"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.

  • Negative Numbers
  • Real numbers (formal definition!)
  • Complex Numbers
  • Groups
  • The Lebesgue Measure (work in progress)
  • The Pythagorean theorem
  • Solving the quadratic equation
  • Solving the cubic equation
  • Solving polynomials
  • Metric spaces
  • Topological Spaces
  • The Calculus
  • Modular Arithmetic
  • Amenable groups (Sam? Ask Dan)
  • Axiomatization of Set Theory
  • Logarithm
  • Trigonometry
  • Analysis of trigonometric functions
  • ...many more