Metaphysical Numbers

A metaphysical number is one whose displacement never closes. Every digit placed narrows the interval but the interval never reaches zero width. No finite length perfectly describes the number — it can only be approached.

The Same Process on $\pi$

Apply the iterative zoom to $\pi = 3.14159265\ldots$:

Zoom level	View window	Displacement
$1/1$	$[3, 4)$	width $1$
$1/10$	$[3.1, 3.2)$	width $0.1$
$1/100$	$[3.14, 3.15)$	width $0.01$
$1/1000$	$[3.141, 3.142)$	width $0.001$
$1/10000$	$[3.1415, 3.1416)$	width $0.0001$
$\vdots$	$\vdots$	never zero

The window shrinks at each step but never closes. $\pi$ has no finite address. $\frac{d}{dN}$ is always nonzero — adding more length always yields more precision, but the destination keeps retreating. Length and precision are directly proportional and neither reaches its end.

$1/3$ in Base 10

The same phenomenon appears with a much simpler number: $\frac{1}{3} = 0.3333\ldots$ in base 10.

Iteration	Value	Displacement
1	$0.3$	$0.0\overline{3}$ remaining
2	$0.33$	$0.00\overline{3}$ remaining
3	$0.333$	$0.000\overline{3}$ remaining
$N$	$0.\underbrace{33\ldots3}_{N}$	always nonzero

The displacement shrinks geometrically but never vanishes. In base 10, $\frac{1}{3}$ behaves as a metaphysical number — it demands infinite length to be perfectly stated. This is a property of the base, not the number itself (see Physicalization & Metaphysicalization).