Perhaps I did not express myself clearly. My definition is explicitly aimed at doing mathematics at a high level, i.e. professional mathematics. (I dropped out of professional mathematics, but since I'm now working in Functional Programming I did not stray too far.)
What I want to say is, that the process of doing mathematics is solving and understanding problems. And since we are rarely interested in concrete solutions to concrete problems, we build up theories and algorithms to solve whole classes of problems. Building up those theories is a highly creative process, but ultimately, a problem can only be seen as `solved' if we find a mechanical process for eliminating the need for creativity.
Let me give you an example: There are lots of interesting questions you can ask about linear recurrence sequences (e.g. the Fibonacci sequence), like what happens if we add to sequences? Or when we interleave them? Or when we only pick every n-th element. Or when we want to find out the i-th, without having to calculate every element that comes before.
Solving those kinds of problems requires lots of thinking.
But if you apply even more thinking, you can come up with generating functions. They are a tool that will enable you to solve all those problems really easily. (And enable you to spare your creativity for much harder problems. That's progress!)
What I want to say is, that the process of doing mathematics is solving and understanding problems. And since we are rarely interested in concrete solutions to concrete problems, we build up theories and algorithms to solve whole classes of problems. Building up those theories is a highly creative process, but ultimately, a problem can only be seen as `solved' if we find a mechanical process for eliminating the need for creativity.
Let me give you an example: There are lots of interesting questions you can ask about linear recurrence sequences (e.g. the Fibonacci sequence), like what happens if we add to sequences? Or when we interleave them? Or when we only pick every n-th element. Or when we want to find out the i-th, without having to calculate every element that comes before.
Solving those kinds of problems requires lots of thinking.
But if you apply even more thinking, you can come up with generating functions. They are a tool that will enable you to solve all those problems really easily. (And enable you to spare your creativity for much harder problems. That's progress!)