What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What is the philosophical implication or significance of computable (and non-computable) numbers?

What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What are the philosophical implications or significance of computable (and non-computable) numbers?