Googology

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Googology is the study and nomenclature of large numbers.^[1]

One who studies and invents large numbers and large number names is known as a googologist. A mathematical object relevant to googology is known as a googologism; the term googolism is similar but only applies to numbers. Googology is known for the rather comic names given to the googologisms, such as " googonkosol", "gralgathor", "gongulus", and " tethrisexticulcubor".

Googology is not to be confused with googlology, the study of the Google search engine and its various other services.

The antithesis to googology is ultrafinitism, which states that large numbers simply do not exist.

Etymology

The term was coined by André Joyce according to the fictional story by Michael Halm, formed by combining googol (the classic large number) + -logos (Greek suffix, meaning "study"). Joyce's googology involved devising a system of names for numbers based on wordplay and whimsical extrapolation. Joyce coined several other related words: googologue (dissertation of googology, especially when long and confusing; not to be confused with the number), googomania (obsessive, compulsive googology), googonaut (non-googologist fan of googology), googophobia (fear of large numbers; also see googolophobia), googophiliac (lover of large numbers), googosophy (philosophy of large numbers), and googoscopic (combining art and science using googology).^[2] These words have not caught on like "googology" has.

History

See also: Timeline of googology

Although the term googology is modern, the subject has existed for as long as humans have been fascinated by large numbers.

The earliest known work by a "googologist" is probably the Sand Reckoner written by Archimedes, a Greek polymath, sometime in the 3rd century B.C. In it he develops a system of numbers extending to 10^{8 × 10¹⁶}. There are other examples in ancient history that illustrate mankind's fascination, and even adeptness, with large numbers. Some religious texts contain some very large numbers. Although the Bible contains no definite numbers greater than 10⁸, it uses figurative language in many places to relate very large numbers such as "the stars in the sky" or "the sands of the sea".

With the advent of modern mathematics, and the impending invention of the computer, mathematicians of the 19th and 20th centuries had access to numbers larger than ever before. This fascination was relayed to the laymen through popular books on mathematics. "Googol", "googolplex", and "mega" were all introduced in books of popular mathematics, written by mathematicians who wanted to explain to the laymen what mathematicians meant when they invoked infinity.

Eventually, the fascination of large numbers spread to a class of amateurs who took it upon themselves to extend the ideas hinted at in these popular books on mathematics. These became the early googologists. This took on something of a form of a hobby that still continues today, with amateurs writing papers claiming to have "invented the largest number ever". That being said, not everything produced is brilliant, nor is it all crank mathematics. There is a variety of skill levels, and some of googology actually comes from professional mathematicians, not amateurs. In particular, there seem to be three classes:

Googologisms that arise in professional mathematics as side-effects of more serious math problems, such as Graham's number and Skewes' number.
Googologisms devised recreationally by professional mathematicians, such as chained arrow notation and Steinhaus-Moser Notation.
Googologisms created solely by amateurs, such as array notation and Hyper-E notation.

During most of the 20th century, early googologists worked in isolation. Since the advent of the internet however, there has been a greater confluence of ideas, and several websites have sprung up to gather the loose bits of information that form the body of knowledge, methodology, and conventions known as googology. Perhaps the most important of these sites are Googology Wiki, Robert Munafo's site, and One to Infinity.

Furthermore, from 2002 through 2013, a loosely knit community of large number enthusiasts, dubbing themselves googologists, has emerged, building websites, sharing information, and developing a culture with a unique approach to one particular challenge: "What is the largest number you can come up with?" Googologists generally avoid many of the common responses such as "infinity", "anything you can come up with plus 1", "the largest number that can be named in ten words", "the largest number imaginable", "a zillion", "a hundred billion trillion million googolplex", or other indefinite, infinite, ill-defined, or inelegant responses. Rather googologists are interested in defining definite numbers using efficient and far reaching structural schemes, and don't attempt to forestall the unending quest for larger numbers, but rather encourage it. So perhaps a more accurate description of the challenge is: "What is the largest number you can come up with using the simplest tools?"

As far as mathematical fields go, googology is an oddball. It precariously teeters on the edge of what we call "science", becoming more of an art form as opposed to a mathematical study.

Although googology remains, and shall probably always be, an obscure, esoteric, and impractical study, it at least now has a name, a history, and a community.

List of googologists

See also: Category:People

Below is a list of people who have contributed significantly to large number studies.

Hugo Steinhaus and Leo Moser created Steinhaus-Moser Notation.
Donald Knuth invented arrow notation.
Ronald Graham defined Graham's number.
John Conway invented chained arrow notation.
Tibor Rado devised the busy beaver function, the original uncomputable function.
Harvey Friedman investigated various combinatorial functions, including n(k), TREE(k), and SCG(k).
Jonathan Bowers created BEAF; he is regarded as the founding father of modern googology.
Chris Bird invented Bird's array notation and helped investigate and develop BEAF.
Hyp cos invented strong array notation, arguably the fastest growing computable (= recursive) function in googology.
Nathan Ho founded Googology Wiki and republished BIG FOOT on his site.
Lawrence Hollom invented hyperfactorial array notation.
LittlePeng9 defined what was supposed to be the largest named number at the time, BIG FOOT. Later it was found to be ill-defined.
Agustín Rayo invented a famous function named after him, one of the fastest-growing functions known.
Sbiis Saibian invented the Extensible-E System, and is writing a web book about googology, titled One to Infinity: A Guide to the Finite.
Aarex Tiaokhiao invented various extensions to other notations.
P進大好きbot invented the current largest named number, Large Number Garden Number.

Googology

Contents

Etymology

History

List of googologists

Sources

See also