The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What Do Mathematicians Do? by A.J. Berrick.) Both surveys have - notwithstanding their merits - a somehow narrow view on what working mathematicians actually do: somehow too general and somehow too specific. And I am not really happy with both of them.
What I am looking for is an (as comprehensive as possible) list of catchy descriptions of what working mathematicians actually do (i) in laymen's terms, (ii) in working mathematician's terms, and (iii) in philosopher's of mathematics terms.
The laymen's list would include:
- propose (definitions)
- conjecture (theorems)
- prove (theorems)
- understand (proofs)
- classify (objects)
- characterize (objects)
- calculate (objects)
- count (objects)
- "visualize" (objects = structures)
- represent (abstract objects by concrete objects)
- construct (new objects out of given objects)
Question 1: How should/could this list be extended? And how could it be organized, with regard to the fact that - for example - to propose a definition and to conjecture a theorem is somehow related, and that to count (i.e. to give the cardinality of some set of objects) is to prove a theorem?
Concerning the (idealized) working mathematician's list, I am insecure how she would describe her daily work in abstract terms.
Question 2: Some examples how working mathematicians would describe their work in abstract terms other than those above and below?
Concerning the philosopher's (of mathematics) list, it might contain items like
- find natural isomorphisms
- find pairs of adjoint functors
Question 3: How should/could this list be extended? Especially without the categorical bias?