# [Atlasphere] Rand-inspired Non-mathematician Discovers Hypercomplex Numbers, with Possible Implications for Mathematics and the Philosophy of Science

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I haven’t the foggiest idea whether this would bear scientific scrutiny, but he certainly deserves an A for creative self-promotion.

From a new press release which appears to have been authored by Atlasphere member Rodney Rawlings:

(PRLEAP.COM) A Toronto, Ontario, writer and editor has arrived at a system of creating hypercomplex numbers”numbers that extend the complex number system to more dimensions using only high school algebra, as viewed through the lens of Ayn Rand's philosophy of Objectivism. He contends that this has implications for mathematics and the philosophy of science.

<p>Rodney Rawlings calls his multidimensional numbers RADN numbers for rotating any-dimensional numbers, because they have a property of rotation exactly analogous to that of the complex numbers. They are also commutative and distributive like them.

<p>He says that he arrived at this result by asking himself what exactly numbers are, how they arise in the human mind, and what their relationship to reality is. But these questions were only so fruitful because he used a correct philosophy, he claims Ayn Rands. Any other philosophy, such as the currently influential one of Karl Popper, he says, would not have led to such a result. This has two implications: first, that Rand's philosophy has a strong element of truth, at least in the area of epistemology; and second, that the type of numbers I discovered must have a special significance, seeing as how they are intimately related to the basic nature of numbers.

All I set out to do was to understand, and perhaps explain, the complex numbers, which are two-dimensional, in a more concrete way, says Rawlings. Then, at one point, I realized that my reasoning might apply with equal force to three dimensions and beyond. When I finally turned to that question, I was led step by step to what I thought was a novel method for creating new systems.

Numbers of more than two dimensions are called hypercomplex numbers, and there are many types, developed over the past century or so. One of the most famous are the quaternions four-dimensional numbers developed by Irish genius William Rowan Hamilton that can be used to represent space and time, and have found a niche in 3D graphics applications. Quaternions, however, are only one result of the Cayley-Dickson construction, a program that can be used to create systems of any dimensionality. And there are many types of such programs, for example hypernumbers and œmulticomplex numbers.

Rawlings contends that the RADN program (which, he hastens to add, is not a new one but already known to mathematicians under a different name) must have a unique status among the hypercomplexes, because of the way he, a non-mathematician, arrived at them by means of extremely simple algebra absent any of the tools of modern analysis, but armed with a philosophy that takes a particular and unconventional view of the nature of concepts and of mathematics.

Accordingly, Rawlings decided to write up his thoughts and reasoning in an essay entitled Understanding Imaginaries Through Hidden Numbers, which he is currently offering at a low price on Lulu.com (go to www.lulu.com/content/750696).

He is hoping the essay will stimulate thought on the nature of mathematics, of addition and multiplication, of dimensions, of imaginary units, and of multidimensioned numbers, and lead to a fresh, and more positive, assessment of the ideas of Ayn Rand.</p>

It remains to be seen whether my conjecture that this class of numbers has a unique status holds any water, says Rawlings. But, he adds, at the very least the mere fact that a layman, using Ayn Rand's philosophy, could invent on his own such an idea as hypercomplex numbers is worthy of note to anyone interested in the philosophy of science and mathematics.

Contact Information

R. Rawlings

Perhaps someone could page Objectivist Philosopher Scientist-in-Chief Harry Binswanger for his opinion.

http://www.theatlasphere.com/metablog/699.php

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<p>I haven’t the foggiest idea whether this would bear scientific scrutiny, but he certainly deserves an A for creative self-promotion.</p>

We have a case of galloping balderdash. Whoever this joker is he didn't bother to do the background research in the subject. Shame on him!

Hypercomplex numbers were developed in the 19-th century as a natural extension of the complex numbers over the real number system. Aside from the usual complex numbers there are only two extensions that produced reasonable algebraic systems: the quaternions (a 4 dimensions extension of the reals) and the octonions (an 8 dimension extension of the reals).

See the Wiki article on hypercomplex number at

http://en.wikipedia.org/wiki/Hypercomplex_numbers

Note the following:

Quaternion, octonion, and beyond: Cayley-Dickson construction

All of the Clifford algebras Cℓp,q® apart from the complex numbers and the quaternions contain non-real elements j that square to 1; and so cannot be division algebras. A different approach to extending the complex numbers is taken by the Cayley-Dickson construction. This generates number systems of dimension 2n, n in {2, 3, 4, ...}, with bases \{1, i_1, ..., i_{2^n-1}\}, where all the non-real bases anti-commute and satisfy i_m^2 = -1.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. However, satisfying these requirements comes at a price: Each increase in dimensionality introduces new algebraic complications. Quaternion multiplication is not commutative anymore, octonion multiplication additionally is non-associative, and sedenions do not form a normed space with multiplicative norm.

Because quaternions and octonions offer a (multiplicative) norm similar to lengths in four and eight dimensional Euclidean vector space respectively, these numbers can be referred to as points in some higher-dimensional Euclidean space. Beyond octonions, however, this analogy fails since these constructs are not normed anymore.

Ayn Rand had nothing at all to do with the above developments since they happened before she was born.

I seriously doubt that any idea of Ayn Rand could produce any worthwhile mathematical construction.

Ba'al Chatzaf

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~ Who says math ain't interestin'?

~ Even cooks can handle the formulaic recipes with the right chef pointin' things out! --- I notice, however, for all the 'nions' thrown in, On(e)ions aren't covered yet. Is the 'one' the ultimate base, maybe, that all are derivatives from? Shades of Unified Math Theory!

LLAP

J:D

PS: Ba'al: Give the guy a break; he's been talking about this on ROR for a while, and he is backing off since a 'flaw' was found in his assumptions, and he accepts it. He's an amateur (self-described, so C'MON!) mathematician who did have a pro or two check his stuff out and they found nothing wrong...'till another later did. Don't get too condescending with non-'Pros.'

Edited by John Dailey

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PS: Ba'al: Give the guy a break; he's been talking about this on ROR for a while, and he is backing off since a 'flaw' was found in his assumptions, and he accepts it. He's an amateur (self-described, so C'MON!) mathematician who did have a pro or two check his stuff out and they found nothing wrong...'till another later did. Don't get too condescending with non-'Pros.'

I would not have said boo! if the guy had not attempted to -sell- his crap! You can get the Good Stuff for free from the Journals or ArXiv. One of the useful protocols is that theoretical results are openly published in the journals (and other archives). They can be read for free. If one cannot afford to subscribe to a journal he can get access (to read but not to borrow) from the local university library. Photo copies os particular articles can be had for as little as a nickel a page which is not a budget buster.

I find it a tad annoying when an amateur who won't even research his subject has the almighty gall to offer his pap for a price. Which is why I posted the wiki page. The wiki page as dozens of references to the professional literature and an interested reader can get to them for little or no cost.

I am not going to give any breaks or slack for that kind of fiddling. Mathematics is MY profession and I will not see it profaned or besmirched.

Andrew Wiles who finally proved Fermat's Last Theorem busted his ass for seven years. When he was done he presented his results to the world at no charge. Was this altruism? It was not. Wiles needed to have the qualified professionals vet his work (an error was in fact found which Wiles corrected the following year). That is how the system works. A person produces a result and others vet it and expand it. Perhaps the original publisher then can use the consequential increments to carry his work even further. It is the Trader Principle pure and simple and it is done at no monetary charge. The real charge is the mental effort required to understand what is produced. Like Richard Halley in -AS- said. What he wanted was for people -to understand- what he was doing. Those were his terms. The no-charge system is optimized for promoting the best results in the shortest time. The coin of the realm is understanding of and admiration for the creator. Respect from one's peers is the payoff.

Ba'al Chatzaf

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~ Ok; you're correct (dammit!) He shoulda been more careful.

~ I give...on that one.

~ "U-N-C-L-E !"

(sheesh.)

LLAP

J:D

Edited by John Dailey

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