large_numbers
1.1.2
The Allure of Large Numbers
"We don't mind where the question leads us, we know nowhere is better than everywhere, we have only one answer: we don't care " -- Tim Exile , Family Galaxy (Lyrics)
What do you consider to be a large number? Is a hundred large, or just moderately sized? How about a thousand, or ten thousand, or a hundred thousand! Surely a million is a large number, but if that is so, than a billion even more so, and a trillion even more so than that!!
That's all good and well as far as "large numbers" go, but just how large can we get? What if we are not satisfied with a number being merely large but want an example of a number which is especially large, astoundingly large, mind-bogglingly large. What then?
Perhaps a hundred thousand million billion trillion. Wow! That's got to humongous, right? But is that the best we can do?
What's the largest number you can think of? Come on, don't hold back. I'll give you some time ...
What! you've got it already?! That was fast. Whatever number you picked I'm sure it's mind-bogglingly unfathomably absurdly vast ... but I just have to ask, exactly how many digits does it have? What! You can answer that question! Sorry, then it's not vast enough! If you can't say how many digits it is, kudos to you, but exactly how many terms are in your power tower ... oh, sorry, if you got an answer to that question it's still not vast enough. If neither of these questions can be answered than congratulations, you have already imagined a number much larger than the majority of humanity ever will.
But regardless of how vast the number you have chosen, it's still not vast enough! For savvy readers you will undoubtedly expect me now to wax philosophical about it's insignificance compare to the infinite, ... but let's not get ahead of ourselves here!
The fact of the matter is you didn't it really answer the question! If you took 5 seconds to answer the question, no matter how large the number was or how intelligent or knowledgeable you are, can you really honestly say that's the largest number you can think of, or is it just provisionally what you could come up with off the top of your head. Even the most modest amongst us would have to announce "of coarse I can do much better", and then ramble off some even more unfathomable unfathomable. But then the number you thought of could not possibly be the largest you could think of. Furthermore even if you spent five minutes, five months, or fifty years, if you're still living and breathing there is nothing preventing you from adding, multiplying, squaring, raising it to some power, or otherwise improving your number. So what then can we say about the largest number any one person can "think of"?
Perhaps you've heard of the googol and googolplex, but what are those numbers and exactly how large are they anyway! And even if we didn't know what those numbers were couldn't we make larger ones simply by adding or multiplying them with other numbers too?
Then of coarse there are larger but even more obscure numbers like Skewes' Number, Second Skewes' Number, Mega, Megiston, Moser, Graham's Number, and many many beyond!!!
Is that the limit of what professional mathematicians have come up with? What's the limit of what they could come up with? What's the largest number ever recorded in human history? What's the largest number humanity will ever devise? What's the largest number humanity can ever devise ...
If you mind is swimming now that's a good sign. It means that you've got some sense of the sheer unfathomableness and unanswerableness of such questions. It's one thing to ask how many grains of sand are in all the beaches on the world, or even to ask how many grains of sand would fit into the observable universe, but it's another thing entirely to ask just how many grains of sand we could imagine. Even Archimedes couldn't answer that one, though he gave us a small glimpse.
Our ability to imagine numbers seems truly boundless, as near to an actual infinity as were ever probably going to get; Certainly far far more boundless than the boundaries of anything known to exist.
Where does this incredible ability come from? Why are we able to understand such numbers at all? Why are they so out of step with our surroundings, so beyond anything we could call a "human scale" of things? And why have human beings being so fascinated by them through the ages? And come to think of it, what do large numbers have to do with us anyway?! What possible relevance could they have to our very small and mundane plane of existence! In short, why should we take up large numbers as a subject matter?
Ironically, it is mathematicians themselves who are sometimes the most dispassionate and disinterested with "large numbers", but when this is so it's usually because they find even more amazingly vast realms of abstract truths beyond the large numbers. So really even the scoffers are actually unknowingly the biggest fans :)
Many kids go through a phase at some point of being fascinated with large numbers. Usually these are kids who have a pretty good grasp at arithmetic, but sometimes not! Surely there was a time in your young life in which you thought a hundred was a pretty big number and tried to grasp it.
Perhaps mathematics and numbers don't appeal to you. But have you never been blown away by the sight of a mountain, or the expanse of clouds during the day, or the expanse of stars during the night. You may have been struck by a sense of unfathomable vastness. Your own eyes can not entirely tell you the size of what your looking at in these instances and you might just stare all the more intently trying to understand what your seeing.
I believe that human beings are naturally drawn to the vast, the tremendous and the unfathomable. We are a species that looks up, and in doing so we find ourselves humbled by the world and the universe. But in doing so we also elevate ourselves, for it is us that "sees", us that is "humbled", and us that "yearns" for more. It is ultimately a desire to elevate ourselves that makes us wonder about our world and our place in it. I don't know if other species know of this sense of wonder, but I believe that it is an important characteristic of self-awareness.
I believe the same thing which is at the root of the religious experience, or the sense of wonder at the scale of our world or universe, is also at the root of our fascination with large numbers. Is it any surprise that religious writings have often appealed to large numbers and infinity to describe God? A day for God is a thousand years, there are a hundred million angels in heaven, and God exists for time indefinite to time indefinite (a euphemism for potential infinity).
But large numbers are more fundamental than that. They represent pure and visceral force. We are simply blown away by them (when earnestly mediated upon) by sheer power. It's that simple. We needn't see anything deeper into it than that, though being human of coarse we will tend to.
In short, studying large numbers is not about achieving some end, it is most emphatically about the experience, the journey.
I first came to the subject of large numbers as a kid attempting to find the exact value of infinity. At the time then, large numbers could be nothing more than way-stations on the railroad to infinity, and yet the large numbers were truly fascinating in and of themselves. I can honestly say that the journey itself was rewarding even if I never "arrived at the destination".
Now that I'm older however, I no longer think of infinity so much as a place as a direction. If infinity is by definition that which never ends, how can infinity be thought of as some kind of place on the number line, or some kind of definite totality. That's the whole point, it's never ending, so why speak of completed infinities. If it could be reached, what then? where else would there be to go? Rather than look at infinity as an insurmountable obstacle, it can be seen as an ever-flowing opportunity! A blessing rather than a curse. It's better to see that with large numbers we can abandon our need for closure, completion, and ending, and embrace our desire for freedom, exploration, and boundlessness. Seen this way the journey to find larger and larger numbers can be seen as absolutely liberating. Toss aside your prohibitions, throw caution to the wind, forget the practical and leave it far far behind, think BIG, really really BIG, forget endings and goals that always seem out of reach, and experience the present, the here and now. Forget the vagaries of the infinite and the indefinitely large and contemplate the concrete, the real, the definite, and yet still radically large.
Ultimately the only justification I can give for the study of large numbers is pure unadulterated human curiosity. Mathematicians are quick to try and come up with justifications for the study of large numbers and especially large infinities, but maybe it wouldn't hurt to fess up: it's just natural curiosity, prior to and outside of any benefits the investigations into large numbers, infinity, and mathematics might lead to.
If what I've said so far has not struck a chord, read no further. From here on out I will be discussing large numbers as exhaustively and discursively as I can. But if your curious, even by a little, then read on. You may well be surprised with what you learn here. You may have already come here with preconceived notions of what is meant by "large number", especially if you've never explored this subject before. If that is so there is a good chance that your notion of "large number" and even infinity will be shattered and will have to be expanded to fit into a new paradigm, ... repeatedly. Along the way you may find a greater appreciation for mathematics and perhaps you'll even learn some practical things about it (I will be reviewing most of elementary arithmetic for the development of the mathematical tools necessary for really large numbers). You'll also get a history lesson, seen from the eyes of intellectuals instead of war-mongering nations. But these are just fringe benefits. The focus here is primarily the development of large numbers. You can read more in depth information on mathematics and history elsewhere, but you'd be hard pressed to find the focus brought to bare on the subject of large numbers elsewhere. There is actually only a handful of really great sites on large numbers, such as Robert Munafo's well-known site, and the Big-Psi project (See Links in the Appendix if your interested), but these are still the exceptions not the rule.
So if you've decided to keep reading allow me now to leave no ambiguity as to our purpose and motivation here. Our motivations for the study of large numbers are:
To satisfy our own curiosity
To explore the answer to the question "What is the largest number we can think of?" through concrete demonstration rather than by abstract speculation, without any false expectation of arriving at some final and definitive numeric answer
The second motivation may seem rather overstated. There is a reason for this however. Most discussions of large numbers are surprisingly brief, usually only a few pages, as if the discussion quickly runs out of steam or the imagination suddenly implodes. There are many common "exits" to the discussion. My statement is meant to have enough qualifications so as not to be susceptible to the usual objections. Here are just some examples of typical exit strategies:
After a brief discussion of large numbers they are all trumped by "the largest number of them all", infinity. This is effectively the end of the conversation and speculation about the large yet finite.
Reach a very high number with a high recognition status such as a googolplex or graham's Number and simply leave it at that. The experts know best. We dare not speak of greater numbers lest we offend them...
After discussing some very simple recursive functions (perhaps still primitive recursive) the conversation quickly switches to the non-computable, typically the busy beaver. In short, it's a mad dash to the punch line of the large-but-not-infinite-number discussion. The impression this leaves for the laymen is pretty much thou shalt not understand these things, they are beyond your reckoning.
Variations on the berry-paradox, which hold that we can't know what the largest number we could know is because it leads to a paradox.
"There is no highest finite number". No really, people use this as a conversation cap more often than I care to recall. This is said so often you'd think it was revolutionary. In truth this is a quick way to get out of the "trap" of thinking about larger and larger numbers. The implication is simply that "there is no point" because you aren't going to reach the end. That, I will argue throughout my book, is completely missing the point.
"there is no largest number we can think of ... we can continue as long as we like", and after this statement of coarse we don't. This is pretty much a conversation killer too. Besides being not much more than a bald assertion (I will argue how there are considerable impediments to continuing indefinitely), there is the issue that just because we can continue unimpeded doesn't automatically make the effort worthless. If we could go out for a leisurely walk as long as we liked, does it mean it's not worth it to even step outside?
"You can always add 1" ... so let's quit while we're ahead. Basically a variation of "there is no largest number"
"God is so much bigger than anything we can imagine" ... as if it were impious to try. When this is a conversation cap it usually indicates that there was an ulterior motive in the discussion of large numbers to begin with.
"the question being asked has no answer". This is typically the answer given by professionals to any of the following questions: "What is the highest number?", "What is the highest named number?", "What is the highest number we can think of?", "What is the highest number known?", "What is the largest number ever defined?", etc. Typically what the questioner is looking for is simply a number much much larger than anything he/she has been previously aware of. This answer comes back as a dull empty response, even if it is factually true. It would be better, from the questioners point of view, not to answer the question being asked, but to answer the question vaguely being hinted at, namely, "can you tell me about a really big number?". Furthermore, it's not entirely clear that all of these questions have no answer. For example, if I ask what the "highest number known" is, and I define this as the largest finite number conceived by any modern human within the last five hundred thousand years, there must be a definite answer, or at least a slightly fuzzy answer, even if it's impractical or impossible for us to determine it. The "no answer" then simply becomes a cop out of sorts, a knee jerk response to any question attempting to elicit a precise numerical response.
I think that covers all the basics. You will find these to represent the majority of swiftly reached conclusions in large number discussions. I will make no such appeals however, which pretty much means there is nothing to terminate the conversation. This is a no holds barred discussion of very large finite numbers. All the stops have been pulled, and while I will inevitably have to stop the conversation somewhere rest assured that when I do I hope that I will have thoroughly answered the question "what are the largest finite numbers humans have actually contemplated?". That comparatively modest question I would think should at least have a general answer.
This brings us to our goals for the remainder of this site:
To expand our notion of "large number" to include as many (but not all) numbers beneath infinity as possible
To gain proficiency and insight into large numbers and working with them
To Develop the mathematical know-how to define very large numbers
To be familiarized with the history of large numbers and the latest developments in the 20th and 21st century
To provide a unified source of information on the subject of large numbers
To develop a narrative of large numbers through the history of thought
To provide my own contribution to the subject
To reach the highest numbers ever conceived of by man
Sound promising? Well then let me offer you these prescriptions before we begin. My site should not be treated as an all-or-nothing contract. Just because everything on this site is not immediately clear to you doesn't mean you could not gain anything from it. You don't need to understand everything, just focus on the parts you do understand and try to expand your knowledge from there. My site is not intended solely for a "professional audience", nor only for laymen, nor some level in between. Rather my site is a more like a stepping ladder (a textbook if you like) to bridge the gap between the novice and the expert. This is a very difficult task which is still being implemented (feedback on how well I'm bridging the gap is appreciated, so that I can improve the transitional material). In short, read this book as casually or seriously as you like and take away whatever you make of it.
I recommend reading things in order for the most part, skipping over inessentials as interest dictates, because I have tried to structure things as cumulatively as possible. Section I mostly lays the foundations for the subject and illustrates a parallel between early numeration systems and the later development of recursive functions, revealing how notation and the ability to describe larger and larger numbers is closely linked. Section II reveals how our universe are full of large numbers, and explores ways of extending ordinary number notation to both describe these numbers and numbers far far beyond. Section III takes the mathematical foundation established throughout the first two sections and runs with it, developing the idea of a recursive function. Section IV takes the idea of a recursive function and takes it to it's natural conclusion with super structures. Section III is probably one of the most important Sections as it forms the foundation for later material.
You might well ask why I just don't get to the flipping point and reveal what the largest number is? You may be tempted to skip to the last chapter of the book and find out. That's perfectly fine. However for many people the answer wouldn't make any sense anyway. If you are not familiar with the mathematics involved, all it will seem like is a lot of mumbo-jumbo wizardry. All sense of scale will be lost on you and you won't be able to fully appreciate just how large these numbers are. That's what the rest of the website is for, to provide foot holds on your climb up towards the infinite. And I do think that the journey is a large part of the fun. The world of large numbers is like a vast realm that you can explore, not just it's highest peaks, but even the spaces between the spaces can expand into mind-numbingly enormous vistas, whole universes to explore in every space imaginable! And by exploring it thoroughly you get only a greater and greater appreciation of just how awesomely transcendent it's highest peaks truly are!!
If your interested in that journey, then continue on to the next article ...