If Six Was Nine

"If Six Was Nine" is the name of a Jimi Hendrix song from Axis: Bold as Love (1967). How could six have been nine? Hendrix's title plays on the symmetry of the Arabic numerals, "$6$" and "$9$": each is obtained by rotation of the other through 180 degrees. But the possibility of converting a representation $r$ to a representation $r^{\circ }$ doesn't automatically correspond to some important relation between what they refer to. That's a use/mention confusion.

Following Frege and Russell, (finite) cardinal numbers are the cardinalities of (finite) sets, and cardinalities are obtained by abstraction over the equivalence relation (on sets) of equinumerousness: i.e., there is a bijection $f:A\to B$. Writing $A\sim B$ to mean this, the guiding axiom is Hume's Principle: $card(A)=card(B)\leftrightarrow A\sim B$. So, for example, $0$ is defined as $card(\mathrm{\varnothing })$.

Suppose that $A=\{a_1,a_2,a_3,a_4,a_5,a_6\}$, with $a_i\ne a_j$ for $i\ne j$, and $B=\{b_1,b_2,b_3,b_4,b_5,b_6,b_7,b_8,b_9\}$, with $b_i\ne b_j$ for $i\ne j$. So $card(A)=6$ and $card(B)=9$. But there is no injection $f:B\to A$. So, $card(A)\ne card(B)$ and therefore six isn't nine.

But could six have been nine, even though it actually isn't? I don't think so, because pure mathematical objects are modally invariant. Unlike "concreta", they don't change their properties from world to world. Concreta have "counterparts". Though Quine-in-the-actual-world $w^{\ast }$ was a logician, for some other world $w$, Quine-in$w$ was not a logician. Quine-in-$w^{\ast }$ and Quine-in-$w$ are mutual counterparts. But abstract mathematical entities like six and nine are just what they are, and couldn't have been different. Concrete worlds are like planets embedded in a fixed background mathematical universe: mathematics is the spacetime of modality.

There are ways, however, of making the linguistic representation "$6=9$" true, if we change the interpretation of the symbols. Suppose we have the ring ${\mathbb{Z}}_3$ of integers modulo three. The ring ${\mathbb{Z}}_n$ of integers modulo $n$ involves treating integers that differ by adding a multiple of $n$ as equivalent. We write: $p\equiv k\text{ (mod }n)$ as short for $\mathrm{\exists }a(p=k+a\times n)$. So, for example, $1\equiv 4\text{ (mod }3)$. Then, if we define terms of the language so that, roughly the term "$+n$" is "$0+1+1...+1$'', with $n$ occurrences of "$+1$" (and similarly for $-n$), then what corresponds to the terms "$6$" and "$9$" both refer to $0$ in ${\mathbb{Z}}_3$. In that sense, "$6=9$" is true in the structure ${\mathbb{Z}}_3$.

Still, the truth of "$6=9$" in ${\mathbb{Z}}_3$ isn't what is meant by wondering whether 6 could have been 9 (or 6 might be 9, even though we don't know). That question concerns whether the finite cardinal numbers 6 and 9 could have been identical, and the answer to that is no.

Unfortunately, there isn't a Youtube video of Jimi Hendrix's "If Six Was Nine", but there is an Eddie van Halen version,