Today, another important discussion… of course, what I say here is rather trivial for mathematicians but it is so unknown to physicists that I start asking myself what physicists actually do? Well, the discussion is about a procedure called “axiomatization”. In essence it is the way in which a notion is reduced and by that generalized to its main properties! To give you an example: what does the derivative, the Poisson bracket and the commutator have in common? Well, if you axiomatize derivation you end up with the fact that the Poisson bracket and the Lie bracket and the commutator are in fact derivatives. There is also the Lie derivative which is another name for our well known commutator in physics. You see, they all share some common axioms and a set of theorems that unify their properties. Believe it or not, many objects and operations we use are in fact at their origin the same thing and have properties that are “covering” all of them. So, the main idea of this post is to think not only in terms of direct applications of some notions you learned but also in terms of how you define them. Extra information is hidden more often than not in the axiomatic definition of the concepts you are dealing with.
And now that the idea of axiomatization is more or less known to my fellow readers, I can end by quoting my favorite childhood book:
“Why is a Raven like a writing desk?”
Alice in Wonderland, Charles Lutwidge Dodgson (aka Lewis Carroll)