No dumb questions: What's the biggest number?

Today we'll address a question that has long occupied the minds of children, mathematicians, and child mathematicians: What's the biggest number?

Whereas the dialogue above illustrates a naive, blustering approach to the question, we will investigate the topic with rigor in the name of science and progress. Together, we'll discover the monstrosities that lie beyond the number of grains of sand on the earth, past the number of subatomic particles in the universe, and even beyond the number of possible arrangements of subatomic particles in the universe.

Ready? Let's begin.

The first thing we need to do is deal with the question of infinity. You'll see plenty of amateurs at local "biggest number" competitions invoke infinity like it's some kind of ace in the hole, but this gambit is easily punished by experience googologists.

Why? For one thing, there isn't one "infinity" in the same way there's only one 5. "Infinity" could refer to the number of rational numbers. Or, it could refer to the number of real numbers, which is infinitely larger than the former kind of infinity. You could also consider the number of possible subsets of real numbers and get an even bigger, infinitely larger infinity. Or maybe you're referring to transfinite numbers?

More to the point, though, we discard infinities because they behave differently than regular counting numbers like 1, 2, 3, etc. Regular old, salt-of-the-earth counting numbers have certain nice properties that you lose with infinities. For example:

Infinity is and has always been slightly problematic for mathematicians because it's so unintuitive. We literally have no intuition about infinity.

And finally, infinity is just a joykill. It's like naming Finchley Station on your first turn of Finchley Station (the game). Naming big numbers is a beautiful sport, and declaring your number to be the biggest by definition is blatantly anticompetitive.

With that out of the way, let's start counting.

If you've ever seen a movie where an ambitious protagonist wants to become the greatest {karate master, jedi, boxer}, you know that you don't get a training montage until you go back to basics. So, let's revisit the fundamentals.

Consider the number 1. We can add one to that and get 1+1, which represents a bigger number (namely, 2). Add another one and we get 1+1+1, or 3. We could keep adding one forever, which clues us into the fact that we can generalize this process. Instead of adding five ones, or ten ones, or a hundred ones, we can summarize the act of summing any number of ones together. Without any further ado, I'm pleased to present you with our first "fast"-growing function:

Okay, I know what you're thinking.

While that may be true, I wanted to start here because it illustrates the absolute simplest case of a surprisingly powerful strategy that will come up time and time again. I'll call it the Cookie Clicker Principle. It goes like this:

If you were on the internet in 2013, you'll remember spending anywhere between a few seconds to a few days of your life clicking a cookie. The hook of the game, though, was that you could "power up" by buying upgrades which (among other things) allowed you farm more cookies per click. This is roughly our guiding principle in the quest for bigger and bigger numbers.

In this case, we noticed that the + operation could be performed any number of times, so we generalized the process of repeated addition with a parameter, n. It just so happens that our "symbol" for repeated addition just is the number itself, so it doesn't look very impressive. The next example will be a little more illustrative.

So, with f(n) = n, we might notice that the function only spits out a single copy of n. Why not two copies? That would correspond to f'(n) = 2n. What about three copies of n? That gets us f''(n) = 3n. Whereas f(10) gave us a measly 10, f''(n) gets us 30. But of course, we could keep going, with 4n, 5n, and so on.

This brings us back to the Cookie Clicker Principle. The number of copies of n could be increased arbitrarily, so why not generalize that idea? We could use a new variable like m, but just to keep things simple, we'll be environmentally friendly and reuse the same n to produce our next fast-growing function:

Now, with that same input of 10, the g function gives us a crisp 100. That's pretty good!

But, we notice that we arbitrarily chose to multiply n by itself only once. Why not multiply n by itself twice? Or three times? That would correspond to g'(n) = n*n*n, g''(n) = n*n*n*n, etc. Well, well, well, we've walked straight into the Cookie Clicker Principle again. So, let's use a different notation to represent the idea of multiplying n by itself any number of times. In keeping with the pattern, we'll just reuse the same n.

Now we're cooking with gas. While f(10) gave us 10 and g(10) gave us 100, h(10) gives us ten billion. That's more than the number of people on earth! Not bad at all.

To illustrate how fast h grows, consider h(100).

That's a one followed by two hundred zeroes. To put that in perspective, there's a popular number out there called a googol which would be written as a one followed by a hundred zeroes. To put that in perspective, there's between 10^78 and 10^82 atoms in the observable universe, so a googol is at least a billion billion observable universes' worth of atoms. Our h(100), on the other hand, is worth a googol googols. That's pretty big.

But will we stop there? Absolutely not. In the grand scheme of things, what's two hundred zeroes? It's nothing. We notice that we only raised n to the power of itself once. Why not raise n to the power of n, and then raise n to that power? Or how about a stack of 4 ns? Or 5? Once again, the Cookie Clicker Principle applies.

Computer pioneer and TeX inventor Donald E. Knuth saw what was going on here, and in the '70s he came up with a new notation to generalize all of this generalization. It's called "up-arrow notation", and it's pretty powerful. Here's how it works:

One arrow (↑) means repeated multiplication, a.k.a. exponentiation. So 2↑3 means "multiply together three copies of two" which becomes 2*2*2 = 2^3 = 8.

Two arrows (↑↑) means repeated exponentiation. So, 2↑↑3 would represent (2^(2^2)), which is 16. The cute name for a stack of exponents like that is a "power tower".

Keeping with the pattern, three arrows means repeated double-arrowing, so 2↑↑↑3 means:

And so on.

I intentionally picked small numbers so that the results are manageable, but these examples don't give a sense of how insanely powerful the up-arrow notation is for expressing big numbers. Let's define j(n) = n↑↑n and consider j(10). This would correspond to a power tower ten 10s tall. It's hard to even find a way to describe how big that number is. We could say it's a one followed by 10↑↑9 zeroes, but that's not very helpful. We could alternatively take a stab at the top part of the stack: 10↑↑4 is a one followed by a hundred million googol zeroes. If you think you can picture that, you can't. There's little point in trying to put the whole number into perspective, because it's so much bigger that than we could ever fathom. And that's just with two arrows. What if we had used three? Or four? Or five? Wait a second...

Just when we thought we had generalized the whole thing, from repeated multiplication to repeated exponentiation to repeated repeated exponentiation, and so on, the Cookie Clicker Principle reappears!

To "automate" the process of up-arrowing, we can use a shorthand to express how many arrows there are, like (↑^4) to represent four up-arrows. Instead of "hard-coding" a specific number of up-arrows, we can "pull it out" as a parameter, just like all the previous times. This would give us our next fast-growing function:

It goes without saying, but if you thought j(10) was big, consider k(10): 10↑↑↑↑↑↑↑↑↑↑10. At this point, I've basically run out of interesting commentary to give for numbers this big. It's just, like, big.

You may have lost your appetite, but there's still more to come.

By now, you should be convinced that there is no limit to how fast we can make a function grow. Each time we come up with a new operator to generalize the act of applying a previous operator, we can always generalize the number of times that the new operator is applied.

Mathematicians have devised many, many notations to describe functions that grow unspeakably faster than k. One example is Conway's chained-arrow notation. Without getting into the weeds of it, in the chained-arrow notation, a→b corresponds to regular exponentiation, a→b→n corresponds to a(↑^n)b, and a chain of four arrows or more is... just crazy. I won't give a full definition since it's not easy to summarize, but feel free to check out the definition on Wikipedia if you're curious.

We can generalize the up-arrow function in a different way, where the number of up-arrows is calculated recursively:

To put that into words: m(1) is already incomprehensibly, inexplicably large. m(2) goes ahead and sticks m(1) up-arrows between the 3s. Then, m(3) sticks that many up-arrows between the 3s, et cetera. If you repeated this process sixty four times, you would get Graham's number.

Graham's number arose when mathematician Ronald Graham was trying to find an upper bound on the solution to a graph theory problem, and it was the largest number ever used in a mathematical publication up to that point. To quote Wikipedia:

It's so dang big that even if you stuck a 9 in every Planck volume in the observable universe and interpreted the whole thing as a power tower (i.e. 9^(9^(9^...))), the result would still be smaller than Graham's number.

But in the grand scheme of things, what's a universe-scale power tower? There's still bigger fish to fry. For example, there's the TREE function, which grows so fast that TREE(3) is already unfathomably larger than Graham's number. Need we go on?

Actually, it turns out that we mortals are forbidden from constructing the fastest-growing functions. Why? To understand that, we need to understand the difference between computable and uncomputable functions.

In broad terms, a computable function is one that could be run by an algorithm (a little circular, I know). By "algorithm", we informally mean "a step-by-step process". Formally, that "step-by-step process" is usually a well-studied model of computation, the most famous example being the Turing machine. If you're not familiar with Turing machines, I always like to recommend this Quanta article. If you're in a "TL;DR" mood, then (a) I'm surprised you made it this far, and (b) if you just know the vibe of a Turing machine, you'll still probably get the gist of this section.

Let's consider a specific function: a function that adds two to an input. Even if you're not an algorithms person, you could probably guess that this function is super computable. In Python, we would just write:

The Turing machine would be similarly simple. I sketched it out below, in case you were curious:

When the Turing machine terminates, it will have left n + 2 1s on the tape.

A slightly more complicated function would be one that computes the nth prime. That is, p(1) = 2, p(2) = 3, p(3) = 5, p(10) = 29, and so on.

Considering how much mystery and intrigue surrounds the primes, you might guess that this function wouldn't be computable, but it is. In plain English, the algorithm could look like this:

It may be a "brute force" approach, but brute force is computable! The corresponding Turing machine for this pseudocode would be a beast, for sure, but it can be constructed, which is all that matters.

Now that we have a sense of what a computable function is, what would it mean for a function to be uncomputable? Just to set your expectations, most functions are uncomputable. There are a few especially notable examples, though. One has to do with the halting problem, which could be loosely stated like this:

In 1936, Alonzo Church and Alan Turing independently proved that it is impossible to write an algorithm for HALTS. Which is a bummer, because this function would be the most godlike superpower for mathematicians and computer scientists. I mean, so many great mathematical mysteries could be solved with this method: take the Goldbach Conjecture, which manages to come up in almost every one of my essays. The conjecture states that every even number greater than two can be expressed as the sum of two primes. There's currently no proof. But, if we had HALTS, then we could do the following: Write a program that counts up and stops if it ever finds an even number that breaks the rule. Then, pass its code to HALTS. If HALTS returns 0, then the conjecture is true, otherwise, it's false!

So, solving the halting problem is uncomputable, and as a result, anything that depends on cracking the halting problem is also uncomputable. This brings me to the Busy Beaver problem, which I described in more detail in my previous essay on unthinkable thoughts. To summarize quickly, we can define a function BB(n) that tells us the longest number of steps that a Turing machine with n states can run (with no input) before eventually halting. By definition, Turing machines that run forever are disqualified, which is where the halting problem shows up.

Currently, we know the value of BB(1) through BB(5). Although we can calculate BB for specific input values, we cannot write an algorithm that computes BB(n) in general, because that would violate the uncomputability of the halting problem.

And here's the real kicker: In 1961, it was proved that BB(n) will eventually grow faster than any computable function. In other words, if we can explain, in a step-by-step manner, how to compute the result of a fast-growing function, then by definition that function doesn't grow as fast as BB(n) (or countless other uncomputable functions).

In other other words, we can't fly too close to the sun.

I've pretty much finished talking shop, but there's another fascinating angle to this story, which is the individual known pseudonymously as Sbiis Saibian. While not the creator of the Googology wiki (the fandom page for people who enjoy hunting down large numbers), Saibian is one of the most prolific contributors. I was immediately enchanted by the seriousness with which Saibian treats the subject. In fact, my introduction to this entire topic was reading his Introduction to Googology page. Just to sample a few highlights:

And my personal favorite:

I was completely fascinated by this anonymous individual and his seemingly unbounded devotion to large numbers. To learn more about the subject, I read through his "online textbook" on large numbers. Saibian begins the first section with an introduction so steeped in religious awe that I hardly knew what to make of it:

First of all, starting with an ellipsis is such a baller move. "Gaping Yawn of Eternity" sounds like the name of an album I would have liked in middle school, back when I planned on naming my first album Pi Cubed. I have this picture of Saibian crafting massive number after massive number, feeling the burden of humanity on his shoulders as he tames the unspeakable horrors that lie beyond our meager world of billions and trillions.

Every fact I learn about Sbiis Saibian makes me even more curious about the kind of person he is. The most interesting tidbit I've discovered about Saibian is his Decimice. Saibian writes that, since childhood, he's had strong color associations with different digits. Wanting to give each digit a distinct personality, he designed and drew ten anthropomorphic mice to represent each of the ten digits, giving each one a unique color theme and even a list of hobbies. He adds:

But even here, in the realm of colored-pencil-drawn numerological mice, Saibian feels the need to define "a few important conventions" he used in creating the Decimice:

It may sound like I'm just making fun of Sbiis Saibian and his obsession with numbers. I will admit that his prose occasionally leans into a bombastic and/or angsty tone that makes it a little difficult to take seriously. But the truth is that, to an extent, I get it.

In my 6th grade math class, there was a poster directly next to my desk with the first hundred digits of pi. When I zoned out in class, I would memorize a few digits, and eventually I had the whole thing memorized. This lead me to read Wikipedia pages about pi and its relationship with prime numbers, which got me into number theory. In high school, I was reading about Russell's Paradox, Gödel's Incompleteness Theorem, and the Riemann Hypothesis. I didn't necessarily understand all of it, but it all felt important in a way I couldn't quite put into words.

I distinctly remember a moment in college when I became hyper-aware of the fact that humans' unique ability to conceptualize numbers as an abstract thing enabled us to describe the laws of physics and even communicate at the speed of light. In that moment, I felt a kind of existential terror. I looked around at everyday objects like lamps and laptops with a new sense of awe and disbelief. Somehow, the world kept turning.

And that's kind of the crux of this essay. The main reason I found Sbiis Saibian (and the entire subject of googology) so fascinating is that it makes me feel, simultaneously, two polar opposite reactions: "Who cares?" and "How could you not care?"

So... what are the practical implications of this? Sure, very big numbers are often useful, like the speed of light or the number of bits in a massive data warehouse. But once we start dealing with numbers like Graham's number, whose size absolutely dwarfs the scale of the universe, what's the point?

Some philosophers have taken this viewpoint to the extreme: to ultrafinitists, if something (e.g. a number) can't be actually constructed, then we're not justified in saying it exists. Since it doesn't seem possible to ever have Graham's number of anything, an ultrafinitist would argue that the number itself hasn't actually been "found" or "discovered". It's just as nonexistent as unicorns. Ultrafinitism takes issue with the circular definition of numbers in classical mathematics: at the lowest level, a number is the repeated application of adding one, so e.g. 3 = 1+1+1. But for numbers like Graham's number, you'd just say "Trust me, there's Graham's number of +1s". Ultrafinitists aren't a big fan of that circularity.

Pedantic? Maybe. But it illustrates the point that the physical, actual, as-it-relates-to-humans usefulness of numbers like Graham's number (or any of the unimaginably larger numbers googologists have cooked up) seems like basically zero.

Furthermore, we know that there's no pot of gold at the end of the rainbow, so to speak. Above every fast-growing function is another, equally pointless faster-growing function, and above that is an infinite number of uncomputable functions that will never be described. So who cares?

First, I'd like to refute the idea that "googology is pointless because there's very limited practical value". As Sbiis Saibian himself stated, googology is a craft. It's an art, it's a challenge. When George Mallory was asked why he wanted to climb Mount Everest, he reportedly said, "Because it's there." Well, googologists seek out bigger numbers because they're there. What's more human than that?

Well put, George. Also, saying that the study of large numbers is useless because we'll probably never use the numbers themselves is kind of like arguing that school is useless because no one remembers what year the French Revolution started, or what the Golgi apparatus does, or what a covalent bond is. Just like school teaches kids how to self-organize and handle social situations (among other things), there's value in the method that googologists use to define larger numbers.

To me, googology is like a dojo where one's capacity for abstraction is pushed to its limits. I very quickly ran into my own limits for abstraction when trying to intuitively understand Conway's chained arrow notation for chains of four or more arrows. The definition itself was simple enough, and I was able to work through a few examples with four arrows. But I still wasn't able to grok it; I didn't fully see what the chained arrows were supposed to represent in the same way that I deeply understand what addition, multiplication, and exponentiation represent.

The study of large numbers pushes this understanding of numbers to the absolute extreme; to the scale of the universe, and then so far beyond it that the observable universe becomes a rounding error. It's so irreverent, I totally love it. But, as Saibian pointed out, crafting these massive numbers requires much more skill than "smooshing powerful notations in a haphazard manner"; each new notation requires effort and ingenuity, holding in one's head multiple dizzying towers of abstraction, and generalizing those towers with an even grander layer of abstraction.

Humans have a unique power for identifying abstract patterns and generalizing them. It's that very skill which has allowed us to generalize, say, counting numbers to rational numbers, to irrational numbers, to negative numbers (and zero), to complex numbers, to multidimensional vectors, and so on. Each one of those breakthroughs improved (or enabled the existence of) all kinds of human endeavors from accounting and steam engines to GPS and quantum computers.

One of my favorite examples of this principle is the work of Claude Shannon, who realized that the amount of "information" in a message could be measured, regardless of whether that message was a telegraph, a sequence of numbers, or the results of coin tosses. His development of this idea resulted in a theory so general that it revolutionized basically every domain of science, and was the basis for many new disciplines. Just to name a few: electrical engineering, statistics, computer science, neurology, physics, cryptography, biology, and artificial intelligence. That's a pretty powerful abstraction.

Today, we're seeing computer scientists attempt to generalize thought itself as a massive, immensely complex function. If such a thing were truly possible, would that mean that we generalized our own ability to generalize? What does that even mean for human intelligence? It's all very Gödel-y.

In other words, humans have used abstractions like numbers and information theory to harness the laws of reality and propel ourselves into the next technological age. But in the face of all of that technological progress, I think there's something deeply artistic about discovering higher abstractions just for the sake of doing it, despite the Sisyphean nature of the task. In my mind, googologists are like martial arts masters, honing their craft in the art of self defense without expecting a fight. Beyond the realm of practical value, I salute the googologists building their towers in the sky. Damn, now I'm starting to sound like Sbiis.

Probably ten. ■