Some mathematicians do not believe that a distinguished world of mathematics should be sought at all, but rather that the multiplicity of such worlds should be looked at simultaneously. A major result in algebraic geometry, due to Alexandre Grothendieck, was the observation that every commutative ring may be viewed as a continuously variable local ring, as Lawvere would put it. In the same spirit, an amplified version of Gödel’s completeness theorem would say that every topos may be viewed as a continuously variable local topos, provided sufficiently many variables (Henkin constants) are adjoined to its internal language. Put in more technical language, this makes the possible worlds of mathematics stalks of a sheaf. However, the question still remains as to where this sheaf lives if not in a distinguished world of mathematics or—perhaps better to say—metamathematics.
These observations suggest that the foundations of mathematics have not achieved a definitive shape but are still evolving; they form the subject of a lively debate among a small group of interested mathematicians, logicians, and philosophers.
Zenos-paradox-illustrated-by-Achilles-racing-a-tortoiseFigure 1: Zeno’s paradox, illustrated by Achilles racing a tortoise.[Credits : Encyclopædia Britannica, Inc.]
Contrasting-triangles-in-Euclidean-elliptic-and-hyperbolic-spacesFigure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.[Credits : Encyclopædia Britannica, Inc.]
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