I come from an engineering background and so I am used to things being a particular way just because that is the way that happens to work. However when I started looking at abstract mathematics I was hoping there would be more of a reason that things are done in a certain way.
As I read about the subject [I have been reading textbooks by Joseph Gallian, Serge Lang, etc.] I keep asking myself - Where did that come from? Why choose that structure instead of some other structure?
It seems to me that as we get more abstract then the complexity and the apparent abitryness just stays the same (just like a fractal).
I have written down the sort of questions that occur to me below, I am interested to know the answer to these, I am also interested to know if these are even reasonable questions to ask. I'm sure I can't be the only one to ask these type of questions, are there any books or websites that answer these type of questions?
If we start with addition and do that 'n' times we get multiplication, if do that 'n' times we get powers and so on?
Q1) Are powers associative?
Q2) What properties do 'n' powers have?
Q3) What properties do powers of powers have?
Q4) How far can we go down this sequence? What mathematical properties would such mathematical operations have?
Compound Objects:
Q5) In the sequence: with complex numbers whose elements are reals, then complex numbers whose elements are complex numbers (quaternions), then quaternions whose elements are complex numbers (octonions) the properties degrade. Does this usually happen when the elements of an algebra are other algebras. What happens when the elements of polynomials are themselves polynomials?
Groups and Rings:
Q5) Groups have one operation, Rings have two operations, are there interesting mathematical structures with 3 or more operations (where all 3 products interact with some sort of extension of the distributive law)
Q6) Groups have a binary operation are there interesting mathematical structures with a 3-way operation?
Q7) Could such a structure with 3-way operation be represented, in general, by a sequence of binary operations?
Q8) The definition of a ring or a field does not seem to define multiplication in terms of 'n' additions? Is this a subtype of a ring? How is this related to the distributive law?
Generators:
Q9) We can define groups like <generators | relations> Is it possible to separate out the generators and the constraints so that a given group could be defined both by starting with nothing and adding generators or by starting with a completely general structure and adding constraints?
Q10) Can field extensions be defined in this generator and constraint way?
Q11) How many ways are there to 'generate' mathematical structures? Can mathematical structures be generated from nothing or do they need some structure to start with?
Sets:
Q12) Can we define arbitrary mathematical structures both as sets with internal structures and as sets with hidden structures and morphisms/arrows between them.
Algebra:
Q13) Are the commutative, associative and distributive laws unique or are there equivalent laws that could be used to define algebras?
Equations
Q14) I have read that we don't know if most equations are solvable (symbolically) and its not even possible to completely categorise equations? Is this true? What is possible?
Geometry:
Q15) Are there spaces where 'distance' is not a single real quantity?
Q16) In higher dimensional or alternative spaces are there additional transforms to translate, rotate and mirror that preserve distances?