Comments on equality

This semester, I am assigned as a grader for 21-355 Principle of Mathematical Analysis I since I did really bad in the iTA test last year and ended up in the category 3 which doesn’t allow me to hold any recitation for undergraduate students. Anyway, I need to find some fun in the given situation. That’s my principle for life.

The textbook for the course is Rudin’s book, one of the most classic introductory books in analysis. As I need to grade several problems from the exercises part, I read through the first chapter and kept in mind what results students can use in their proofs. But quickly, I was stuck at the point where students started to use things like, if a=b, then ac=bc or if a=b then a+c=b+c.

Then I questioned myself, how can I use the axioms given in Rudin’s book to justify those arguments? I know I was being nerdy at that time, but it turned out that those field axioms tell us nothing about the interplay between the equality and the two operators, namely addition and multiplication. Moreover, in the proof for some basic properties of field Rudin took those arguments as granted.

One might say “It is obvious, isn’t it?”. But my response is “No.”. In first order logic with equality, those arguments actually used the axioms of equality.

To exaggerate a little bit, how does maths scare a lot of people? I can imagine a scenario that someone is looking up a solution using this kind of ‘tricks’ and get depressed since he didn’t think of such an apparent way of solving the problem.

But in fact, that’s not this guy’s fault, it’s the deficiency hidden in maths education. In my opinion, there are no magic in mathematics, but axioms and rules that people use them unconsciously. So maths educators, especially for those in a primary level, should pay more attention to these obvious issues. Sometimes, one really need to emphasis the trivial things at the beginning level.