^> 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 indc7c: lent axiomatic (nontestable) assumptions. It is suggested that a finite field representation should be used as the fundamental basis of a physical representation. Jj ■ ; i ; ' , i Rapioduod by NATIONAL TECHNICAL .NFORMATION SERVICE Sprin»fl.W. V.. MtSI DISTRIBUTION STATEMENT Thi* document ha« been approved (or public reloam« and sole; its distribution is unlimited. Dm-2 ~i to 200 x 10 and declare the answer to be Ik x 10. In this way, replacement permits our mathematical operations to continue and avoids cessation due to uncomputability or lack of performable instructions. One couid also construct his system to reset itself to some arbitrary point-say zero-whenever an impasse is reached. However, the rationale for replacement is clearly preferable because it seeks a "nearby" problem and we shall adopt it.

The Einstein-Podolsky-Rosen paradox continues to be the center of great controversey and illumination within quantum theory. In this paper we examine Einstein's scientific epistemology in order to discover the basis for his objections to quantum theory. Then we discuss the consequences and implications of this for the EPR paradox and the quantum mechanical method of state representation.

The state representative for a composite system cannot be uniquely reconstituted solely from knowledge of the state of the constituents. Spin and angular momentum examples are used to illustrate this behavior. The need and motivation for holistic principles to allow the determination of a single state is discussed.