I had the good pleasure to watch Batman vs. Superman this past weekend (I loved it)

with my niece and her father. In the conversation preceding it she asked about Knuth arrows. I had shown them to her previously and she was trying to remember the details to show her boyfriend something cool in math. *As a disclaimer*: I’m not informed enough to comment on where these are actually used in mathematics. I can say things like

“Ramsey Theory” and what not but I would just be pretending to understand any of the

research that goes on such areas. Regardless, Knuth arrows are a neat notation that

allows us to write down numbers of unimaginable magnitude. This is a neat feature of

mathematics, maybe particular to western mathematics, to exceed our own imaginations.

We stretch them constantly with new questions and then seek to create a road from here

to … where ever those questions take us. Notation is a powerful tool for exploring

new lands. Let’s get into it.

Multiplication is a shorthand for repeated addition. So when we see 3×4 we know it

means we should add the first number,3, to itself 4 times:

Exponentiation is just repeated multiplication so we know that 3^{4} indicates that we

should multiply the first number, 3, by itself 4 times:

While multiplication is commutative, 3×4=4×3, exponentiation is not, 3^{4}≠ 4^{3}. What if

we want to repeat exponentiation? We could write something like:

Where we start by evaluating it at the top, so:

Yeah that’s big. In fact if you’re familiar with logarithms you can change the

expression to a power of ten and you’ll find out that the number above has over three

trillion digits in it (this is already difficult to comprehend. For example, using typical margins, Arial font, 11pt, how many 8.5″x11″ pages would be required to type this number out?). The above notation isn’t that bad for repeating exponentiation

a couple of times but what if you’re serious about this? It’s easy to repeat

addition and multiplication a silly number of times, consider the meaning of something

like 3×100 or How do we repeat exponentiation a hundred times without

taking up the hold page with a stack of 100 threes (often called a *power tower* ).

Knuth arrow notation allows us to repeat exponentiation more than almost anyone could

want. It starts with repeated multiplication. So instead of writing for

we write

The 4 tells us how many threes we’re going to use and the single arrow tells us we’re going to multiply. Nothing special yet. The point of the arrows is that they are recursively defined. That means that the double arrow operation is defined using the single arrow

operation. So an expression like is understood as

repeated single arrow.

The parantheses are necessary because exponentiation (single arrow) isn’t commutative.

We’ve already worked this one out but let’s do it again:

So what should we do first? Make the 4 bigger (i.e. try something like ) or add any extra arrow (i.e. try something like )? Let’s try both and compare, you may want to try this on your own first.

A bit hard to grasp eh? . So this boils down to multiplying a lot of threes (many many more

than a googol, what do you call a number with over three trillion digits?).

vs

Whoa. alone represents a power tower

with 7,625,597,484,987 threes in it!! But that’s just the number of the number of

threes in the power tower that represents !!

If that’s not enough for you you can use exponents in arrow notation, for example:

. If you’d like to learn more listen to an actual mathematician (of well earned fame) talk about how he’s used Knuth arrows.

I could end here after introducing a curious bit of mathematical notation but I feel

there’s more to say. This notation allows us to capture numbers that are beyond

imagining. Numbers that would be too large for the universe to hold them (quite possibly literally in the sense that the universe, at least the observable universe, is finite) and **yet** we can represent them with some marks on a piece of paper. We can prove things about

them, manipulate them even though we can never see them in their entirety, that’s

incredible but it doesn’t just happen in math.

“The Library of Babel” by Jorge Borges is a fantastic story concerning a very special library.

This library contains every possible 410 page book that could be created using 25 symbols. That description is mundane but the consequences are anything but. I won’t spoil the fun of figuring things out for you, but there are more books in such a library than there are atoms in this universe. What if we conceive of the universe as a sphere with radius 14 billion light years (or approx meters) which gives a volume of around cubic meters. What if we consider the number of Planck volumes (cubes meter on a side so cubic meters. Then there are around plank volumes in the universe. If we put a book in each one, would we have enough space? Laughable. You’ve not even dented the library, it’s a laughable attempt! What percentage? There aren’t words to describe what a miniscule portion of the library

represents.

I include this brief description because this is another way of capturing something

unimaginably large. At five pages of text it’s not the most efficient way to denote

the number which represents the amount of books in the library. Certainly five pages

of arrows would create some more fantastical monster of unimaginable magnitude. I

like the story for many reasons but one is certainly how the magnitude of the library

takes some work to appreciate. It begins as large and as the story progresses becomes

hauntingly vast and perilous in its unfathomability. The more you try to crunch the

numbers the more the implications unfold. There are many other questions you can ask

and try to answer about the library. There’s a lovely book about the many

mathematical consequences by William Bloch called the Unimaginable Mathematics of Borges’ Library of Babel.

It’s almost paradoxical. We can imagine that which is unimaginable. Keep in mind I have written nothing here about the many levels of infinity that mathematics has studied. These unimaginable quantities I’ve written about today are *merely finite*.