googology

4.1.1

A Purview of the World of Googology

What is Googology?

What is googology? I've used this term loosely throughout parts of Section III without really explaining what is meant by it, and where the term comes from. Before we delve into the work of Jonathan Bowers, perhaps the greatest googologist of our time, let's first define what googology is, and explore briefly its history.

Simply put, googology is the unadulterated "study of Large Numbers" for their own sake. The Phrase "Study of Large Numbers" is one that I have used frequently to encompass everything that we discuss in relation to large numbers on this site. Until only very recently, there was no "official" name for this study. I was initial reluctant to adopt the term, however it has surged in popularity in the large number community (with a few notable exceptions), and so I too have found the label useful.

But what is meant by the "Study of Large Numbers"? This simple phrase doesn't quite capture some of the nuance that distinguishes googology from other areas of mathematics.

For example, unlike professional mathematics, googology does not eschew the inventing of oodles and oodles of ridiculous, strange, or technical sounding names for large numbers. This is taken as something of an art form, and it is but one of the creative aspects of googology that makes it more recreational, than "serious". Professional mathematicians prefer to use jargon which is concise and to the point. This can be seen in terms used in higher geometry. Professional mathematicians refer to the surface of a 4-dimensional sphere as a "3-sphere". In polytopist circles though, this is known as a "glome". Polytopist's, like googologists are usually not professional mathematicians, but rather interested amateurs.

Googology, in a strict sense, is more than merely a "study". It also is a body of conventions. Numbers are given names, and these assignments are made largely on ascetic grounds. Sometimes these names are part of a larger system, but even so, googology is often willing to strain language in order to keep up with the numbers. These "named numbers" are more than mere objects of study. They act a "beacons", "guideposts", "markers" in an ever expanding and evolving landscape made up of such beacons. It would be more accurate to describe them as objects of veneration, rather than study. Googologist's only study numbers in the sense, that they try to grasp their size, understand their properties, estimate, approximate, or bound them, learn about their digits, or compare them with other numbers.

Googology is also, about more than mere "largeness". Infinity is "large" in the usual sense, but googologist's avoid the use of it as a "Number cap". There are larger infinities still, such as Aleph-one, Aleph-two, etc. and one can make a point of making larger and larger infinities, and in a somewhat different sense these can all be thought of as "large numbers" ... VERY large numbers! But in another sense these aren't numbers at all, as we have nothing for them to measure or count, except for themselves. Even the smallest infinity, Aleph-zero, really is defined in a circular manner: Aleph-one is the number of positive integers, and the number of positive integers is Aleph-one. Yet there is no positive integer we could say is the number of positive integers, so the idea that Aleph-zero is "countable" is somewhat academic. In some ordinary sense, no infinity is countable, because no infinity can be put in one to one correspondence with any set of the form " {1,2,3, ... ,Z} ", where Z is any positive integer.

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Typically when the question "what is the large number you can think of?" or even "what is the largest number?" is asked, a typical laymen response is to say "infinity", because "infinity" is the "largest number". This answer however doesn't leave much room for the imagination. If "infinity" is the largest, than perhaps that's the end of it and there is nothing more to say about large numbers. Googologist's however, follow another line of thought. What do we mean by "infinity"? How BIG is infinity? To answer that, in any meaningful and non-circular way, and to really capture the deep endlessness of it, we must explore that vast and infinite reaches of the finite. The googologist revels in the freedom of unbounded exploration, and it is that exploration which is just as important as the largeness. The reason googologist's avoid infinity, despite it being "larger" than anything they generally study, is because there is no sense of exploration in declaring a "largest number". For the googologist, there is no such thing as a "largest number", and if there was, they would come up with something greater!

So if googology is NOT simply mathematics, NOT simply a study, and NOT simply about coming up with the "largest number" and being done with it, what IS IT?

I would define googology as the exploration and creative cataloging of the infinite expanse of finite numbers. The goal of googology is to gather all the information that has been accumulated about large numbers over human history, and expand it to the limits of human knowledge, by both extending its range arbitrarily, and also by filling in as many gaps within this range as possible. Googology, as a subject, is the body of gathered facts and knowledge about specific large numbers, properties of large number arithmetic, and size comparisons of large numbers. It is also the body of naming conventions, traditions, and history of the human fascination with large numbers. Lastly, aside from the "facts of googology" (analogous to the "facts of science"), aside from the "conventions and history of googology" (analogous to conventions in engineering such a "conventional current"), there is also the body of methodology, theory and philosophy that provides the means and impetus for moving forward (analogous to the "scientific method").

I would hope that the above definition is comprehensive, but googology is such a sprawling subject it's somewhat difficult to say. Googology, by aiming to explore, understand, and expand upon our understanding of large numbers, inevitably gets involved in many other areas of mathematics tangentally, including: set theory, recursion theory, computability theory, modular arithmetic, theory of functions, etc. Because these become foundational to further progress, googology can begin to seem to be about anything BUT large numbers. Any subject it touches upon however, is all ultimately towards the goal of making a larger number, and this is the single unifying facet of the subject.

I myself have tried to devise some names for it, although I've made virtually no use of them. I have used the term "arithmology" privately from time to time. Literally this translates as "The Study of Numbers", and in a broader sense I'm arithmologist as well as a googologist. I am interested in numbers for their own sake, and large numbers are just a particular part of that fascination. But if arithmology is literally the "Study of Numbers", wouldn't that also include things like small transcendental numbers, Complex Numbers, transfinite numbers, Number Theory etc. None of these however, completely overlaps with what is meant by googology, yet everything in googology deals with involves numbers, so googology seems to be a subset of arithmology. This is one of the reasons I did not use the term for the "Study of Large Numbers". Other names I've devised include "algorithmology" (the study of algorithms), "aritology" (the study of arity), and "megalo-arithmology" (the study of excessively large numbers). While the study of large numbers, does lead naturally to the study of algorithms and arity, only the last term really captures the spirit of the "Study of Large Numbers".