Deriving the Order of Operations: The Foundations of Mathematics at the Arithmetic Scale

In mathematics the order of operations is taught as a convention. Parentheses, exponents, multiplication, division, addition, subtraction. The sequence is enforced but never derived. No account in the standard curriculum or the foundations literature explains why the operations must be performed in that order rather than another. The order of operations is the dependency chain of arithmetic operations read backward. Counting is the first operation, corresponding to logic at the foundations layer, the binary distinction that admits something rather than nothing. Addition is the second, corresponding to set theory, two things brought into relation. Multiplication is the third, corresponding to type theory, repeated addition resolved into a single determined quantity. Exponentiation is the fourth, corresponding to category theory, the operation composed with itself, carried forward. Each operation depends on the prior. Addition presupposes counting, multiplication presupposes addition, exponentiation presupposes multiplication. The dependency chain is the foundational ordering applied to mathematical operations. The chain closes at the scale of arithmetic because the output of exponentiation is a count. The fourth operation's output is the first operation's input. 4³ = 64 is a power that produces a number. The closure has been deployed in related work (*The Arithmetic of Scale Invariance*, Stewart, 2026n) reading powers of four as counts across domains. The correspondence between the four operations and the four foundations, and the order among them, is derived at the foundations scale in *A Forced Operational Ordering of Logic, Sets, Types, and Categories* (Stewart, 2026r), and this paper instantiates it on arithmetic's own operations. The order of operations is the dependency chain read at the scale where mathematics operates on itself. **Keywords:** order of operations, PEMDAS, BODMAS, arithmetic, foundations of mathematics, logic, set theory, type theory, category theory, counting, addition, multiplication, exponentiation