Are all good mathematicians fluent in computational aspects of mathematics

Solution 1:

To be a successful mathematician, is it necessary to be good at doing computations, to be able to calculus inverses of linear transformations quickly,integrate some complex functions,solving some difficult differential equations etc. or it depends on one's own choice?

Yes and no. Strictly speaking if we go by your definition of "quickly invert linear transformations, computing integrals, solve differential equations", then the answer is no. I study partial differential equations for a living, but if we use Arnold as a measuring stick, by no means am I any good at any of the above.

But for every field you choose to study, there would be some computations involved, and those you will have to get good at. For example, by necessity due to the field I study, I've gotten pretty good at parsing and simplifying tensorial expressions using symmetry properties and doing certain arithmetic computations related to dimensional analysis. Some of my friends in algebraic topology are very good at computing fundamental groups. I would also consider Diagram chasing a computational skill. And while we are on the topic of computations without numbers, some of the modern papers in low-dimensional-topology and knot-theory contain some amazing "computations". And I haven't said anything about the stuff that analytic number theorists regularly deal with.

However, I didn't get very good in computing things like derivatives/integrals etc. using them, and in fact, I skipped those parts to some extent that required computations or manipulations of things but I did get to see things more clearly and deeply and enjoyed writing few proofs and reading many.

Are you sure you did get to see things more clearly and deeply?

In every branch of mathematics, the mathematicians often walk around carrying a list of fundamental (counter)examples in their heads. When faced with a conjecture, we often quickly test it against our known fundamental (counter)examples to assess the likelihood of it being true. You don't get intuition about those examples by pure abstraction! You get intuition by playing with the proofs of theorems, nudging them to see when and where they break, and by computing explicit objects to develop your heuristics.

The exercises in books like Artin's Algebra are not just there to fill up space. Let me quote Jordan Ellenberg for a recent example: he and a coauthor was able to generalise a result in arithmetic geometry in part because

All of us who did the problems in Hartshorne know about the smooth plane curve over $F_3$ with every point an inflection point. ... I have certainly never needed to remember that particular Hartshorne problem in my life up to now.

Those kinds of nuggets build-up over time in your development as a mathematician. I would rather not recommend skipping exercises and computations as a habit.

Solution 2:

I think that, no matter what kind of math you do, you will eventually get down to ugly-looking huge formulas. And you will have to work with these formulas, and not just dismiss them as "computations".

It won't necessarily be a complex integral, a PDE or whatever; maybe it will be a huge commutative diagram, or a spectral sequence, I don't know. But the ability to focus, do big computations with a minimal amount of errors, and not give up even when the thing looks very complicated, is very important.

Re: your comment, abstract objects and theorems don't pop out of nowhere. It's usually after a tremendous amount of work that you find the right definition, the right theorem to prove, the right proof... And that work involves computations.

And this is the kind of things you can only learn by practising. As it turns out, integrals, DEs... are problems accessible at an early stage and that can train you in that respect. So yes, you should do your Calc III homework.

Solution 3:

When you start learning maths, you get the illusion that there is a beautiful theory, which is somehow hampered by calculations. You can get to the core of the theory without doing the calculations. If you only want to learn to read mathematics, this is true to a large extent. Especially in (linear) algebra, most calculations seem unnecessary to understand the underlying beauty of the theory.

However, when you learn more and you want to actually DO mathematics, things will get ugly, if you fear calculating an integral or following the steps of a long estimate, because this is what you will find. Most results are not refined, yet, they are ugly, nobody knows, what the real underlying structure is. Only with time and sweat will people slowly distill the essence (maybe you want to call it the "intuition") behind the theory and identify the parts that are "just calculations". Then, one can go and teach a computer to do it.

Especially in the analytical areas of mathematics, you will find whole papers devoted to finding estimates or improving estimates, etc. and this is all work that, if you want to prove an estimate yourself, you need to know how to do, and it's all basically calculations and tricks that you acquire over time. There is a whole toolbox, every mathematician carries with him and most of them involve calculations. Even in subjects that seem more remote from calculations, you will typically start understanding things by sitting down and working through simple examples, tweak the number a bit, etc.

All working mathematicians I know only use computers in two situations: - they need to get an idea of what might be going on and write simulations. - they need to calculate something, but the details are tedious, i.e. they already know how the result should look like, because they did a rough calculation/estimate, but getting all the constants right is just a mess, so they pluck it in a computer.