User:Boris Tsirelson/Sandbox1

Univalent or multivalent

Plane geometry (called also "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe. The question "which plane?" is inappropriate, since planes do not differ in their geometric properties. Every two planes α, β are isomorphic, that is, there exists an isomorphism f between α and β. Treating α and β as sets of points one defines isomorphism as an invertible (one-to-one and onto) map f : α to β preserving all primitive relations. Namely: f maps lines into lines; the distance between f(A) and f(B) on β is equal to the distance between A and B on α; etc. The same is required of the inverse map Failed to parse (unknown function "\be"): {\displaystyle f^{-1}:\be\to\al} .

Axioms of the plane Euclidean geometry leave no freedom, they determine uniquely all geometric properties of the plane. More exactly: all Euclidean planes are mutually isomorphic. In this sense we have "the" Euclidean plane. In terms of Bourbaki, the plane Euclidean geometry is an univalent theory. In contrast, axioms of a linear space (called also vector space) leave a freedom: a linear space may be one- dimensional, two- dimensional, three- dimensional, four- dimensional and so on (infinite dimension is also possible). The corresponding theory is multivalent.

According to Bourbaki, the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.

A similar idea occurs in mathematical logic: a theory is called categorical if all its models are mutually isomorphic. However, for Bourbaki a theory is embedded into the set theory, while in logic a theory is stand-alone (embedded into predicate calculus).

   Motivated or indiscriminate

Monkeys could type into a computer a sequence of hardware instructions; the computer could execute them; but the result of such "programming" has almost no chance to be fascinating or useful. Fascinating computer games reflect human's predilections. Useful programs reflect human's needs. A computer is dull for humans unless its software reflects human's life in one way or another.

Likewise, a theorem is of no interest for humans unless it is motivated in one way or another by human's life. The motivation may be quite indirect; many theorems "only" help to prove other theorems, many are appreciated "only" for their aesthetic value, etc. But some kind of motivation is vital. Indiscriminate stream of logical consequences of the axioms is not publishable in the mathematical literature.