What does it mean to study mathematical practice? Cultures of Mathematics IV

1. What does it mean to study mathematical practice? Cultures of Mathematics IV www.herts.ac.uk/philosophy

2. Who studies mathematical practice? Why?The promise-blurb Plus bonus: swipe at Andrew Pickering! 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.

7. Wholly Partly

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 diversityLittle 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.SchatzkiThe Practice Turn in Contemporary Theory (Routledge 2001) Almost everybody on the philosophy of maths practice circuit would dismiss or resist much of this “…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.”

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 (What is a ‘field of agency’?) “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.”

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 Vive la Resistance There is a respectable precedent for replacing humanism with vitalism or panpsychism… And Pickering is not alone in taking this sort of line E.g. Latour’s Actor-Network Theory However…

24. Let’s do some philosophy on Pickering Accommodation: I like a lot of what he has to say! “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) 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!

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

26. 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!