New Objectivist thinking on mathematics

[radn]

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:

• How do we arrive at concepts of numbers?

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

• How would one define "number"?

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

The 38-page essay is dense with ideas, many of them different from those of other Objectivist thinkers on the topic. 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 perhaps 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 intellectual steps by which this result was achieved.

For a while after this discovery, I thought I might have arrived at 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 consider this particular form of the numbers that I had reinvented to be especially important--a conclusion that I saw would follow from the reasoning that had brought me to that point.

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

Understanding Imaginaries Through Hidden Numbers

Edited April 30, 2007 by radn

[LaszloWalrus]

I have a problem with you calling it "Objectivist thinking on mathematics." Objectivism is a closed system. At most, you could perhaps call it "An Objectivist's thinking on mathematics."

Also, you should not sell it as an "appendix to IOE," without Ayn Rand's permission (which would now be impossible to get).