A few years ago famed MIT computer science professor Scott Aaronson proposed a contest on who could name the bigger number.

Its a simple premise, two people go up to a whiteboard and take turns trying to write a bigger number with the winner being the one who wrote the biggest number. There are some ground rules like you can’t write “the other guys $number + 1$” and no infinities but for the most part you can do as you please as long as the definition of your number is clear.

You might try writing a whole bunch of 9’s until you run out of room. That’s a big number. If you were smarter you would write a whole bunch of 1’s because they take less space and you can fit more on the board.

If you were a mathematician or a philosopher you might write something like $3\uparrow\uparrow\uparrow\uparrow3$ or $Tree(3)$. Those are both functions that create pretty freaking big numbers. And if you wrote $Rayo(100)$, well you just won the game.

What does an \$latex \uparrow\$ and $Tree(3)$ mean and why should I care about big numbers? Good question. I didn’t care about big numbers until about a year ago when I stumbled on to a curious entry in the Guiness Book of world Records for the largest number used in a mathematical proof. The number was called Graham’s number and it was used in a proof in the 1970’s. That led me to a YouTube video on the Numberphile channel all about Graham’s number.

Graham’s number

Graham’s number has so many digits in it that if you tried putting all of them into your head it would collapse into a blackhole.

At first I didn’t get it. How could your head collapse into a blackhole from putting a big number into it?

There is a maximum amount of entropy that you can store in something the size of your head and the amount of information it would take to write out Graham’s number is greater than the amount of information in a blackhole the size of your head.

That just blows my mind. Numbers like Graham’s number are so big that they can’t be represented EVER because even if you wrote one digit on every single particle in the universe you would run out of particles before even making a dent in these numbers. And those are small numbers compared to some of the colossal numbers formed from functions like $Tree(3)$ and $Rayo(100)$.

That little discovery led to a new fascination of mine in big numbers. What are these big numbers and why are they useful in philosophy and mathematics? If I were going to win the big number contest what number should I write down and why?

Ridiculously Huge Numbers

First I wanted to learn a little more about big numbers. I wanted to try to understand or rather conceptualize how big these numbers are. I started with a video series from David Metzler a professor from New Mexico that over the course of 14 videos takes you from addition all the way to the fast-growing hierarchy.

That series gave me a pretty good idea of what it means to be a big number. It also allowed me to sort of visualize what it means to be a big number.

Big numbers are important, they are used in mechanical statistics, cosmology, and cryptography. One of the coolest uses is the Poincaré recurrence time

which corresponds to a time in a model where our universe’s history repeats itself arbitrarily many times due to properties of statistical mechanics, this is the time scale when it will first be somewhat similar (for a reasonable choice of “similar”) to its current state again.

That number is calculated to be:

$10^{10^{10^{10^{10^{1.1}}}}}$ years

Think about that for a moment. Given all of the random elements in our universe after Poincaré recurrence time they will repeat themselves in exactly this same manner.

Big numbers are a new hobby of mine. Trying to understand them is a challenge in itself and seeing the application of them has introduced me to new areas of science like quantum mechanics and cosmology.

Lastly I’ll leave you with the definition of Rayo’s number. This is the number that you would write if you were going to win the biggest number contest:

“the smallest positive integer bigger than any finite positive integer named by an expression in the language of first order set theory with a googol symbols or less.”

You might need to take a look the explanation of Rayo’s number on Googology to see why it is so big.