Non-Mathematician Devises Hypercomplex Numbers, with Possible Implications for Mathematics and the Philosophy of Science

A lay individual, by combining high-school algebra and Objectivist philosophy, hit upon an advanced math concept that extends the complex numbers.

Toronto ON, Canada, December 25, 2007 --( 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.

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 associative like them.

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 Rand’s. 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 “multicomplex 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 (go to

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.

“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.

Rodney Rawlings