Studing mathematics is sometimes extremely difficult and needs a lot of concentrating and logic, of course… ^_^
But this days I was just thinking about AXIOMS and how important are them for everything in this world (in my mathematical world). All the topics in mathematics need this wondeful thing(s) and it is like we cannot live without them.
It seems that the first person that came with this genial idea was Euclid when he wrote the set of axioms for the plane geometry, around 350 B.C. And after some many years we are now depended of them and it’s not a normal dependence, it’s truly and strongly dependence. This is not like the need to have a mobile phone or a smartphone, because we can survive without it, but without axioms mathematics is null.
It is really strange to think that we depend on some things for all our work, and how devastating it is when our lovely tower is just blown out by someone that claims that our axioms are not good enough and it becomes more badly when he/she can prove it. And this is what happened with Bertrand Russel and the inconsistency of set axioms, the Russell paradox is wonderful and so devastating at the same time. “Of all the things that can be wrong with an axiom system, inconsistency is absolutely the worst. It is possible to work with axioms that are hard to understand. It is possible to work with axioms that are counterintuitive. And all might not be lost if your axioms don’t accurately describe the structure you intended to capture – maybe they will find some other application, as has happened on more than one occasin. But an inconsistent set of axioms is completely useless.” (Keith Devlin, “The Language of Mathematics – Making the invisible visible”)