The fundamental axiom of mathematics is as simple to state as the fundamental axiom of Objectivist philosophy:
1=1 (Source: Ron Pisaturo)
What is meant by thissimple statement is that the unit measure of any class of entity, quality, or relationship is invariant across the domain of its measurement. The first apple in a barrel is the same (qua apple) as the 10th, 100th or 1000th apple. The distance measured by the first mile on the commute to work is the same as the distance measured by the 13th.
The concept of a unit,abstracted from that which the unit measures, is the concept which Mathematics labels as the number 1.
The concept of equality implies commensurability. By commensurable I mean that the referent which the symbol on the left side of the equation represents is of the same kind as the referent represented on the right side of the equation. Further, the precise definition of commensurate is “measured by a common standard”, here meaning measured by a common, universally constant, unit.
This needs slight elaboration. The “kind” which must be constant between sides of an equation is precisely the kind for which the unit is defined. This may or may not mean that the entities represented on both sides are of fully identical classes, but only that at the level of representation in the equation the classes are identical. Lets say we have 10 moose and 10 monkeys (because, as my kids know, I love talking about moose and monkeys). As moose and monkeys, these entities are not commensurable. The expression 10 Moose = 10 Monkeys is therefore not an equation, because the units do not match. However, in the abstract category “Mammals” to which both moose and monkey belong, we have an abstract unit “Mammal” which can measure both moose and monkey. The expression 10 Mammals = 10 Mammals is a valid equation, and in this context we can properly write 10Moose = 10 Monkeys and imply this equation.