There have been several threads lately about "constructing" our mathematical numbers and operations. I wanted to create a new discussion in hopes of getting the brilliant community actively engaged in a foundational talk about mathematics. I take most of the following questions from Calvin's comment in this thread. You may want to read up on this thread and the recent thinking like a theorist. Here are the things to think about:
1) When we are first introduced to addition and multiplication, we can only add finitely many of them together. What does an infinite summation (or multiplication) represent? Is it 'legal'? Does it obey the commutative and distributive laws?
2) What is the concept of infinity? Is it simply "bigger than any number"? Are there different concepts of infinity? How does infinity affect addition and multiplication? What is ∞/∞? Which is 'bigger', ∞+∞ or ∞×∞?
3) What is addition and multiplication? Are there other ways that we can 'add' and 'multiply' in other systems (say colors)? What is the essential characteristics / qualities of addition and multiplication?
4) Is there mathematics that no system can handle? Can we prove or disprove every statement?
5) Is there an 'innate' idea of mathematics? Certain concepts which we take as obvious truths actually took a long time to develop. The concept of 0 was only introduced in 9th century AD by India. Using symbols to represent equations (i.e. algebra in high school) only developed in 16-17th century.
6) Is there math that an isolated civilization will never discover because of the way that they decide to interpret the world? How different is the world if we started off learning about complex numbers immediately?
7) Is there more than one way to approach mathematics? With set theory, logic, intuitively, algebraicly, geometrically, practically, etc. Is there a best way?
I hope these questions spark a few thoughts in all of your brains. If you have any other interesting questions, feel free to comment away. Post your thoughts below.