So I've been reading the proofs for e^(i*theta) = cos(theta) + i*sin(theta) (2) and wondering what is actually going on.
I know that one of the ways this was derived by finding the solution of the differential equation dz/dtheta = iz (2).
Firstly I know that complex numbers have been proved to follow the normal rules of algebra (associativity, commutativity, etc.) and so normal operations are justified. The solution to (2) uses these rules to get (1) as a particular solution. All the formal proofs are rigorous and that is not what I am thinking about.
Addition, multiplication, exponentiation, etc. are abstractions that humans invented to model real life scenarios. You can therefore visualise what is going on, e.g 6^3 = number of possible outcomes when rolling a six sided die 3 times.
It seems to me that after discovering complex numbers and playing around with them, some very beautiful truths were found in abstraction land like (1).
Because this was a reality found only in abstraction land is there no "real life" way to think about this?
Or is the famous proof(it represents a point on the complex unit circle) the only way to visualise it because it is inherently defined as such?
It has many real life applications (i don't know what they are exactly but i know they exist) - is this a coincidence or are the fundamental axioms of maths linked to nature?
Moving away from complex numbers,
Do abstractions provide meaning to all levels below them? E.g differential geometry is used to study general relativity, general relativity is a good model for many realistic scenarios; do the truths found in differential geometry land have some significance in real life?
If there was an intelligent alien species, would they have these same ideas just with different notation?
If we had 7 fingers and so worked with base 7 numbers would maths change? Because maths is used to model physics, would physics models change?
These are just some questions that I'm writing down - they're not particularly well organised and I don't full know what I'm asking but I want to hear your thoughts.