# The Anthropology of Numbers.

Ever since Kronecker (a 19th century German mathematician) declared
that 'God made the integers, all the rest is men's work',
it should have become easier to accept that mathematics is the product
of human endeavour and is open to anthropological inquiry in its
entirety. Just as no culture falls outside the scope of anthropological
reflection, so no mathematic does, whether scientific or otherwise.

As for God's share, here too anthropologists are better suited to critically illuminate the problems of the foundations of number, the most elementary structures of mathematical activity, than the phalanxes of cognitive psychologists, philosophers, mathematicians, linguists, computer scientists, cosmologists, theologians, etc. Anthropologists know well that there is more than one way of being God. The one who was bodyless, of vicious temper, but literate, and who, in a special numinous stunt, appeared to Moses as a burning bush which does not incinerate, giving him ten rather than some other number of commandments, is a very specific cultural and historical specimen. Contra physicists and cosmologists, anthropologists have known for a long time that there are Gods who indeed play dice, loaded and unloaded, who are subtle and crass, that some are both numerate and literate, while others prefer spectacular sexual over intellectual modes of creation of the universe and life; that some are malicious scoundrels (as Yahweh of the Book of Job), and so on.

As a man knowledgeable of God (to the best of my knowledge he was formerly an Anglican priest) and as an anthropologist 'who started his scientific career as a mathematician' Crump would appear to be eminently equipped for a project such as the anthropology of numbers. But in the preface Crump writes: 'at the end of my first supervision as an undergraduate reading mathematics at Cambridge, the late Prof. A.S. Bescovitch -- whom I later came to know well -- asked me, "Mr Crump, please tell me one thing: why is it you read mathematics?" I do not think he found my reply entirely convincing, but I hope this book answers his question'. This book is indeed written with the pretence, unease, and reverence which so many mathophobic Westerners feel for and unduly accord to mathematics and to those who, like the author's undergraduate instructor, aspire to authority by virtue of being academic mathematicians. Although they may be gratifying, such internalized authority figures tend to becloud consciousness and damp critical spirit. And for a project of this scope a minimal requirement is to keep the Western ideology of mathematics uncompromisingly at a critical distance.

At the fundamental level of human numeracy anthropological research provides vital evidence for the diversity of the modes of construction of the basic category of mathematics, the number. Indeed, a closer inspection shows that in all known cultures humans know how to count in one fashion or another. True, there are academic bigots who claim that much of the so-called primitive counting may be nothing more than a recital of number-words (numerals) rather than a properly numerical activity. But a sensitive empirical study of cultures with such counting systems will show that when people actually count, as opposed to mechanically recounting a set of labels, then they do mind and body work, often of a most complex cognitive kind, precisely because constructing numbers, even with God's help, immediately requires human work. This is so because in many of these systems the constitution of their systematicity is not dependent on lingual subsidies, for language too, like anything human, is the product of human activity and the flux of experience, not an autonomous, pre-constituted foundation of human rationality and mind, delivered gratis by God or Ms. Evolutionary Process.

In the first three chapters where Crump touches on some of these issues his discussion is characterized by an airy style lacking in analytical accuracy, clarity and depth. Specifically, I find his dichotomization of the natural numbers into the symbolic, 'belonging to language and defined by the local culture', and 'the natural numbers defined by the mathematics or logic' unsustainable. It privileges modern Western mathematics in the ontological determination of numbers on the pretext that 'the natural numbers |of mathematics and logic, JM~, are an infinite series, which the symbolic numbers, by force of circumstance, can never be'. This conceptualization lacks any historical perspective on the relation between numbers and the infinite (potential and actual) in Western mathematical thought, and their systematization throughout its history. This dichotomization obfuscates the fact that logical-mathematical theorizing itself is a cognitive and symbolic activity which is as dependent on the specific local culture as is any 'counting' activity. This is so regardless of the 'force of circumstance' which makes a group of people aware of the infinite in general, and a notion of an infinite series or progression of natural numbers in particular.

With Crump the theoretical consequences of a detailed examination of the inner structures of counting activities and their correlate, the natural number cannot be developed because of his acceptance of Lancy's view that 'symbolic counting is a universal skill that develops gradually with age and is relatively unaffected by variations in the counting system employed, culture, cognitive development or schooling' (ibid.). The presumed universality of counting skill is a convenient abstraction, yet every system and activity of counting is always and only a concrete cognitive-bodily construction. More significant and informative for an anthropological inquiry into the cognition and ontology of number is the examination of the specificities and the local character of the structuring of numbers in particular cultures. A more involved examination of counting systems around the world will make the investigator aware that there is more than one way of constructing a series of natural numbers and that a seemingly single mode (eg. binary, quinary) can be articulated into a variety of counting schemata. This is to say that the common concept of base is not informative as to the actual structuring of a counting system. The development and transformation of the mathematical domain with the inception of self-aware logico-mathematical thought (modern mathematics) is an event in human history which followed long after that crucial, foundational activity where humans themselves are actually doing God's work (I always knew God was a parasite) or, at best, there is no distinction between the two. But in this situation there is always and only a particular counting activity and system; the universal is contingent upon and made actual by and in the particular human cultural situation (see Mimica Intimations of Infinity, 1988).

The problem, then, is that the author does not seriously scrutinize 'primitive' counting in an effort to understand numeracy in radically different cultures. He hides behind the common Western usage of mathematics as the ideology of complexity and arcane intellectuality accessible only to the fraternities and sororities of the initiated. Thus, having made a potentially productive observation on the relevance of Lancy's work for the evaluation of Russell's philosophical examination of the foundations of mathematics, Crump retreats into the gullible rhetoric which authorizes modern mathematics on the basis of culturally shared ignorance and mathophobia: 'On the other hand, the elementary cognition of a pre-literate population in Papua New Guinea can hardly be taken as a confirmation, let alone a proof, of advanced and esoteric propositions in mathematical logic (which belong to a conceptual domain beyond any such cognition), nor even as sufficient to falsify the results of experimental work in development psychology, such as that done by Brainerd...'.

On my reading of modern mathematical logic and philosophy of mathematics I find nothing 'esoteric' about mathematical propositions. This applies even when some authors, eg. Cantor, appeal to Plato or, for good measure, God. All these 'advanced' propositions and the reasoning behind them are comprehensible if one uses imagination and is willing to explore the world of modern mathematics and science as a human activity. Furthermore, a critical study of the history of mathematical logic in the last hundred years and the construction of rationality in this 'conceptual domain' will show all the antinomies of human creative genius, especially its misery, preposterous pretensions and the gullibility of those commonly taken as the champions of 'advanced' cognition.

On the other hand in reading Iamblicus (a Pythagorean) on the theology of arithmetic, one will indeed encounter an esoterica which many anthropologists would find quite familiar, although modern mathematical rationality is useless when it comes to the interpretation of either the mathematicity or the sanctity of the ancient Pythagorean mathematics. This is so because the two form a single system of rationality, a single 'cognitive domain'. As for 'the elementary cognition of pre-literate' New Guineans in the domain of numeracy, I am sure that Crump does not know what this actually is in concrete terms, for his sources are as unhelpful as many developmental studies of cognition tend to be. The obvious question to ask is has he actually learned to think-act, i.e. count, in such a counting system? He might be surprised to learn how much cognitive ingenuity goes into such a simple counting as adding digits. But to make the counter-point, when one learns about Einstein's actual thinking activity in which imagistic-muscular rather than verbal components predominated, does it follow that, therefore, the theory of relativity was a product of an 'elementary cognition' which does not belong to it? Here I am merely indicating the complexity of the dialectics of human thinking and concrete knowledge forms, and their ideological ossification in academic disquisitions on 'higher' cognitive domains -- mathematics, physics, and science in general. Familiarity with Max Wertheimer's classic gestaltist account of primitive number concepts (in W.D. Ellis A Sourcebook in Gestalt Psychology, 1938) might have alerted Crump to the need for a more cogent reflection on the 'elementary cognition' of number, or the place of this cognition in human activity in general, esoteric or not.

The upshot of my critique is that the whole book is symptomatic of the following: the anthropologist Crump did not critically reassess the impact of his commonsensical flirtation with modern mathematics and the local cognitive-cultural milieu of his undergraduate days. And conversely, in coming to embrace anthropology as an approach into other cultural life-worlds he has, it seems to me, failed to take other modes of numeracy seriously enough to understand their mathematicity. But this also holds true of his assumptions about the nature of mathematicity of 'advanced' Western mathematics. I cannot see in the book any evidence that Crump has taken the history of that 'self aware' mathematics, in search of its foundations, as an anthropological problem. It is these epistemological and ontological quandaries within Western mathematics that make it most interesting for any anthropologist concerned with the formation of cognitive styles and modes of ir/rationality. In this regard 'the anthropology of numbers', to be worthy of the name, should not be restricted to 'other' mathematics but should also aim at the mathematicians' mathematics.

Finally, consider the belief of the proverbial American good citizen that only the American dollar is the real money. In spite of the currently unenviable state of the American economy the US dollar is indeed the 'real' money, but for reasons of human power relations which make human economies real and not because they simply are one of many human creations occurring in different cultural life-worlds. Power relations do not determine in any absolute sense the cognitive basis of mathematics, or cognition in general, but they certainly shape it. And this is yet another reason why the anthropology of numbers has to be about all mathematics. Crump's survey of the use of numbers in other human domains (cosmology, time, money, music, art, and architecture) does not go beyond the limitations of the position and style evident in the first three chapters. Even so, any reader who takes anthropology seriously can see that anthropological modes of interpretation of all things human can also make intelligible numerical rationalities in all their diverse refractions, although this book falls short of achieving this. The 'self awareness' of modern mathematics can only become more authentic by reflecting upon its deeper cultural and historical character as the work of the human embodied mind which is of and in the world. Having read this book I think I know why Crump left mathematics. But I am tempted to echo his mathematical mentor from his undergraduate days: 'Mr Crump, please tell me one thing: why do you study anthropology?'

JADRAN MIMICA University of Sydney

As for God's share, here too anthropologists are better suited to critically illuminate the problems of the foundations of number, the most elementary structures of mathematical activity, than the phalanxes of cognitive psychologists, philosophers, mathematicians, linguists, computer scientists, cosmologists, theologians, etc. Anthropologists know well that there is more than one way of being God. The one who was bodyless, of vicious temper, but literate, and who, in a special numinous stunt, appeared to Moses as a burning bush which does not incinerate, giving him ten rather than some other number of commandments, is a very specific cultural and historical specimen. Contra physicists and cosmologists, anthropologists have known for a long time that there are Gods who indeed play dice, loaded and unloaded, who are subtle and crass, that some are both numerate and literate, while others prefer spectacular sexual over intellectual modes of creation of the universe and life; that some are malicious scoundrels (as Yahweh of the Book of Job), and so on.

As a man knowledgeable of God (to the best of my knowledge he was formerly an Anglican priest) and as an anthropologist 'who started his scientific career as a mathematician' Crump would appear to be eminently equipped for a project such as the anthropology of numbers. But in the preface Crump writes: 'at the end of my first supervision as an undergraduate reading mathematics at Cambridge, the late Prof. A.S. Bescovitch -- whom I later came to know well -- asked me, "Mr Crump, please tell me one thing: why is it you read mathematics?" I do not think he found my reply entirely convincing, but I hope this book answers his question'. This book is indeed written with the pretence, unease, and reverence which so many mathophobic Westerners feel for and unduly accord to mathematics and to those who, like the author's undergraduate instructor, aspire to authority by virtue of being academic mathematicians. Although they may be gratifying, such internalized authority figures tend to becloud consciousness and damp critical spirit. And for a project of this scope a minimal requirement is to keep the Western ideology of mathematics uncompromisingly at a critical distance.

At the fundamental level of human numeracy anthropological research provides vital evidence for the diversity of the modes of construction of the basic category of mathematics, the number. Indeed, a closer inspection shows that in all known cultures humans know how to count in one fashion or another. True, there are academic bigots who claim that much of the so-called primitive counting may be nothing more than a recital of number-words (numerals) rather than a properly numerical activity. But a sensitive empirical study of cultures with such counting systems will show that when people actually count, as opposed to mechanically recounting a set of labels, then they do mind and body work, often of a most complex cognitive kind, precisely because constructing numbers, even with God's help, immediately requires human work. This is so because in many of these systems the constitution of their systematicity is not dependent on lingual subsidies, for language too, like anything human, is the product of human activity and the flux of experience, not an autonomous, pre-constituted foundation of human rationality and mind, delivered gratis by God or Ms. Evolutionary Process.

In the first three chapters where Crump touches on some of these issues his discussion is characterized by an airy style lacking in analytical accuracy, clarity and depth. Specifically, I find his dichotomization of the natural numbers into the symbolic, 'belonging to language and defined by the local culture', and 'the natural numbers defined by the mathematics or logic' unsustainable. It privileges modern Western mathematics in the ontological determination of numbers on the pretext that 'the natural numbers |of mathematics and logic, JM~, are an infinite series, which the symbolic numbers, by force of circumstance, can never be'. This conceptualization lacks any historical perspective on the relation between numbers and the infinite (potential and actual) in Western mathematical thought, and their systematization throughout its history. This dichotomization obfuscates the fact that logical-mathematical theorizing itself is a cognitive and symbolic activity which is as dependent on the specific local culture as is any 'counting' activity. This is so regardless of the 'force of circumstance' which makes a group of people aware of the infinite in general, and a notion of an infinite series or progression of natural numbers in particular.

With Crump the theoretical consequences of a detailed examination of the inner structures of counting activities and their correlate, the natural number cannot be developed because of his acceptance of Lancy's view that 'symbolic counting is a universal skill that develops gradually with age and is relatively unaffected by variations in the counting system employed, culture, cognitive development or schooling' (ibid.). The presumed universality of counting skill is a convenient abstraction, yet every system and activity of counting is always and only a concrete cognitive-bodily construction. More significant and informative for an anthropological inquiry into the cognition and ontology of number is the examination of the specificities and the local character of the structuring of numbers in particular cultures. A more involved examination of counting systems around the world will make the investigator aware that there is more than one way of constructing a series of natural numbers and that a seemingly single mode (eg. binary, quinary) can be articulated into a variety of counting schemata. This is to say that the common concept of base is not informative as to the actual structuring of a counting system. The development and transformation of the mathematical domain with the inception of self-aware logico-mathematical thought (modern mathematics) is an event in human history which followed long after that crucial, foundational activity where humans themselves are actually doing God's work (I always knew God was a parasite) or, at best, there is no distinction between the two. But in this situation there is always and only a particular counting activity and system; the universal is contingent upon and made actual by and in the particular human cultural situation (see Mimica Intimations of Infinity, 1988).

The problem, then, is that the author does not seriously scrutinize 'primitive' counting in an effort to understand numeracy in radically different cultures. He hides behind the common Western usage of mathematics as the ideology of complexity and arcane intellectuality accessible only to the fraternities and sororities of the initiated. Thus, having made a potentially productive observation on the relevance of Lancy's work for the evaluation of Russell's philosophical examination of the foundations of mathematics, Crump retreats into the gullible rhetoric which authorizes modern mathematics on the basis of culturally shared ignorance and mathophobia: 'On the other hand, the elementary cognition of a pre-literate population in Papua New Guinea can hardly be taken as a confirmation, let alone a proof, of advanced and esoteric propositions in mathematical logic (which belong to a conceptual domain beyond any such cognition), nor even as sufficient to falsify the results of experimental work in development psychology, such as that done by Brainerd...'.

On my reading of modern mathematical logic and philosophy of mathematics I find nothing 'esoteric' about mathematical propositions. This applies even when some authors, eg. Cantor, appeal to Plato or, for good measure, God. All these 'advanced' propositions and the reasoning behind them are comprehensible if one uses imagination and is willing to explore the world of modern mathematics and science as a human activity. Furthermore, a critical study of the history of mathematical logic in the last hundred years and the construction of rationality in this 'conceptual domain' will show all the antinomies of human creative genius, especially its misery, preposterous pretensions and the gullibility of those commonly taken as the champions of 'advanced' cognition.

On the other hand in reading Iamblicus (a Pythagorean) on the theology of arithmetic, one will indeed encounter an esoterica which many anthropologists would find quite familiar, although modern mathematical rationality is useless when it comes to the interpretation of either the mathematicity or the sanctity of the ancient Pythagorean mathematics. This is so because the two form a single system of rationality, a single 'cognitive domain'. As for 'the elementary cognition of pre-literate' New Guineans in the domain of numeracy, I am sure that Crump does not know what this actually is in concrete terms, for his sources are as unhelpful as many developmental studies of cognition tend to be. The obvious question to ask is has he actually learned to think-act, i.e. count, in such a counting system? He might be surprised to learn how much cognitive ingenuity goes into such a simple counting as adding digits. But to make the counter-point, when one learns about Einstein's actual thinking activity in which imagistic-muscular rather than verbal components predominated, does it follow that, therefore, the theory of relativity was a product of an 'elementary cognition' which does not belong to it? Here I am merely indicating the complexity of the dialectics of human thinking and concrete knowledge forms, and their ideological ossification in academic disquisitions on 'higher' cognitive domains -- mathematics, physics, and science in general. Familiarity with Max Wertheimer's classic gestaltist account of primitive number concepts (in W.D. Ellis A Sourcebook in Gestalt Psychology, 1938) might have alerted Crump to the need for a more cogent reflection on the 'elementary cognition' of number, or the place of this cognition in human activity in general, esoteric or not.

The upshot of my critique is that the whole book is symptomatic of the following: the anthropologist Crump did not critically reassess the impact of his commonsensical flirtation with modern mathematics and the local cognitive-cultural milieu of his undergraduate days. And conversely, in coming to embrace anthropology as an approach into other cultural life-worlds he has, it seems to me, failed to take other modes of numeracy seriously enough to understand their mathematicity. But this also holds true of his assumptions about the nature of mathematicity of 'advanced' Western mathematics. I cannot see in the book any evidence that Crump has taken the history of that 'self aware' mathematics, in search of its foundations, as an anthropological problem. It is these epistemological and ontological quandaries within Western mathematics that make it most interesting for any anthropologist concerned with the formation of cognitive styles and modes of ir/rationality. In this regard 'the anthropology of numbers', to be worthy of the name, should not be restricted to 'other' mathematics but should also aim at the mathematicians' mathematics.

Finally, consider the belief of the proverbial American good citizen that only the American dollar is the real money. In spite of the currently unenviable state of the American economy the US dollar is indeed the 'real' money, but for reasons of human power relations which make human economies real and not because they simply are one of many human creations occurring in different cultural life-worlds. Power relations do not determine in any absolute sense the cognitive basis of mathematics, or cognition in general, but they certainly shape it. And this is yet another reason why the anthropology of numbers has to be about all mathematics. Crump's survey of the use of numbers in other human domains (cosmology, time, money, music, art, and architecture) does not go beyond the limitations of the position and style evident in the first three chapters. Even so, any reader who takes anthropology seriously can see that anthropological modes of interpretation of all things human can also make intelligible numerical rationalities in all their diverse refractions, although this book falls short of achieving this. The 'self awareness' of modern mathematics can only become more authentic by reflecting upon its deeper cultural and historical character as the work of the human embodied mind which is of and in the world. Having read this book I think I know why Crump left mathematics. But I am tempted to echo his mathematical mentor from his undergraduate days: 'Mr Crump, please tell me one thing: why do you study anthropology?'

JADRAN MIMICA University of Sydney

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Author: | Mimica, Jadran |
---|---|

Publication: | Oceania |

Article Type: | Book Review |

Date: | Jun 1, 1993 |

Words: | 2062 |

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