much more connectivity can fit into a cranium small enough to pass through a birth canal? To get around the physics of brain size and functions, we have the Internet as a qualitative leap -like writing and printing - for the storage of information and for its retrieval. Instead of computers replacing human intelligence they are prosthetic devices. One alternative is for all babies to be grown out the womb. For nearly a millennium, spectacles have been helping individuals to see better. Science and technologies spur on the artificial selection upon which Darwin relied to persuade his readers about natural selection.[56]

New York Museum of Natural History curator Niles Eldredge in his aptly titled Dominion presented a variation on Huxley’s outlook:

Without isolation, there can be no true evolutionary change awaiting Homo sapiens – still less of a chance that Homo sapiens will produce a novel, different-looking descendant species. Once again, provided our population size remains in the same ballpark (or greater) as it is right now, there is no realistic chance for anything much to happen to us in an evolutionary sense.

The simple truth is that in stepping away from local ecosystems and in substituting cultural devices for physiological and anatomical adaptations, we have unwittingly changed the rules of the evolutionary game.[57]

Capitalism grew on the trade in spices, sugar, tobacco, coffee, tea – and slaves. From the late nineteenth century, our daily bread ceased to be sourced locally, undermining the rhetoric of ‘blood and soil’ as wheat crossed the Atlantic and arrived from Australia.[58] Railways and steamships helped to market tinned and frozen meats over continents and oceans.[59]

The extraction of our species from locality as the source of our food has consequences for the rest of life-forms on the planet by altering their environments, and by depriving many of them of the isolation required for speciation through natural selection. Commerce transported creatures great and small out of their habitats, whether rats escaping trading vessels or goats being put ashore to feed shipwrecked survivors. Ferals are ubiquitous but no more so than sheep and cattle. These changes amount to a third nature, encouraging some earth scientists to speak of the present era as the anthropocene.[60]

The more we understand about how human actions affect the rest of nature, Engels observed,

the more will men not only feel but also know their oneness with nature, and the more impossible it will become the senseless and unnatural idea of a contrast between mind and matter, man and nature, soul and body, such as arose after the decline of classical antiquity in Europe and obtained its highest elaboration in Christianity.[61]

The decline of religion among the intelligentsia saw mathematics take its place as the human activity most severed from worldly needs.

Not all fingers and thumbs
The Australian-born Marxist who founded the discipline of pre-history, V Gordon Childe, wrote in Man Makes Himself (1936) about how the forming of clay pots by hand possibly had contributed to the plasticity of the brain:

The constructive character of the potter’s craft reacted on human thought. Building up a pot was a supreme instance of creation by man. The lump of clay was perfectly plastic; man could mold it as he would. In making a tool of stone or bone he was always limited by the shape and size of the original material; he could only take bits away from it. No such limitations restrict the activity of the potter. She can form her lump as she wishes; she can go on adding to it without any doubts as to solidity of the joins. In thinking of ‘creation’ the free activity of the potter in ‘making form where there was no form’ constantly recurs to man’s mind; the similes in the Bible taken from the potter’s craft illustrate the point.[62]

Childe’s hypothesis must remain speculative since no observations survive from that era. The strongest support for his suggestion is how our species has developed. Something like his depiction must have happened. We become what we do, as a species and as individuals, which is the shortest statement of materialist dialectics. The opposable thumb is the greatest of all ‘inventions’ because it increased the ways in which we can apprehend objects through fashioning them to our needs.

French mathematician-philosopher Henri Poincare (1854-1912) made a parallel point on how we acquired our capacities for one area of mathematics:‘if there were no solid bodies in nature, there would be no geometry.[63] That rare bird, a materialist mathematician, Charles Davis, glossed Poincare’s precept by saying that ‘that the daily needs of humans to handle rigid bodies is related to our great insight into Euclidean geometry of two or three dimensions’.[64] Poincare himself had proposed that although

the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, but the mind only makes use of this faculty if experience furnishes it a stimulus thereto. … this stimulus was the notion of the physical continuum, drawn from the rough data of the senses …

But apart from the data of sight and touch, there are other sensations which contribute as much and more than they to the genesis of the notion of space. These are known to everyone; they accompany all our movement, and are usually called muscular sensations … sight and touch could not have given us the notion of space without the aid of the ‘muscular sense’.[65]

This way of apprehending reality resulted from protracted interactions between natural selection and cultural transmissions. Our capacity for geometry, Poincare observed, ‘is, like all associations of ideas, the result of a habit; this habit itself results from very numerous experiences’.[66] Or, in the words of A N Whitehead: ‘Civilisation advances by extending the number of important operations which we can perform without thinking about them’.[67]

For habituation to produce a propensity for Euclidian geometry, the environment needs to be stable. Poincare underlined this conclusion by raising the prospect of an alternate planet:

… without any doubt, if the education of our senses had been accomplished in a different environment, where we should have been subjected to different impressions, contrary habits would have arisen and our muscular sensations would have been associated according to other laws.[68]

Engels would have agreed with this point about the impress of external realities on patterns of perception:

But it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is, to perform the first arithmetical operation, are anything but a free creation of the mind.[69]

If we had evolved with twelve fingers, and not ten, the majority of humankind would compute from base twelve, not base ten.[70] Engels continued:

Like the concept of number, so the concept of form is derived exclusively from the external world and does not arise in the mind as a product of pure thought. There must have been things which had shape and whose shapes were compared before anyone could arrive at the concept of form.[71]

Engels is not saying that the concepts of number and form are limited to counting or to touching. Indeed, he is clear that

Counting requires not only object that can be counted, but also the ability to abstract from all properties of the objects considered except their number – and this ability is the product of a long historical development based on experience.[72]

Hence, the otherwise judicious Carl B Boyer is wrong to allege that Engels and other ‘Marxist materialists will not grant mathematics the independence of [from] experience necessary for its proper development’.[73] Over and again, Engels discussed how that independence operates, for good and ill. To write that an ‘ability’ is ‘based on experience’ is not to say that it is limited to those experiences. Indeed, Engels is clear that he is concerned with the ability to abstract from experience. Moreover, he knows that a different order of experience was essential to the development of calculus. Every advance required the experience of making calculations. As Mao Zedong put it: ideas neither fall out of the sky, nor are they innate in the mind.[74]

Engels knew full well that the sources of mathematical axioms were more likely to have been exogenous than were their developments since Descartes. What he rejects is that the initial notions derived ‘from the free imagination of the mind’ and not from ‘sturdy reality’. Even more, he dismisses those Platonists who claim that

the first line came into existence through the movement of a point in space, the first plane through the movement of a line, the first solid through the movement of a plane, and so on.

Boyer is right to claim that the ‘denial of independence’ for mathematics would make ‘impossible the concept of the derivative and the scientific description of motion in terms thereof’. This is not a happy choice for Boyer given that Marx and Engels - as dialecticians - championed motion in every domain of existence and became addicted to derivatives. On receipt of some of Marx’s writings about calculus Engels replied: ‘The thing has taken such a hold of me that it not only goes round my head all day, but last week in a dream I gave a chap my shirt-buttons to differentiate, and he ran off with them’.[75]

Boyer next attributes to Engels the prejudice that ‘The mathematical infinity is, in accord with this view, a contradiction of the ‘tautology’ than the whole is greater than any of its parts’. Here, Boyer has failed to distinguish Engels’s paraphrasing of Herr Duhring from the mockery with which he assails those ‘most ridiculous delirious phantasies’.[76]

Of course, Engels did not get everything right even according to the most advanced understandings of his time, still less for the discoveries made since. A sympathetic critic, Robin Hirsch, shows that Engels’ commentary on the infinite ‘is only marginally less confused than that of Duhring’ while his treatment of infinitisimals ‘cause much confusion’.[77] To suppose otherwise is to be anti-materialist.

Over millennia, our species has gone beyond sensate experiences of the world and also beyond the reflecting on them that enabled us to reach generalisations. Scientists needed a way past induction which can never provide enough instances to exclude the possible discovery of the single ‘black swan’ that refutes the conclusion drawn from all observed swans having been white.[78] Like Engels, Poincare recognised that more than contact with solid objects and subsequent induction were needed to develop the concepts underlying geometry:

None of our sensations, isolated, could have conducted us to the idea of space; we are led to it only in studying the laws, according to which these sensations succeed each other.[79]

Although we nowadays associate those patterns with the work of one man, Euclid, Poincare knew that ‘individual experience could not create geometry’ until ‘by natural selection our mind has adapted itself to the conditions of the external world, that it has adopted the geometry most advantageous to the species’.[80] Euclidian geometry serves the human needs that depend on certain aspects of the way that the world is and how we experience it.

In an overly instrumentalist account, Whitehead proposed that ‘land-surveys had produced geometry’ for the Ancients.[81] Truer to say that the need to determine property rights over land drew on and contributed to advances in the practice of Euclidean geometry.

Its theorems, of course, were only a starting point for geometry. Descartes added direction and time to his equations. These elements are not immediately sensible in the way that handling a brick was in Poincare’s claim about how the apprehension of space contributing to the acceptance of Euclidian geometry. Nonetheless, Descartes’s contributions to pure mathematics were not disassociated from physical realities but took place in a world of increasing travel, as Lancelot Hogben outlined:

In the static figures of Euclid’s geometry we meet only with angles not greater than four right angles (360°). It may have occurred to you that when we introduced the radian or wheelwright’s measure of the angle we made room for angles as big as we like to make them. When the Renaissance geometry came into use people were becoming familiar with two instruments which impress this possibility on the imagination vividly. One was the mariner’s compass. The other was the clock.[82]

The new ways of calculating spatial relations were still assisted by handling objects. Now, they include tools and machines, not just shapes.

Davis cautioned that Poincare’s appreciation of how our handling of solid objects has contributed to our adoption of Euclidian geometry was not the whole story. It

cannot distinguish between what is our common heritage because it was bred into our ancestors by natural selection and what is our common heritage because it is in everyone’s childhood experience.[83]

The level of mathematical abilities throughout the Australian population is a mix of inheritance through natural selection and 140 years of compulsory schooling. New Zealand political scientist James Flynn gathered ‘Data from fourteen nations to reveal IQ gains ranging from 5 to 25 points in a single generation’.[84] The results demonstrate our ability to adapt culturally, proving that learning is not fixed in our genes, still less in our blood.[85]

To spotlight what is being said here about how learning by an individual can affect brain function, we fall back on the experience of London taxi-drivers. MRI scans reveal that one section of their brains operates differently from that in the rest of us. The explanation for this peculiarity is the accumulated experience of their labour. No one suggests that this change is inheritable by their offspring. Indeed, for that to happen, this learned capacity would have to provide some survival advantage. The chances of that happening are nil. In the time that it would require for natural selection to take its effect, taxis will be a thing of the past – and perhaps London as well. However, in the immediate future, some of the children of the drivers are likely to have inherited their parents’ taxi licences and absorbed some of their progenitors’ knowledge by travelling with them. Of course, it is also possible that the drivers’ income and the system of compulsory education will have allowed those children to become stockbrokers and doctors – hence, passengers. Their parents’ brain-scans are but snapshots within shifting socio-economic arrangements.

Richard Dawkins, in one of his earliest marketing ploys, branded cultural transmission as a ‘meme’ in the hope that this neologism would absorb some of the authority of the gene as ‘the new replicator’.[86] Even as a metaphor for socialised learning, ‘meme’ risks being understood as anti-Darwinian acquired characteristics, or the Jungian collective unconscious. Either way, it has no basis in reality and is pseudo-scientific like so many of the claims about genetic determinates for socio-cultural behaviours.[87] What can be demonstrated without recourse to any self-publicising logo is that collective human activities have multiplied our mathematical capacities.

Whether or not the individual is born a blank slate becomes less significant once access to culture opens a treasure house for the extension of human capacities.[88] That prospect will now be explored through the spread of mathematical abilities across time. In identifying that expansion, the next part distinguishes what is not possible through natural selection from the cultural transmissions that are available to individuals because of collective experience. The third section is a Prolegomenon to a materialist history of mathematics.

[1] Steven Rose, The 21st Century Brain, Explaining, Mending and Manipulating the Mind, Jonathan Cape, London, 2005; Scientific American, July 2011, pp. 38-43. The human brain has 100 billion nerve cells and 100,000 billion interconnections.

[2] Martin Campbell-Kelly and William Aspray, Computer, A History of the Information Machine, Basic Books, New York, 1996; Michael Riordan and Lillian Hoddeson, Crystal Fire, The Invention of the Transitor and the Birth of the Information Age, W W Norton, New York, 1997.

[3] Hubert I Dreyfus, On the Internet, Routledge, Abington, 2009, pp. 16-20
[4] Scientific American, June 2011, p. 612.
[5] Scientific American, April 2011, p. 66.

[6] Quoted John C. Greene, The Death of Adam, Evolution and its Impact on Western Thought, Iowa State University Press, Ames, Iowa, 1959, pp. 5 and 10.
[7] John A Stewart, Drifting Continents & Colliding Paradigms, Perspective on the Geoscience Revolution, Indiana University Press, Bloomington, 1990.
[8] F K Barasch, The Grotesque, Mouton, The Hague, 1971; G Harpham. On the Grotesque, Princeton University Press, Princeton, NJ, 1982; W Kayser, The Grotesque in Art and Literature, Columbia University Press, 1981.

[9] Arthur O Lovejoy, The Great Chain of Being, A Study of the History of an Idea, Harvard University Press, Cambridge, Mass., 1936, pp. 189ff.
[10] see my ‘Men of Flowers’, Arena, 104, February-March 2010, pp. 18-22; Collections, 8, June 2011, pp. 10-13.
[11] Immanuel Kant, ‘The Critique of Teleological Judgement’, Great Books of the Western World, volume 42, Encyclopedia Britannica, Chicago, 1952, pp. 569-70.

[12] Simon Garfield, Mauve, W W Norton, New York, 2001; John Buckingham, Chasing the Molecules, Discovering the Building Blocks of Life, Sutton, Stroud, 2005.
[13] Adrian Desmond & James Moore, Darwin, Penguin, London, 1992, p. 545.
[14] John Gage, Colour and Culture, Practice and Meaning from Antiquity to Abstraction, Thames and Hudson, London, 1993; Colour and Meaning, Art, Science and Symbolism, Thames and Hudson, London, 1999.

[15] The upper classes favoured mauve and other pastels, Alexandria Hasluck (ed.), Audrey Tennyson’s Vice-Regal Days, Australian National Library, Canberra, 1978, pp. 183 and 276; for Jewesses, Victorian Banner, 8 October 1881, Hiliary Rubenstein, Chosen, Allen & Unwin, Sydney, 1987, p. 76; for colonials, Alexandria Joel, Parade, History of Fashion in Australia, HarperCollins, Pymble, 1998, p. 27.

[16] Frederick Engels, Ludwig Feuerbach and the End of Classical German Philosophy, Foreign Languages Publishing House, Moscow, 1950, p. 33.

[17] William H Brock, Justus von Liebig, Cambridge University Press, Cambridge, 1997; read by Marx and Engels before they met and referred to throughout their writings, e.g., Capital, I, Moscow, pp. 239, 328, 504-6, 573 and 622; Penguin, pp. 348-9, 446n., 638-9, 718 and 973.

[18] Peter Thompson, Seeds, Sex and Civilization, Thames and Hudson, London, 2011; Julian Huxley, Evolution: The Modern Synthesis, George Allen & Unwin, London, 1963.
[19] Frederick Engels, Dialectics of Nature, Progress Publishers, Moscow, 1964, pp. 24 and 26.
[20] Frederick Engels, Anti-Duhring, International Publishers, New York, 1970, pp. 18-19. The passage continues ‘and if the report is confirmed, also birds that walk on all-fours’. Fossils of a reptilian bird, Archaeopteryx, had been unearthed in Germany in 1861. Engels never strayed beyond the existing evidence, championing empirical labours as firmly as he rejected ‘English empiricism’.

[21] Ann Moyal, The Platypus, Allen & Unwin, Sydney, 2010, pp. 154-5, with evidence from its genome since 2001.
[22] Peter J Bowler, The Eclipse of Darwinism, Anti-Darwinian Evolutionary Theories in the Decades around 1900, Johns Hopkins University Press, Baltimore, 1983.
[23] R C Grogin, The Bergsonian Controversy in France, 1900-1914, University of Calgary Press, Calgary, 1988.

[24] Collected Works, volume 14, Progress Publishers, Moscow, 1972.
[25] Fred Gage, ‘All in the Mind’  22 January 2011
[26] Norman Doidge, The Brain that Changes Itself: stories of personal triumph from the frontiers of brain science, Scribe, Melbourne, 2007, Appendix I.

[27] Rose, The 21st Century Brain, pp. 163 and 178.
[28] Fred H Gage, ‘Neurogenesis in the Adult Brain’, Journal of Neuroscience, 22 (3), 1 February 2002, pp. 612-3.

[29] Alvar Ellegard, Darwin and the General Reader, University of Chicago Press, Chicago, 1990, pp. 293-331; William Irvine, Apes Angels & Victorians, Readers Union, London, 1956; Dov Ospovat, The Development of Darwin’s Theory, Natural History, Natural Theology & Natural Selection, 1838-1859, Cambridge University Press, 1981; Keith Thompson, Before Darwin, Reconciling God and Nature, Yale University Press, Princeton, 2005.

[30] Janet Browne, Charles Darwin, The Power of Place, Pimlico, London, 2002, p. 67.
[31] Engels, Dialectics of Nature, p. 180.

[32] On the dualism associated with Descartes, Stephen Gaukroger, Descartes an intellectual biography, Clarendon Press, Oxford, 1995, pp. 388-94; Thomas Fuchs, Mechanisation of the Heart, Harvey and Descartes, University of Rochester Press, Rochester, 2001, especially pp. 122-5; John Cottingham, ‘Cartesian Trialism’, Mind, 94, April 1985, pp. 218-230; Daniel Garber, Descartes Embodied, Cambridge University Press, Cambridge, 2001, pp. 257-273.

[33] Descartes needs to be read as if in dialogue with Aristotle and Aquinas, F C Coppleston, Aquinas, Penguin, London, 1955, pp. 156-98; Anthony Kenny, Aquinas, OUP, Oxford, 1980, pp. 46-9; Nicholas Jolley, ‘Malebranche on the Soul’, Steven Nadler (ed.), The Cambridge Companion to Malebranche, CUP, Cambridge, 2000, pp. 31-58. Because the notion of ‘soul’ has dropped out of fashion, those philosophers who compile a Syllabus of Descartes’ Errors rarely consider his views in relation to the theological discourse of the seventeenth century with its doctrines of transubstantiation; whether God had created the body and the soul out of utterly distinct substances, as Descartes maintained in arguing for the latter’s immortality; disputes over whether the resurrection of the body and soul was coterminous or did one wait until the Last Judgement, Marcia B Hall, ‘Michelangelo’s Last Judgement as Resurrection of the Body: The Hidden Clue’, Michelangelo’s ‘Last Judgement’, Cambridge University Press, Cambridge, 2005, pp. 95-112.

Margaret Dauler Wilson, Descartes, Ego Cogito, Ergo Sum, Routledge & Kegan Paul, London, 1978, where the index entry for ‘soul’ says ‘see mind’, which reads ‘mind/soul’ as if there were no distinction; Deborah J. Brown, Descartes and the Passionate Mind, CUP, Cambridge, 2006, recognises the differences and the three kinds of soul in her discussion but slips between soul, mind, brain and will without spelling out definitions of each, pp. 34-35.

John Cottingham argues that Descartes uses mind and soul interchangeably but provides only one debatable quotation, ‘ “l’esprit ou l’ame de l’homme (ce que je ne distingue point).” This assertion of the interchangeability of the terms ‘mind’ and ‘soul’ in Cartesian metaphysics appears in the 1647 French version of the Meditations. The original 1641 Latin text refers simply to the mind (mens). ‘Cartesian dualism: theology, metaphysics, and science’, John Cottingham (ed.), The Cambridge Companion to Descartes, Cambridge University Press, Cambridge, 1992, pp. 236 and 253n.

[34] Carl Zimmer, Soul Made Flesh, Free Press, New York, 2004, p. 217.
[35] Gaukroger, Descartes, pp. 278ff.
[36] Richard Rapport, Nerve Endings, The Quest to Find How Brain Cells Communicate, W W Norton, New York, 2005.

[37] Fred Gage, ‘All in the Mind’ 22 January 2011.
[38] Steven Rose, The Making of Memory, Anchor Books, New York, 1992; The 21st Century Brain, pp. 140 and 156-65.

[39] Wal Sutching, ‘Marx’s Theses on Feuerbach: Notes Towards a Commentary (with a New Translation)’, John Mepham and David-Hillel Ruben (eds), Issues in Marxist Philosophy, volume II, Harvester Press, London, 1979, p. 18.

This aphorism is one of the so-called ‘Theses on Feuerbach’ which unfortunately survived the knawing criticism of mice. Their significance requires attending to the first ninety pages of The German Ideology (1845-7) which Peter Singer fails to supply in his scrappy and shallow A Darwinian Left, Yale University Press, New Haven, 2000, and his Marx, OUP, Oxford, 1980.

[40] Karl Marx, A Contribution to the Critique of Political Economy, Progress Publishers, Moscow, 1970, p. 189.
[41] S S Prawer, Karl Marx and World Literature, OUP, Oxford, 1976, pp. 273ff.; that Defoe knew better than the vulgar economists is clear from his manual, The Complete English Tradesman, Alan Sutton, Brunswick Road, 1987.
[42] C B Macpherson, The Political Theory of Possessive Individualism, OUP, London, 1962.

[43] Karl Marx, ‘Marginal Notes to the Programme of the German Workers’ Party’, Karl Marx and Frederick Engels, Selected Works, volume three, Progress Publishers, Moscow, 1970, p. 13.
[44] Engels, Anti-Duhring, pp. 94-105; Lenin, Materialism and Empirio-Criticism, Collected Works, volume 14, pp. 131-7.
[45] Karl Marx, Theories of Surplus Value (TS-V), III, Progress Publishers, Moscow, 1971, p. 259.

[46] Marx to Kugelmann, 27 June 1870, Marx-Engels Selected Correspondence, Moscow, FLPH, n.d., p. 290.

Marx thought that this was not the worst: ‘The last form is the academic form, which proceeds historically and, with wise moderation, collects the ‘best’ from all courses, and in doing this contradictions do not matter; on the contrary, what matters is comprehensiveness. All systems are thus made insipid, their edge is taken off and they are gathered together peacefully in a miscellany. The heat of apologetics is moderated here by erudition, which looks down benignly on the exaggerations of economic thinkers, and merely allows them to float as oddities in its mediocre pap. Since such works appear only when political economy has reached the end of its scope as a science, they are at the same time the graveyard of this science’. Marx, TS-V, III, p. 502

[47] For one survey of this supposed split between science and critique see Alvin Gouldner, The Two Marxisms, Contradictions and Anomalies in the Development of Theory, Macmillan, London, 1980.
[48] Engels, Ludwig Feuerbach and the End of Classical German Materialism.

[49] Lewis Henry Morgan, Ancient Society, New York Labor News, Paolo Alto, 1978; John Mulvaney, ‘The Australian Aborigines, 1606-1929: Opinion and Fieldwork’, Historical Studies, Selected Articles, MUP, Carlton, 1964, pp. 1-56.

[50] Marvin Harris, The Rise of Anthropological Theory, A History of Theories of Culture, Thomas Y Crowell, New York, 1968, p. 248; for a different view of Morgan, Emmanuel Terray, Marxism and ‘Primitive Societies’, Two Studies, Monthly Review Press, New York, 1972, pp. 5-92.

[51] Lamarck’s reputation suffers because of his atheism and as a punching bag for Darwinists. Acquired characteristics was a tiny part of the contribution of the man who coined biology and invertebrate, L J Jordanova, Lamarck, OUP, Oxford, 1984, pp. 1 & 6.

[52] Alfred Russell Wallace, Infinite Tropics, An Anthology, (ed.) Andrew Berry, Verso, London, 2002, pp. 175ff.
[53] Huxley, ‘Introduction’, Evolution: The Modern Synthesis, unpaginated.

[54] Julian Huxley, Soviet Genetics and World Science, Chatto and Windus, London, 1949; V N Stoletov, The Fundamentals of Michurin Biology, Foreign Languages Publishing House, Moscow, 1953; Richard Levins and Richard Lewontin, ‘The Problem of Lysenkoism’, The Dialectical Biologist, Harvard University Press, Cambridge, Mass., 1985, pp. 163-96.

[55] Engels, Dialectics of Nature, p. 183.
[56] Charles Darwin, On the Origin of Species, Gramercy Books, New York, 1979, pp. 71-100; the effects of artificial breeding are contradictory, with the loss of grain species in tandem with a proliferation of breeds of show dogs, Steve Jones, Almost Like a Whale, Anchor, London, 2000, pp. 31-44.

[57] Niles Eldredge, Dominion, Can Nature and Culture Co-exist?, Henry Holt, New York, 1995. p. 139.
[58] Anver Offer, The First World War, An agrarian interpretation, The Clarendon Press, Oxford, 1989.
[59] William Cronon, Nature’s Metropolis, Chicago and the Great West, Norton, New York, 1991; Derek J Oddy, ‘The Growth of Britain’s Refrigerated Meat Trade, 1880-1939’, The Mariner’s Mirror, 93 (3), August 2007, pp. 269-80.

[60] J Zalasieicz, ‘The New World of the Antrhropocene’, Environmental Science and Technology, 47 (7), 1010, pp. 2228-31.
[61] Dialectics of Nature, p. 183.

[62] V Gordon Childe, Man Makes Himself, London, Watts, 1965 edition, p. 93; Childe’s approach is established in the preceding pages. Despite the switch in gender pronouns in this passage, we might be tempted to rephrase his title to ‘Human Beings Remake Ourselves’.
[63] Henri Poincare, Foundations of Science, Science Press, New York, 1921, p. 73.
[64] Charles Davis, ‘Materialist Mathematics’, R S Cohen, J J Stachel and M W Wartofsky (eds), For Dirk Struik, Reidel Publishing, Dordrecht-Holland, Boston Studies in the Philosophy of Science, XV, 1974, p. 54.

[65] Poincare, pp. 49, 69 and 72.
[66] Poincare, p. 69.
[67] A N Whitehead, An Introduction to Mathematics, OUP, Oxford, 1911, p. 61.

[68] Poincare, p. 69.
[69] Engels, Anti-Duhring, p. 45.
[70] Geoffrey de Ste Croix, ‘Greek and Roman Accounting’, A C Littleton and B S Yarney (eds), Studies in the History of Accounting, Sweet & Maxwell, London, 1956, p. 50.

[71] Engels, Anti-Duhring, p. 47.
[72] Engels, Anti-Duhring, p. 47.
[73] Carl B Boyer, The History of Calculus and its Conceptual Development, Dover, New York, 1959, pp. 307-8.

[74] Mao Zedong, Four Essays on Philosophy, Foreign Languages Press, Beijing, 1966, p. 134.
[75] Engels to Marx, 10 August 1881, Karl Marx, Mathematical Manuscripts, New Park, London, 1983, p. XXVIII.
[76] Boyer, p. 307-8; Engels, pp. 46-7.

[77] ‘Is Mathematics a Pure Science?”, Science & Society, 60 (1), Spring 1996, pp. 68-9.
[78] C Smith, ‘Hegel, Marx and the Calculus’, Marx, Mathematical Manuscripts, pp. 260-2.
[79] Poincare, p. 71.

[80] Poincare, p. 91.
[81] Whitehead, 1911, pp. 36-7.
[82] Lancelot Hogben, Mathematics for the Million, George Allen & Unwin, London, 1937 edition, p. 425

[83] Davis, p. 54.
[84] James R Flynn, ‘Massive IQ Gains in 14 Nations: What IQ Tests Really Measure’, Psychological Bulletin, 101 (2), 1987, pp. 171-91; James R Flynn, What is intelligence? Beyond the Flynn Effect, CUP, Cambridge, 2007.

[85] See my Temper Democratic, Wakefield, Kent Town, 1998, pp. 158-68.
[86] Richard Dawkins, The Selfish Gene, OUP, Oxford, 1976, chapter 11.
[87] Temper Democratic, pp. 144-57.
[88] Enjoy Louis Menand’s dissection of Stephen Pinker’s cultural illiteracy, New Yorker, 25 November 2002, pp. 96-101.