I have been advised by a mathematician to publish my results on the Net, so here I present the main latest conclusions I have reached in my application of Ayn Rand’s epistemology to mathematics. My reasoning, and other conclusions to be drawn, will be published later.
These are the parts of my math ideas that will be relevant to non-Objectivists; there are a great many others that will be of interest, I think, to those interested in Ayn Rand’s philosophy.
I would like to say at the outset that the famous mathematician Hamilton, who invented the four-dimensional numbers called quaternions when he could not arrive at a three-component counterpart to the complex number system, would have been very interested in what follows, because it solves this problem, which he spent ten years struggling with, until he decided he needed four components.
My method not only allows the construction of a 3D version of the complex number system (which define a 2D space), it allows construction of a version of that system for any number of dimensions (part of my theory, by the way, involves a certain view of what exactly a dimension is).
It has come to my attention that the reasoning behind the more recent so-called "hypercomplex numbers" is also able to construct number systems of n dimensions. However, on examining this question I find that the hypercomplex equations are slightly different from mine; and since I built my system from the ground up, with knowledge of only basic algebra and of Ayn Rand’s epistemology, I have to conclude that only my system is capable of "versioning" the complex numbers into higher dimensions properly. That is, my reasoning establishes the basis of complex numbers in physical reality, and extends it. The authors of hypercomplex numbers chose their rules of operation in a somewhat arbitrary way, and I strongly suspect that this misleads them a bit as to which operations are the "proper" ones, and which are tweakings or modifications, so to speak. This is not to deny that their numbers may have many applications. (It is a question I have not looked into.)
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To construct an n-dim’l space using n-dim’l nos.:
-Define two entities, labelled plus and minus, such that e+^n = +1 and e-^n = -1
-Label n axes of a grid, all crossing at their zeros, as follows:
e+^n ---------0--------- e-^n
e+^n-1*e-^1 ---------0--------- (-e+^n-1*e-^1)
e+^n-2*e-^2 ---------0--------- (-e+^n-2*e-^2)
:
:
e+^1*e-^n-1 ---------0--------- (-e+^1*e-^n-1)
(so that all combinations of powers are represented).
-Then the grid forms an n-dimensional space whose points are defined by n-ary numbers of the form
Ae+^n + Be-^n + Ce+^n-1*e-^1 + D e+^n-2*e-^2 + … + Ze+^1*e-^n-1
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When n = 2, we get the complex number system. I call the plus element "h" and I refer to multiples of h as "hidden numbers."
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With n = 3, the rotator factors, which touch all 6 axis branches in succession in a spiral or twist, are j^2*k and -jk^2, where j^3 = +1 and k^3 = -1.
Rules for when n = 3:
j^3 = +1
k^3 = -1
j^2*k × k^2*j = -1
(j^2*k)^2 = k^2*j
(k^2*j)^2 = j^2*k
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Operations in 3-plex:
Let x = j^3, k^3 = -x, y = j^2*k , z = k^2*j.
Then
x^2 = x
xy = y
xz = z
yz = -x
y^2 = z
z^2 = y
mult’g opposites or complements gives k^3.
sq’g opposites gives j^3.
sq’g complements switches them.
mult’g by j^3 gives no change.
mult’g by k^3 changes the sign.
These rules follow from algebra on j and k, with the lopping-off of 3-powers. (But careful to keep minus if lopping k^3! And if done twice, it becomes plus again.)
I.e., j^4 = j, j^5 = j^2, etc.
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To reflect a vector, in the xy plane: reverse the sign of the z term. And analogously for the other planes.
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With n = 4, you get f^3g as a rotator--have not yet tested the other axes to see if they are rotators (I expect they are).
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To convert the j^3 into conventional "complex" notation, simply use these rules:
y^2 = z
z^2 = -y
yz or zy = -1
and use the format A + By + Cz, where A, B, and C are real numbers. The same may be done with any number of higher dimensions.
Note that the mult’n is commutative in every case, just as for complex numbers.
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Rules for 4D space:
u2=w
v2=-w
w2=-1
uv=-1
vw=u
uw=v
uvw=w
u2=uvw (=w)
u3=v
v3=u
w3=-w
u4=-1
v4=-1
w4=1
In 4D space, one rotator is f^3g, if the entities are f and g. (I suspect all the "imaginary" axes are full rotators at any dimensional level.)
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Rules for 5D space:
r2=s
s2=u
t2=-r
u2=-t
rs=t
rt=u
ru=-1
su=-r
st=-1
tu=-s
tr=u
rst=-r
rsu=-s
tsu=-u
urs=-s
rstu=1
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I shall call all such dimensioned numbers RADN numbers, for "Rotating Any-Dimensional Numbers."
Rodney Rawlings
(Edited by Rodney Rawlings on 8/13, 1:03pm)
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