A new Objectivist perspective on mathematics

[Rodney]

I am offering an essay that brings Objectivist epistemology to bear on mathematics in a novel way, leading to new conclusions. Its title is "Understanding Imaginaries Through Hidden Numbers." It is available here:

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

The article's title refers, among other things, to the imaginary unit i used in complex numbers, but all the more basic ideas of math are covered as well. Most importantly for Objectivists:

o How do we arrive at concepts of numbers?

o Why do we learn mathematical ideas at roughly the same time as the rest of our early concepts?

o How would one define "number"?

o What is the meaning in physical reality of the so-called imaginary/complex numbers?

The 38-page essay is packed with fresh ideas, many of them different from those of other Objectivist thinkers. I present my reasoning step by step in a manner that encourages introspection, and lay out the fundamental aspects of human life that give rise to mathematical concepts. I believe readers will find the presentation persuasive, and will begin to see mathematics in a new way.

There is another factor that might commend my ideas to those who accept Ayn Rand's epistemology: in the course of pondering the deepest meaning of numbers, to my great surprise I discovered a new and immensely useful kind of number that I had never known existed--multidimensional numbers analogous to the (two-dimensional) complex numbers, with no restriction on the number of dimensions.

The essay presents the main steps by which this result was achieved.

For awhile after this discovery, I thought I might have discovered an entirely new mathematical idea; but protracted Internet research on my part gradually established that the numbers I had invented were already known to higher mathematicians as a certain class of what are called hypercomplex numbers.

However, I also found out that some mathematicians considered these numbers to be especially important--a conclusion that I saw would follow from my reasoning.

Which, of course, should be highly interesting to Objectivists, since it goes to prove Ayn Rand's hotly disputed contention that philosophy can serve as a guide to science.

Understanding Imaginaries Through Hidden Numbers

[BaalChatzaf]

I am offering an essay that brings Objectivist epistemology to bear on mathematics in a novel way, leading to new conclusions. Its title is "Understanding Imaginaries Through Hidden Numbers." It is available here:

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

The article's title refers, among other things, to the imaginary unit i used in complex numbers, but all the more basic ideas of math are covered as well. Most importantly for Objectivists:

o How do we arrive at concepts of numbers?

o Why do we learn mathematical ideas at roughly the same time as the rest of our early concepts?

o How would one define "number"?

o What is the meaning in physical reality of the so-called imaginary/complex numbers?

First, what is a "hidden number"? In a mathematical theory developed according to contemporary standards of rigor a mathematical theory will set forth its undefined terms and the postulates which give the undefined terms their semantics. Nothing is "hidden". The method of development is transparent, straightforward and logical. This is how the theory of (so-called) real numbers is presented nowadays. Likewise for complex numbers. A complex number is an element of the extension field over the real numbers generated by the irreducible polynomial equation x^2 + 1 = 0. The root of this equation is the (so-called) imaginary unit, usually designated by the letter i.

Second, we arrive the concept of numbers by counting. Kids can do it by the age of two usually.

Third, we learn mathematical ideas the same way we learn the names of things, the names of actions, the meaning of relations etc.. Kids can do this by the age of three usually. There is no magic here.

Fourth. How would one define number. What kind of number? Integer, rational, real, complex, quaternion, octonian? Which? Please be specific. Use standard mathematical terms if you are going to discuss mathematics.

Fifth, what is the physical meaning of the so-called complex numbers. The usual physical interpretation is that of angular phase. Complex numbers are given by an angle and a magnitude (relative to some frame of reference). This is very nifty for discussing harmonic oscillators and is this interpretation is even more nifty in formulating the concept of a quantum state. Quantum states are unit vectors in a Hilbert Space. The set of operators with quantum significance on this space are the so-called Hermite Operators.

Now let me ask you something. Has your magnum opus added one substantial theorem to the theory of analysis of complex valued functions that is not already in the literature?

If not, what is it you are bringing to the table?

Ba'al Chatzaf