Physicalization & Metaphysicalization

Whether a number is physical or metaphysical is not always intrinsic — it can depend on the base system used to describe it. A number that demands infinite length in one base may terminate cleanly in another. Physicalization is the process of choosing (or moving to) a base in which a number becomes physical. Metaphysicalization is the reverse — a number that was physical loses its finite terminus.

Base as Prime Factorization

The key is prime factors. Base 10 has prime factors $\{2, 5\}$. Any number whose denominator (in lowest terms) contains only these factors will terminate in base 10. Any other denominator produces a repeating expansion — a metaphysical number in that base.

$\frac{1}{3}$ in base 10: denominator $3$ is not a factor of $10$, so the expansion never terminates — $0.3333\ldots$

$\frac{1}{3}$ in base 12: base 12 has prime factors $\{2, 3\}$. Now $3$ is a factor of the base, and $\frac{1}{3} = 0.4$ exactly — a physical number, perfectly described in a single digit.

Number	Base 10	Base 12
$\frac{1}{3}$	$0.\overline{3}$ (metaphysical)	$0.4$ (physical)
$\frac{1}{5}$	$0.2$ (physical)	$0.2497\ldots$ (metaphysical)
$\frac{1}{6}$	$0.1\overline{6}$ (metaphysical)	$0.2$ (physical)

The Implication

The base system can occlude a number from its true physical potential. A measurement taken in base 10 may look irrational while the same quantity in a base whose prime factors include 3 — such as base 12 or base 60 — terminates cleanly. The ancient use of base 60 (Babylonian) is notable here: $60 = 2^2 \times 3 \times 5$, which covers far more denominators than base 10.

MMP observes that the physical or metaphysical character assigned to a number by a given base is a property of the description, not the number. The number itself only has one true character — it is the base that may be failing to express it.