In a recent conversation, my friend Sagar Kolte, currently working towards his Ph.D. in the TIFR School of Mathematics, raised some questions:

- What distinguishes mathematical maturity from mathematical proficiency?
- What factors control the number of insights per unit time for a person learning mathematics? Is the number of insights one has about a concept a true measure of how well one understands the concept?

These are very profound questions to which I don’t have clear answers myself. Nonetheless, I’ll try to explore answers to these questions in this blog post.

First, the question on mathematical maturity. The term maturity suggests that it is the product of mathematical experience, as opposed to mere skill. However, I contend that it is not so much the amount and time duration of what one reads but the number of perspectives from which one reads that determine one’s mathematical maturity. Let me begin with some examples.

Let’s begin with the simplest examples: addition and multiplication. We have seen these operations right from kindergarten, where we were made to observe some basic rules (commutativity, associativity, distributivity etc et era). Since then, we have been practising addition and multiplication practically throughout our student life. And yet, some of the real significance of these operations becomes evident only once we start studying more generic stuff such as: addition of vectors, addition and multiplication modulo prime numbers, pointwise addition of functions, and so on. In fact, genuine appreciation for commutativity, associativity and distributivity is built when we encounter natural structures where these laws do not hold.

Those of us who took mathematics in high school probably broke our heads with indefinite integration. Indefinite integration is, by and large, not easy. It requires a lot of tricks, and it requires an instinct as to which trick will help simplify the given integrand. With a combination of practice and skill, one can get a fairly good mastery of indefinite integration. We’ve probably also seen a bit of “differential equations”, which again have a range of solution approaches and a collection of heuristics as to which approach to follow.

However, the high school student who understands all of indefinite integration and differential equation types is unlikely to have the same kind of maturity dealing with these things compared to the college graduate, who may have forgotten some of the methods, but understands what a differential equation essentially stands for. The more mature student sees the link between differential equations and families of curves, the link between indefinite integration and exactness of 1-forms in one variable, the link between the equation of the circle and the integration of inverse quadratic radicals. The mature student can thus appreciate more easily issues of unsolvability just as much as solvability. In fact, one of the most important aspects of maturity is understanding the inherent limitations of an approach: a person who know why a fifth degree equation cannot always be solved is more mature mathematically than a person who knows all the ways to solve third and fourth degree equations and naively hopes to extend the solution method to higher degrees.

Another example. Before I joined CMI for the B.Sc. (Hons) programme, I had already started reading a lot of group theory. I read a lot of results, solved a lot of problems, and got a lot of insights. Despite all this, I had no clear understanding of the raison d’etre (reason for existence) of groups. This came to me only after I joined CMI, when I started picking up a little here and a little there on Lie algebras and Lie groups, on algebraic geometry, and on representation theory. I particularly started appreciating groups when I realized how crucial they were in describing actions on sets.

Maturity is linked with the ability to see connections with other material. Of course, reading a wider arena of subjects helps towards getting mathematical maturity, but the process of gaining maturity can be speeded up by continually asking oneself the following questions:

- What is the raison d’etre of the subject I am studying? Had this subject not been developed, how would mathematics be affected? What areas of mathematics would get held up? If the subject is developed further, what areas of mathematics will benefit?
- What are the big questions in the subject, and how has the development of the subject been determined by attempts to solve these big questions?
- Which other subjects have made unexpected entry into this subject, and why? Can the connection be justified post facto?

Now for the question of insight rate.

The way I interpret insight rate is somewhat different from the way I interpret maturity. When I talk of mathematical maturity, I typically think of the ability to connect across adverse range of mathematical areas. On the other hand, when I talk of mathematical insight rate, I think of the ability to go deep into a particular area and come up with more and more insights about it.

I’ve had the privilege of interacting with a large number of people involved in teaching and researching mathematics at various levels. What I’ve noticed is that while a large number of people know a lot of stuff and can see a lot of connections, there are only a few who are gifted with the ability to come up with new insights, new patterns.

Coming up with insights is not so much based on experience as it is based on the willingness to really explore and ask searching questions, and persist in attempting to answer them. Unfortunately, while many of us are trained to ask practically all sorts of questions, we are no trained to pick on the relevant and important ones and keep thinking about them to get new insights.

Here are the typical words that I carry around in my mind to generate insights:

- Properties: The strongest and weakest property for a certain purpose. The logical operations on properties. More can be found in my Property theory page.
- Separation versus indistinguishability: A very important concern in mathematics is separating and distinguishing objects.
- Invariance under operations and symmetry
- Completeness of a system of invariants
- Natural correspondences
- Transfer of structure from one object to another, via a map.

These words and ideas have often guided me towards interesting questions. By interesting I don’t necessarily mean the questions that were particularly difficult to answer or the questions that are still unsolved. Rather, I mean the simple conceptual questions whose answers clarified the strands of theory. For instance, here are some conceptual questions I asked (and answered easily):

- Which automorphisms of a group extend to inner automorphisms for some embedding in a bigger group? Answer: all. A consequence of the holomorph or semidirect product of a group with its automorphism group.
- Given two conjugacy classes in a finite group, when is there a character of a representation that takes different values on the conjugacy classes? Answer: always because the characters form a basis for the space of class functions.
- Given an automorphism that preserves conjugacy classes, does it extend to an inner automorphism for every faithful linear representation? The answer is yes because the character determines the representation.
- What property must a subgroup have in a group such that every normal subgroup of the subgroup is also normal in the whole group?
- If the product of a space with every paracompact topological space is paracompact, must the space be compact? I believe the answer to be no.
- Is the space of continuous functions from the unit interval to reals, a Jacobson ring? This questions interestingly can be solved using some basic analysis and the presence of nonzero functions whose derivatives of all orders are zero.

Here’s another important point I want to make based on the above examples. The true relevance and importance of the small questions we raise comes up when we start seeing that resolving these questions requires the big and grand results, in different, new and subtle ways.