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Science and an African Logic
Does 2 + 2 = 4? Ask almost anyone and they will unequivocally answer yes. A basic equation such as this seems the very definition of certainty, but is it?
In this captivating book, Helen Verran addresses precisely that question by looking at how science, mathematics, and logic come to life in Yoruba primary schools. Drawing on her experience as a teacher in Nigeria, Verran describes how she went from the radical conclusion that logic and math are culturally relative, to determining what Westerners find so disconcerting about Yoruba logic, to a new understanding of all generalizing logic. She reveals that in contrast to the one-to-many model found in Western number systems, Yoruba thinking operates by figuring things as wholes and their parts. Quantity is not absolute but always relational. Certainty is derived not from abstract logic, but from cultural practices and associations.
A powerful story of how one woman's investigation in this everday situation led to extraordinary conclusions about the nature of numbers, generalization, and certainty, this book will be a signal contribution to philosophy, anthropology of science, and education.
In this captivating book, Helen Verran addresses precisely that question by looking at how science, mathematics, and logic come to life in Yoruba primary schools. Drawing on her experience as a teacher in Nigeria, Verran describes how she went from the radical conclusion that logic and math are culturally relative, to determining what Westerners find so disconcerting about Yoruba logic, to a new understanding of all generalizing logic. She reveals that in contrast to the one-to-many model found in Western number systems, Yoruba thinking operates by figuring things as wholes and their parts. Quantity is not absolute but always relational. Certainty is derived not from abstract logic, but from cultural practices and associations.
A powerful story of how one woman's investigation in this everday situation led to extraordinary conclusions about the nature of numbers, generalization, and certainty, this book will be a signal contribution to philosophy, anthropology of science, and education.
- GenresAfricaAnthropology
285 pages, Paperback
First published December 15, 2001
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May 18, 2023Comps reading. When a professor described this book in a class, it sounded like it would be interesting. When I read the synopsis, I became a little skeptical. After reading it, I think all I can say is that I didn’t really get it. I couldn’t really follow Verran’s arguments. Even as I re-read the summaries of the chapters that Verran includes at the opening of each section, I feel like I’m staring at a wall. She writes like a certain type of philosopher where everything they write feels like its been translated numerous times through google translate, and comes out impenetrable. Maybe I am not reading patiently enough, or maybe I am too dumb or philosophically illiterate to follow the odd syntax of Verran’s prose. Perhaps if I had read any philosophy of mathematics beforehand, I could at least grasp the basic issues at stake. Unfortunately this is not the case.
One thing that I had to keep in mind throughout was that different parts of the book were written at significantly different times, sometimes a decade apart. So it is not written in a unified voice. She labels some sections her ‘relativist papers’ and critiques them throughout. She writes:
“The relativist papers failed to deliver a useable critique; nevertheless, they are significant and important, and not only because they embody a considerable creativity and were born from dedicated scholarship.”
Verran starts in Chapter 1 with asking: Is it possible African numbers are different than scientific numbers? I just find the phrasing of that sentence really weird, but I tried to enter into it with an open mind. She details how she arrived at this question when struggling to teach science to Yoruba school teachers. Verran proposes two interpretations of the pedagogical struggle of the “African classroom” (which is already a troubling category in my mind):
“Rightist and leftist politics identify quite different causes for the conceptual difficulties that can arise in teaching and learning mathematics and science in African classrooms. Adopting one or the other of these has a theorist distributing praise and blame in particular ways, and not only to the student teachers featured in these stories and to their lecturer. The contrasting analyses also distribute praise and blame to African communities, and to their European colonizers. In epistemological terms, this is the familiar contest between universalism and relativism.”
So far I get this. There are racists who blame students and their culture for holding back their intellectual capacities. She claims this rightist perspective holds a universalist epistemology and this is the perspective allied with institutional power and orthodox teaching of the canon. (I think this is a dramatic framing, and I am skeptical already, but let’s continue).
She puts forward a critique of natural numbers and universalist epistemology where numbers are universally real even if abstract. Numbers are abstractions that contain the structure of the physical world. She then turns to logical positivists in particular, and takes Carnap as their exemplar. As an aside, Carnap, along with Neurath, were I think the only left-wing logical empiricists of the Vienna Circle who engaged to some extent with Marx’s work. But Carnap, leaned more towards libertarian socialism, at least eventually.
Anyway, according to Verran, Carnap sees numbers as a “universal form embedded in the structure of experience of the physical world” and three concepts in the logic of numbering: classificatory, comparative, and quantitative. For Carnap, “A quantitative concept is the extent of a quality; it expresses an objective value of a particular body… Qualities are the first abstract entities of quantifying, and numbers are a second-order abstract entity. Numbers arise in the analogy between the extension of qualities held by the perceived spatio-temporal entities and the numeral sequence expressing linear extension recursively.”
She then corresponds different things she witnessed in the classroom with each of these perspectives. All the teachers she describes here are African, just to be clear. The first teacher uses an uninterrupted straight line to convey length and this line can afterwards be divided into smaller sections to derive units, which can be accumulated to get a plurality. Another teacher uses a bundle of string to convey length, which for some reason Verran calls the “primitive” way. Perhaps I’m not getting this, but how is using string to measure length primitive? People use tape measures all the time, and those rolled up rulers they have at Ikea which I used growing up to measure my waist so my parents could buy clothes for me. The whole notion of waist size or circumference. I’m not sure I understand this point here.
Verran then says the leftist perspective is a relativist one that prioritizes language in numbering:
“Instead of positing a prelinguistic experience of physical world as universal, this theory of knowledge points to social practices like linguistic methods, as the origin of categories in knowledge. Such a position would maintain that Yoruba and English languages create alternative starting points for numbering, resulting in different quantifying logics.”
She goes on to write:
“mathematics and science educators puzzle about what "Yaruba mathematics" and "Yoruba science" might be… This leftist critique presents itself as an appropriate anticolonizing move. It promotes liberation of the Yoruba community, beginning in
its schools, which have been laboring under the burden of a European curriculum for over a century”
I agree that the legacy of colonialism in school curricula is a bad thing, but I don’t see how making the numbering systems of other cultures as radically other is a useful solution. I fail to see the distinctions Verran is making. Again I am not good at math and also don’t understand the epistemological stakes here, but I’m struggling to see what Verran sees. If you understand, please enlighten me.
The next chapter involves Verran trying to deal with this problem of explaining away the experiences she had as a teacher which initially provoked this question. She again tells stories of her experiences, and to me, it seems like a lot of this stems from local teachers presenting mathematical problems (like measuring length or calculating volume) and proposing solutions in a way that is very counter-intuitive to Verran and seems pedagogically fraught. From these issues, she formulates these sorts of philosophical and epistemological distinctions that seem very overblown to me. But again, I think this is because I am not actually able to follow her arguments. I don’t understand so many of the sentences in this book.
In Chapter 3 she compares Yoruba and English number systems, how numeral sequences are both generated and used, and offers a relativist theory of the origins of number in practice.
In Chapter 4 she discusses how numbers were deployed in the project of colonialism, and critiques her previous chapter for treating Yoruba numbers as found objects rather than something she just objectified in her writing. She contextualizes her relativist comparison with previous universalist studies of Yoruba numbers, and that she arrived at those differences by “literalizing the figure of recursion” (why must people write like this, I don’t understand.)
In Chapter 5, she thinks about the relationships between unity and plurality that numbers enable, and then somehow enters into the domain of Althusserian interpellation. She emphasizes that numbers are both multiple and simultaneously singular and definitive, and somehow connects this with a politics of difference that numbering systems embody. I am having a very difficult time as imagining numbers as subjects to make this Althusserian thing work here. Verran writes:
“Interpellating is a form of hailing, a hailing or greeting that "drives."14 Etymologically the term "interpellating" incorporates the Latin verb pellare, pellere (to drive). We can imagine interpellation "driving" to effect outcomes. Interpellation, in the way I use it, is as much involved with the production of"objects" as with "subjects," in that both are seen equally as outcomes or effects of myriad enacted associations. Along with this change, the specific Marxist notion of ideology,15 understood as interpellating individuals as subjects, is remade as the much more diffuse and general notion of enacting. Althusser's ideology is both much more and much less than enacting as I use it, and the individuals/subjects contrast/ connection that he uses to illustrate interpellation by ideology is both more and less than the numerals/numbers contrast/connection I use here to illustrate interpellation by enacting. Yet the arguments he makes about the ways ideology interpellates individuals as already always subjects translate easily as arguments that I want to make to reentangle numerals as numbers. I re-present Althusser's arguments here-suitably translated to do the work I want them to do… The sounding, or appearance in writing, of a numeral initiates the rituals of its recognition as a number, guaranteeing that numerals are indeed already always numbers.”
I just cannot seem to understand what Verran is saying half the time. This is a later summary of Chapter 5 that Verran provides and it is still such a struggle for me:
“In introducing the notion of interpellation in chapter 5, I pointed to the ways that ritual both generates and resolves tension. I used Althusser's example of a handshake as a ritual that, momentarily transcending difference, unites two men as collective subject, at the same time as it separates them as differently embodied subjects. Althusser's example of a hand- shake points to the creating and resolution of a tension in which the generalization "collective identity" is effected.
We can recognize that a handshake effects and is effected as ritual in an ordered/ordering microworld. The participants must stand just far enough apart, not too close, not too distant. Right arms must be proffered across the participants' bodies so hands clutch in a particular manner. During a handshake, it is as if the two men are enclosed in a bubble- the confusing complexity of their separate lives is momentarily excluded. Simplification is achieved in a foregrounding. The generalization effected in the doing of the ritual is internally complicated, embedding in unambiguous ways the often highly ambiguous complexity that the microworld structures in ritual. In the case of Althusser's handshake, the complexity that is remade as manageable complication in the ritual is the contradictory separate yet co-constituted and singular identit(y)ies.”
In Chapter 6, Verran shows how the generalizing that occurs in Yoruba quantification is just as logical and abstract as the quantification that happens in English. This is done to criticize the claims of cross-cultural psychologists that blame local knowledge systems for educational outcomes. This chapter makes sense.
In Chapter 7 criticizes the previous chapter and says the comparison between the two is performed through ‘literalizing’ which denies the difference between the two systems as real and manageable. (Did not understand this.)
In Chapter 8, Verran asks what are objects or concepts like ‘volume’? She tries to provide an alternative explanation for generalizations that did not understand them as abstractions but rather products of collective acting. From there she interrogates how differences work in this domain of generalizing.
The last three chapters are on predicating-designating — where in questions like “Is it water?” the “is” is the predicating term and the “it” is the designating term, and the “water” is a characterizing term. Verran says this process is done differently in English and in Yoruba (Chapter 9), and then ‘decomposes’ her argument (Chapter 10), and then offers a new account of predicating-designating as a way of figuring relationalities through collective acting and “articulating the ontological politics implicit in differences in generalizing logics.”
I wish I had time to make a serious attempt to understand this book, but I just have too much to do. Perhaps someone will one day explain to me what this book was about.
One thing that I had to keep in mind throughout was that different parts of the book were written at significantly different times, sometimes a decade apart. So it is not written in a unified voice. She labels some sections her ‘relativist papers’ and critiques them throughout. She writes:
“The relativist papers failed to deliver a useable critique; nevertheless, they are significant and important, and not only because they embody a considerable creativity and were born from dedicated scholarship.”
Verran starts in Chapter 1 with asking: Is it possible African numbers are different than scientific numbers? I just find the phrasing of that sentence really weird, but I tried to enter into it with an open mind. She details how she arrived at this question when struggling to teach science to Yoruba school teachers. Verran proposes two interpretations of the pedagogical struggle of the “African classroom” (which is already a troubling category in my mind):
“Rightist and leftist politics identify quite different causes for the conceptual difficulties that can arise in teaching and learning mathematics and science in African classrooms. Adopting one or the other of these has a theorist distributing praise and blame in particular ways, and not only to the student teachers featured in these stories and to their lecturer. The contrasting analyses also distribute praise and blame to African communities, and to their European colonizers. In epistemological terms, this is the familiar contest between universalism and relativism.”
So far I get this. There are racists who blame students and their culture for holding back their intellectual capacities. She claims this rightist perspective holds a universalist epistemology and this is the perspective allied with institutional power and orthodox teaching of the canon. (I think this is a dramatic framing, and I am skeptical already, but let’s continue).
She puts forward a critique of natural numbers and universalist epistemology where numbers are universally real even if abstract. Numbers are abstractions that contain the structure of the physical world. She then turns to logical positivists in particular, and takes Carnap as their exemplar. As an aside, Carnap, along with Neurath, were I think the only left-wing logical empiricists of the Vienna Circle who engaged to some extent with Marx’s work. But Carnap, leaned more towards libertarian socialism, at least eventually.
Anyway, according to Verran, Carnap sees numbers as a “universal form embedded in the structure of experience of the physical world” and three concepts in the logic of numbering: classificatory, comparative, and quantitative. For Carnap, “A quantitative concept is the extent of a quality; it expresses an objective value of a particular body… Qualities are the first abstract entities of quantifying, and numbers are a second-order abstract entity. Numbers arise in the analogy between the extension of qualities held by the perceived spatio-temporal entities and the numeral sequence expressing linear extension recursively.”
She then corresponds different things she witnessed in the classroom with each of these perspectives. All the teachers she describes here are African, just to be clear. The first teacher uses an uninterrupted straight line to convey length and this line can afterwards be divided into smaller sections to derive units, which can be accumulated to get a plurality. Another teacher uses a bundle of string to convey length, which for some reason Verran calls the “primitive” way. Perhaps I’m not getting this, but how is using string to measure length primitive? People use tape measures all the time, and those rolled up rulers they have at Ikea which I used growing up to measure my waist so my parents could buy clothes for me. The whole notion of waist size or circumference. I’m not sure I understand this point here.
Verran then says the leftist perspective is a relativist one that prioritizes language in numbering:
“Instead of positing a prelinguistic experience of physical world as universal, this theory of knowledge points to social practices like linguistic methods, as the origin of categories in knowledge. Such a position would maintain that Yoruba and English languages create alternative starting points for numbering, resulting in different quantifying logics.”
She goes on to write:
“mathematics and science educators puzzle about what "Yaruba mathematics" and "Yoruba science" might be… This leftist critique presents itself as an appropriate anticolonizing move. It promotes liberation of the Yoruba community, beginning in
its schools, which have been laboring under the burden of a European curriculum for over a century”
I agree that the legacy of colonialism in school curricula is a bad thing, but I don’t see how making the numbering systems of other cultures as radically other is a useful solution. I fail to see the distinctions Verran is making. Again I am not good at math and also don’t understand the epistemological stakes here, but I’m struggling to see what Verran sees. If you understand, please enlighten me.
The next chapter involves Verran trying to deal with this problem of explaining away the experiences she had as a teacher which initially provoked this question. She again tells stories of her experiences, and to me, it seems like a lot of this stems from local teachers presenting mathematical problems (like measuring length or calculating volume) and proposing solutions in a way that is very counter-intuitive to Verran and seems pedagogically fraught. From these issues, she formulates these sorts of philosophical and epistemological distinctions that seem very overblown to me. But again, I think this is because I am not actually able to follow her arguments. I don’t understand so many of the sentences in this book.
In Chapter 3 she compares Yoruba and English number systems, how numeral sequences are both generated and used, and offers a relativist theory of the origins of number in practice.
In Chapter 4 she discusses how numbers were deployed in the project of colonialism, and critiques her previous chapter for treating Yoruba numbers as found objects rather than something she just objectified in her writing. She contextualizes her relativist comparison with previous universalist studies of Yoruba numbers, and that she arrived at those differences by “literalizing the figure of recursion” (why must people write like this, I don’t understand.)
In Chapter 5, she thinks about the relationships between unity and plurality that numbers enable, and then somehow enters into the domain of Althusserian interpellation. She emphasizes that numbers are both multiple and simultaneously singular and definitive, and somehow connects this with a politics of difference that numbering systems embody. I am having a very difficult time as imagining numbers as subjects to make this Althusserian thing work here. Verran writes:
“Interpellating is a form of hailing, a hailing or greeting that "drives."14 Etymologically the term "interpellating" incorporates the Latin verb pellare, pellere (to drive). We can imagine interpellation "driving" to effect outcomes. Interpellation, in the way I use it, is as much involved with the production of"objects" as with "subjects," in that both are seen equally as outcomes or effects of myriad enacted associations. Along with this change, the specific Marxist notion of ideology,15 understood as interpellating individuals as subjects, is remade as the much more diffuse and general notion of enacting. Althusser's ideology is both much more and much less than enacting as I use it, and the individuals/subjects contrast/ connection that he uses to illustrate interpellation by ideology is both more and less than the numerals/numbers contrast/connection I use here to illustrate interpellation by enacting. Yet the arguments he makes about the ways ideology interpellates individuals as already always subjects translate easily as arguments that I want to make to reentangle numerals as numbers. I re-present Althusser's arguments here-suitably translated to do the work I want them to do… The sounding, or appearance in writing, of a numeral initiates the rituals of its recognition as a number, guaranteeing that numerals are indeed already always numbers.”
I just cannot seem to understand what Verran is saying half the time. This is a later summary of Chapter 5 that Verran provides and it is still such a struggle for me:
“In introducing the notion of interpellation in chapter 5, I pointed to the ways that ritual both generates and resolves tension. I used Althusser's example of a handshake as a ritual that, momentarily transcending difference, unites two men as collective subject, at the same time as it separates them as differently embodied subjects. Althusser's example of a hand- shake points to the creating and resolution of a tension in which the generalization "collective identity" is effected.
We can recognize that a handshake effects and is effected as ritual in an ordered/ordering microworld. The participants must stand just far enough apart, not too close, not too distant. Right arms must be proffered across the participants' bodies so hands clutch in a particular manner. During a handshake, it is as if the two men are enclosed in a bubble- the confusing complexity of their separate lives is momentarily excluded. Simplification is achieved in a foregrounding. The generalization effected in the doing of the ritual is internally complicated, embedding in unambiguous ways the often highly ambiguous complexity that the microworld structures in ritual. In the case of Althusser's handshake, the complexity that is remade as manageable complication in the ritual is the contradictory separate yet co-constituted and singular identit(y)ies.”
In Chapter 6, Verran shows how the generalizing that occurs in Yoruba quantification is just as logical and abstract as the quantification that happens in English. This is done to criticize the claims of cross-cultural psychologists that blame local knowledge systems for educational outcomes. This chapter makes sense.
In Chapter 7 criticizes the previous chapter and says the comparison between the two is performed through ‘literalizing’ which denies the difference between the two systems as real and manageable. (Did not understand this.)
In Chapter 8, Verran asks what are objects or concepts like ‘volume’? She tries to provide an alternative explanation for generalizations that did not understand them as abstractions but rather products of collective acting. From there she interrogates how differences work in this domain of generalizing.
The last three chapters are on predicating-designating — where in questions like “Is it water?” the “is” is the predicating term and the “it” is the designating term, and the “water” is a characterizing term. Verran says this process is done differently in English and in Yoruba (Chapter 9), and then ‘decomposes’ her argument (Chapter 10), and then offers a new account of predicating-designating as a way of figuring relationalities through collective acting and “articulating the ontological politics implicit in differences in generalizing logics.”
I wish I had time to make a serious attempt to understand this book, but I just have too much to do. Perhaps someone will one day explain to me what this book was about.
December 17, 2020
If you pour water from a tall narrow tube into a wide shallow tub, is it the same amount of water? Many kids will say no, because it's shorter, even though we know it's the same volume.
So if such conceptions of "more," can depend on development, why shouldn't they depend on language and culture? Such interconnections are roughly what this book takes up. Verran presents a number of experiments and observations from her time teaching in Yoruba schools, and relates her finding to various theories of what language means and how math is thought about.
Overall, her writing clearly presents fascinating main points. By the end, a few rhetorical styles and references get a bit weary. Each section of the book contains an older paper where she presents an experiment, followed by a critique of how she hid certain assumptions that she now disagrees with. It's an effective strategy. But there's enough shared content between the essays, i.e. outlining Yoruba's 20-based number system, or comparing linguistic and mathematical/set-theoretic conceptions of numbers, that the repeated invocations of the same move start to feel a bit stale by the end.
So if such conceptions of "more," can depend on development, why shouldn't they depend on language and culture? Such interconnections are roughly what this book takes up. Verran presents a number of experiments and observations from her time teaching in Yoruba schools, and relates her finding to various theories of what language means and how math is thought about.
Overall, her writing clearly presents fascinating main points. By the end, a few rhetorical styles and references get a bit weary. Each section of the book contains an older paper where she presents an experiment, followed by a critique of how she hid certain assumptions that she now disagrees with. It's an effective strategy. But there's enough shared content between the essays, i.e. outlining Yoruba's 20-based number system, or comparing linguistic and mathematical/set-theoretic conceptions of numbers, that the repeated invocations of the same move start to feel a bit stale by the end.
December 12, 2024
I really kept thinking this was going to culminate in some brilliant materialist analysis and instead it kept stopping short at a point of pretty basic philosophical critique, with a few new random and vaguely articulated categories thrown in there. Also self-flagellating to the point it seemed like navel-gazing. 3 stars though (meaning I generally liked it somewhat), because there was some interesting material for thought.
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