Are all good mathematicians fluent in computational aspects of mathematics

Question

I am in first year at college. I like studying Abstract Algebra from Michael Artin's book and Calculus from Apostol's books. 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.

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

Answer 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.

Answer 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.

Answer 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.

Answer 4

I think the answer is yes and no.

For yes, computation is a basic procedure throughout every branch of mathematics. It is difficult to admit that one knows a result while not knowing how to get it. For beginners, computation is important because doing computation helps one touch it and feel it in each step and then establish an intuition and belief that the thing is precisely right. How can one run without walking first? Also, detailed computation contributes to making induction and forming abstract things.

For no, once understanding the procedure of computation and believing you can do it right sooner or later, computation is actually an unnecessary thing because there is so much software doing that already. It is helpful for mathematicians to think of only the big ideas and ignore the detailed computations.

In all, mathematicians may not be good at it, but must be ever.

Answer 5

If you include applying axioms or transformations as part of derivations or proofs as calculations then, yes, you should get good at doing them. If one is poor at copying long expressions with the proper substitutions then there is a modern answer. That is to use an application such as Mathematica (the one I'm familiar with and reference, but there are others).

The trick is to replace one skill, error-free (and neat) copying of long expressions with substitution or transformation, with another skill, writing rules and routines that specify the axioms and transformations and applying them. I argue that the second skill may not be faster to use, but it is less error-prone and more concentrated on the actual mathematics.

A second trick, when using Mathematica, is not to regard it as a scratchpad, or a super graphical calculator, or a programming worksheet, or a mathematical word processor (although it is in part all of these things), but as a blank sheet of paper on which you are developing/learning and writing your mathematics. It could, should actually, contain sectional organization and textual description and discussion as well as calculations. It is a rather magical sheet of paper because it has memory, can do active calculation, can accumulate knowledge (in the definitions, rules and routines you specify), can contain beautiful graphics and dynamic displays of various types.

The next trick is to make a sincere attempt to calculate everything. Don't fill in with word processing. I don't claim, or am not certain, that this can always be done but a pretty wide and deep swath of mathematics can be done by computer calculation. This has several advantages: the entire calculation, derivation or proof is largely self-proofed in the sense that the calculations won't work with input errors; the starting point (axioms, theorems, and transformations used) must be present, sometimes a confusing issue with students; the gap between the starting point and the desired result is more clearly defined for a researcher or student.

Like any computer document one can revise and edit. You could have Try 1, Try 2, etc., in separate sections and then throw away the failed tries. You don't have to keep copying the starting expression by hand. The finished document is useful and can be referred to in the future or added to. Knowledge accumulated could be passed to other notebooks or upward to packages used by other notebook documents. There are tremendous advantages in this approach.

The disadvantages are that you do need to have the Mathematica application and there is an extended learning curve. It is unlikely that you could buy it and then start using it off the self for any significant mathematical problem.