1. Mathematical Cognition as Distributed among Mathematicians and Computers Christophe Heintz Institut Jean Nicod – EHESS

2. Context of the project Natural studies of Mathematical knowledge Cognitive and social factors in mathematical knowledge production Using Cognitive Anthropology  Distributed Cognition (cf. Ed Hutchins’ Cognition in the Wild)

3. Aim of the communication: Presenting an analysis of a change in the way Mathematics is done, and introducing a theoretical framework for such analyses, where the change is the introduction of computers in the work of Mathematicians and the theoretical framework is the diachronic analysis of distributed cognitive systems

4. Mathematical Cognition as Distributed Cognition (1) The idea of distributed cognition “Distributed cognition looks for cognitive processes, wherever they may occur, on the basis of the functional relationships of elements that participate in the process.” “- Cognitive processes may be distributed across the members of a social group - Cognitive processes may involve coordination between internal and external (material or environmental) structures” (e.g. adding large numbers)” Hollan, J., E. Hutchins & D. Kirsh (unpublished)

5. Mathematical Cognition as Distributed Cognition (2)  Cognitive systems (singling out a cognitive process and the means it uses)  Cognitive functions (who and what does what)  Cognitive (social) architecture (trajectory of information) Mathematical knowledge as the output of a cognitive system

6. The organisation of mathematical knowledge production Distribution of cognitive tasks (specialists in sub-disciplines, professional hierarchy, … determinate cognitive tasks) Collaborative work in Mathematics (increasing number of co-authored papers) The role of artifacts: pen and paper, compass… computers (Mathematical cognition is situated, it takes place in the environment and uses external devices)

7. Development of cognitive systems Cognitive system: a dynamic but synchronic notion The role of trust in the social repartition of cognitive functions - The ‘cement’ of distributed cognitive systems - Trust ascription (who/what, for which task) as the principle of change in distributed cognitive systems

8. Mathematicians don’t trust nobody… or do they?  ‘Surveyability’ in mathematics … is not attained by single individuals– TRUSTING OTHER MATHEMATICIANS “Specialization and teamwork are thus inescapable features of much modern knowledge acquisition…. [These] apply in the context of justification as well [as in the context of discovery]. It is likely that no one mathematicians has or will ever have the logical justification for each step in Brange’s proof ” J. Hardwig ‘The role of Trust in Knowledge’ (1991)

9.  ‘Surveyability’ in mathematics … is not attained by single individuals – TRUSTING ARTIFACTS Trusting computers The four-colour theorem: Appel&Haken, 1976. After which computers continued to be used in mathematical proofs with increasing frequency.  Is one prepared to concede that surveying a proof can be done by a combination of human being and machine? + Increasing use of computer graphics

10. Significance of the four-Colour Theorem Traditional analyses: The meaning of ‘proof’ has changed. (Tymoczko, 1979; McKenzie 1999) I would add: The 4CT led a revolution in mathematical knowledge production: the structure of the distributed cognitive system that allows the production of mathematical knowledge has changed, incorporating new artifacts, viz. computers

11. In Brief 1. I have used the notion of distributed cognition in order to apply it to mathematical knowledge production.This allowed me to observe that the distributed cognitive system for mathematics include, nowadays, computers

12. 2. I have introduced the notion of trust in order to account for the historical distribution of cognitive functions, thus allowing diachronic analyses of cognitive systems. If mathematicians trust computers for some specific tasks, then they ascribe to them a cognitive function, a place in their cognitive system. They henceforth change the organisation of the production of mathematical knowledge.