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I got to know that there is a branch in mathematics that studies the application of logic to mathematical structures . Before , I proceed I would like to say that I have seen the questions on model theory in Maths.SE as well MO.

I have always wondered that how a concept like "Group " , which is nothing but a collection of elements abiding by few conditions, can reveal this insight about roots of polynomial equation which would have been quite difficult without them . So ,can we explain by model theory how is it easy to study about roots of polynomial equations by groups ? Does model theory help in choosing the right mathematical structure for a given problem or which mathematical structures will be useful ?

I was having another question about model theory applications , so to save one more post , I would like to ask here itself , Is model theory related to symbolic dynamics ? Intuitively from definition of symbolic dynamics I can roughly think of it as representing a system by group of symbols and hence if we think the inclusion of something in a set as symbol "1" and exclusion as symbol "0" , then set theory can be sketched as a system in symbolic dynamic or maybe ,the strings in the theory of languages or the statements about a particular structure can be viewed as a dynamic system with certain rules to assign truth value to the strings or statements being generated . Hence , I can anticipate a connection between symbolic dynamics and model theory .

Are there any works on connection between symbolic dynamics and model theory ?

itp dusra
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    Thinking about groups as objects defined by some random collection of axioms is missing the point. Groups are very much not just a random algebraic structure: they naturally occur as groups of symmetries of concrete mathematical objects all over mathematics, this is how they were originally discovered, and the relevance to polynomials is that the roots of polynomials also admit such symmetries. – Qiaochu Yuan Mar 08 '17 at 04:33
  • "Does model theory help in choosing the right mathematical structure for a given problem or which mathematical structures will be useful?" No. In fact, no mathematical theory does those things. What you're asking about is the art of problem solving. There is no general theory that takes as input a problem $P$ and tells you which mathematical structures $S$ will be useful for solving it. Perhaps you should read Pólya's books. – symplectomorphic Mar 08 '17 at 06:47

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