Googology, apparently, is a subfield of mathematics dedicated to the study of large numbers. It has its own wiki. I found this wiki after attempting to read an old article by mathematician Scott Aaronson about big numbers. Actually, what surprised me about some of the material on the googology wiki more than anything else was that, in fact, I found myself making some effort to understand it, despite the dense mathematics.
I more-or-less understood the idea behind hyper-operators (and up-arrow notation), but became lost by what was called BEAF (a sort of systematic way of specifying functions with hyper-exponential growth, I guess), and I was eventually sidetracked by the plethora of whimsical terminology: big numbers beyond - way beyond – googol, with names like boogagoogolplex or meameamealokkapoowa oompa (which is defined by {{L100,10}10,10&L,10}10,10 , in case you were wondering – and no, I don't understand that).
There's a nice glossary of recently-coined, really big numbers (many created in response to Aaronson's original article) at an aesthetically-challenged web page called Infinity Scrapers. Note that the "meameamealokkapoowa group" appears at the bottom of the list (does this mean that it is really the biggest-of-the-big numbers? or just the most recently to be characterized?).
It is worth noting, for the uninitiated, that the absolute smallest of these numbers (but the largest which I can be assured that at least a few of my middle-school students, for example, might be aware of) is googol (= 10100), yet that number is still greater than the estimated number of elementary particles in the observable universe, 1086.
It's rare that I've tried so hard to penetrate a mathematical concept since my first year in college, when, after a semester of trying to make sense of the number-theoretical foundations of calculus under the unkind tutelage of Professor A. Wayne R__, nicknamed "B" Wayne R__ since he never gave A-grades. I concluded I wasn't cut out to be a math major, and abandoned ship for the more hospitable fields of the humanities surfing around to Religious Studies and English Lit before landing in Linguistics, which was a semi-return to the more rigorous fold. It's one of my few genuine regrets in life, I suppose. Not a regret at having jumped ship – rather, a regret to having found myself obligated to do so… which is to say, it's not really regret, more like disappointment with myself.
[daily log: walking, { {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}} , {{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}} } (=amount in km, represented set-theoretically using Von Neumann ordinals)]
Great to see that people do occasionally stumble across the googology community! 🙂
Meamealokkapoowa oompa is ill defined (BEAF in general beyond X^^X&n is ill defined and ambiguous), but even under the most optimistic definition it isn’t the biggest number (by the way, that site just categorizes numbers from BEAF, not all numbers).
The currently largest number named (that is, disregarding naive extensions such as applying hyperoperators or something to that number) is BIG FOOT.
It’s quite complicated to understand though. A good place to start (which is smaller than BEAF) is chained arrow notation, which is a series of numbers separated by a right arrow -> and has the following definition (# means the rest of the numbers, “a” and “b” are any natural numbers):
a->b=a^b Base rule: exponentiation
#->1=# Chop off ones at the end
#->1->a=# Another chopping rule
#->(a+1)->(b+1)=#->(#->a->b+1)->b Defines recursion
This grows quite fast. a->b->c is exactly equal to a^^^…^^^b with c up arrows.
a->b->1=a^b
a->b->2=a->(a->(a->(a->…(a->1->2)…)))->1=a->(a->(a->(a->…(a)…)))=a^a^a^a….a^a=a^^b
And so forth. a->b->c->2 iterated the numbers of up arrows, and produces pretty huge numbers. However, a->b->c->3 iterated over the iterations of the number of up arrows O_o. Just imagine a->b->c->d->2, which iterated over that whole hierarchy of iterations!
That is very small, googologically speaking, but is a good place to start. It introduce the whole concept of array notations: A list of numbers which produce a single, very big number.
For a number bigger than meamealokkapoowa oompa though, BB(n) (the busy beaver function) is probably the easiest to understand.