About Googology
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What is googology? Basically googology ( alt. megalo-arithmology) is the practice of systematically and exhaustively defining larger and larger numbers and giving names to the them. A "googolism" is what we call a "name" coined for a specific large number. An alternative term is "arithmonym" which literally means "number-name". A googologist is a person who practices googology. The googology community is the community of googologists and enthusiasts who have joined together to gather, discuss, and expand upon the existing knowledge of large numbers.
The term "googology" was coined by an "Andre Joyce" by combining the word "googol" (the quintessential "googolism"), with -ology, meaning the study of. So googology literally means the "study of a googol", but it's typically cited as "the study of large numbers". However, as I have pointed out before, this definition is apocryphal as we don't actually "study" the large numbers themselves, but rather study the methods by which large numbers may be defined and generated, with the aim of coming up with the largest numbers possible. Googology is also unusual in that it's nomenclature is not purely prefunctory, but is a point of interest in itself. Creating "names" for the numbers is a traditional part of the practice of googology, even though admittedly not absolutely essential, so that the "names" become an act of creative and artistic expression in themselves, not merely as decorations or functional handles of the "math" involved. A relatively complete definition of googology may be stated as the following:
googology is the art and practice of defining and generating numbers as large as possible, giving names to these numbers, either systematically or on a case-by-case basis, the study of means of generating these numbers as well as techniques to determine which of two numbers is larger, as well as a body of conventions and knowledge maintained by the googology community.
A key feature of googology is that it's more than a "study" of something, but is better understood in the language of a "trade" or "practice". This is why googologist's are sometimes referred to as googolsmiths, as in forgers of googolisms. A theoretician studies the principles of a craft but does not actually produce anything with it. A crafter depends on the theoretical aspects of his/her craft but has not done any actual work until something has been produced by applying that theory.
Nathan Ho, the founder of the googology wiki, the main hub for the googology community, described googology as a never ending quest to outrun itself, which is an apt description.
Googology is marked by a focus on trying to reach numbers as large as possible as an objective in its own right. This is distinguished from deriving large numbers as a mere corollary or by-product of some other pursuit. Googology creates large numbers intentionally for the sake of being large and nothing else. Googology is also often marked by a desire to systematically name these numbers, bringing some sense of order and creation into the exploration. Even in the absence of names for these numbers, there is usually a systematic way to at least generate them, and there is typically a series of "sign post" numbers as if to mark the way.
Googology involves a lot of mathematics, and yet is of a completely different character than "mathematics" in the usual sense. Most people think of mathematics as a tool of practical calculation, as a way to solve problems. Googology isn't created for any practical purpose or to solve any "problem", real or imagined. It's abstraction for abstraction sake. Basically we have this idea called numbers, that were created for the practical purpose of counting and inventory. For theoretical purposes it's important that the concept of numbers be open ended. As a general principle we always need the possibility of more numbers, just in case, we need them. Googology turns this on it's head. Rather than wait for numbers to be useful it turns the numbers into a focus of intense theoretical interest, with the goal being to take the principles of naming and describing numbers and number systems to their logical extreme. There can be no "end" to googology in the proper sense short of oblivion or madness because this goal acts as an absolute imperative without any preconceived boundary to reach. In fact googology can only make sense if it is boundless from the outset. If we have some "boundary" we are trying to "reach" at the outset, we could have only specified such a boundary if we already had the means to define it, in which case we have already accomplished our goal. So instead of the goal of googology being to reach some boundary it is instead to strive to surpass all boundaries. Googology is therefore fueled by a theoretical curiosity rather than a practical necessity.
Mathematicians have a different understanding of what "mathematics" means than the average person. Mathematics is NOT primarily a tool of practical calculation. Instead it is a study of theoretical structures and patterns. These may be put to practical ends, but at their heart they are simply ideas with consequences which may be explored. The main aim of professional mathematicians is to come up with some general truth about a mathematical structure, and to provide satisfactory "proof" of the correctness of this truth. These proofs are what mathematicians call "theorems". So you might say the business of mathematicians is to prove theorems, preferably interesting and difficult theorems. Googology at it's heart is NOT about proving anything. It is primarily a mix of exploration and creativity. The exploration is the striving to define, in a mathematically sound way, larger and larger numbers, while the creativity is the naming of these numbers, creating notations to define them, and other acts of idea-smithing involved in the process. Googology only involves proof as a corollary of (1) ensuring that a number is in fact well defined mathematically and (2) demonstrating that one number is larger than another, or more generally that a certain system is stronger than another. But using "mathematics" and "proving things" does not make googology "mathematics" in the way that mathematicians understand the term. For mathematicians theorems are the goal in and of themselves, any results arising from them, save further theorems, are merely secondary. In googology theorems and "theory" are just useful means of ... you guessed it ... making larger and larger numbers. The fundamental difference is one of focus. You might say googology is math turned on to the particular goal of making numbers as large as possible. As you will learn, "large as possible" turns out to be truly mind-boggling and other worldly to the point where it might radically change the way you look at numbers and mathematics in general.
To learn more about the historical development of googology and the googology community proceed to the next section...