4.1.1 - bowers
4.1.1
The Infamous Jonathan Bowers
Introduction
The googological work of Jonathan Bowers is notable for going much further than would be deemed sensible by most, and also for transcending all mainstream discussion of large numbers, and for its own peculiarities particular to Bowers himself. Amongst the googologist's, he is something of a founding father. He refers to himself, in turns, as an amateur mathematician and other times as a mathematics guru. The truth seems to be somewhere between these. Jonathan Bowers obtained a Masters degree in mathematics, and his genius and vision is clearly evident. At the same time, Bowers work is often highly imaginative but difficult to interpret. Bowers work lacks the polish, clarity, and depth that his fantastic ideas would seem to demand. Yet before Bowers there was nothing quite like it from amateurs, and even to this day his work continues to be the inspiration for further research into large numbers, and the recursive functions needed to define them. We will be exploring in some depth, both the merits, and shortcomings of Bowers work. Although we will mainly be interested in Jonathan Bowers googological work, he really has two great passions, the other being multi-dimensional shapes. Just as he has a significant reputation amongst the googologist's, he has a similar reputation amongst the "polytopists", amateurs and professionals who have tried to completely enumerate the uniform figures in higher dimensions (greater than 3). Jonathan Bowers interest in multi-dimensional shapes began back in 1990, specifically "polychorons", 4-dimensional analogs of polyhedrons. A natural question is: How many polychorons are there? This number depends a great deal on how we define a "polychoron". Bowers had developed his own definition, and with it he began investigating the different ways that polychorons could be formed, using his own techniques. Eventually he found 8190 distinct "uniform polychorons". If this was not enough, Bowers also developed an unusual naming scheme for naming all of these objects now known as the "Bower's style acronyms".
It is not known exactly when Bowers became interested in large numbers, but it seems to have been close to the time that he got interested in multi-dimensional geometry. Bowers, like most googologists, was always interested in large numbers, but his interest really took off sometime around 1987 when he read somewhere about the hyper-operators. What Bowers did next would reveal both his extreme ingenuity, and his own particular brand of genius. He combined both his interest in higher-dimensions and large numbers and invented a special set of notations he called "array notation". Array notation goes well beyond popular and well-known notations for large numbers such as Knuth Up-arrows, Steinhaus-Moser Polygons, and Conway Chain Arrows. In fact, Bowers system provides a powerful and ascetically pleasing alternative framework to the mathematicians equivalent: the fast-growing hierarchy. What's more remarkable is he achieved this level of sophistication just from reading about the hyper-operators!
Jonathan Bower's first went public with his work in 2002 when he published an aol hometown website. The website has long since been down, although it can still be visited via the "wayback machine", and I've also gotten permission from Bowers himself to archive it on my site. My archival version can be found here. He provided a complete catalog of the 8190 polychorons he had discovered at the time, along with his own designations for them, and also introduced the world to array notation. Bowers work did not immediately garner notice, but gradually people seemed to pick up on this strange site and the strange content contained there-in. Bower's met with great success with his polychoron research. He came in contact with other professional and amateur mathematicians who were also interested in the polychoron count, and together they formed a kind of collaborative group. This is probably Bowers greatest claim to fame. His array notation took hold in a slightly different manner. Jonathan Bower's system allowed him to easily pass up Graham's Number and Conway's 3 3 3 3, which he stated "are the largest numbers I've ever seen in the professional literature". While these are not strictly speaking the "largest numbers in professional mathematics", both of these having been long surpassed by the work of others, it is still a significant achievement to surpass both of these as thoroughly as Bowers did. Because of the high perceived status of Graham's Number, many independent googologist's have thought that merely surpassing it was enough to go beyond the numbers of professional mathematics. Consequently most amateur googologist's (is that redundant?) have used Graham's Number as their benchmark, and have been satisfied with barely pushing the envelop. Bowers was different, in that he didn't stop, just as soon as he got a little past Graham's Number ... his system sprawls on and on, until Graham's Number seems like an appallingly early point to stop at!
Because Bowers work went so much further than other amateur's, and because it was frankly so much more bizarre it got people interested. On a handful of blogs, people would mention the strange numbers Jonathan Bowers had come up with like, gaggol, tridecal, boogol, xappol, collosal, gongulus, and above all else the guapamonga and guapamongaplex, and would hail them as the largest named numbers in the world. Other people would use the numbers for humor, such as claiming their rank or power level to be a guapamongaplex^guapamongaplex. Slowly, Bowers work on large numbers began to gain recognition. When large number competitions would start up, or people would mention Graham's Number, there was always some smart ass there to mention Bowers work.
Gradually other amateur mathematicians with varying levels of ability began to recognize some merit to Bowers work. Once you got past a purely superficial appreciation of Bowers work however it became quickly apparent that Bowers was vaguely talking about something that was actually quite complex. Bowers originally had only 3 webpages covering his entire array system, and in this tiny space he had gone from an understandable recursive function, to an ungodly bit of bizarre abstraction that practically no one could understand! But no matter, the seeds of the googology community had been planted. In subsequent years, interest in large numbers in the Bowers vein would grow steadily forming a small community of sorts. I don't think it's an exaggeration to say that Bowers made googology as we know it today possible! He completely changed the field of large numbers for lay men from something only professionals could understand, into something that the amateur could experiment with. In a real sense Bowers is the founding father of the "amateur large number scene". When Bowers first went public in 2002 there was nothing like this. There weren't really whole websites devoted to large numbers (with the rare exception being Robert Munafo's Site). You could find a few isolated pages on large numbers on a number of personal websites, usually of math professors, but that was mostly it. Today we have many fine large number sites and blogs, and all of it revolves around the revolution that Bowers started. There is, my site, the Big Psi project, the googology wiki, googology 101, Jonathan Bower's new site, Chris Birds site, Peter herfords blog, Sam Hughes everything 2 articles, etc. All of this is motivated by the same fundamental impulses: To provide a more complete description of Bowers system and how it works, to connect his work to the body of professional mathematics on the same subject, to discover it's place in the larger scheme of numbers proposed by professional mathematicians with non-computable functions, and lastly to take Bowers system to the next level. These still remain difficult problems in googology that I feel have not yet been adequately addressed. There does however seem to be a growing focus on these problems and a network of googologists has been gradually forming around these problems.
Ironically, beyond his work, little is known about Jonathan Bowers himself. He was born on November 27, 1969 in Tyler, Texas. Bowers was therefore only 21 when he began his work on polychorons and large numbers. Bowers can be seen to have a somewhat strange sense of humor from his website. Bowers seems to be obsessed with making up words and seems to have a knack for coming up with strange sounding ones. The names he has chosen for polychorons are really just as strange and sometimes even stranger than his number names. It is mentioned on his site in passing that he has a brother, and his original site mentions that he has a M.S. Degree in mathematics. Beyond these small biographical details nothing is really known about him other than his eccentricity.
Bowers work is so important for the rest of our discussion, because the new developments in googology only make sense in the context of what Bowers started. Bowers actually has two systems: his original 2002 array notation, and his new 2007 array notation. The differences are almost purely academic, however in the interest of not being anachronistic I will be discussing his original 2002 system first. This is really where the story begins.
In the next article we begin where Bower's began ... by extending the established Hyper-operators ...