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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?

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Community wiki? – Jeel Shah Sep 19 '13 at 0:47
I agree! But where do I switch to community wiki? – Hans Stricker Sep 19 '13 at 0:54
In laymen's term, a mathematician sits in front of a monitor and presses keys on a keyboard. Slightly more detailed, he probably reads and understands many papers, both mathematical papers and papers from the discipline he's currently working on. Then maybe he's explaining what he learned from these papers to other people, and how relevant it is or could be for what they are doing. Independent of whether what I wrote above is true, I hope it gives some idea what "laymen's terms" could mean. – Thomas Klimpel Sep 19 '13 at 1:02
Hans, if you simply edit your post, there should be, at the bottom right, a little box that you can click on, it should say Community Wiki. I just checked, it did not give me the option for your post, I guess because I'm not a moderator. Anyway, you should be able to do it yourself. Checked on one of my own, between the text you are editing and the preview box, on the right, in lower case, community wiki... – Will Jagy Sep 19 '13 at 1:13
Collect grant money. – Potato Sep 19 '13 at 1:26

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