archive.is | Musean hypernumbers
are an algebraic concept envisioned by Charles A. Musès
(1919–2000) to form a complete, integrated, connected, and natural number system.[1][2][3][4][5]
Musès sketched certain fundamental types of hypernumbers and arranged them in ten "levels", each with its own associated arithmetic
and geometry.
Mostly
criticized for lack of mathematical rigor and unclear defining
relations, Musean hypernumbers are often perceived as an unfounded
mathematical speculation. This impression was not helped by Musès'
outspoken confidence in applicability to fields far beyond what one
might expect from a number system, including consciousness, religion,
and metaphysics.
The term "M-algebra" was used by Musès for investigation into a subset of his hypernumber concept (the 16 dimensional conic
sedenions
and certain subalgebras thereof), which is at times confused with the
Musean hypernumber level concept itself. The current article separates
this well-understood "M-algebra" from the remaining controversial
hypernumbers, and lists certain applications envisioned by the inventor.
"M-algebra" and "hypernumber levels"[edit]
Musès was convinced that the basic laws of
arithmetic
on the reals are in direct correspondence with a concept where numbers
could be arranged in "levels", where fewer arithmetical laws would be
applicable with increasing level number.[3]
However, this concept was not developed much further beyond the initial
idea, and defining relations for most of these levels have not been
constructed.
Higher-dimensional numbers built on the first three levels were called "M-algebra"[6][7]
by Musès if they yielded a distributive
multiplication, unit element, and multiplicative norm. It contains kinds of
octonions
and historical quaternions
(except A. MacFarlane's hyperbolic quaternions) as subalgebras. A proof of completeness of M-algebra has not been provided.
Conic sedenions / "16 dimensional M-algebra"[edit]
The term "M-algebra" (after C. Musès[6]) refers to number systems that are
vector spaces
over the reals,
whose bases consist in roots of −1 or +1, and which possess a
multiplicative modulus. While the idea of such numbers was far from new
and contains many known isomorphic number systems (like e.g.
split-complex
numbers or tessarines),
certain results from 16 dimensional (conic) sedenions were a novelty.
Musès demonstrated the existence of a logarithm and real powers in
number systems built to non-real roots of +1.
