2010.12.11

2010.12.11 22:37 Saturday

A New Approach and New Content for 2011

I have decided to revamp Chapter 1-1, the first chapter of my web book for two important reasons. 1. When my googlepages site got transfered to google sites, most of the pages in this chapter got partially scrambled, and not all the links work properly since the change. I have postponed correction of these problems mainly to work on new content in chapters 2-4 and 3-1. However I have decided to fix these problems to improve overall site functionality. 2. I was never very happy with my introduction of the counting numbers. I am trying to create an entirely new approach in introducing and developing the topic of large numbers.

Originally I took a more traditional approach to how to introduce the subject. In order to speak about large numbers it is usually a good idea to provide some background on numbers and mathematics to begin. The problem is that this can prove to be a daunting detour for some, and it only tangentially relates to large numbers. This is because we already possess an intuitve understanding of such concepts. Another important aspect of my original approach was to treat the development of large numbers as mainly historical; that as time progressed people were able to come up with larger and larger numbers. While this is certainly true it is not entirely inline with the main impetus of this site. I did not create this site to discuss historical developments alone. My main objective for this site is to show how large numbers can be generated and to develop methods and theory to extend further and further. The historical approach distracts from this main objective and also does not give the reader an "active" role in the development of large numbers. For these reasons I think that chapter 1-1 needs some revision to bring it more in line with the main purpose of this site.

Another impetus for the revision is that I always felt dissatisfied with the first article as an introduction to the subject. The use of "number sense" as foundational concept didn't seem to entirely work. It was based on the "historical approach" I had originally took. "Number Sense" would simply be the earliest conception of numbers in modern humans, before the development of language or even symbolism. However this fails to define number. Recently after some thought on foundational issues, and doing some reasoning, I came to what I feel is a much more intuitive and clear way to define counting numbers, that doesn't even require me to use tally marks, number names, formal set theories, functions, or sophisticated and subtle definitions ! The most straightforward way to define the counting numbers is as the size of all collections ( for simplicity sake I at first ignore a distinction between finite and infinite sets. I later use this to create a paradox related to the number of such numbers. Since such a paradox arrises it necessates the usual distinction. This seems the more logical approach since infinity is a derivative concept of the finite). I will admit that even this first article is somewhat technical in detail, but it tries to base things only on familiar and intuitive concepts, rather than heavy formalism. In this sense it is relatively approachable, but it does require some thought, especially the section discussing the principle of equivalent orderings, and proving that (N) is greater than any counting number, yet is not itself a counting number. I still feel it is a much more clear and decisive introduction to the counting numbers. Better yet it provides the real impetus for an interest in large numbers: a desire to better understand the distinction between the finite and the infinite and how they are related. My original introduction "One-to-One" never even mentions infinity, and at the time that was intentional because I wanted to discuss the finite in isolation from the infinite. However I now am coming to the opinion that the infinite is an important motivator into large number research. Jonathan Bowers even refers to very large finite numbers as "Infinity Scrapers". Doesn't that suggest that we really are trying to reach for the infinite, even if we know such a task is futile?

I am also very pleased with the following article "Jacob's ladder". What better way to introduce large numbers than by generating some. Because I only use the most basic concepts I needn't set up so much foundational material before getting into large numbers. I can almost immediately start generating very large numbers with only the most basic ideas as a base. The difficulties in conceptualizing such large numbers also provides a motivation for the reader to gain a better understanding of numerical and mathematical concepts.

In any case, I plan to revamp chapter 1-1 and correct all the technical problems as well as improve the flow of the writing to better suit my new design goals. I also am considering the creation of new chapters for section I. I noticed when I started working on chapter 3-1 that I hadn't really provided an adaquate foundation for functions and operations in the previous two sections. I simply assume this is well understood. In Section II, I make extensive use of scientific notation and exponential concepts, but nowhere do I really provide good grounding in how these things work (especially the compounded versions of stacked exponentials) and I don't really provide a way to conceptualize just how large these numbers are. I came to the realization I would have to introduce the basic operations to provide some kind of foundation for my later discussion of recursive functions in Section III. I haven't entirely decided how to reorganize Section I, so it's still a work in progress. Most likely there will be an addition of at least one more chapter, and the last chapter "Encyclopedia of Numbers" is probably going to get restructured to better suit my new design goal.

I also plan to return to working on Chapter 2-4 in the near future, probably after most of Chapter 1-1 is properly corrected and enhanced. 2-4 is especially important because I think the -illion series is one of those very popular topics in large numbers circles (that and the googolplex series), and I have delayed it for so long. I had originally promised to complete it within a month of my websites first release, just over 2 years ago today, and I still have not completed it. I plan to remedy this soon.

I must admit that the time I have available to work on this site is quite limited. I am currently attending college and working part-time. However the semester will be ending soon and I will have about a month off of work and school. I am hoping this provides me at least alittle more time than usual to work on this website as well as my other personal projects.

In any case, you can expect new content, fixed links, and better flow of ideas in the coming months and year. Stay tuned and thanks for your interest and patience.

Sincerely,

-- Sbiis Saibian