Saturday, 23 April 2011

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 $f : A \rightarrow B$. 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 $card(\emptyset)$.

Suppose that $A = \{a_1, a_2, a_3, a_4, a_5, a_6\}$, with $a_i \neq a_j$ for $i \neq j$, and $B = \{b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9\}$, with $b_i \neq b_j$ for $i \neq j$. So $card(A) = 6$ and $card(B) = 9$. But there is no injection $f : B \rightarrow A$. So, $card(A) \neq card(B)$ 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 $\exists a(p = k + a \times n)$. 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,

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