Addition comes from rules for counting

Addition is literally “counting on”.  It is defined that way in the Peano Axioms.

2+0 = 2

2+3′ = (2+3)’ = 5′ = 6

where n’ means count on by one from n.

The count on symbol lets us recognize that addition is defined and its properties are derived from the rules for counting on.  These rules are called the Peano Axioms.

The rules for fractions are not derived from rules for counting or dividing, but are themselves invented first as objects with rules and then applied to geometry and spatial objects.

In effect, fractions are part of physics when applied to objects in space.  If we think of actual objects in space as geometric physics, then fractions are a method that is applied to modeling geometric physics.



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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