Googology101 : Part II - Initiation
The Two Principles and The Golden Rule
Welcome back to Googology101. In the last part we covered the definition of googology and it's prime objective. In this part, we will cover the Seven Deadly Sins of Googology. Essentially these are the Seven rules that you have to abide by in order for a googological discussion not to dissolve into a mindless shouting match! As Sharkee put it, before we start discussing large numbers we have to set up a few ground rules first so things don't go "crazy". My ground rules differ considerably from Sharkee's. There is no need for the number to be "useful" in any way. If it's a number then good. This is more a concern about always keeping the game "going". We don't want any game-stoppers. To that end there are basically two guiding principles we want to consider in designing googology proper (1) It should never be a matter of opinion what the larger of two numbers is and (2) it should never be the case that there is an "unbeatable" number. This leads to what I'll call the Golden Rule of Googology...
there is no such thing as an unbeatable number
( Alternatively ... you can't win at googology)
This means that you can't ever declare your number as something which can't be beaten. In googology, there is no such thing. At very least you can always add 1, and this must always result in a larger number. This means, in a manner of speaking, that by design, every number in googology must be built-to-fail. If somehow your "number" isn't "built-to-fail", then you are no longer playing googology. Your just declaring yourself all-time champion. But what you will find is that such a "victory" is entirely vacuous. The thing that makes googology meaningful as an endevour is that it actually takes hard work to master. It may be hard to imagine how a simple task like naming the largest number you can could be anything requiring mastery. Googology is so esoteric that I can't at the outset explain to you why it's a subject requiring a lot of study. But just trust me on that. Your view of infinity, largeness, number, and mathematics will be radically changed from your exposure to googology. It may seem that this is asking for some kind of surrender, and it is. But all your surrending is your right to "pretend" your the best, that your "unbeatable", which is really an empty boast. But I promise you what you will get in return for genuinely engaging in googology is infinitely more valuable. You will get to experience, and even understand in a tiny way, the tremendous vistas that lie in the heart of the infinite, and that it is far more awe inspiring than any mere word like "infinity" could ever describe.
Googology, is NOT about winning, at least not in the sense of there being any final accounting where we can say ... ah, here's the winner! Googology is an ungoing "game" with many players, that come and go as they please. It is both a personal pursuit and a communal enterprise. It is both competitive and cooperative by its nature. If your coming into this just thinking your going to find the largest number and add 1 to it, then your coming in with the wrong idea. That is neither competitive nor cooperative. It doesn't contribute anything to the furtherance of googology and it doesn't demonstrate competance in googology. It merely proves you know how to add ...
Am I saying there is no glory to be had? Not at all! But you will only be given credit where credit is due. Do not expect people to be impressed when all you can do is borrow and imitate the work of others, but you have nothing to contribute yourself. "Winning" at goology isn't so much about winning, but about being a good player. By constantly increasing your knowledge and understanding of the subject and putting it into practice to create new googological "products" all of your own you will become a recognized and important part of the googological enterprise. Remember this: You can't win at googology, but there is playing well and playing poorly. Adding 1 is playing poorly. Are you interested in learning how to play well? Good. Let's get started by learning the basic rules of the game...
The Seven Deadly Sins of Googology
The seven deadly sins of antiquity are luxuria, avaritia, superbia, ira, invidia, gula, and acedia. I have paired these up with the Seven Deadly Sins of Googology. The Seven Deadly Sins of Googology are:
I. Invocation of the Infinite
II. Invocation of the "Last Number"
III. Touting Nonsense.
IV. Going Meta
V. Standing on the Shoulders of Giants
VI. Going Ballistic
VII. Lack of Specificity and/or Unverifiability
In what follows I will elaborate what these sins mean and the kinds of things they entail. In the process you'll gain a better understanding of what we might call the googologist's perspective...
I. (Luxuria) Invocation of the Infinite
It should go without saying, except that apparently, it doesn't. It is almost a reflex response, seen as early as grade school, that when a large number competition gets started someone will want to end it , or alternatively, simply declare themselves supreme-champion by envoking infinity. There is no taboo greater in googology than envoking the infinite. This is why it is listed at the top of the list.
It's not that googologist's hate infinity. At least that's not quite how I see it. Rather it's that infinity is simply too sacred to be invoked. It is particularly impious for someone who lacks even adaquate knowledge of the finite to proudly proclaim they know the infinite. But there are more fundamental, less idealogical reasons why infinity must never be envoked in googology. Firstly, it's overkill! It's gauranteed to be larger than any finite number. It's the ultimate cheese combo, and it's a total game-breaker! To make matters infinitely worse, it is the easiest of all things to envoke. It's just about as easy to envoke as zero, and that's about how much recognition you'll get for envoking it in googology. Because it is so easy to envoke, because it is such a conversation killer, it must be disallowed if googology is to be any sort of engaging and illuminating enterprise. In googology the most interesting numbers are the ones we know the least about. But infinity is perhaps one of the best known concepts in mathematics outside of zero. So in that sense ... it's boring. It's already known. Googology is about exploring the unknown space between zero and infinity. (Or more commonly One to infinity, which I consider to be the interval of the large numbers).
Just to clarify, this isn't just about using the word "infinity", and it doesn't have to do with infinity being somehow ill-defined. Sometimes fledging members think that if they can just provide a precise definition of infinity, then they will get to claim it as their number. Hence you'll see things like 1/0 , or lim(x-->0)1/x, or -log0 or lim(x-->0) -logx, etc. This won't help you. Attempting to set up a non-terminating process is no better than trying to use limits. In this case any non-terminating process is considered to have no output. So it will be about as good if you hadn't tried anything at all!
In short, it's not an issue of needing a definition of the infinite or a way to get infinity as an output. It's that you can not envoke something which is automatically larger than all numbers, and infinity almost always an attempt to do just that. So it's simply a violation of the golden rule of googology.
But guess what ... even if you allowed infinity in the game ... the game still wouldn't end. If infinities allowed why not infinity+1? No bigger you say? Not a problem. Did you know that Cantor proved that there are ways to get infinities larger than infinity! All such infinities have the common property that you can't exhaust them even in principle. But it turns out you can have infinities so large that they can't even be exhausted by other infinities! So guess what happens when you add infinity into the game ... you just get even more insanity! So infinity doesn't even actually serve the purpose you think it does! It won't end the large number competition.
Some of the more knowlegable here will now bring up absolute infinity, the supposed largest infinity. But one fundamental problem with it is that it's so large that it leads to a paradox, and the other is, that even if we granted it was a number of some sort, it would violate the golden rule of googology. So again, we can not invoke the infinite, or else we invoke the death of googology proper! Suffice it to say that ALL of these different kind of infinities are not allowed in the game. All we want is an honest-to-goodness finite number.
(psst... in case you are interested in learning more about the infinite, no worries. In googology we also study infinite numbers, albeit in an indirect way, so you really lose nothing by playing fair here)
II. (Avaritia) Invocation of the "Last Number"
Alright. So as weird as it seems sometimes people try to come up with a last number to solve this "dilemma" called googology once and for all. First and foremost this is distinct from calling out "infinity". People who violate this principle are already working under the assumption that "infinity" isn't a number, so it's not the right answer to finding the "largest number". But rather than just admit there is no such thing as a "largest finite number", these people simply declare that there IS, and that THAT is there number.
It's difficult to tell whether this is trolling or it's sincere. If it's trolling this also kind of falls under the third sin of touting nonsense, but if it's sincere it might be a form of the fourth sin of going meta. One can argue it constitutes a lot of things, but it's different than the other rules as well. This is about a fundamental violation of mathematics as well as googology. It's a stubborn "wish" for there to be a final number so that there is an easy way out of the game. I believe the idea is possibly born out of desperation. It also may be born out of confusion. Some people seem either incapable or unwilling to understand what the infinite really means. This may seem incredible to some, but there really are people who seem to genuinely believe there is a "largest number". The argument usually boils down to something along the lines of "comprehension". What I want to emphasize here is that numbers exist whether we can understand them or not. Why should some massive mysterious number be any more or less real than say three. Because we know 3? But how can the existence of something depend solely on whether we know about it? In this way I'm definitely a platonist. Googology lends itself to platonism. We will be encountering numbers so large that you may pinch yourself to see if your still living in reality. It can be disorienting. So disorienting in fact, that I think some people outright reject it. NO numbers can't be THAT BIG! They must be fictions of the mind! There is a largest finite number. These are what are known as the ultra-finitist's. They are rare but very fanatical. They are convinced they are right. But Ultra-finitism stands against everything that googology represents. Googology is about doing away with artificial "boundaries" to explore the world of numbers in the spirit of exploration and mystery. But in order for that to make sense, the numbers, all the numbers, have to already be out there for us to find.
Suffice it to say, if you try to come up with some convoluted way to "arrive" at the last number, then your breaking the rules and not playing googology anymore. Remember, every number, no matter how large, is never, ever, the largest, period. That kind of open-endedness can make anyone feel uneasy at times, but if you want to be a googologist, persevere! Face the dragon head on and embrace the never-ending. I promise the rewards will be plenty!
III. (Superbia) Touting Nonsense
What exactly is touting nonsense? To "tout" means to basically solicit aggressively for the purposes of making a sale. In this case the person who is "touting nonsense" is trying to convince you that it isn't nonsense at all! I'll explain the motivations for this and the forms it can take, but let's first clear up an underlying assumption that leads to this violation: the assumption that googology is somehow nonsensical.
It may seem that since googology is, at it's heart, a frivolous endevour, and admittedly so, that therefore there is no need to take it "seriously". That it's "all nonsense anyway". However there is a difference between whether something is "practical" or not and whether it is "sensible" or not. For something to be practical it only need to appeal to ends of practical utility, ie. staying alive for example. But just because something is not towards some practical end does not mean that it is nonsense. Being sensible, in the sense of being logical and coherent is not directly related to practicality, it's just that being logical and coherent is often conducive to bringing about practical ends. But one can think logically and coherently about things which are entirely abstract and very far removed from practical utility. Googology almost inevitably heads in that direction because, if we were to consider the practical utility of our subject it would only inhibit our furtherance of the prime objective of googology. We must toss notions of usefulness asunder if we are going to persue the project of googology meaningfully. Because it's really a question about what we can do, about what is possible. It's not about why we do them. Basically if we don't take googology seriously, if we don't take the project on with some sense of inner purpose, then we probably won't get very far. Worse yet, we run the risk of being sloppy. Sloppy to the point of literally spouting gibberish, and if we aren't talking about numbers, then what are we talking about? What is the point then? It may seem odd to speak of googology in these terms when it is so easy to dismiss as a "pointless" endevour. But I think the important thing here is to understand that the very idea of something being "pointless" is a relative concept. What people really mean when they say that is that it doesn't serve some preconcieved purpose, usu. practical utility. But what does it really mean for something to have a "point". It simply means that the activity has a certain aim. An activity is only pointless if it does not aspire to fulfill it's alleged aim. Therefore googology is only pointless if we do not aspire to speak of numbers, but only to impress with nonsense!
And this brings me to what touting nonsense is really all about! It's a form of charlatanism. Basically someone will spout nonsense because they think ... no one will no any better. They use a blanket of false authority, and the principle that the "less people understand the more they are impressed". Unfortunately for the non-expert this is basically true. Without knowledge it can be hard to distinguish real googology from pure nonsense, and so it's very easy to be swindled by this. But the same tell tale signs in all forms of charlatanism can be seen here as well. You'll notice harshness towards any criticism, insulting your intelligence, implying that you only question it because your "ignorant". Simultaneously there will be a refusal to provide any proof of product. Details will be handwaved or claimed to be "too complex" to be understood, even when the definitions provided are clearly very simplistic and likely misguided. So basically touting nonsense is a kind of scam and confidence game where someone will simply claim that there number is so much larger than anything anyone has ever come up with because it diagonalizes over all nth dimensional mathematics, which is so big that only our "sage" (charlatan) can understand it!
A second motivation for touting nonsense is to lapoon the entire enterprise of googology all together:
" What poppycock! Here let me show that I could come up with this nonsense too! Now be impressed with my incredible googology! " the scoffer says.
What the scoffer probably doesn't realize is that what they are touting really IS nonsense, and what googologist's are saying actually isn't. So the scoffer instead of demonstrating the googology is silly actually only demonstrates his own ignorance of the subject.
Can one tout nonsense sincerely? Yes I suppose that is possible. There are some very weird ideas which their creators are adamant "make sense". But scrutinize the idea just a bit ... and you discover it's all fuzzy thinking, just an elaborate mental mirage.
At it's heart, touting nonsense is about being ill-defined or not well-defined enough, to even clearly know what your talking about. In googology you must be precise when you define numbers. You have to be able to pin point precisely what number your talking about so that others can gauge and compare it to other numbers. Failure to do so, whether intentionally or unintentionally, makes googology impossible. If we can't even clearly express the idea for the number, how on earth are we suppose to compare it to other numbers?! If someone actively keeps making there number somehow "larger" everytime someone seems to finally pin down what they are talking about, then they are engaging in goalpost moving. This is definitely nonsense. Remember, the correct response of the challenge of googology is either to show your hand or fold. There is no bluffing in googology.
The lack of a clear definition can happen in a variety of ways, but I think 2 are so prevalent that they deserve mention of here.
One form is the litany of examples. If you can not provide a clear way for any other reasonably intelligent person to always determine how your number or notation works for any given situation ... if they always have to ask you to tell them how to do things, then you are potentially at risk for tauting nonsense. This is not absolutely gauranteed though. The point of something being nonsense, isn't just about the idea being simply unclear, or needing to still be worked out. It's about the idea being fundamentally unworkable, about literally being nonsense that only makes sense when you squint your mental focus. If as soon as you bring the idea into focus you discover that there is some fatal flaw in the idea, then guess what, you were tauting nonsense all along. This is why it is paramount that you understand your own ideas clearly before you go presenting them to others. And if you really did understand your ideas you wouldn't have to go on a case by case basis. You'd be able to identify the principles at work for the general instance.
A litany of examples, is essentially a liability. How do you know that it all works? Can you demonstrate this to others? A litany of examples is usually only part of the testing phase for a new googological idea (more on that in a later part). But there is a serious danger here (explored more in the sixth sin and seventh sin) of going through "examples" so fast, that you fail to realize a crucial snag that only occurs between the examples! More on this later. For now this is merely a cautionary note that a litany of examples might lead to touting nonsense.
Another problem I'll call the mirage sequence. Here's how it usually works...
" Let the sequence be defined as ... 1, googolplex , Rayo's Number, ... hmm what's the 4th member of this sequence? It must be something really huge right? Well my number is the 1000th member of this sequence. It is impossible to beat this sequence! My number is far to big to be computed, it's the biggest number of all time! "
The key thing to understand here is that a list does not imply a sequence. To have a sequence you have to actually specify a rule by which you can always determine the nth member of that sequence. "AH! That's no fun!" some of you will be saying, but that is because you know that you don't possess the skill to make a very powerful sequence. So you wold rather just "pretend" you have such a sequence, by allowing everyone else's imagination to fill in the blanks for you. This is an illusion, a mirage. In fact, the best example I know of this is something I once read about called illusory space. The author claims it to be something which always beats any other sequence, because it just keeps diagonalizing over itself and becoming more powerful than the rule which establishes what the next term is. It sounds impressive, it even sounds promising, but try to formalize this concept! Try to actually specify what that rule is that makes the whole sequence. You can't! Because if you could it would just surpass this too! Don't be fooled by this sort of thing. It's a mirage. It is not productive. Commit to an actual sequence, and build on that sequence. That's how you'll make real progress. At it's heart the mirage sequence is just yet another attempt to trump all of googology. You can't do that anymore than you could fit all of mathematics in a jar.
One last thing that touting nonsense entails is all forms of circularity. For example...
ultimatillion = 10^ultimatillion
or Kedrillion = 47 gazoogol
and gazoogol = 846 kedrillion
Again, specify an actual number. This sort of stuff won't be recognized as anything but nonsense.
If you want to avoid the third sin, always make sure your definitions are clear and "well-defined". We will go more into what that means in later parts, but for now, we may informally define it as a definition that others can inspect and draw the same conclusions about. If there is any kind of debate about what you even mean or how it works then ... your very much in danger of touting nonsense. Correct the matter as soon as possible.
IV. (Ira) Going Meta
Although I have ranked this as the fourth sin, it is perhaps one of the most irrating of all sins! First, what does "meta" even mean? The word meta is a prefix used in english which literally means to go beyond. There is metaphysics which goes beyond the physical world, and there is metafiction, which is aware of itself! When we "go meta" we trascend one level of understanding and find ourselves at another. Going meta is actually a fundamental part of the googological enterprise. Despite being a sin you will be "going meta" a lot when doing googology. So why is it a sin? Because there is one meta you must never do ... you can't meta googology. This should be for obvious reasons at this point. What would it mean to meta googology? It would mean to somehow jump outside of all googology and then turn around and make something bigger. This can take a variety of forms, but on it's most basic level it's something of the form...
" My number is the smallest number not expressible in all googology"
There is nothing more annoying and aggravating than this. This is again an attempt to somehow "win" googology and come up with a "final" answer. Note only is this phrase entirely vague (violation of sins 3 and 7), but even if we were to take this at face value it's fundamentally self-contradictory. Why? Because if you expect your number to be recognized in googology, then this could not be the smallest number not expressible in googology. In order for this work your number would have to somehow "not be" part of googology?! But if that were so you can't claim to have won googology! @_@ Dizzy? Yeah, so don't do this sort of thing. It almost always leads to paradox.
Some other variations...
"My number is the largest number in all mathematics" (there is no largest number in mathematics!)
"My number is the largest number anyone could think of plus one" (then how did you think of it?!)
The oldest variation of the going meta is basically "whatever number you can think of +1". Now what is kind of odd about this, is that this is usually declared before the other person has declared a number. The idea here is that since the other persons number is well defined, then your number has to also be well defined and larger, therefore you win by default. But winning by default is antithetical to what googology is about. This is a form of piggybacking on someone elses hard work. As it turns out however there is a good reason to consider such metas as not being numbers at all! Why? Because a number needs to be something specifc and fixed at the moment of declaration. That is what it means to have a valid entry. What if the other person simply refuses to provide a number (as they should). Then what you just said becomes invalid. It needs a reference number in order to have meaning. The trick here is what I call a "dynamic reference". You try to have a number which changes based on something else. For example, say someone posted this as an entry on googology wiki...
" [Username]'s Number is equal to the sum of all numbers on this wiki plus 1"
We'll leave aside the obvious feedback loop that happens when this number is also a number on the wiki. A more basic problem is this: it's value, excluding itself, would depend on what the other numbers on the wiki are ... which is always in flux. Numbers are constantly being added, deleted, and candidate "numbers" are constantly being scrutinized and under dispute. Therefore even the question of what that sum is, is something debatable! And even that aside, even if we could determine the precise some of all well defined numbers on the wiki (aside for you entry) plus 1, the fact is you haven't actually specified a specific number! That's because the phrase ambiguously refers to any point in time, usu. now, in the wiki's history. So you never really had a specific number to begin with. So this is again just a form of goalpost moving so you can always say "this is the largest number on this wiki". Are you beginning to see how many problems this causes and how silly it is.
Now mind you going meta doesn't always have to be paradoxical, and it isn't excluded solely because of that. It's excluded because it's not conducive to the googological enterprise. It is actually possible to turn this paradoxical meta into a totally valid one. How? Simply do this:
"[Username]'s Number is the sum of all valid numbers on this wiki as of [Insert Specific Date] plus 1"
Just make sure you add this entry at any time after the specific date. Whalah! You have a totally valid entry! But guess what. If you did so there is a good chance that sooner or later this entry will be obsolete anyway, because now you have given it a fixed value. The really funny thing is that all one would have to do to beat this number is simply do 2*the largest number on the wiki as of [Insert Specific Date] and it would be larger than your number, and without even referencing your number!
You see the problem with this isn't just that it's paradoxical, it isn't just that it's a game-breaker. It's also a very naive approach. All you've demonstrated is that you understand what a sum is. Such an entry does nothing to expand upon googology. Such expressions, which I call vapid forms, can never do more than simply take the maximum of whatever they are attempting to meta. But in truth they don't contribute anything new to what they meta. Add 1, take the sum, multiply, raise one to the power of another ... w/e. None of this is new to googology, and so you aren't really working sincerely towards the aim of googology, which is to grow in knowledge and add that to the ever growing body of knowledge that is googology. With that being said we move on to ...
V. (Invidia) Standing on the Shoulders of Giants
This is closely related to going meta, but here we don't usually strictly run into contradictions. This has more to do with naive extensions, and salad numbers. More on that in a later part. The idea of "dwarves standing on the shoulders of giants" is an old metaphor that suggests the idea of new successes being built on top of older successes. This can be seen in both the enterprises of science and mathematics. It is also true in googology. We are indebted to those who have gone before us, who have paved, at least some of the way forward; and it is up to us to pick up where they left off to continue along the infinite trail of numbers. So once again we will be doing a lot of this. Why call it a sin then? Because it's how you go about it that is important. There is a dark side to this old saying ... it suggests that the dwarves are just usurpers. Undeserving whelps who benefit from the work of those who proceeded them but contribute little themselves. And it is here that we must be vigilant.
Now let me clarify ... it's okay to stand on the shoulders of giants. But googology is an enterprise very different than other enterprises. For example you can't simply add 1 to the previous efforts and expect that to count for anything. If you are a dwarf standing on the shoulders of giants, it means that you have done no more than replicated the results of your forebearers, even if you should take the largest number they came up with and extend upon it. To truly avoid this sin you must be a giant that stands on giant shoulders, you must in fact be a giant so much larger than the giant on which you stand that it is the dwarf by comparison! o_0;
When you have done this ... you have transcended your forebearers and contributed something actually new to the enterprise! Sound like a tall order? It is. But that's an important part of what googology is about. But if your content to be the little dwarf standing on the shoulders of giants you aren't going to get very far.
Okay, so metaphors aside, what does this mean in practical terms? Well a common thing that fledging googologists like to do is jump straight to the biggest number they can find. That is, the biggest number someone else came up with already. Understandable. The key thing here is that they don't try to understand that number, they just know what it's called (it's name). Then what do they do? Stuff like this...
Rayo's Number +1
Rayo^N(10^100) where N is Rayo's Number
Again, this isn't actually the most egregious part of the sin. It's okay to experiment like this when your starting out. It is practice. But you shouldn't end it here. The key impediment you actually have to further progress isn't that you can't keep extending the number ... it's that you don't understand the Rayo function to begin with! How then, how then can you expect to really trump it? Wait isn't Rayo's Number+1 bigger? Yes. But it's not big enough. It's not big enough to really get anyone to notice. But why? Doesn't Rayo's Number get alot of recognition? Sure. But Rayo didn't get there by repeatedly adding 1s ya'know :P
Now even so, this still isn't a sin. It's okay to not know everything. At the outset no one expects you to know how Rayo's Number, or TREE(3) or even Graham's Number completely works. But you shouldn't be satisified with simply borrowing these numbers and then extending them. You should be seeking to understand them for yourselves. Then before you know it you won't even need Graham's Number. You'll be making larger numbers from scratch in your sleep! And once that happens you'll quickly find that Graham's Number shrinks to being completely obsolete when you start working with an even larger class of numbers. It will take work to be able to say the same thing for the next major googolism, and the next, but this is how you'll actually make progress in googology. Borrowing is uselss unless you can understand what your borrowing. But if you understand then there is no longer a need to borrow. Eventually the thing that once seems like a giant to stand on, will be become just a dwarf to crush! You'll find you don't need to stand on it's shoulders anymore! Making you dizzy?! Welcome to the weird world of googology.
Oddly enough it's by becoming a giant yourself that you will do the most honor to those who have come before you, because you will indeed be taking the enterprise to the next level.
But this is something which is very counter-intuitive to most beginners. For whatever reason, people come in with the preconceived notion, that if you want the largest number, you just grab it off the shelf and add something to it.
But the real sin, isn't so much grabbing the number off the shelf. The real problem is the false pride that comes with it.
Let's just say for argument sake we consider the typical case. Someone takes the number that everyone seems to be touting as the largest currently (typically Rayo's Number, but at one point the Xi-function was touted as the best, much to the determinant of some hapless giant shoulderers). They make a larger number. It maybe trivially larger, or it might be a lot larger ... in ordinary terms. But in googological terms it hasn't even come close to clearing the large numbers "local" neighborhood. In otherwords, to seasoned googologist's it's going to look like your numbers are standing still. And then comes the boast "I have come up with the largest number in the world!". But does it really count? Do you understand how the initial number you borrowed works? Can you explain it? Do you know how to improve it? Or is everything you did to make the number bigger already a construction that was already utilized in the construction of said number. If so it will appear to most googologist as if your a teeny tiny dwarf standing on the shoulders of an enormous giant. Forgive us if we are less than impressed by your exclaimations of being king of the world. Where you'd you be, we'll think, if you didn't have that giant on which to stand on.
And this is why Rayo's Number is considered the biggest even though, technically, you can find "slightly" larger variations on it all over the internet. They don't really contribute anything new so they are just ... meh, forgettable. Why is Rayo's Number remembered? Because it was a giant that towered over a entire world of giants when it was first devised. Because it doesn't just beat the other competition, it completely trounces it.
This might seem unfair. But consider the alternative? What if adding 1 was all it took? Well we wouldn't get very far would we! And what would be the incentive to do better than add 1, if someone else was just going to take all the credit when they inevitably, added one to your number. The end result: the game would end. People would get bored, they'd go home, and that would be the end of it.
So to preserve the spirit of googology we must give credit where credit is due ( Thanks Rayo, for an awesome number! ) As long as you do that you won't be commiting this sin. Strive to learn from your borrowing, and seek to build upon what you know. Grabbing numbers from other sources and mashing them together is the lowest form of googology.
VI. (Gula) Going Ballistic
Going "ballistic" is just a way to say going "crazy", going off the "deep end". Again it might seem, to the casual observer, that all googology is about "going ballistic", but I'd say that googology, at least as I practice it, is a kind of controlled madness. Mind you that is more a matter of taste and style. One can also go at googology in a very meticulous and serious manner. There are some that feel that the whimsical aspect, such as coming up with the crazy names, is superfluous. I myself have sometimes expressed that it is not the most essential aspect. None the less it's become more or less part of the tradition. And with the sheer size of the numbers we work with it's no wonder we haven't all gone nuts! But it's one thing to be whimsical and quite another to be actually insane. The key thing here is that all of the "crazy off the wall" stuff in googology is kind of like the icing on the cake. It adds flare, but without the cake itself, it's all just a lot of fluff. Even while making it seem to most observers that your "barking mad", behind the scenes you still must always ensure that the mathematics, the foundation of your googology, is indeed sound.
But sometimes people take the superficial silliness as something much more deep seated. As something fundamental to googology. But one must be careful not to get too whimsical. This invites sloppy thinking, sloppy mathematics, and ultimately sloppy googology. You must make sure that under all the pomp that your googology is indeed producing real honest-to-goodness numbers. Don't lose site of the primary objective! You notation/system w/e must actually work, be well defined, and produce unambiguous numbers.
But going ballistic encompasses more than taking your googology too lightly. There is another drive that leads to sloppiness in googology and it's pretty basic to googology. There is a certain "giddiness", a "hysteria" that large numbers can produce. As a fledging googologist you are likely to get your first experiences of this very soon as you delve into a world far vaster than your imagination ever previously dared to imagine. After the initial "fear" wears off, a "hunger" can emerge. A primal greed for larger and larger and larger numbers! You'll also begin to see patterns ... very very complex patterns ... in your minds eye, long before you can even properly process what you are even thinking about. As a young googologist, learn to ride the wave, but also learn when to hemm it in. There is a time for ferverish dreams of grandier and there is a time to cool off, reflect, process, analyze, and then build. But all too often fledging googologist are tempted by the ferverish dream, and they don't slow down, cool down, or reflect. They blast on ahead to sights unseen, adrenline rushing, thoughts racing, chasing after big numbers, and feeling like they are on top of the world. You will probably experience this a few times in your "career" as a googologist, and that's okay. It's happened to me before. It will be like that a lot in the beginning. As your mind adjusts to the bigness you naturally cool down, but don't be surprised if you have a sleepless night. It's a normal reaction to radically new ideas. I had a similar experience the first time I grasped what was meant by 4-th dimensional space. It was a revelation. Googology will be like that. A new way to look at the world. And this is a good thing...
But don't let your imagination run away with you! This is when your are in danger of "going ballistic". It has been seen many times in this wiki. Someone describes a notation. It starts out sane enough. The rules are completely worked out. Everything is great. Not to powerful, but it works. Then the notation is taken a little further ... the rule set has some ambiguities. Time to iron them out? Nope, not for this googologist ... now it's a mad rush to the next notation and the next and the next ... and soon any attempt to explain it is abandoned. Examples are all that's given, and the vague assurance (or vague wish really) that somehow, someway, this still makes sense ... and what lies beyond that ... and that ... and that ... eventually you begin to "hallucinate", mathatically hallucinate that is. You start to see patterns that aren't there, you make grandiose assumptions, extrapolate beyond what can really be extrapolated ... and congradulations ... you have now gone completely ballistic!
Is it always bad to go ballistic? Well it depends, how ballistic are we talking? Sometimes going ballistic can lead to important future insights. But that only works if you temper it with patience to go back and do the hard work of formalizing your idea. If you insist of flailing your arms in the air and screaming like a mad man that your notation is somehow the most powerful, but refuse to make it comprehensible to anyone besides yourself (and really do you even understand what your shouting about?) don't expect anyone else to be impressed.
Whimsy is welcome in the googology community, encouraged even. You can express your excitement about large numbers, we get it, it's cool. But don't allow the hysteria of it all to keep you from thinking straight. Remember the last point about mirage sequences. Well you'll find that there are a lot of fuzzy ideas out there just waiting to pop into your head. If your going to be successful at googology you need to learn how to recognize the good ideas from the bad ones. An even a good idea won't be much good unless you calm down and do the somewhat less fun grunt work of summarizing your results and making a coherent system out of them.
I'm fond of analogies, so here's a good one. I sometimes like to think of googologist's as googolsmiths. We forge "googolism's", not from fire and metal, but out of pure imagination. But going ballisitic is the equivalent of a mad metal worker who has this idea for a fabulous sword, A sword to end all swords! In his fervorish imagination the sword just keeps getting more and more insane, and he hammers away like a madman, convinced he's a genius, convinced it will be the greatest thing ever concieved. But he never quenches the fire, he never cools the metal, because it's never "enough". Instead he keeps the fire hot and burning and the metal glowing white and malliable. As long as he never stops hammering, never goes to "finish" his masterpiece, he can maintain his delusion that he is on to the world's most wonderous thing! This is what going ballistic in googology is like! But remember until you quench your steel, you do not have a marvelous work of metal, just a glowing amber of an idea. Reigning in the madness means, putting the finishing touches on your work, and cooling off the metal. Don't worry, this sword does not have to be your last. You will make another one later, a better one.
So what's a good example of what going ballistic looks like? Cookiefonster suggested this might be something like a croutonillion, a notorious "super-salad-number". But going ballistic isn't so much about salad numbers and naive extensions. That was already covered basically in standing on the shoulders of giants. No going ballistic is really the opposite of that. It's not a naive extension, is a huge extension, following by a mind boggling extension, following by an incomprehensible extension, followed by ... it's about racing on a head before you even understand the notation a few notations back. I'll be a good sport here an give one of my own notations as an example. At the end of my Hyper-Extended Cascading-E Notation, I create in rapid succession a series of increasingly unwieldy and unspecified notations culminating in something I called Solidus-Extended Cascading-E Notation. Rules went out the window, a litany of examples flooded the gates, and I ignored ambiguities as if they weren't there and plowed on ahead! The result is that the apotheosis of my system ends in something bordering on madness. But this was more a case of deliberately going ballistic. I warn several times in the development that if we try to move forward ... without an understanding of the underlying principles at work, then the notation will only develop in an ad hoc way and will eventually become untenable. And that is exactly what happened. I pushed the primitive approach as far as it could go and it eventually broke down. But all is not lost. When this happens what you should do is TURN BACK, carefully retrace your steps, and start again where you last made sense. Build a better and a stronger path forward. You will be doing yourself a much bigger favor in the long run. People will be much more impressed with a polished sound result, even if it's "smaller", then a supposedly "larger" result that is on such shaky grounds that it might just topple at any minute! (In fact as Vel is sure to point out, if it's not well defined it's not "large" at all. It's nothing at all! ). Chris Bird, another esteemed googologist, had this to say on the art of building googology: "Googology is like chess. Sometimes you have to plan your moves ahead of time". Unlike chess though, in googology you get to take some of your moves back. Sometimes THIS is a better move that to plow on ahead with a poor foundation...
and with that we finally reach our last sin ...
VII. (Acedia) Lack of Specificity and/or Unverifiability
This sin covers a lot of vague and undesirable things in googology, but the main feature of it is that it involves something which impedes the ability to percisely pin point the number your talking about.
One very clear sub-heading of the sin is the use of physical constants. Your googolism, should never, I repeat, should never include a reference to a physical measurement. What we want is a number defined purely in mathematical terms, terms that don't change are not subject to debate, and can be directly scrutinized by all interested parties. Let's say your number is "All the grains of sand in the sahara raised to the power of all the dust on the surface of the moon". Now informally, we can definitely provide upper-bounds to a number like this. In googology there is always ample room for over estimation. We could take the entire volume of the solar system measured in plank volumes and raise it to it's own power and be sure that that would be a much bigger number. But the problem is that we can never pin down such a number precisely, we could never directly verify tight bounds for it's value, because guess what ... your definition ... is the sum total of all the information contained in the sahara and the surface of the moon! No, seriously, think about it. Think of yourself as a standards bearer for your own googolism. How would we really verify the exact size of your number. Well, first you'd have to precisely define what you mean by the "sahara". Where exactly does the sahara begin, and where does it end. Then you'd need to determine what counts as a grain of sand. Aren't there particles so small you can't see them? Are they still grains of sand? If you do say there grains of sand as long as they are of the same composition, then guess what, now we have to comb the sahara with an electron microscope! And don't even get me started on moon dust. How in heck is anyone here suppose to verify that? We can't exactly book a ticket to the moon at the moment, ya'know! And that doesn't even address problems about the fact that in the real world sands would be drifting in and out of the sahara, however you define it. Suppose your going to need to time stamp it as well. But if you do that how is anyone suppose to verify how many grains of sand were in the sahara at some random date in the past! I'm not even kidding here. The problem is the only real standard by which we could say such numbers are defined is by the actual physical artifacts you are referring to. In fact it is this innate belief in us that there is a precise answer even though we can't verify it, that makes us prone to adopt such definitions. This isn't even all that new! The bible uses "the grains sand in the ocean" and the "all the stars in the heavens" as ways to describe really big numbers. Early man would have relied to what he could point to to get large numbers because at the time he lacked both the notation to describe such numbers clearly, and the mechanical means to verify precise counts of such huge quantities. Hence all the waxing philosophical about the "stars that can not be counted for number", impiously inferring that because a finite number is beyond our ability to count that it is somehow comparable to the infinite!
In short, this sort of tack is one born of desperation for a quick and easy way to summon a really big number! No math, no notation needed. Just point to something in the physical world you THINK is really big, and whalah! But there are two main reasons why this is taboo in googology (1) Nobody is going to be able to precisely verify your number and such a definition is an extremely inconvenient standard bearer and (2) It doesn't advance the knowledge of googology. It's just like standing on the shoulders of giants, except you look to the physical world to supply you would the ready made numbers. As it turns out, and we will see later, even if this were allowed you'd always get trounced if you relied solely on large numbers in the real world because numbers in googology turn out to be so unbelievably more massive!
When it comes to bearing a standard, a definition which requires a physical artifact is highly inprecise, inconvenient for independent verification, and subject to all manner of variations. Consider the standard bearer for the kilogram. Unlike other physical units, the kilogram standard is kept by a chunk of metal known as Le Grand K defined as precisely 1 kilogram in mass. Apparently despite being kept under 3 bell jars alone with 6 other "identical" sister kilogram standards in a controlled environment there was a variation of 0.000005% (see the video here for more details on this topic). Point being, that if even when scientists have a physical standard as a definition and they can't keep it perfectly straight, how much hope do we have of using the grains of sand in the sahara as a perfect standard. On the other hand, if you define a number with mathematics, that constant is preserved as long as you preserve the information composing it's definition.
But that is only one thing I want to address under this sin. Another thing that this sin encompasses are what are known as pseudo-quantifiers. We have a lot of these in english (I just used one in this sentence!). For example , a few, several, some, a lot , many , and the unusual zillion. All these words have in common that they are meant to be treated like numbers, but they are not meant to refer to any number in particular. Rather they only suggest a rough order of magnitude. A few is not many, several is probably in the ballpark of seven, give or take a few, some is not most , and probably less than many, and a zillion is a whole hellva lot! But in googology we might as well be talking about ducks, because we don't deal with pseudo-quantifiers. If we did we'd all just be arguing about whose pseudo-quantifier was suppose to represent the largest of numbers ... roughly speaking. That invites opinion, and as I pointed out at the outset, the larger of two numbers should never be a mere matter of opinion. So vague not-so-exact quantities are not allowed. Mind you we sometimes use vague pseudo-quantifiers ourselves! For example "googologically large" is itself a pseudo-quantifier. One can also use the concept of on the order of such as such ordinal in FGH as a way to vaguely quantify the size of numbers we are talking about when we either don't have a specific one in mind, or the actual value is not known. But these are only as a matter of speaking. They should never be considered themselves as entries in a large number competition. You will find that ... eventually the numbers get so big that we really can afford to be very imprecise. None the less, on this point we must be pedantic to keep the game stable. Everyone is required to provide precise definitions for there numbers, down to the integer. It's the only way we can properly conduct things.
On the subject of unverifiability I'd also like to point out this is why we should reject things in googology like "The largest number that can be concieved of by the human mind", or "the size of the actual universe", or "The largest number anyone will ever think of in the next billion years". In each of these instances we have no way to actually carry out an investigation into these things in any meaningful way, and that's assuming they are even meaningful in the first place!
So there you have it. These are the seven sins you should avoid like the plague in googology. They won't make you look smart, they won't help you become a better googologist, and they will probably give you a bad reputation. So avoid them. Instead focus on good practice. In a later part we will consider what good practice involves, and the incredible results they lead to. Before we get to that we have one more major thing to discuss before talking about googolsmithing. See you in Part III ... (Coming Soon)