Numbers and imagination: notation and literature

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:

3\times 4 = 3+3+3+3

Exponentiation is just repeated multiplication so we know that 34 indicates that we
should multiply the first number, 3, by itself 4 times:

3^4 = 3\times(3\times(3\times 3))

While multiplication is commutative, 3×4=4×3, exponentiation is not, 34≠ 43. What if
we want to repeat exponentiation? We could write something like:


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

3^{3^{3^3}} = 3^{3^{27}} = 3^{7 625 597 484 987}

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 3^{100} 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 3^4 for
3\times 3\times 3\times 3 we write

3\uparrow 4 = 3\times 3\times 3\times 3

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 3\uparrow\uparrow 4 is understood as
repeated single arrow.

3\uparrow\uparrow 4 = 3\uparrow (3\uparrow (3\uparrow3))

The parantheses are necessary because exponentiation (single arrow) isn’t commutative.
We’ve already worked this one out but let’s do it again:

3\uparrow\uparrow 4 = 3\uparrow (3\uparrow 27) = 3\uparrow 7,625,597,484,987

So what should we do first? Make the 4 bigger (i.e. try something like 3\uparrow\uparrow 5) or add any extra arrow (i.e. try something like 3\uparrow\uparrow\uparrow 4)? Let’s try both and compare, you may want to try this on your own first.

3\uparrow\uparrow 5 = 3\uparrow (3\uparrow (3\uparrow (3\uparrow 3))) = 3\uparrow (3\uparrow 7,625,597,484,987)
A bit hard to grasp eh? 3\uparrow 7,625,597,484,987\approx 10^ {3,638,334,600,000}. 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?).


3\uparrow\uparrow\uparrow 4 = 3\uparrow\uparrow (3\uparrow\uparrow (3\uparrow \uparrow 3)) = 3\uparrow\uparrow (3\uparrow\uparrow 7,625,597,484,987)
Whoa. 3\uparrow\uparrow 7,625,597,484,987 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 3\uparrow\uparrow\uparrow 4!!

If that’s not enough for you you can use exponents in arrow notation, for example:
3\uparrow^{8} 4 = 3\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow \uparrow 4.  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 10^{26} meters) which gives a volume of around 10^{79} cubic meters. What if we consider the number of Planck volumes (cubes 10^{-35} meter on a side so 10^{-105} cubic meters. Then there are around 10^{184} 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
10^{184} 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.


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