notes/chapter-2.tex

\newpage
\section{Chapter 2: Symbols}

\subsection{Review}

In the second chapter, Knuth doesn't add much new information. The only
interesting bits I see are the following two quotes:

\begin{framed}
``The Stone's version is a little different, but $x \leq y$ must mean the same
thing as $y \leq x$.''
\end{framed}

\begin{framed}
``On the first day of creation, Conway ``proves'' that $0 \leq 0$. Why should
he bother to prove that something is less than or equal to itself, since it's
obviously equal to itself.''
\end{framed}

Why, indeed? It seems our prior efforts are headed in a useful direction.


\subsection{Exploration}

This week I'm interested in defining an addition operation on the surreal
numbers generated during my Chapter 1 explorations.

When crafting generation-2, we noted that $\surreal{}{} \similar
\surreal{-1}{1}$, leading us to conjecture that each number takes on a value
directly between the value of its left and right sets, ignoring the null set
for now, and assuming the number is in reduced form.

Let's begin by attempting to create an addition rule satisfying

$$-1 + 1 = 0.$$

Since $-1 = \surreal{}{0}$ and $1 = \surreal{0}{}$, and since $\surreal{0}{0}$
is not a valid number per Axiom \autoref{ax:number-definition}, we're pushed
toward a definition for

$$x + y = z$$

which discards $X_R$ and $Y_L$. This rules out simple definitions like $Z_L =
X_L \cup Y_L$ and $Z_R = X_R \cup Y_R$.

We could try something like $Z_L = X_L \cup y$ and $Z_R = Y_R \cup x$, but then
addition is non-commutative and non-closed, creating non-numbers if $y > x$.
But, if we could rework that definition to function regardless of the relative
magnitude of $x$ and $y$, then we might be on to something useful.


\subsection{Conjecture}

Geometrically, and allowing myself a recursive definition, I want something
like $Z_L = x$ and $Z_R = x + y + y$, but expressed in terms of set operations.