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Saturday, August 12, 2006 - 5:37pmSanction this postReply
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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.)


------------------------------------------------------


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


------------------------------------------------------


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."


------------------------------------------------------


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


------------------------------------------------------


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.


------------------------------------------------------


To reflect a vector, in the xy plane: reverse the sign of the z term. And analogously for the other planes.


------------------------------------------------------


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).


------------------------------------------------------


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.


------------------------------------------------------


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.)


------------------------------------------------------


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


------------------------------------------------------

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)




Post 1

Saturday, August 12, 2006 - 7:48pmSanction this postReply
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Interesting.



Post 2

Sunday, August 13, 2006 - 6:09amSanction this postReply
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The above will be presented in better format in due course. I wanted to get it up quickly, though.




Post 3

Monday, August 14, 2006 - 9:10amSanction this postReply
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I'll now be concentrating, not on dealing with questions or criticisms that may come up, but on writing my account of how I got from AR's epistemology to a method of generating equations for multiple dimensions. The real story here is the power of philosophy to guide science, whether or not some of the above has been thought of by higher mathematicians (of which I am not one).

However, at present it appears that the following aspects are original with me:
  1. The reasoning about concepts and math that brought me (unexpectedly) to the point of hypercomplex equations.
  2. The method of generating those equations.
  3. Some of the equations themselves.
  4. The connection of that method to Ayn Rand's epistemology.
  5. The identification of these numbers, what I call RADN numbers, as the true extensions of complex numbers to higher dimensions--in contradistinction to other number systems, with different equations, that hypercomplex-number thinkers may have devised to extend the idea of complex numbers. (I make this claim only because my reasoning leads to these sets of equations and no others.)
The paper is half written, and when I can get the time to finish it, I'll offer it on Lulu.com. (My thanks to Mr. Gennady Stolyarov for mentioning this outlet to me a year or two ago in connection with my music.)

-----

PS: I will first put up a clearer version of my initial post--and I am hoping there are no important typos in the initial post that are preventing anyone from following the presentation.

(Edited by Rodney Rawlings on 8/14, 9:25am)




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Monday, August 14, 2006 - 12:31pmSanction this postReply
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IMPORTANT CORRECTION TO EQUATIONS:

For 3-plex numbers,

(k^2*j)^2 = -j^2*k

or

z^2 = -y

 

There should be a minus there in both cases!

(Edited by Rodney Rawlings on 8/14, 12:34pm)




Post 5

Tuesday, August 15, 2006 - 2:10pmSanction this postReply
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ANOTHER TYPO: APOLOGIES. The first post should read:

------------------------------------------------------

-Then the grid forms an n-dimensional space whose points are defined by n-ary numbers of the form

Ae+^n + Be+^(n-1)*e-^1 + Ce+^(n-2)*e-^2 + … + Ze+^1*e-^(n-1)

remembering that e-^n = -e+^n.

------------------------------------------------------

 

I know this notation is hard to read, but just keep in mind that (e+) and (e-) are self-contained expressions meaning the "plus entity" and the "minus entity."

I will soon recast [DONE NOW!] the whole explanation in an easier-to-read format, using just single letters for the entities.

-----

OK, I can still edit this post so I'll do it here:

Define the two entities:
img124/5223/defineentitieseb9.gif

Then construct the axes of the n-space as follows:

Axis 1:
img124/2389/axis1sh1.gif

Axis 2:
img108/7034/axis2yu1.gif, with the right end being the minus of the left (because, sorry, in the graphic I forgot to put the minus sign in!).

Axis 3:
img124/7194/axis3rj2.gif, with the right end being the minus of the left.
:
:
:
Final axis:
img420/5489/axislastjx3.gif, with the right end being the minus of the left.
 
Then the grid forms an n-dimensional space whose points are defined by n-ary numbers of the form:
img108/8984/narynumberformatua8.gif, remembering that m^n = -p^n. Together, p^n and m^n make up the "real" value of the RADN number by addition.

The rules of addition are the same as for complex numbers. The rules of multiplication follow from algebra on p and m, with the lopping-off of n-powers. (But be careful to keep the minus if lopping off m^n! And if you do it twice, it goes plus again.)
 
Once the rules of multiplication have been worked out, the axes in, for example, 3-space may be labeled with single letters, except this is not necessary for Axis 1, which consists of "real numbers." Thus, for 3D numbers we have the format:
 
1 + y + z
5 + 2y + 7z
etc.

(Edited by Rodney Rawlings on 8/16, 11:35am)




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Saturday, August 26, 2006 - 9:17amSanction this postReply
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Oops, corrections to the RADN-4 equations (sorry, but my formula-generating process, described above, is very tricky!). The following has been tested more thoroughly. (Hopefully the RADN-5 group has no errors--I did test that one more.)

I have also switched v and w, which makes a bit more sense alphabetically.

------------------------------------------------------

CORRECTED Rules for 4D space (with v and w switched.):

 

In 4D space, one rotator is f^3g (called u here) if the entities are f and g.

 

(The 2's, 3's, and 4's are exponents of course.)

 

u2 = v

w2 = -v

v2 = -1

 

uw = -1

vw = -u

uv = w

uvw = -v

 

u3 = w

v3 = -v

w3 = -u

 

u4 = -1

v4 = 1

w4 = -1

------------------------------------------------------

 

However, anyone who applies my process will avoid such errors by being careful, especially when powers are lopped off--the changes of sign must be preserved.




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Saturday, August 26, 2006 - 10:51amSanction this postReply
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I can vaguely grasp what you're doing since it reminds me of an issue of how data structures have to be nested in some programs and what not. In this case, I'm guessing that the previous n-dimension is corollary to the next higher n-dimension? Meaning, the next one suggests the other?

-- Bridget
(Edited by Bridget Armozel
on 8/26, 10:52am)




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Saturday, August 26, 2006 - 10:57amSanction this postReply
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Just off the top of my head, I'd say no. The first one, RADN-2, which corresponds to the familiar complex number system, was merely used as a model to create higher "versions." (Bringing a Randian epistemology to the question.) The systems seem quite separate.

However, I am thinking about chaining them somehow. And one idea I have is to nest one inside the other.

(Edited by Rodney Rawlings on 8/26, 3:12pm)




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Tuesday, October 24, 2006 - 11:23amSanction this postReply
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“Why are these called ‘rotating’ systems?” some might be wondering. The following diagrams illustrate this feature:

img70/1176/compare2d3ddy0.jpg

The first diagram geometrically represents the behavior of RADN-2, familiarly known as the complex numbers. Multiplying any number by the imaginary unit i causes the point represented by the number to rotate (actually “revolve” if we look at it this way) about the origin by 90 degrees. In four steps the point is back where it started.

The second diagram, which represents RADN-3, shows the result of six successive multiplications by the first of the two imaginary units, y. The value touches every axis in turn, once only, on both sides (plus and minus), just as it did in RADN-2. Multiplication by –z, in other words by the minus value of the other imaginary axis, does the same thing in the opposite direction. Six steps completes the cycle. (In general, n × 2 steps, of course.) 

A RADN system of any dimensionality shows the same property, and other interesting ones that I am writing about. 

(A word on the very great mathematician Hamilton. I have since learned that no 3D system would have satisfied him, since he was seeking a “division algebra” in three dimensions, which later mathematicians have discovered to be an impossibility.)

(Edited by Rodney Rawlings on 10/24, 11:35am)




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Friday, November 24, 2006 - 6:33pmSanction this postReply
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The title of my essay will be "Understanding Imaginaries Through Hidden Numbers" and I will try to sell it on Lulu.com. The numbers have been hidden for philosophic reasons, but they have real consequences and implications, as I will explain. I hope this will serve as an example of the necessity of being continuously guided by philosophy while seeking scientific knowledge of any kind.

(Edited by Rodney Rawlings on 11/24, 7:18pm)




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Wednesday, November 29, 2006 - 7:16pmSanction this postReply
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Success! I have learned since writing this that the RADN systems described above are isomorphic to those particular hypercomplex numbers known to mathematicians as multicomplex numbers. However, because my method of generating them involves philosophic considerations of the basic nature of numbers, and therefore implies that these number systems have special importance, and because my approach may bring to light some novel angles and aspects regarding them, I will continue to use the term “RADN,” both here and in my forthcoming essay recounting how I arrived at them on the basis of Ayn Rand’s epistemology. I have also thought it wise to stick to my terminology for the reason that some differences may yet come to light between the multicomplex and the RADN numbers as I have conceived them.

 

It does appear that some mathematicians are beginning to realize a special significance and importance in multicomplex/RADN numbers.

(Edited by Rodney Rawlings on 11/30, 6:05am)




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Friday, March 2, 2007 - 10:56amSanction this postReply
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Since I am not even a mathematician (having arrived at hypercomplex numbers just by epistemological thought), I’m quite pleased with myself that I have been able to find an error in the Wikipedia article “Multicomplex number.” Its author recently posted the following comment in reply to a criticism I had made, but which I had then deleted due to a small lapse of confidence. This just goes to show how proper philosophy can increase the intelligence and productivity of workers in any scientific field:

Talk:Multicomplex number

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Hello, 8OYscLu9. Your earlier remarks are fully justified in my eyes. The statement that the bicomplex numbers are a special case holds true; however, both articles use different reference material I believe. Well, and then I may have made a mistake altogether, who knows. While I was glad to create the multicomplex number stub from the reference given, the bicomplex number article uses a web reference with different definitions. In short, if instead of real number coefficients you use another multicomplex number as coefficients, you’ll again end up with a multicomplex number. So, I find your “clean-up please” remark appropriate and would leave it out here ... until someone has the time to pick it up ... Thanks, Jens Koeplinger 14:57, 20 January 2007 (UTC)

Retrieved from “http://en.wikipedia.org/wiki/Talk:Multicomplex_number




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Saturday, March 3, 2007 - 3:38pmSanction this postReply
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Rodney:

     I find your subject and arguments (to the very limited extent I can follow, esp. the math), well, my favorite 'F'-word (no, not the 4-letter one). I'm barely familiar with Hamiltonians, and the i-axis use, but...

     Keep posting!

LLAP
J:D




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Saturday, March 3, 2007 - 4:39pmSanction this postReply
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"Font"? Oh, wait, that's four letters! I hope you mean fascinating! Thanks if so.

I am hopeful that the final essay, which I plan to sell on Lulu.com, will be more understandable to nonmathematican Objectivists, since it is based on AR's epistemology and builds from there. I am sure I have found the link between conceptual and mathematical development that she sought, and have taken some steps down the road she had to abandon in her final illness.




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Wednesday, March 7, 2007 - 11:31pmSanction this postReply
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Rodney:

     You DO pick ambitious projects.

     F***n'  Fascinating.

LLAP
J:D




Post 16

Friday, March 9, 2007 - 4:00pmSanction this postReply
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It's more like the projects picked me. I started out thinking about something else!



Post 17

Wednesday, March 21, 2007 - 9:08amSanction this postReply
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My essay is now published on Lulu.com:

http://www.lulu.com/content/750696

The title is "Understanding Imaginaries Through Hidden Numbers."

It is priced at US$5. The PayPal account has not been fully set up yet, but in a few days it should be possible to purchase the essay as a 38-page PDF download.

 

I rushed finalizing the piece so it could come out today, which is the first day of spring. Consequently, as an essay it is not as polished as I’d like; but intellectually it is complete.

(Edited by Rodney Rawlings on 3/21, 9:29am)




Post 18

Sunday, March 25, 2007 - 1:30pmSanction this postReply
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Here is an annotated Table of Contents to my essay. I hope some readers will be interested enough to buy it at Lulu.com.

Understanding Imaginaries Through Hidden Numbers

With the Introduction of the RADN Hypercomplex Numbers

By R. Rawlings

The Purpose of This Essay

“The purpose of this essay is to bring philosophy to bear on the subject of mathematics, and show the connection of mathematics to concept formation and to reality. Since the philosophy I employ is not yet widely held, I believe it leads to a different understanding of what in mathematics are called imaginary quantities and of the hypercomplex numbers they generate. As a secondary consequence, the conclusions imply that there is a special importance to a certain class of hypercomplex numbers that I have called RADN systems, and that are subsumed in conventional math under the more general concept of multicomplex numbers.”

How This Came to Be Written

“The impetus for this discussion was actually my curiosity about a different question that occurred to me while participating on an Internet message board. … Thinking about this eventually brought me to the subject of mathematics.”

The Nature of Mathematics and Numbers

Preliminary Notes on Ayn Rand’s Ideas About Mathematics

Summary of Rand’s thinking on the topic.

My Own Conclusions About Concepts, Mathematics, and Numbers

A 33-point list of my own ideas, including a new theory of how numbers and mathematics develop in man (different from the ideas of Pisaturo and Marcus), why our conceptual and our numerical abilities have a parallel development, and a new definition of number.

The Problem of the Imaginary Unit

Outline of the first issue, the puzzle of what “the square root of minus one” means.

The Quadratic Equation

Delving into the heart of the quadratic equation, in which the issue arises.

Numbers, Addition, and Multiplication

What these operations mean, according to Rodney Rawlings.

Quadratic Equations and Their …

(Full title withheld.) The deep structure of the quadratic equation.

Rules for the …

(Full title withheld.) How the structure works.

Hidden Numbers

The key to understanding imaginary ones.

Operations with Hidden and Imaginary Numbers, … , … , and Beyond

(Full title withheld.)

More on Multiplication and Powering

Deeper into these operations.

Implications for the Imaginary Unit

What this all means for “the square root of minus one.

Building the RADN Systems

How the foregoing leads to imaginaries, and therefore number systems, in more than two dimensions.

Meaning of the RADN Systems

The significance and importance of RADN systems, in relation to other multidimensional numbers.

Understanding Imaginaries

How imaginary numbers in general are to be understood.

Relations to Previous Objectivist Ideas

Similarities and differences of my ideas to/from those of (1) Ayn Rand and (2) Ronald Pisaturo and Glenn Marcus. (With regard to the latter, I have a different idea about how numbers and math arise, and a consequent new definition of number.)

Appendix A: More on the RADN Systems

Properties I have noticed and that may be significant.

Notating and Diagramming

A simple, clear way of notating and picturing RADN numbers.

Avoiding Subscripts and the Shift Key When Typing

(For typists.)

Rotations

How multiplication rotates numbers.

nth Roots on the Axes

Additional Square Roots of –1

Operations Within and Across Systems

How to add and multiply between RADN systems of the same or of different dimensionality.

Collapse Equivalents

A concept that supports my interpretation of the nature of these systems.

Collapse Increments

A way to calculate the Collapse Equivalents.

Appendix B: Taxonomy of Concept Types

Tentative classification of concept types with regard to how flexible their definitions are.

Annotated References

Works mentioned, or that I wish to mention.

Copyright Notice

 © 2007 Rodney Rawlings. All rights reserved. Text passages may be reproduced to the usual extent for purposes of a review or discussion; otherwise, this material may not be reproduced, displayed, modified, or distributed without the express prior written permission of the copyright holder. For permission, contact [by email]. For other inquiries you may either email me or call [phone number].

---------------------------------------

I hope all my formatting of the heading levels shows up. Let me know if it doesn't.

(Edited by Rodney Rawlings on 3/26, 1:35pm)




Post 19

Wednesday, November 14, 2007 - 10:20amSanction this postReply
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Reference hypercomplex numbers on Wikipedia. There you will find out that only certain dimensions will extend the complex numbers to algebraic structures with addition, multiplication and (sometimes) division. In general there is no extension of complex numbers to a division ring with arbitrary dimension over the real numbers. See also Cayley Algebras

Bob Kolker




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