I was talking with a friend about Graham’s number. For those who are not familiar, here’s a (somewhat) brief explanation.
What even is Graham’s number?
Graham’s number is an extremely, extremely large integer. It is completely theoretically countable, as in if you had enough time you could count all the way from 1 to Graham’s number, and then you’d be done. It’s a finite task for a finite number. Many people are already familiar with very large numbers. Think of 52 factorial, (also known by its property as the number of possible orders you can put a deck of cards in) or a google (10100, ie a one with one hundred zeroes). These numbers are effectively 0 compared to Graham’s number.
Multiplication is quite integrated into our daily lives, so it is easy to forget that it is actually just a shortcut to express repeated addition (3 * 5 is really just 3 + 3 + 3 + 3 + 3, or 5 + 5 + 5 if you prefer). Exponentiation (raising something to a power) is the same thing, just a way to express repeated multiplication. We will use both superscript and up arrow notation for exponentiation, for example:
2↑4 = 24 = 2 × 2 × 2 × 2 = 16
The base (the number before the arrow, 2 in the above example) is being multiplied by itself repeatedly a certain number of times. That number of times is the index (also called power or exponent, 4 in this case). With this slightly-more-rigourous-than-an-algebra-I-class definition of exponentiation in place, why not take it a step further?
We can define a new operation, called tetration, that is to exponentiation what exponentiation is to multiplication — it’s just repeating it n times, with n being the index. We’ll use two up arrows instead of one to define tetration. So let’s try 2↑↑4.1
2↑↑4 = 2^2^2^2 = 2^2^(2*2) = 2^2^4 = 2^(2*2*2*2) = 2^16 = 65536.
You might be asking, why can’t we create another operation, but with three arrows, signifying repeated tetration? And to that I would answer “of course we can, that’s called pentation”.
This will be the final example we attempt to fully spell out, and it will become obvious why.
2↑↑↑4 = 2↑↑2↑↑2↑↑2 = 2↑↑2↑↑(2↑↑2) = 2↑↑2↑↑(2^2) = 2↑↑2↑↑4 = 2↑↑(2↑↑4) = 2↑↑65536 (we substituted 2↑↑4 for 65536 because we calculated 2↑↑4 above, but we also could have continued expanding via the same process and gotten the same answer).
Now I don’t know about you but I don’t feel like writing 2^2^2^2^2^2^2^2^2… with 65536 ^2s. As you can probably imagine, that number is enormous. In addition, I used 2 as a base, which is very small and has the property that 2↑↑2 is equal to 2^2, whereas for any integer base greater than 2 will not be even remotely close to true (3^3 = 9 and 3↑↑3 = 7.6 trillion. Now try working out 3↑↑↑3).
So of course, upon discovering a way to almost instantly create numbers so large we couldn’t hope to express them with regular integer or even exponential notation, we will push this much further.
Hexation comes after pentation, as in it is repeated pentation and is denoted by four up arrows. Let us define a number, g1, that is simply 3↑↑↑↑3. This number is already huge – a googol is 0 compared to it. A googleplex (10 to the power of a googol) is 0 compared to it. Now let’s define g2, which will be 3↑↑↑↑…↑↑3. How many arrows are there? g1 arrows. Recall that 3↑3 = 27 and 3↑↑3 = 7.6 trillion and 3↑↑↑3 is unwritable with integer or exponential notation. When you add more arrows, numbers get bigger very quickly. So let us continue along this path, and define g3 = 3↑↑↑↑…↑↑3 with g2 arrows, g4 = 3↑↑↑↑…↑↑3 with g3 arrows, etc. Graham’s number is g64.2
If you tried to conceptualize Graham’s number, your brain would become a black hole. By this I mean if you stored every digit of Graham’s number in your brain, the information required (about 3.3 bits per digit, if you were curious) would carry so much mass-energy that the Schwarzschild radius3 for your brain would be larger than your brain, and thus your brain would be smaller than its Schwarzschild radius, and therefore would be a black hole. That would actually happen long, long before you finished encoded the digits of Graham’s number. Also, this is completely hypothetical (really for so many reasons) partly because the number of particles in the observable universe is effectively 0 compared even to g1, much less Graham’s number. Despite this, we still think about Graham’s number. We even think about g65, or ggoogolplex, or even such an abomination as:
gg64↑gg64 gg64.
These numbers are not mathematically useful, nor are they meaningfully numerically different from each other in any way that a human (or any sort of being bound by the laws of physics in our Universe) could possibly conceptualize, so what is the point?
The idea that we can understand that something is inscrutable is inherently paradoxical – not in the sense that it breaks any logical axioms, but just that it is a sentence whose meaning folds in over itself when viewed in the context of the human condition. We are always in the pursuit of knowledge, sure, but more so understanding. I don’t care to know things, I care to apply knowledge in order to prove and gain understanding. It provides no inherent pleasure to me to tell you that the left ventricle is the strongest chamber of the heart, but it is a fact that is very important to me insofar as its function within my personal understanding of the heart, and of the body, and of cardiac muscles, and of muscles in general, and of electrochemistry, etc. So why would we care to explore something that we are aware from the get-go that we cannot comprehend or understand?
There is a pulling that creates a stress on the mind when we stretch our cognitive limits. The image that always comes to mind for me is from a Foxtrot daily strip. Jason is warming up for the new school year by tensing up his face and trying to imagine negative and positive infinity. It is a more abstract form of human curiosity, because it does not have a goal. You will never understand infinity as a positive or negative number because it is by definition not finite. You will never fill your mind with every digit of Graham’s number, or understand the magnitude of its size in any meaningful comparison or abstraction. But we reach out to the stars not to climb into the sky but to reach.
Often it is said that a mark of disintelligence is an inability to imagine hypotheticals. For example, it is said that someone of significantly below baseline IQ would answer a question such as “how would you feel if you didn’t eat breakfast this morning” with an answer such as “but I did eat breakfast this morning”. While this may be a somewhat reliable indicator for assessing certain types of cognitive disabilities, I don’t believe that the converse is true. It is not someone’s ability to imagine the purposeless that makes them intelligent; it is their desire to fixate on it. It is not our ability to fashion complicated tools, or to create arbitrary social hierarchies, or to climb into the heavens that makes us human. It is the blessing that we are crazed and dedicated enough to be satisfied with the Odyssey to the stars.
- Let’s remember that exponentiation always works from the top down. For example, if we had 3^2^3, we would not say 3^2^3 = (32)^3 = 9^3 = 729. We would start from the top down, so 3^2^3 = 3^(23) = 3^8 = 6561. This tends to produce much bigger results and follows from how exponentiation and order of operations are defined. ↩︎
- If that seemed extremely arbitrary, I actually have good news for you – Graham’s number was actually created as an upper bound of dimension number for a specific problem in Ramsey theory. If you don’t like pure math, you might still call that extremely arbitrary. ↩︎
- The radius for any given amount of mass that if compressed into that radius would form a black hole. ↩︎