  Proof theory can be described as the study of the general structure  of mathematical proofs, and of arguments with demonstrative force as  encountered in logic. The idea of such demonstrative arguments, i.e., ones the conclusion of which follows necessarily from the assumptions made, is central in Aristotle's Analytica Posteriora: a  deductive science is organised around a number of basic concepts that are assumed understood without further explanation, and a number of  basic truths or axioms that are seen as true immediately. Defined  concepts and theorems are reduced to these two, the latter through  proof. Aristotle's account of proof as demonstrative argument fits  very well to the structure of ancient geometry as axiomatized in  Euclid. The specific form of   Aristotle's logic,  the theory of syllogism has instead, so it seems, almost nothing to  do with proofs in Euclidean geometry. These proofs remained intuitive for more than two thousand years.