Looking From The Inside And From The Outside

@article{Carbone2004LookingFT,
  title={Looking From The Inside And From The Outside},
  author={Alessandra Carbone and S. Semmes},
  journal={Synthese},
  year={2004},
  volume={125},
  pages={385-416}
}
Many times in mathematics there is a natural dichotomy betweendescribing some object from the inside and from the outside. Imaginealgebraic varieties for instance; they can be described from theoutside as solution sets of polynomial equations, but one can also tryto understand how it is for actual points to move around inside them,perhaps to parameterize them in some way. The concept of formalproofs has the interesting feature that it provides opportunities forboth perspectives. The inner… 
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