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
$\endgroup$ 15Question 3: How should/could this list be extended? Especially without the categorical bias?
1 Answer
$\begingroup$This is more of a philosophical answer and I don't know if this is what you were looking for, but in general what a mathematician does is finding an abstract representation of reality (or some reality). This is why mathematics finds so many applications in physics, for example.
Through symbolic manipulation, mathematics creates representations that, respecting some fixed rules (axioms), define the world and make predictions about its laws.
Please, note that "reality" and "the world" do not necessarily imply our reality and our world. I can conjecture a reality in infinite dimensions for example. However, even if that reality does not exist, I can still create its representation, and it will hold true should that world actually exist. Minkowsky created a metric used by Einstein way before it was needed, for example.
So, mathematics abstracts concepts creating symbols that can be manipulated, and by so doing we can derive rules and make predictions.
Everything else is a redefinition of these concepts, or an application.
As another mathematician, whose name I forget, said regarding mathematical physics: "To summarize, the end of mathematical physics is not merely to facilitate the numerical calculation of certain constants or the integration of certain differential equations. It is more, it is above all to disclose to the physicist the concealed harmonies of things by furnishing him with a new point of view".
This holds for other sciences besides physics, of course.
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