1. Who studies mathematical practice? Why?
19 March 2014 UCL STS Seminar Series
www.herts.ac.uk/philosophy

2. Who studies mathematical practice? Why?
The promise-blurb
The philosophy of mathematical practice
has grown rapidly in the last decade and
now has the beginnings of canonical
literature to match its conferences and
edited volumes. However, it is still better
described as a movement than a field, and
there are reasons to doubt whether its
practitioners have seriously reckoned with
the consequences and connotations of the
word ‘practice’. This talk will survey the
current condition of the philosophy of
mathematical practice and consider the
methodological and ideological challenges it
faces.
Plus bonus: swipe at Andrew Pickering!

8. Can you sum this up?
Yes
• English-speaking philosophy of
mathematics has a lot of Enlightenment
assumptions built in
• No surprise if your history of philosophy
stops at Kant…
• …And your history of your problem
starts with Russell
• (I ended up banging on like Collingwood
about history and change)

15. Would you say the same today?
Sort of, but…
• ‘Mainstream’ philosophers of
mathematics have responded to the
charge that they are distant from
mathematical practice
• There is a rise in ‘foundational’
philosophy that does not map on to the
old ‘foundations’ problem-geography
• I’d say more about maths education
(and the strange birth of the APMP)

16. Much diversity
Little unity
• Most of these people need places to
speak and publish, but do not need
interdisciplinary connections
• Only the philosophers are intellectually
needy
• Moreover there are serious disciplinary
barriers to overcome
• Maybe some intellectual glue is latent in
the common words. E.g. ‘practice’

17. ‘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.”
Almost everybody on the philosophy of maths
practice circuit would dismiss or resist most of this

18. 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.”
(One of my aims today is to get some help with these words)

19. 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.
– on the next page, we learn that “Bridging and filling are free moves” (so human action is
not always automatic & machine-like)

20. 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.”

21. Let’s do some philosophy on Pickering
Vive la Resistance
Pickering invokes agency to explain “Why
concepts are not mere putty in our
hands?”
As if resistance to my will can only be
explained by an opposing will
Passivity and agency are not the right
concepts! Flexible/ rigid, (un)determined
or horticultural metaphors may apply
Conceptual structures, notations, etc.
cannot be agents because they do not
have metabolisms.

22. 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)

23. 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)

24. Let’s do some philosophy on Pickering
Why does he say this stuff?
Too much Wittgensteinian rhetoric about
rule-following?
Mesmerised by symmetry?

25. Let’s do some philosophy on Pickering
What do I miss if I reject his agency-symmetry idea?
Maybe we’re just rejecting different orthodoxies
(phil of maths does not treat the subject as a decider
but rather as a reasoner)
All determination is negation!

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b.p.larvor@herts.ac.uk