On and off over the years I’ve been thinking about the concept of understanding, most particularly in the field of mathematics. When one encounters a new branch of mathematics, one is (conceptually) presented with the definitions, axioms and theorems in the field. Of course, one doesn’t necessarily understand them at first. However, after delving into the field, studying the axioms and definitions, and following through the proofs one soon gets the general feel for the field.
I might for instance read a ‘popular’ book about mathematics and read about ‘hyperreal numbers‘. By a ‘popular’ book, about maths, science or practically anything else, I mean a book which provides a non-rigorous introduction to a field, intended to give a feeling for it. If a reader of such a book requires an in-depth treatment of the subject, the reader would no doubt have to spend a long period studying the necessary maths to even start to address the topic. Such a ‘popular’ book could not possibly give one an understanding of ‘hyperreal numbers‘ of course. It’s more of a geography lesson, in that it would provide an understanding of where the topic fits in to the topography of mathematics, just as one might read about Greece, but one would not understand from the guide book what Greece is really like. For that one must go there.
So you need to study something in maths to understand it. When do you know that you have understood it? I believe that there are at least three stages or signs that you understand it.
In the first stage, you have studied the axioms or the basis for the field of mathematics that you are trying to understand and you have seen and understood a few of the proofs of theorems in the field and you can reproduce them if asked. You have arrived in Greece, you are getting to grips with the money, you know how to purchase something in a cafe or restaurant, to extend the analogy.
In the second stage, you understand the ‘why’ of the proofs, you start to get a feel for the way that proofs are put together in the field and what can be achieved in the field. The concepts are sinking in. In Greece, you know how to find the best bars and restaurants where the Greeks drink and eat, you know what to order from the menus, and you are picking up a smattering of the language.
In the third and final stage, you are using the mathematics in the field with confidence, either to develop your own ideas in the field or in further study. When you look back at the difficulties you used to have in the field you are astonished that it took so long for the concepts to sink in. They seem obvious now. In the Greece analogy, you speak the language fluently, you have married a Greek person and you have lived in Greece for years. You mock the foreign tourists, and also you realise that there is more to Greece than your particular corner of it.
So, my point is that to understand something you need to immerse yourself in the topic. I don’t mean to say that you won’t have “Eureka Moments“, but even Archimedes had to immerse himself in his topic of study to come up with his Principle!
Me on the net 
I would like to point out another way in which “understanding” comes in mathematics. When I studied physics it really shocked me how Riemannian geometry could encompass not only all of geometry, but how it actually turned differential operators into vectors. What I mean is Riemann took two completely separate areas and merged them into a coherent whole, just like complex numbers gave a different meaning to the relationship between exponentials and trigonometric functions. This kind of understanding, that superceeds all theories and goes beyond them to a more simple, elegant and unified approach really baffles me. It is to me one of the great joys of learning mathematics.
Exactly. I especially like the ‘isomorphism’ between geometry and algebra.