I take the view that mathematics is simply an extension of language, made more precise by use of abbreviated symbolism. This doesn't seem to be the same as either "formalism" or "logicism", though it has aspects of both. on this basis mathematics is language developed to describe "structures" from simple things like sets up to complex scientific theories. This approach makes no presumptions about the nature of what is being described, i.e. the world, other than that it can be described. Mathematics is indeed inadequate to describe many aspects of the world. It is effective in areas where elements can be clearly defined. On this view "pure" mathematics is an outgrowth from this practical process of developing methods of mathematical description, abstracting it to describe formalisms not yet having any application in the world other than to mathematical ideas themselves.
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