Hypernumbers: Real or Imaginary?

Orchard-Hays, W. (1976). Hypernumbers: Real or Imaginary? IIASA Working Paper. IIASA, Laxenburg, Austria: WP-76-012

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Abstract

On 22 August 1975, Dr. Charles Muses gave a talk at IIASA on hypernumbers, in which he has worked extensively. The talk was fascinating to me and he expressed many viewpoints with which I very much agree and which are not often voiced. He left reprints of two papers which I was eager to read. In general, his discussion and some of the claims he put forth stimulated me to re-examine a subject to which I had not given serious thought for perhaps fifteen years or more.

On reading the papers, however, I found myself confused by the notation, especially by apparent inconsistencies. Also, the proliferation of "species and subspecies" of numbers without any apparent motive, and the incomplete development of the theory were troublesome. Furthermore, the use of exponential forms and the introduction of "bimatrix arithmetic" before the set or sets of numbers and their arithmetic (on which Muses lays great stress, properly I think) are rigorously defined, gives an impression of sleight-of-hand. Finally, the statement that "specific details of method cannot be discussed explicitly at this time because of negotiations in progress" is a virtual invitation to examine the subject critically. This paper does so from one viewpoint, namely the use of matrix arithmetic as a convenient mechanism for calculation with quantities which are noncommutative and, in some ways, non-unique.

Rather than attempt to "straighten out" Muses' notation, I will simply start from the beginning with notation of my own, standard in so far as applicable. Accidental similarities with other parts of Muses' notation should not be assumed to imply equivalence.

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