"Mathematics is an organism for whose vital strength the indissoluble union of the parts is a necessary condition"
David Hilbert
Why are certain ideas and questions studied in Mathematics? It is more complicated than we might be taught to think.
Let's start with an example. Please read this before the others. Complex Numbers
This is an attempt of mine to describe the vision behind this project: Behind the conjectures and definitions
Now let's go through all of Mathematics and do this for each concept and question. The best answer is not always historical, but history must be involved in the answer (the first people studying a subject, in most cases, are the best to know what the motivation is). Here are some guidelines as to when Mathematical concepts should be introduced.
Real numbers (formal definition!)
The Lebesgue Measure (work in progress)
The Pythagorean theorem
Solving the quadratic equation
Solving the cubic equation
Solving polynomials
Topological Spaces
The Calculus
Modular Arithmetic
Amenable groups (Sam? Ask Dan)
Axiomatization of Set Theory
Logarithm
Trigonometry
Analysis of trigonometric functions
...many more