Many people seem to be convinced that in maths, there’s always one right answer. But really, given a well-defined question, well-defined terms and enough information or assumptions, there are always right and wrong answers, regardless of the subject.
The ill-defined definitions are often less ill-defined and more disagreed upon. In fact, many people find themselves arguing despite having the exact same opinion, all because they were using different definitions. And arguing about the “correct” definition is futile.
The lack of information shouldn’t really be an issue. We can consider all the possible answers assuming all the different states of the missing information and give answers of the form “Assuming X, this should be the case, but assuming Y, that’ll hold instead.” But schools often want students to take a side.
This is because in the real world, there is one true state of the missing information. There is one reality. And many subjects are necessarily and directly relevant to the real world.
In maths, no-one argues about definitions. Does the natural numbers include 0? Doesn’t matter, no big deal. Just go along with whoever’s talking at the moment. And it’s not intrinsically attached to the real world, so instead of trying to find The Best Answer, we can just look at Any Interesting Answers.
Well, that’s one reason I love maths. Another is just how little information you need to get started. For instance, imagine some bunches of stuff (sets). What can you do with them? You can combine them (unions), you can take all the things the bunches have in common (intersection), you can take one thing from one bunch, one thing from another bunch, and make a list of all such combinations of things from the two bunches (Cartesian products). You can assign everything from one bunch to something in another bunch (functions). You can compare things in a bunch (relations). What if there’s some sort of interesting pattern in a bunch of stuff? Can you associate the things in that bunch to things in another bunch in such a way that the patterns in the associated things match up (homomorphisms)?
All that, just from bunches of stuff. And the more you construct, the more you can construct with the things you construct.
Okay, there’s one more reason I want to mention. Am I the only one who really likes the pretty, pretty notation? There’s something strangely comforting about upside-down A’s and back-to-front E’s …
Yao Tang was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.