Culture Lisbon 30 June 2022.pptx

Why would anyone not take a cultural approach to mathematics? 1
Brendan Larvor 30 June 2022

Brendan Larvor 30 June 2022

Why would anyone not take a
cultural approach to mathematics?
Why indeed? Three reasons:
• Metaphysics
• Essentialism
• Relativism

The plan:
• How I became an expert
• Metaphysics: how natural is naturalism?
• Essentialism: what is the unit of cultural
analysis? What is a culture anyway?
• Relativism: difference and change

How I became an expert
With help from
Benedikt Löwe &
Thomas Müller

Prehistory: education and anthropology but no philosophy
Bishop, A. J. (1988). Mathematical enculturation: A cultural perspective on
mathematics education. Dordrecht/Boston: Kluwer Academic Publishers.
Kline, M. (1953). Mathematics in western culture. New York: Oxford
University Press.
Raum, O. (1938). Arithmetic in Africa. London: Evans Brothers.
Vithal, Renuka & Skovsmose, Ole (1997) The End of Innocence: A Critique
of Ethnomathematics. Educational Studies in Mathematics. 34(2), 131–157.
White, L. (1947). The locus of mathematical reality: An anthropological
footnote. Philosophy of Science, 14(4), 289–303.
Wilder, RL (1981) Mathematics as a cultural system. Pergamon Press.

Metaphysics: how natural is naturalism?
“[Natural] Science is the measure of all things, of what is that
it is, and of what is not that it is not.”
Wilfrid Sellars Science, Perception and Reality (London:
Routledge & Kegan Paul Ltd, 1963) p. 173.
Note the conscious echo of Aristotle’s definition of truth
(Metaphysics 1011b25).
…or rather, philosophers’ convictions about what natural
science says.

“[Natural] Science is the measure of all things, of what is that
it is, and of what is not that it is not.”
Wilfrid Sellars Science, Perception and Reality (London:
Routledge & Kegan Paul Ltd, 1963) p. 173.
Note the conscious echo of Aristotle’s definition of truth
(Metaphysics 1011b25).
…or rather, philosophers’ convictions about what natural
science says.

Things natural science cannot detect:
• Propositions
• Arguments
• Validity of arguments
• Academic status
• Money
All essential for the doing of science

“The first rule for understanding
the human condition is that men
live in second-hand worlds.”
C. Wright Mills The Cultural
Apparatus

LA White: when you give cultural accounts, the species
and the individual both drop out. (Multiple discoveries,
etc.)

Section summary:
Taking a cultural view seems to entail giving metaphysical
priority to a level of reality—the social—that
contemporary anglophone philosophy usually considers
to be derivative or fictional.

Essentialism: what is the unit of cultural
Chemla & Fox Keller against
cultural essentialism (even though
actors are often keen on it esp.
national or religious cultures).
Cultures are internally diverse,
changed by their own products
and interactions with others,
dynamic and actors are often
relatively self-aware.

Fox Keller: the endless slicing of
identity politics leads to strategic
essentialism.

Gerovitch, S. (2016). Creative Discomfort: The Culture of
the Gelfand Seminar at Moscow University. In: Larvor, B.
(ed.) Mathematical Cultures. Trends in the History of
Science. Birkhäuser, Cham.
One seminar is too small to constitute a culture, but the
existence of the Gelfand seminar requires explanation in
terms of Soviet mathematical culture.
For more on Soviet maths, see José Ferreirós in this
series

D’Ambrosio: any collection of people can be an ethnos for
the purposes of ethnomathematics—if they share some
mathematical practices.

Makenzie (in Chemla & Fox Keller p. 38) presents
four criteria for cultural distinctness:
Different practices associated with different
 Ontologies
 Processes of socialisation
 Mechanisms of interaction among participants
 Path-dependent patterns of change (i.e. different
histories)
Makenzie requires all four…

Makenzie requires all four, but is this too much? The
last one is probably redundant—if you have the first
three, there must be different histories.
If you had two, would that be enough?
Do we need different ontologies every time?
(Notice: all Moscow maths seminars had the same
four Makenzie elements.)

All determination
is negation:

Section summary:
• Taking a cultural view smells of ethnic
essentialism or even apartheid to some and
‘woke’ identity politics to others.
• Mathematics as trans-cultural, colour-blind
shared knowledge is part of its appeal.
Culturalism builds walls.

Relativism: Alan Bishop’s mathematical values
Ideology: Rationalism & Objectism
Sentiment: Control & Progress
Sociology: Openness & Mystery
Bishop AJ. Western mathematics: the secret weapon of
cultural imperialism. Race & Class. 1990;32(2):51-65.

Rationalism
• In maths, truth is determined by proof—but this can be
divisive and some cultures value social ease and respect for
hierarchy more than they do logic. If the authorities in your
life say contradictory things, accept all of them.
• Rationalisation in the sense of Weber. Formatting the world.

Objectism
• Mathematics prefers objects to processes (and makes
objects out of processes, e.g. mapping, permutation)
• Material objects are the paradigm—stable, detachable from
context, unlike the unstable, processual world of social
relations and semiosis
• Prefers artefacts to nature

• Control: through technology and bureaucracy
• Progress: mathematics is part of a scientific set of values
in which structures of knowledge and authority are
provisional, in contrast to cultures where authority is
eternal and unchanging

• Openness: in principle, anyone can learn mathematics. It’s
difficult but not intrinsically esoteric (the high-school
student at the Gelfand seminar)
• Mystery: mathematics associated with other-worldliness
and mysticism

• Much of this literature depends on a problematic
Western/indigenous contrast
• Nevertheless, Bishop is right that mathematics is associated
with Enlightenment, anti-aristocratic, anti-clerical and
cosmopolitan values. A cultural approach seems to
contradict these.

Relativism: difference and change
L. E. J. Brouwer: bare two-oneness of temporal
experience
Brouwer, L. E. J. (1913). Intuitionism and
formalism. Bulletin of the American Mathematical
Society, 20(2), 81-96.

Relativism: difference and change
Everybody counts
and almost
everybody measures
(See also Ferreirós)

How I became an expert
(2012) “The Mathematical Cultures Network Project” Journal of Humanistic Mathematics 2
(2): 157-160.
(2016) Mathematical Cultures: The London Meetings 2012-2014 Edited collection,
Birkhäuser, Basel
(2016) “What are Cultures?” in Cultures of Mathematics and Logic Selected papers from the
conference in Guangzhou, China, 9-12 November 2012. Shier Ju, Benedikt Loewe, Thomas
Mueller, Yun Xie (eds.) Birkhäuser, Basel
(2016) with Karen François “Cultural and Institutional Inequalities: The case of mathematics
education in Flemish schools” Journal of Mathematics and Culture, 10(2), 37-54.
(2018) with Karen François “The concept of Culture in Critical Mathematics Education”. In P.
Ernest (Ed.), Philosophy of Mathematics Education Today Springer Verlag.

Thank you
Brendan Larvor
b.p.larvor@herts.ac.uk

Culture Lisbon 30 June 2022.pptx

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