(12/18/2022, 09:08 AM)MphLee Wrote: Also Qiaochu Yuan makes some good points that any foundational work on hyperoperations needs to be pair with Devlin's critique of "multiplication as repeated addition"...

Devlin is wrong. He claims that multiplication is scaling, but scaling only works as multiplication in \(1D\).

If you multiply (make repeated additions) a line (real numbers), you get the same length as scaling by the same factor.

However, this does not hold in higher dimensions. When you scale by \(s\) in \(D\) dimensions, you multiply by \(s^D\).

For example, if you scale a square (\(2D\)) by \(s\), you multiply its area by \(s^2\).

The concept of repeated addition works in any dimension, but scaling does not.

Additionally, in complex numbers, a product involves both multiplication (by the modulus of the complex number) and rotation of angles.

Therefore, the product should be understood as encompassing different concepts: multiplication, scaling, and rotations. Maybe each rank adds more meanings for the product, which we do not notice due to symmetries, that higher hyperperations break.