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Historical Precursors of Googology

Although it might seem like a hobby as odd as googology must only be the invention of modern eccentrics, it turns out that the essence of "googology" is probably as old as civilization itself. Given the natural human propensity for curiosity and bringing order to things it was probably inevitable that after they started dabbling around in number systems they might next consider the theoretical implications of their practical invention.

The earliest known exploration of just such a sort happened sometime in the 3rd Century B.C. It was the commonly held wisdom that human ingenuity was not up to the task of encompassing something as massive as the grains of sand in all the deserts of the world. It was supposed that no number system had yet or could be devised that could name a number larger than the number of these grains. The ancients seemed to have used nature as a way to talk about very large numbers. The bible many times represents very large numbers using ideas like "as numerous as the stars in the sky", "as numerous as the sands on all the earth", or "as numerous as all the grains of sand upon the ocean floor". These kinds of quantities were often hailed as "innumerable", that not only could one not find the exact number of such quantities, but in fact one could not even describe numbers that big with any human notation! It should be noted that this was at a time when common number systems didn't typically extend beyond millions. Apparently this was such a common misconception that even the King of Syracuse was puzzled by it. So he apparently commissioned Archimedes, one of histories greatest minds, to set his mind to bear on this question: Could one bound, with any certainty, such large quantities, and if so how large were they? Archimedes did that and much more: he completely floored these kinds of quantities by devising perhaps the first example of a number system designed specifically just to be frighteningly large! In a work popularly called "The Sandreckoner" Archimedes begins, in a grandiose tone any modern googologist would recognize:

" There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its magnitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the Earth, including in it all the seas and the hollows of the Earth filled up to a height equal to that of the highest mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken.

But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the Earth filled up in the way described, but also that of the mass equal in magnitude to the universe "

-- Archimedes of Syracuse

Archimedes goes on to create an extended number system based on the existing greek one that extended the numbers all the way up to 10^(8*10^16). He then went to show, by way of comparison, that even if the entire universe (as understood at the time) was filled up to the brim with sand this would only amount to a paltry sum of 10^64. In so doing Archimedes handily demonstrated that abstract reasoning could dream up monsterous numbers far larger than any practical endeavor could ever hope to employ. In so doing he also created what many would recognize as the first known work of googology. A number system created purely just to see how far number systems could be taken!

There are several instances like this in history. Large Numbers also played a large part in the religious texts of the ancient indians. There are accounts of the Buddha naming powers of ten up to 10^421 in a contest against the mathematician Arjuna. Even more staggering numbers were employed as a way to try and describe the indescribable size of the cosmos. In a text translated in english as the "Flower Ornament Sutra", in book 30, it is said that the buddhas speak of numbers called incalculable, measureless, boundless, incomparable, innumerable, unaccountable, unthinkable, immeasurable, unspeakable, and finally untold. The Buddha then proceeds to explain these numbers by beginning with 10^10 and beginning a mind-numbingly long succession of iterated squares. By the end of this the Buddha reaches an untold and it's equal to 10^(10*2^122). This number even surpasses Archimedes benchmark. The Buddha then goes on to use the untold to describe the cosmos, in which everything one can imagine has untold multitudes across all space and time. There are untold buddha's each in untold buddha-lands speaking untold virtues for untold eons. These passages suggest a poetic continuation of the numbers beyond the untold as successive powers of untold. The purpose of this text differs significantly from the Sand-Reckoner. Here large numbers are used to titillate the reader into a kind of cosmic ecstasy, to literally blow the readers mind away. It should be noted that the numbers here are no mere speculation but appear to be regarded literally as, at least a hint, of just how grand the cosmos may be. So was the text about large numbers or just how great the buddha-verse is? Probably both. The text has a definite googological flare to it. I think it's fair to say that book 30 is a mediation on the magnificent scales that exist within the infinite cosmos of the buddha. In this sense it is not so much an attempt to bound this cosmos as it is to explore the portion of it that can be understood. In other words, the writer(s) attempted to take the numbers as far as their imagination could carry them in order to convey to the reader the wonder and terror of the cosmos fully realized. So once again we have an isolated incident of ancient speculation on the very large.

But there is a decidedly elitist flavor to this narrative. Was it only elite intellectuals and spiritual sages who contemplated such large numbers, even if only briefly? Who knows. Perhaps peasants sometimes dreamed of such things. We wouldn't know but for what was written and preserved. But one gets the very clear indication that "large numbers" were the stuff of personal musings. Thoughts about large numbers appear to be just flickers of light in a great darkness. After all, life presents many more pressing concerns than sheer abstract contemplations. This is as true of the present as it was of the past no doubt. It would take something decidedly modern to trigger the birth of googology as we understand it today.

The ancient world rarely had need for very large numbers, but as science grew and our knowledge of the actual cosmos with it, a need for larger and larger numbers was born ... up to a point. The name million was at one time itself a neolism, formed from the word "mille" for thousand and adding the superlative "-ion" for great. Thus a million is literally the "great thousand". Then around the 15th century the french mathematician Nicholas Chuquet devised an extension upon the million by naming successive powers of a million by prefixing them with the latin orders, thus billion, trillion, quadrillion, quintillion, etc. This too is recognized as an early ancestor of googological practice. It involves the invention of names for successively larger numbers, systematically created, just for the sake of working out such a system. Chuquet would hardly have known that these numbers would one day actually be of use in science, in fact, that numbers in science would eventually surpass these numbers so named. Chuquet only went up to nonillion (10^54), but left it up to others to provide the obvious continuation using further latin numbers. And here another theme of googology emerges. One in which systems are regarded as open ended, the torch to be handed down for further extensions as needed or desired.

In 1904 a professor Henkle took up the challenge of taking Chuquets idea to its logical conclusion by extending, for purely theoretical purposes, the system of -illion names up to the limit of the latin language. Henkles system is perhaps the first example of a publicized system to name the first million -illions, a notion as loopy as it sounds. The millionth -illion incidently is only 10^3,000,003 in the commonly used short scale, the one used by Henkle which developed well after Chuquets long scale system. Later still John Conway and Richard K. Guy would revisit this topic and create a system for naming arbitrarily large -illions, the only limitation being when the names would get prohibitively long. With thousands of letters available this system can express numbers as large as 10^10^3000.

Despite this interesting line of development from the humble million, we still see such systems existing in great isolation of each other through out time and space, and worked on in a mostly individual basis. But this development was a definite precursor to googology.

But perhaps one of the most foundational events to what we might call the modern googology movement, obscure as it is, is when in 1940 Edward Kasner introduced the googol and googolplex to the popular imagination in a book titled, fittingly, "mathematics and the imagination". Much like the million, billion, trillion, this left the reader with an urge to take such number names even further. No one really knows where names like googolduplex, googoltriplex, etc. come from but it can be surmised that such names were probably coined independently many times over until they caught on. Also more obscure names such as googolplexian, and gargantugoogolplex exist for these numbers. Another part of googological lore from around this time is the coining of mega and megiston by Hugo Steinhaus in 1939 in his book "Mathematical Snapshots". These, extremely brief, excursions deep into the number line and the fascinating world of large numbers would spark popular interest in large numbers in the 20th century and sow the seeds for the googological community.

But the most important developments for googology came not from popular discussions but rather deep in academia in the study of recursion and the so called "computable functions". Here we finally see the development of a systematic way to generate successively larger and larger numbers through combinations of iteration and diagonalization that make the untold vanish in the twinkle of an eye. In fact we see in retrospect that the attempts of the ancients were just the very beginning of a fantastic world of numbers that had until the modern age had literally been untold! In 1928 Ackermann invented the first function proven to grow faster than any primitive recursive function, leading to tremendous numbers that leave even the modern googolplex and mega in the dust! On a much smaller scale the Skewes' Number and 2nd Skewes' Number made it into the Guiness Book of world records as the largest specific numbers to have a definite purpose in a mathematical paper. This was eventually supplanted by Graham's Number in 1980. Graham's Number is so large that even the Ackermann function can not handlely describe it!

The googolplex and later Graham's Number became the benchmark by which amatuers of this emerging hobby (as of yet without a name, of coming up with record large finite numbers and coining funny names for them), measured their work. Yet still, these people worked in isolation. Meanwhile professional mathematics had already well superseded these kinds of numbers with such monstrosities as TREE(3) and SCG(13), not to mention uncomputable functions like BB(n).

It took the internet age to really get the amateurs together into a cohesive community bent on gathering all this information into one place and taking stock of it. To learn more about the development of the googology community read "About Community".

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