The googological work of Jonathan Bowers is notable for going much further than would be deemed sensible by most, and also for transcend

Large Numbers - 4.1.1 - bowers

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2024-12-12 02:00:05

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!