We all the time write and talk about mathematical problems in plain English. For example, we define "the traveling salesman problem". But what is the actual, underlying structure of a "mathematical problem"? Can arbitrary mathematical problems be expressed exactly, in a formal language? Is it meaningful to talk about an algorithm that solves any arbitrary problem? Of course, that wouldn't decidable, but can we express it? For example, can we write a program which accepts, as input, a problem (example: "find an algorithm that sorts a list optimally") and outputs a solution (example: the merge sort algorithm)?

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    $\begingroup$ I'm not sure that the term "problem" has a clear, precise, & standardly-accepted definition. Take a look at cs.stackexchange.com/q/13669/755 for some explanation of standard usage of that term. $\endgroup$ – D.W. Dec 19 '16 at 3:07
  • $\begingroup$ Are you looking for a formalization of mathematics as a language? $\endgroup$ – Raphael Dec 19 '16 at 12:16

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