When my mathematics teacher first introduced Algebra to me, it was looking a bit similar to a fairy-tale. The grandmother is always true! Then during the second session in Ravenshaw (remember each session is of 45 minutes), he asked us (we were a class of hundred students) to ponder if we can prove that the number one is greater than zero (1 > 0). There was a smearing laughter from the back-benches whereas the front-sitters, including me, were pretending that we have taken the task seriously.
I was feeling helpless. This statement of teacher had forced me to rethink my basic belief. I had been counting 1, 2, 3…. And zero is the starting point before 1. Zero was nothing before. And then, zero had been a special number as proclaimed by my teacher.
Teacher had already gone ahead with his customary way of writing the theorem and proving it with algebraic operations and logical deductions. He had reached the conclusion that one(1) is indeed greater than zero(0) in a minute. We were relieved — simple task, again indeed! However, this statement had created a lasting impression, and I was trying to cope up with the justification of the statement. Is it a necessity to prove a structure (the number system and ordering of whole number set) when we have created it to start with?
After a deep introspection, I got a clue towards this: Proving 1 > 0 is just an assertion, and we are not discovering any new territory of algebraic system. We are just trying to see if we still comply to the ground axiomatic structure as we wander in the fairy-tale landscape of abstract algebra. Yes, that is a requirement of any science, and of any intellectual endeavour since last few centuries of human civilization.