Read on Twitter
About the exercise on abelian groups, I hope you noticed how many people started tinkering with it. Can we skip/weaken one or the other assumption? Will it work for non abelian groups? Some related the problem to other mathematical structures.
As I often do, I learned from the thread, both in language ( @benjamindickman raised a question on closure) and in maths ( @ProfKinyon knows what a quasigroup is, I didn't — I still don't but at least I googled it).
More importantly for this thread, @yet_so_far and @thewordninja_bk discussed whether having a definition with more assumptions is better or worse than a streamlined one. Which leads us to the heart of this thread:
It is very, very useful to know as many equivalent definitions as possible. In research papers, I am often tempted to have Theorems-Definitions "The following properties are equivalent (...); if any <=> all hold, the object is called ADJECTIVE".
Good mathematical knowledge isn't necessarily, or even often, good exposition or good pedagogy. As a student, I benefitted greatly from having ONE definition of group, and a small list of exercises showing what could be shaven off.
But later, for research? The more, the better. In particular, it might be you're interested in a property you can't prove, but you can prove all the assumptions of a (non trivially equivalent) definition. The more equivalences you know (or can google), the easier your life.
As a beginner, feel free to tinker with definitions! Build examples, counterexamples (check the answers! there are plenty of cool ideas) and feel free to see if, by changing something a bit, you still get a useful notion... but here I stop, we'll go on in thread #3
TL;DR while it is often good pedagogy to start with ONE definition, ultimately the collection of all the equivalent ones you know is more important than any specific one.
Definitions aren't given from above: "The Book" is one we humans write. https://www.goodreads.com/book/show/696238.Proofs_from_THE_BOOK
One way you recognize a really good definition is because it has many, often not obvious* equivalent forms.
*Thank you @wtgowers for explicitly stating this.
One way you recognize a really good definition is because it has many, often not obvious* equivalent forms.
*Thank you @wtgowers for explicitly stating this.
