Logic in Computer Science (LICS 1995)

Paper: The Stone Gamut: A Coordinatization of Mathematics (at LICS 1995)

Authors: Vaughan R. Pratt

Abstract

We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete self dual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a two dimensional space called the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from -1 for sets to 1 for complete atomic Boolean algebras (CABA's), and vertically by complexity of language. Complexity 0 contains only sets, CABA's, and the inconsistent empty set. Complexity 1 admits noninteracting set CABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories are all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts Stone Pontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework

BibTeX

  @InProceedings{Pratt-TheStoneGamutACoord,
    author = 	 {Vaughan R. Pratt},
    title = 	 {The Stone Gamut: A Coordinatization of Mathematics},
    booktitle =  {Proceedings of the Tenth Annual IEEE Symp. on Logic in Computer Science, {LICS} 1995},
    year =	 1995,
    editor =	 {Dexter Kozen},
    month =	 {June}, 
    pages =      {444-454},
    location =   {San Diego, CA, USA}, 
    publisher =	 {IEEE Computer Society Press}
  }