My post is about arithmetic and defining operations.
Philosophy is down the hall and politics is in building B.
I have defined multiplication for the rationals as mk = the combination of m either added to or subtracted from zero k times
Scaling can give you the same numeric answer as multiplication, yet that does not make the operations the same.
The scaling symbol is : and it results in a change of magnitude not change of multitude. Toy planes, trains and automobiles are scaled as you know. Therefore the concept is already familiar to children.
My definition of mathematical scaling is 'a change of magnitude created by making the multiplier the unit of measure.
Multiplication involves many instances of the same object in units or parts and relates to multi-tude.
Scaling involves one instance of the object with various units of magnitude.
Multiplication makes same more or less.
Scaling makes same, bigger or smaller.
Professor Devlin and I both agree Euclid's* definition of multiplication from 2300 years ago should not be used today.
Neither Euclid nor anybody since has provided precise definition(s) because operations are what they are.
Saying multiplication is not repeated addition does not define what multiplication is.
Proving multiplication is not ONLY repeated because it does not apply to scaling does not define multiplication.
My goal is to make arithmetic simpler for children and that is why simple, precise and correct definitions for the rationals are essential.
Stan Dehaene and others have proven babies have both number and magnitude or size sense.
Doctors who perform life and death operations say cells multiply and organs enlarge or shrink.
Anyone that cannot understand the difference between multiplication and scaling deserves sympathy and respect as a person, yet not a mathematician.