PASML2018 (Brussels, September) celebration of the work of Jean Paul van Bendegem

1. The material-ideal dyad of culture and the revolutionary
materialism of practice studies
PASML2018 (Brussels, September)
Brendan Larvor

2. Mathematical practice is quite hard to capture on film

3. The material-ideal dyad of culture and the revolutionary
materialism of practice studies
According to Theodore R. Schatzki (The Practice Turn in Contemporary Theory Routledge 2001), theorists in the humanities
and social sciences who thematise practices have some features in common: they are usually opposed to understanding
human activities in terms of relatively rigid abstract structures (be they conceptual, grammatical or social). Practices are
preferred because they are materially present, and because they are historical—that is, they change. Practice-theorists also
tend to deprecate explanations that appeal to individual volition. Putting these elements together, it seems that a turn to
practice is a turn to post-humanist materialism. Raviel Netz, for example, argues that practices rather than concepts are the
proper objects of study for historians of ancient mathematics and for ‘cognitive archaeologists’. I shall illustrate this
observation and suggest that it presents philosophers of mathematical practices with some dilemmas.
Turning from practice to culture, I shall present and briefly argue for a model of culture that I developed with mathematical
cultures in mind. On this model, culture is a mass-noun rather than a count-noun. Culture is the expression of values in
practices. Thinking this way has some advantages over the tradition of treating cultures as anthropological units (Chinese
culture, millennial internet culture, early modern Dutch print culture, etc.). It allows us to recognise multiple values realised in
a single practice, and to see the same values expressed in different practices. This helps us to understand cultural change
and to facilitate communication between cultures. It also allows us to embrace the benefits of the turn to practices in
humanities and social science without falling into materialism or post-humanism.

4. The revolutionary materialism of practice studies
According to Theodore R. Schatzki, theorists in the humanities and social sciences
who thematise practices have some features in common: they are usually opposed to
understanding human activities in terms of relatively rigid abstract structures (be they
conceptual, grammatical or social). Practices are preferred because they are
materially present, and because they are historical—that is, they change.
Practice-theorists also tend to deprecate explanations that appeal to individual volition.
Putting these elements together, it seems that a turn to practice is a turn to post-
humanist materialism. Raviel Netz, for example, argues that practices rather than
concepts are the proper objects of study for historians of ancient mathematics and for
‘cognitive archaeologists’.
I shall illustrate this observation and suggest that it presents philosophers of
mathematical practices with some dilemmas.
A spectre is haunting the humanities:

5. ‘Practice’?
Theodore R.Schatzki The Practice Turn in Contemporary Theory
(Routledge 2001)
• “…philosophical practice thinkers such as Wittgenstein, Dreyfus, and Taylor contend that
practices at once underlie subjects and objects, highlight non-propositional knowledge, and
illuminate the conditions of intelligibility.
• For their social theoretical brethren Bourdieu, Giddens and the ethno-methodologists, talk of
practices bespeaks desires… to free activity from the determining grasp of objectified social
structures and systems, to question individual actions and their status as the building-blocks of
social phenomena, and to transcend rigid action-structure oppositions.
• For cultural theorists [e.g.] Foucault and Lyotard,… to speak of practices is to depict language as
discursive activity in opposition to structuralist, semiotic, and poststructuralist conceptions of it as
structure, system, or abstract discourse.
• And among,… the purposes animating the practice-theoretical study of science and technology
(e.g., Rouse; Pickering) are the development of concepts of science as activity as opposed to
representation and the reconsideration of humanist dichotomies between human and nonhuman
entities.”
How many people on the philosophy of
maths practice circuit fit this picture?

6. Let’s do some philosophy on Pickering
‘Concepts and the Mangle of Practice: Constructing Quaternions’
18 Unconventional Essays on the Nature of Mathematics Hersh (ed.) Springer 2006
“An asymmetry exists in our accounts of
scientific practice: machines are located in
a field of agency but concepts are not.” (p.
250)
“Why concepts are not mere putty in our
hands?” (p. 251)
Answer: “We should think of conceptual
structures as themselves located in fields
of agency, and of the transformation and
extension of such structures as emerging
in dialectics of resistance and
accommodation.”
(What is a ‘field of agency’?)

7. Let’s do some philosophy on Pickering
Wittgenstein + Collins = ?
“Every sign by itself seems dead. What gives it life? –In use it is alive.’ (LW quoted p. 252)
– (note the biological metaphor, sustained in Lynch’s gloss (which is reproduced in
Pickering’s footnote) in contrast to Pickering’s machine-metaphor taken from Collins)
“such uses are disciplined; they are machine-like actions… Just as in arithmetic one
completes ‘3+4=‘ by writing ‘7’ without hesitation, so in algebra one automatically multiplies
out ‘a(b+c)’ as ‘ab+ac’.”
– charity requires us to read a tacit ‘sometimes’ in front of these automaticities. Otherwise,
he’s writing patent falsehoods. E.g. a(b+c) – d(b+c)
– on the next page, we learn that “Bridging and filling are free moves” (so human action is
not always automatic & machine-like)

8. Let’s do some philosophy on Pickering
The Dance of Agency
“Such disciplines [as the automaticities mentioned above] … carry human conceptual
practices along, as it were, independently of individual wishes and intents. The scientist is,
in this sense, passive in disciplined conceptual practice. … I want to redescribe such
human passivity in terms of a notion of disciplinary agency. It is… the agency of a
discipline – elementary algebra for example – that leads disciplined practitioners through a
series of manipulations with an established conceptual system.” (p. 252)
“Conceptual practice therefore has,… the form of a dance of agency, in which the partners
are alternately the classic human agent and disciplinary agency.”

9. Let’s do some philosophy on Pickering
Vive la Resistance
“the agency of a discipline – elementary
algebra for example – that leads
disciplined practitioners through a series
of manipulations”
Note the ambiguity of ‘lead’. A
topographical feature, a path and a human
guide can lead the traveler. Only one of
these is properly speaking an agent.
In fact, elementary algebra leads in only a
weak sense (you always have indefinitely
many moves available)

10. Let’s do some philosophy on Pickering
Accommodation: I like a lot of what he has to say!
The historicism! The focus on the unique
and unpredictable particular situations! The
rejection of ahistorical explanatory essences
and structures! (Collingwood…)
Notations, concepts, etc. are indeed kinda
like machines (J-P Marquis)
Action is indeed constrained by custom &
practice (and the material environment,
tools, machines, notations, etc.); sensibility
is shaped by same.
But not as a potter shapes clay!
“production not only creates an object for the
subject but also a subject for the object”
– Marx, quoted in Pickering ‘Practice and posthumanism’
(The Practice Turn p. 172)

11. Raviel Netz
“There has to be a widely shared
understanding of what is meant by ‘the
number of all prime numbers is greater
than any assigned number’, as well as a
widely shared understanding of what
constitutes proof of such a claim, even
without there being a widely shared
understanding of what is meant by
‘number’”
“Science is what people do together, not
what they each have privately in their
heads.”
(Mathematical Concepts? In What is a
mathematical concept? p. 49)
Concepts are immaterial

12. New materialist ontologies in mathematics education:
the body in/of mathematics
“learners’ bodies are always in the process of
becoming assemblages of diverse and dynamic
materialities, including physical objects such as
pencils, compasses and calculators, and also
non-visible mathematical concepts.” (p. 454)
“Roth speaks of the ‘body’ in terms of the isolated
human body, while the cube is that which is acted
upon. In this paper, we ask: what happens when
we consider the cube itself as a material agent or
recognize the way that degrees of agency
saturate the situation and all its “actants”?” (p.
457)
(Educ Stud Math (2013) 83:453–470)
Elizabeth de Freitas & Nathalie Sinclair

13. Turning from practice to culture, I shall present and briefly argue for a model of culture
that I developed with mathematical cultures in mind.
On this model, culture is a mass-noun rather than a count-noun.
Culture is the expression of values in practices.
Thinking this way has some advantages over the tradition of treating cultures as
anthropological units (Chinese culture, millennial internet culture, early modern Dutch
print culture, etc.). It allows us to recognise multiple values realised in a single
practice, and to see the same values expressed in different practices.
This helps us to understand cultural change and to facilitate communication between
cultures.
It also allows us to embrace the benefits of the turn to practices in humanities and
social science without falling into materialism or post-humanism
The material-ideal dyad of culture
A bourgeois idealist / left Hegelian plea

14. This bit is from a joint paper… and benefitted from VUB funding

15. How to think about culture
• Not as distinct, discrete, fixed units e.g. the
Navajo culture, the Mundukuru culture, British
culture, etc..
• As a material-ideal pair: artefacts & practices ~
ideals
• As a mass-noun, not a count-noun
• As a task for the individual, not a property of the
individual
• Why?
 It’s the truth!
 Allows change
 Allows recognition of shared values
Dialectically!

16. Ethnomathematics
"the mathematics which is practised among
identifiable cultural groups“
Culturally differential pedagogy
But how is this not apartheid education?
In any case, it explodes ‘mathematics’
Falls into essentialism…
And appropriates cultures for state ends…
So Maths Ed learned to be cautious about
culture
Great research, problematic pedagogy

17. So what is culture?
“Culture consists of patterns, explicit and
implicit, of and for behavior acquired and
transmitted by symbols, constituting the
distinctive achievements of human groups,
including their embodiments in artifacts; the
essential core of culture consists of
traditional (i.e. historically derived and
selected) ideas and especially their
attached values; culture systems may, on
the one hand, be considered as products of
action, and on the other as conditioning
elements of further action.”
Kroeber, A.L., & Kluckhohn, C. (1952) Culture: A
critical review of concepts and definitions.
A pair of oppositions

18. So what is culture?
There is a dialectical relationship between
the material aspects of culture (the
artefacts and practices) and the ideas and
values, such that culture is unintelligible
without reference to both.
There is a relation between descriptive and
normative approaches to culture such that
the normative aspect can never be
eliminated. To understand a culture is to
understand its values, but this always
involves some dialogue with the
researcher’s own values (even if these are
solely norms of scientific practice).
A pair of oppositions

19. One of many reasons why the ‘new materialism’ is incoherent
Imagine future archaeologists
finding a crucifix.
Our chances of understanding
prehistoric cave art are the same
as their chances of guessing the
narrative and meaning that goes
with a crucifix.
You can’t understand the practices without reference to ideas

20. The material-ideal dyad of culture and the revolutionary
materialism of practice studies
According to Theodore R. Schatzki (The Practice Turn in Contemporary Theory Routledge 2001), theorists in the humanities
and social sciences who thematise practices have some features in common: they are usually opposed to understanding
human activities in terms of relatively rigid abstract structures (be they conceptual, grammatical or social). Practices are
preferred because they are materially present, and because they are historical—that is, they change. Practice-theorists also
tend to deprecate explanations that appeal to individual volition. Putting these elements together, it seems that a turn to
practice is a turn to post-humanist materialism. Raviel Netz, for example, argues that practices rather than concepts are the
proper objects of study for historians of ancient mathematics and for ‘cognitive archaeologists’. I shall illustrate this
observation and suggest that it presents philosophers of mathematical practices with some dilemmas.
Turning from practice to culture, I shall present and briefly argue for a model of culture that I developed with mathematical
cultures in mind. On this model, culture is a mass-noun rather than a count-noun. Culture is the expression of values in
practices. Thinking this way has some advantages over the tradition of treating cultures as anthropological units (Chinese
culture, millennial internet culture, early modern Dutch print culture, etc.). It allows us to recognise multiple values realised in
a single practice, and to see the same values expressed in different practices. This helps us to understand cultural change
and to facilitate communication between cultures. It also allows us to embrace the benefits of the turn to practices in
humanities and social science without falling into materialism or post-humanism.