liberte@uiucdcs.UUCP (04/30/84)
#N:uiucdcs:28200034:000:8032
uiucdcs!liberte Apr 30 02:02:00 1984
This is a preliminary attempt at describing the hypernumber concept of
Charles Muses which I introduced in an earlier note. This time I quote
primarily from "The First Nondistributive Algebra, with Relations to
Optimization and Control Theory" which is in "Functional Analysis and
Optimization", edited by E. Caianiella, Academic Press, 1966. My notes
are enclosed in braces {}.
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What are hypernumbers and why are they significant?
"Matrices are isomorphic with hypercomplex numbers; and projective geometry,
affine geometry, the theory of vector spaces (hence that of function spaces),
and linear algebra are all isomorphic, forming one of the most significant and
profound convergences of meaning in all mathematics." p 171
"...every matrix represents a hypercomplex number." p196
"Thus hypercomplex numbers can be regarded as embedding in themselves
entire systems of linear equations." p 197
"The matter goes even deeper {out of context}, since matrices may be treated,
often with great gain in succinctness and elegance, as hypercomplex numbers."
p 202
That is, you can do arithmetic of matrices using hypernumbers instead and
apply the algebraic rules of hypernumbers instead of doing many calculations.
Pretty potent, it would seem. Although this is part of Muses' style, what
I have learned seems to indicate that hypernumbers or hypercomplex numbers
do have great applicability and power as well as being somewhat revolutionary.
How do hypernumbers relate to normal complex numbers?
"Just as real algebra is not closed, but open at the operation "square root
of minus unity," so are Gaussian and quaternion algebra similarly open, for
sqrt(-1) {=i} is multivalued, ie, j, k, etc, also satisfy it, leading us from
real (R) to Gaussian (G) to quaternion or Hamiltonian (H) to Cayley (C)
algebra, and beyond. {Gaussian is the normal complex algebra. The
hypercomplex algebras are beyond that.}
"Thus hypercomplex algebra contains complex or Gaussian algebra, and the
latter contains real algebra; or R < G < H < C < N. The last term in this
sequence is new, and previous attempts to extend Cayley algebra have failed
because of not realizing the theorems that: (1) any extension of R must
have nonamalgamative addition {that is, can't express in one number - need
2 or more as in a + bi or (a, b)}; (2) any extension of G must have non-
commutative multiplication; (3) any extension of H must have nonassociative
multiplication; and (4) any extension of C must have nondistributive
multiplication." p 172-173
Five Complete Linear Algebras, Involving Three Kinds of Hypercomplex Numbers
Number of Algebra elements No. of Units Characterization
Real (R) 1, 0 T1=2 am, co, as, di
Gaussian (G) 1, 1 T2=4 -am, co, as, di
Hamiltonian (H) 1, 3 T4=24 -am, -co, as, di
Cayleyan (C) 1, 7 T8=240 -am, -co, -as, di
Nondistributive (N) 1, 15 T16=4320 -am, -co, -as, -di
am = ammalgamative
co = commutative
as = associative
di = distributive "-" means not
"Thus R, G, H, C, and N may be considered to have Euclidean dimensionalities
of 1, 2, 4, 8, and 16, respectively, Tn being the maximal number of equal
tangent hyperspheres that can be fit about another such sphere of the same
radius in Dn, a parabolic space of n dimensions." p 174
"There is a close connection between the groups of tangent hyperspheres
mentioned in the last paragraph and error-correcting computer codes, which
in itself suggests the importance of the present theme for optimization and
control theory." p 175
Does anyone know more about algebras, in particular, "complete linear
algebras" to shed more light on the above? I am not sure what the number
of elements means. The first unit is 1. G has 1 and i. H has 1, i, j, and
k. Beyond that a i(n) notation is used where i(0) = 1.
"A characteristic of complete algebras is successively to require a more
precise formulation of what is meant by addition or multiplication, and
that each embeds in itself all the complete algebras below it, thus
preserving the self-consistency of mathematics. The rules of arithmetic
do not "break down"; they merely become more sensitive, taking more
distinctions into account in higher algebras. A complete algebra is also
one whose elements form a kind of multiplication loop, regenerating each
other, except for zero formation when the number of elements exceeds 8
{N-algebra}. Finite factor (ff) multiplication is that which does not
entail the equation a*0 = b, where a, b != 0 or infinity, although it may
involve a*b = 0, a != b, as in N-algebra." p 173
What is beyond N-algebra?
"Beyond the first nondistributive algebra (i.e. N) there lie at least two
higher forms of nondistribution, which may be symbolized as providing the
results a*0 = b and a*a = 0, respectively, where neither a nor b is zero
or infinity." p 197
Some historical perspective:
"The Greeks considered suspect and abnormal any number such that k*x < 0
where k was any positive number. {x is negative} Renaissance man, though
he had long accepted negative numbers as just as natural as positive numbers,
still balked at x where x*x = -1, although he used such numbers to solve some
quadratic equations.
"It took until the 19th century until man's mind could regard these numbers
too as nonpathological, although the designation "imaginary" still clings
to them.
"In the 20th century, Eduard Study first considered a number x not equal
to zero and such that x*x = 0; although Study still had no realization that
this implies also x^0 = 0, and an advanced form of nondistributive
multiplication...." p 209
"Since mathematics itself may be defined as the science of numbers and their
operations, it is clear that mathematics may be essentially enlarged and
deepened only by enlarging and deepening our notion of number. In this
sense all of mathematics after the ancient Greeks grow out of minus 1 and its
square root, function theory included. Turning to physics, we now see why
quaternions are becoming increasingly important despite their comparative
neglect, although they were actually introduced through the back door as the
basis of the vector product, the rules of which for 3-space repeat exactly
the rules for quaternion multiplication of the unit vectors, except that
their squares are zero instead of minus 1. p 202
"... It is quite understandable
that ordinary physics would find it inconvenient that the square of an
operator should become negative. However, in quantum mechanics that is
not so inconvenient,..." p 203
He writes more on applications to error-correcting codes, algebraic
number fields and function theory. There is a lot of material that
I dont understand much of, needless to say.
Skipping forward to his conclusion...
"It thus turns out that there are eight possible basic kinds of number
(each with their own infinities), plus zero.
"The higher kinds of number for the first time yield concrete hope of
placing the profound and subtle characteristics of bio-, psyco-, and
socio- transformations and processes on an adequate mathematical basis.
Such kinds of number would thus introduce the humanities to their
appropriate mathematics, which will not do them the grave and unscientific
injustice of forcing them to fit some Procrustean bed of inadequate
hypothesis or reductive definition. Man and man's sciences are now ready
to go beyond the square root of minus one. With each new and higher kind
of number a new and deeper algebra and arithmetic become possible, and
hence a new and deeper functional analysis." p 211
There are 27 references, only three of which are Muses'. I will submit
a list of references to more recent material later.
If anyone has a complete record of discussions on the net regarding this
topic, I would appreciate a copy since I missed most of it.
Daniel LaLiberte, U of Illinois, Urbana-Champaign, Computer Science
(uiucdcs!liberte)
{moderation in all things - including moderation}