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 (I like that word) of the object with various units of magnitude.
Multiplication makes same, more or less.
Scaling makes same, bigger or smaller.
Neither Euclid nor anybody else since to my knowledge has provided precise definition(s) because operations are what they are.
Saying multiplication is not repeated addition does not define multiplication.
Proving multiplication is not ONLY repeated addition 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 sense. They are different thoughts and conceptually, different operations.
Doctors who perform life and death operations [sic] say cells multiply and organs enlarge or shrink.
Children understand big and little before they understand number. Big kids and little kids are either allies or enemies! Whatever number their ages are is less relevant.
Animals calculate fight or flight by estimating survival against 3 small enemies or perhaps one big enemy.
The amygdala subitises number and interprets size (relative to distance) before it gets to the neocortex.
We are essentially born with an operating system that relies on discriminating number as different to size.
Our language is not as developed as our brain, which is why definitions DO matter.
The 'red herring' I alluded to is that Professor Devlin objects to teaching multiplication is (only) repeated addition, not because is is wrong, but because it doesn't scale.
Children have more than enough brainpower to play with integers via bumps and holes as a model that later evolves into the arithmetic of rationals.
As toddlers we knew two fingers on our left hand and one finger on our right hand is the same as one finger on our left hand and two finger on our right hand. We also knew two fingers three times counts up to the same as three fingers two times.
We also knew 4 fingers take away one finger leaves 3 and that we could subtract by adding on which is how we give change in a shop.
So having started school to learn arithmetic eager and empowered, it all ends up too many kids fail and dislike math, not because the rules change later on. Kids hate and fail math because it doesn't make sense.
It doesn't make sense because while the laws of math have been established on my fingers, 'your' definitions appear silly!
How can 2 x 3 be 2 added to itself three times? It isn't! It is added to zero three times as the number lines starts from zero these days - unlike in Euclid's day.
Just because Euclid failed the arithmetic logic test, we shouldn't inflict the same on our children 2300 years later.
With my definition of multiplication children understand -2 x 3, 4 x -2 and -3 x -3. they also understand (maybe a little older) exactly how the same multiplication algorithm applies to fractions whatever their sign.
Children can fold paper to change unit size. It's fun! They can also discover how to double the size of a square.
To ignore scale simply results in people not understanding ratio. One unit multiplied by 3 results in 3 units. 1 unit scaled by 3 results in 1 unit three times bigger.
I explain this to kids with pictures of toy cars being multiplied in number or being scaled to be as big as the real thing. They think scaling is more fun.
Unless we address the confusion between multiplication and scaling we will have children thinking + and - are both multiplication and scaling because that's what is on the television remote for the 'zoom' or size button.