How do Pisaturo and Marcus explain complex and hypercomplex numbers?

[Starblade Enkai]

I know that in mathematics it becomes a necessity to conjure up units such as i, j, k, and the like in addition to the unit 1. However, I don't see how nonreal numbers can pertain to something physical. I have heard that Pisaturo and Marcus have attempted to explain nonreal numbers, but I can't quite find their arguments. Now I know such numbers are USEFUL. I just don't know how to derive these from first principles.

[Thales]

Their arguments are available in back issues of The Intellectual Activist.

I believe they refer to these numbers as concepts of method. They employ certain operations when you multiply them, namely changing your orientation in three space. However, the author builds up to an explanation of complex numbers after they've established other more fundamental mathematical concepts. Can't recall whether the author was Pisaturo or Marcus, but I believe both were involved along the way.

[softwareNerd]

I have heard that Pisaturo and ... attempted to explain nonreal numbers, but I can't quite find their arguments.

He sells his lectures here.

[radn]

What a coincidence! Yesterday I posted these messages on Objectivist newsgroups:

Those who are puzzled by, or curious about, hypercomplex numbers might be interested to read my article "Understanding Imaginaries Through Hidden Numbers," at

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

in which I show how I "invented" hypercomplex numbers on my own, never having heard of them, or indeed of any extensions of the number system!

How? Well, I had a powerful tool: Ayn Rand’s philosophy of Objectivism, specifically her epistemology. I was trying to understand complex numbers according to her system of thought, and my conclusions led me to think the idea behind the imaginary unit i might be extended to more dimensions. Tackling this, I eventually arrived at a way of doing just that.

I did not know, until I later researched the topic on the Net, that "hypercomplex numbers" had already been discovered by a long line of geniuses in the past. But apparently there were many types: quaternions, Cayley-Dickson constructions, etc.--and I had come up with one, and only one, type of such numbers! The difference was that my ideas were generated by investigating the basic nature of numbers as such, according to a philosophy.

What particular hypercomplex number system did I end up with? What were the steps in my reasoning? What was it about Ayn Rand’s philosophy that made it so productive in this instance? The answers are all in my essay. I hope you’ll at least consider buying and re

http://' target="_blank">ading it.

-----

I believe that the essay mentioned in my signature significantly builds on Introduction to Objectivist Epistemology in these ways:

1. I offer an alternative explanation to that of Ronald Pisaturo and Glenn Marcus, which as far as I know is the only currently recognized Objectivist theory on the topic, of how number concepts arise in the human mind.

2. I account for the fact that mathematical development occurs mostly parallel to the rest of conceptual growth, by pointing up a new connection between the two realms.

3. I tie "imaginary" and "complex" (two-dimensional) numbers to reality in a different way than Pisaturo and Marcus.

[radn]

Now I know such numbers are USEFUL. I just don't know how to derive these from first principles.

If you mean rationalistically deducing from a priori axioms, no science works that way, including philosophy. What you do is observe reality first, then integrate your knowledge into concepts and continually update your concepts and definitions as you discover new connections and causes. (Perhaps you know this, and it was merely an unfortunate wording on your part. If so, sorry!)

[Robert J. Kolker]

What a coincidence! Yesterday I posted these messages on Objectivist newsgroups:

You wrote: "I did not know, until I later researched the topic on the Net, that "hypercomplex numbers" had already been discovered by a long line of geniuses in the past. But apparently there were many types: quaternions, Cayley-Dickson constructions, etc.--and I had come up with one, and only one, type of such numbers! The difference was that my ideas were generated by investigating the basic nature of numbers as such, according to a philosophy."

I am troubled by the phrase "numbers as such". There are all sorts of numbers. There are numbers which, as a set, form locally compact topological spaces (for example the real numbers). There are numbers which form a linear dense set, but which do not contain all their limit points (for example, the rational numbers). Then there are numbers which form an linearly ordered but discrete (non-dense) set (for example the integers. These systems of numbers are related but they form topologically and algebraically distinct systems. So I would contend that there are no numbers -as such-. There is this kind of number and that kind of number and which kind it is depends are which postulates they satify.

Bob Kolker