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  2. Synthetic geometry - Wikipedia

    en.wikipedia.org/wiki/Synthetic_geometry

    The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favor of a purely synthetic development of projective geometry. For example, the treatment of the projective plane starting from axioms of incidence is ...

  3. Foundations of geometry - Wikipedia

    en.wikipedia.org/wiki/Foundations_of_geometry

    Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have. The laws of logic. The theorems [4] are the logical consequences of the axioms, that is, the statements that can be obtained from the axioms by using the laws of deductive logic.

  4. Hilbert's axioms - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_axioms

    This was, in considerable part, influenced by the example Hilbert set in the Grundlagen. A 2003 effort (Meikle and Fleuriot) to formalize the Grundlagen with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions ...

  5. Building (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Building_(mathematics)

    In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.

  6. Space (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Space_(mathematics)

    One of the building blocks of a scheme is a topological space. Topological spaces have continuous functions, but continuous functions are too general to reflect the underlying algebraic structure of interest. The other ingredient in a scheme, therefore, is a sheaf on the topological space, called the "structure sheaf". On each open subset of ...

  7. Duality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Duality_(mathematics)

    Affine schemes are the local building blocks of schemes. The previous result therefore tells that the local theory of schemes is the same as commutative algebra, the study of commutative rings. Noncommutative geometry draws inspiration from Gelfand duality and studies noncommutative C*-algebras as if they were functions on some imagined space.