Googology101 : Part I - Orientation
So you wanna be a googologist?
So you want to be a googologist? In this series of blog posts I will tell you the things you need to know to get started, common pitfalls to avoid, and how to create successful googology.
Who am I? In the community I go by the name "Sbiis". I've been working on googology for quite a few years now. I am the inventor of the popular ExE system. I'll be giving you the tips and tricks from my experience working in googology and in the googology community.
What is Googology?
First we establish a working definition of googology and what it's about:
Googology is the practice and art of defining large numbers and giving names to them, the body of facts, conventions, and knowledge related to this enterprise, and the study of ways to generate and compare large numbers, as well as the study of fast growing functions and growth rates.
Although not a very concise definition, this covers pretty much everything we might call part of the googology enterprise. As a new googologist you will of coarse want to create your own googology. This means creating your own large number notation and coining names for numbers generated with this notation. But this is not all that googology is about! Remember that analysis also plays a major role. How are you going to know how powerful your notation is? How competitive is it compare to other existing notations and numbers? You also need to familiarize yourself with the conventions of googology and the community ethos. Knowing these things will make you much more effective at communicating your ideas and getting others interested in them.
The Prime Objective of Googology
The first and most important thing you need to know about googology is that it's ultimate aim and goal (it's prime objective) is simply to define as large a number as possible. But googology responds to this question very differently than anywhere else. You can't simply cop out of the question by saying "infinity", or by saying "my number is the largest possible number", or by saying "there is no largest possible number". None of these are acceptable responses to the challenge of googology!
You must understand that the goal of googology IS NOT to define the largest number. That is, it is not our aim to find some way to define a "last number" that "ends" the game of googology. If that is your aim then this is definitely not a place for you. As anyone worth their salt here will confidently tell you there is no largest number. This may seem to like a contradiction. Didn't I just say our goal was to define as large a number as possible. So how could our goal not be to define the largest number. Well think of it this way. Say we did define the largest number period. What would there be left to explore in googology ... nothing! Googology is a never ending enterprise, it's about the expansion of knowledge, not about it's "completion". In fact as a googologist you will begin to appreciate that the very idea of a totality of mathematical knowledge is in fact ... nonsense. Mathematical knowledge has no border because it can always expand upon itself. This is the essense of googological progress.
But then, some of you will think, the goal of googology is unobtainable! Because at the outset we already know we can not "define as large a number as possible", because there is no largest number possible! This is the kind of response your likely to meet in professional mathematics. If you ask a professional mathematician what the "largest number" is, he will take the question at face value and simply say there isn't one. But while true, and an important insight for the practice of googology, this is not a very interesting response. Why? Because it kills the discussion. What we do in googology is we answer the question behind the question. That is, the thing that really motivates the question in the first place! When a child first asks the question "what is the largest number?", it's because they are coming to the realization that numbers can keep on growing and there doesn't seem to be anything to stop them. When the immensity of this dawns on the child what they are looking for is ... just how insanely big is this! They are looking for a very large number in other words. So large in fact, that it transcends their current understanding. So what does the answer do. It says that there is no answer ... it takes that ever-inflating hope of larger and lager numbers and it pops it and says ... there is no answer to this question! So where did the tremendous numbers go? NOWHERE! The important take away from "there is no largest number" is not that the pursuit of large numbers will be fruitless, but rather that it is ever abundant. You CAN HAVE really really large numbers, precisely because there is no artificially fixed "limit" to the process of number formation. So the next step shouldn't be to have your brain shut off. Rather it should be to take this as a grand opportunity to explore the endless world of numbers more fully than ever thought possible! This is the true motivation of googology.
Of coarse we already know at the outset that numbers are never ending so ... don't we already know everything we need to know about numbers? From the googologist's perspective the answer is definitely no. There is no such thing as knowing too much about numbers, or having a number which is "big enough". If that already scares you, googology is probably not for you. Googology is not about needing to know, but rather about wanting to know. We won't be studying large numbers because it has some practical application. In fact you will discover early on that we can create numbers which exceed any practical purpose. Numbers in astronomy, cosmology, physics ... we can get well beyond all of this, and with only the tiniest application of large number theory. What lies beyond this is a boundless world of abstraction like nothing you've ever seen.
But what "exploration", some say. We know what numbers are, right? Well think about this. If the numbers are indeed infinite, then there must necessarily be a largest number anyone has thought of in the history of humanity. Why? Because human beings have only been around for a finite amount of time, there has only been a finite number of human beings, and they could only have had a finite number of thoughts in that period of time. Therefore amongst all the numbers ever imagined by man there must be a largest. What about infinity? This is excluded from consideration. Only finite numbers are allowed. We also discard anything which is ill-formed or self-contradictory (We will gain a better appreciation of what this means in Part II). What's left are just finite numbers. Since numbers are infinite the conclusion is always ... there is a number so large no one has ever thought of a larger one! The point is, that there is ALWAYS something left unexplored in the world of numbers, precisely because numbers are never ending. Even more incredible, between 1 and the largest number ever conceived, almost all of these integers have never, could never, have been thought of. Even if humanity thought up a million billion trillion quadrillion unique numbers within this range ... still ... still far FAR FAR more than I could possibly explain to you would exist in this range that were never defined! So not only can we not claim to have explored the entire length and breath of the number realm, but we have only ever, and will only ever, explore an infinitesimal fraction of it! THAT is the adventure we are embarking on. Interested?
But how are we suppose to accomplish the "goal" of googology to "define as large a number as possible", if it's already impossible at the outset!
There are two underlying assumptions which make this demand sound contrary to what I've been saying about numbers being inherently boundless.
One assumption, which turns out to be false, is that all numbers are equally easy to define. Therefore, the unspoken reasoning goes, the request to "define as large a number as possible" is a fundamentally trivial one. It's so easy to define 10 or 100 or a million or a billion or a trillion, and a trillion, as anyone here will tell you, is already a frightfully large number. Therefore all numbers must be easy, right? WRONG! The scary answer is ... it's only because your not thinking big enough! This false intuition that all numbers are very easy to define is the result of the simple fact that for almost ANY purpose in real life you'll have for numbers, you'll never need a number with more than a handful of digits. Even the largest numbers in science are EASY. So just how BIG are we really talking then?! To get to numbers that are HARD to define you have to go to numbers so large that your very idea of "largeness" and "infinity" will have to be radically expanded to understand. Sound scary? Maybe googology isn't for you. Are you intrigued? That's the ticket!
The first thing to understand is that all numbers are not created equally. Some numbers are easy to define, and some numbers are very very difficult, and some numbers ... lie far far out of mortal reach. Furthermore, when we restrict ourselves to just whole numbers (numbers without a fractional part, more detail on this later), we find a general trend. The larger the number, the harder it is to define. We can observe this even with fairly small numbers if we train ourselves to pay attention to small details. Notice how it's easy to write out 10, but harder to write out 10000000000. You can probably imagine now that with a big enough number it will have enough digits that it will be impossible to write out even in a life time! That's got to be a HUGE NUMBER in googology right? Wrong. You would never really be able to write something like 1 followed by 10 billion zeroes in a life time ... in decimal notation, but as you'll learn in googology, decimal notation is just ONE notation we can use out of an infinite variety of notations we can devise. With exponential notation we can write 10^10000000000, and we now have a number so big it could never be written out by a person in a life time in decimal. The key thing to understand here is that the difficultly of a number IS NOT DIRECTLY proportional to its size. If we double a number it is not necessarily twice as difficult to define. A number with twice the number of digits is also not necessarily twice as difficult to define. Difficultly increases very very slowly and irregularly. But the general trend is ... as the numbers get larger it will get ever so slightly more difficult to define. Just consider how much bigger 10^10000000000 is than 10000000000. Yet it only took 3 more characters to define . . . gulp . . . so how far does one have to go to get numbers that are really hard to define . . . . . . o_0;
So we come to a radically different conclusion as googologist's: the quest to define as large a number as possible is anything but trivial ... because the question becomes harder to answer the further you push. In other words the problem is as difficult as your willing it make it!
If your not convinced this is fine. It will take a while to become accustomed to this view, but you will experience it first hand if you continue to progress in googology. It's also important to understand that the difficultly doesn't strictly increase. Just because A > B doesn't mean A's difficultly is larger than B. In fact we can make a general conjecture, that for every sufficiently large A that we can name, there is always a B which is smaller but more difficult to define. The difficultly of defining numbers is seemingly haphazard and irregular. It's only the general trend which is increasing. What is up with this strange phenomenon? It has to do with the complexities and regularities contained within the integers themselves (more on that later).
The second assumption that leads to confusion over the demand to "define as large a number as possible" is that people take the question on principle. By that I mean that in principle, there is no largest number that can be defined. But what about in practice? The thing about googology is that we don't attack the problem on a purely theoretical basis. This isn't about trying to estimate the largest number you could theoretically come up with given unlimited time, or even given a limited time like let's say your entire life time. Googology is about putting this endevour literally into practice! This means you don't speculate about defining numbers, you actually go out and come up with a number.
Do not think of "define as large a number as possible", as a question. Think of it as a request. And it's a request you are compelled to provide an answer for as a sincere googologist. You can only answer it by giving a number, no excuses, no cop outs. It doesn't matter if it isn't literally the largest number you can come up with. For an analogy, if your boss asked you to "complete as much of the assignment as possible today", are you going to respond to him that that's "Impossible, because the odds of me working at maximum efficiency is virtually zero". No. You'd do the "best you can", which is just a way of saying your going to work really hard. At the end of the day, could you have done a little better? Probably. Did you really literally do your best? Probably not. Does it matter: no. Why? Because what is put into practice is infinitely more valuable than a mountain of speculation and theorizing. If you want to be a googologist attack it like that! Go in, work hard, and at the end of the day have something to show for your work. That is all that matters ( Of coarse there are some guidelines about how to go about your "work". More on that in Part II ) What if your number isn't the "biggest" ever? Just go at it again. The nice thing about googology is that you can keep at it as long as you like. The game only ends when you quit.
Get it? Good.
That's what googology is like. And the cool thing is you won't be a lone frontierman in this endevour. Many other people will also be working on making larger and larger numbers. Furthermore there is already plenty who have come before all throughout history. If your serious about understanding large numbers then you will want to learn from humanities collective knowledge of the art and science of creating large numbers. I'm not even kidding. Human kind has been trying to come up with large numbers and ways to write and understand them for almost as long as we've even known what a number is!
That being said, if you want to get started, then there are some things your going to want to know first.
In Part II, we will go over the basic "rules" of googology, and some basic community guidelines. If googology is a game, then there is certain rules that we abide by to play it. These rules do away with a lot of bad habits that get in the way of actually doing googology. See you there...
Jump to Part II