Geometry Over a Finite Field
Abstract
The development of certain aspects of a physically interpretable geometry defined over a finite field is presented. The concepts of order, norm, metric, inner product, etc. are developed over a subset of the total field. It is found that the finite discrete space behaves locally, not globally, like the conventional 'continuous' spaces. The implications of this behavior for mathematical induction and the limit procedure are discussed, and certain radical conclusions are reached. Among these are: (a) mathematical induction ultimately fails for a finite system and further extension leads to the introduction of formal indeterminancy; (b) finite space-time operations have inherent formal properties like those heretofore attributed to the substantive physical universe, and (c) certain formal properties attributed to continuous spaces cannot be developed from successive embedding in finite space of finer resolution--but must be based on independent axiomatic (non-testable) assumptions. It is suggested that a finite field representation should be used as the fundamental basis of a physical representation.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1969
- Accession Number
- AD0714115
Entities
People
- Donald L. Reisler
- Nicholas M. Smith