Arithmetic

Elements

The fundamental element of arithmetic is the number — an abstract form that represents quantity. Knowledge here is the recovery of these forms: the unit (the indivisible One), and the primes, the indivisible atoms from which every other number is composed.

Number Systems

Numbers form a hierarchy of increasing generality:

• Even / Odd — parity classification
• Prime — divisible only by 1 and itself
• Integer — whole numbers including negatives
• Rational — expressible as a ratio $p/q$
• Irrational — not expressible as a ratio (e.g., $\sqrt{2}$)
• Real — all points on the number line
• Complex — numbers of the form $a + bi$

Essential Properties

A number is a discrete quantity that can be counted.

• Every positive integer greater than 1 is either prime or a product of primes.
• For any two distinct positive numbers, there exists a number between them.

Axiomatic Construction

Under the deductive lens, arithmetic is not observed but derived. Following Peano, everything is built from two primitives — zero and the successor $S$ — and five axioms:

1. $0$ is a natural number.
2. Every $n$ has a successor $S(n)$.
3. $0$ is not the successor of any number.
4. $S(m) = S(n) \implies m = n$.
5. Induction — if $P(0)$ and $P(n) \Rightarrow P(S(n))$, then $P$ holds for all $n$.

Operations as Definitions

Addition and multiplication are defined recursively, then their laws are proven:

$a + 0 = a, \qquad a + S(b) = S(a + b)$ $a \times 0 = 0, \qquad a \times S(b) = (a \times b) + a$

Commutativity, associativity, and distributivity are theorems, established by induction — certain, not empirical.

Measurement & Cause/Effect

Before it was axiomatized, arithmetic was the first measurement: a tally placed in correspondence with a flock, a harvest, a debt. Knowledge is sampled from the world by counting, and its laws appear as regularities in what we measure.

• Counting assigns numbers to discrete objects in sequence — the original act of measurement.
• Adding quantity to a heap causes its total to rise ($+$).
• Partitioning a fixed quantity into more equal groups causes each group to shrink ($-$).

These cause/effect relationships are the empirical shadow of the operations the deductive lens defines abstractly.

Procedures

The algorithmic lens asks only: what is the effective procedure? A number is what a procedure produces; an operation is a finite sequence of steps.

Column Addition

1. Align digits by place value.
2. Add column-wise from the least significant digit.
3. Carry any overflow into the next column.

Cost

• Addition of two $n$-digit numbers: $O(n)$.
• Schoolbook multiplication: $O(n^2)$; Karatsuba: $O(n^{1.585})$.

Here “knowing arithmetic” means possessing a terminating procedure that computes it.

Arithmetic as a System

Seen cybernetically, a number system is a system: a set of elements (numbers) closed under operations that transform them, held in balance by special equilibrium elements.

• Operations ($+$, $\times$) are the transformations — the system’s processes.
• The identity element ($0$ for $+$, $1$ for $\times$) is a fixed point: applying it changes nothing.
• The inverse restores equilibrium — $a + (-a) = 0$.
• Closure is the system’s boundary: operating on members yields members.

Order and composition are the dynamics; identity and inverse are the equilibria they settle toward.

Computation Under Constraint

The engineering lens treats arithmetic as something to be realized on a machine, optimized and made robust against the failures of finite hardware.

• Floating point trades range for precision; the 53-bit mantissa bounds accuracy.
• Round-off error accumulates — summation order matters; Kahan summation recovers lost bits.
• Overflow / underflow must be guarded against at the edges of the representable range.

The objective is maximal accuracy and stability for a fixed bit budget — knowledge as control.