Cardinal addition order ambiguity

If we use cardinality to define addition, what is the meaning of the order of addition?

2+3 = 5

v

3+2 =5

In counting on, we start with 2 and count on 3 in the first case.  In the second, we start with 3 and count on 2.

But for cardinality as the definition of addition, we find two sets in bijection with 2 and 3 that have no common elements, take their union and count it.

What does the order of 2 and 3 correspond to in this definition?

Especially, if we think of time as not being part of math.

So the bijections and the union are not done as steps in time, but already exist. They are pre-existing.

So what does the order of the addends mean in the first place in terms of the definition of addition by cardinality?

This ambiguity may seem to make addition easier to prove being commutative.  However, it makes it harder to show how cardinal addition corresponds to right and left additive identities?

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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