Math Without Variables
Idea
What would math be like without
variables?
[1][1] Ironically, much of this article makes use of variables to communicate the idea of not using them.
More specifically, what is the
canonical data structure behind mathematical concepts, independent of variables?
Example
Each of $f(x)=x$, $a\mapsto a$, and $\lambda y. y$ communicate the same concept, namely
$\text{Id}$, regardless of what symbol we use for variables.
Polynomials and linear operators
We may consider a
polynomial of one variable and of
degree $n$,
$$c_0+c_1x+c_2 x^2+\cdots+c_n x^n,$$
as a
vector of their
coefficients,
$$\langle c_0, c_1, c_2, \dots, c_n \rangle.$$
Functions which are
analytic at $x=0$ may be seen as the limit of $n\to\infty$. For instance,
$$\exp = \left\langle 1, 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \dots\right\rangle.$$
However the
ellipsis is hiding the fact that direct notation like this is formally inadequate to represent infinite structures. In any case, this has the added benefit that
matrices are
linear operators, where operation is multiplication. For example, the
differential operator can be realized as the matrix
$$D = \begin{bmatrix} 0 & 1 & 0 & 0 & \cdots \\ 0 & 0 & 2 & 0 & \cdots \\ 0 & 0 & 0 & 3 & \cdots \\ 0 & 0 & 0 & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}.$$
First order logic
Consider the
first order logic statement,
$$\forall x (x\in x \to \exists y(y \in x)).$$
With the mild assumption that the
universe is non-empty, we may convert this into
prenex-normal form
$$\underbrace{\forall x\exists y}_{\text{prenex}} \underbrace{(x\in x \to y \in x)}_{\text{matrix}}.$$
We may consider the prenex as an
tuple of quantifiers, $(\forall, \exists)$ and the matrix as the set of all
binary matrices such that $M_{1,1}=1\implies M_{2,1}=1$. All together, we find an equivalent interpretation of this statement as,
$$\begin{align*}
\bigg((\forall, \exists), \bigg\{&\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix},\\
&\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix},\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},\\
&\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix},\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix},\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix},\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\bigg\}\bigg).
\end{align*}$$
Although we have successfully represented the original concept without variables, there are a few problems in extending this.
- This becomes exponentially inconvenient as the number of quantifiers increases.
- Not all statements have a unique prenex-normal form, which would affect the order of the quantifiers, and the rows and columns of the matrices.
Articles referencing here
