With the advent of the most sophisticated computers and software, I find over the years that the appetite of solving problems analytically is virtually non-existence among students both with the UG and PG level particularly in applied mathematics. The problem is acute with the students of mechanics who apart from their class notes are unwilling to attempt any unseen problem from any basic text book of mechanics. Even, the teachers feel it is absolutely meaning less to sit and think about solving anything analytically when there are computers to solve those. Almost 90% teachers are keen to complete PhD guidance within 3 years and the easiest path is to go for the computational domain with the help of computers; Being a students of mechanic it hurts me more and I would like to know the scenario in other countries, is it not true that a pillar of mathematics viz mechanics is getting a jolt? When will we realize that idealized analytical mechanics may never replace computational mechanics which encompasses broad range of parameters but it would be a disastrous for applied mathematics vis-à-vis mechanics if we loose the skill of analytical thinking?
I agree with you that analytic skills are declining among the student body, but there are a number of reasons. For one, research in some areas is getting much more complicated than what it used to be in the past, which is due to an increase in computer power as well as advances in the theoretical aspects. As a result, much more reliance on computers is needed. I think it would be impossible for analytic skills to disappear completely though because even programming required for research requires knowledge of analytic methods and how to apply them.
I'm not sure if it's just me but I have noticed that there has been a shift in what mainstream science is about. It used to be about gaining knowledge, but it seems to be shifting more towards innovation, or discovering new ways of solving problems.
In fact, this trend is here to stay, and unfortunately we can hardly do anything about that. Indeed, not just mechanics, this trend is visible in all branches of mathematics in which computational matters are involved. Take the case of Statistics for example. Earlier, for data analysis, the users from outside the domain of Statistics needed to know what sort of analysis would be proper to analyze their data. Now software are available. Therefore, without actually trying to know whether a particular statistical tool would be proper, they just use the software. Unfortunately, most of the statistics fraternity too do the same thing now!
I don't know how it is world-wide but one of the main problems I see in the US is the decay of literacy. Even the better students seem to have less basic language skills. It is tough to analyze any problem without a solid understanding of language. In my experience, students who struggle with math are students with poor reading and writing skills. The next time you have a student who says "I just don't get it", have them read a paragraph from their textbook which has mathematical notation. I would bet that they cannot read it in a meaningful way.
All the voices above, unfortunately, seem very well based on reality. My impression is even worse: more and more people are using software like a "monkey equipped with a razor", they simply don't know what they are doing. On the other hand, there is no way back from widespread computers' usage. As for today, there are still the people producing software, who know what they are doing and why this very way (at least I hope so). Will it be true tomorrow? Who knows ...
But going specifically to mechanics: there are problems requiring optimization of non-convex (even non-differentiable, like "min" or "max") functions on constrained domains. Are they really solvable by classical methods? Yet interval math is of great help here. But the truth is your life is too short to solve such problems manually, you have to resort to computers.
@Marek.... Today I had the opportunity to defend a research proposal that I submitted to some funding agency here at my country, India. I work on Tsunami wave dynamics, the so-called experts could not able to realize that there are plenty of things that have to be done analytically to achieve a benchmark. The experts unfortunately have fallen into the loop of ignorance dominated by software ... nevertheless I tell you one thing, it's my conviction, that analytical skill is the basis of Mathematics vs-e-vis Mechanics , indomitable spirit of Mathematics, you cannot thrive ignoring your spirit..
@Arghya: By no means I am against analytical methods. Even more: they should be definitely preferred whenever they exist and are applicable. They give insight into problem in wide range of parameters, often even in infinite range, in contrast to numerical computations which only describe a given particular case. But your voice only adds to the sad conclusion: the ignorance proliferates all over the world.
A related problem is that there are highy skilled analysts with minimal technical skills. Also, there are many tech-savy individuals with little to no analytical skill. I'd rather hire the former and train them in technical skills than hire the latter and be doomed because they will never have high level analytical skills. Unfortunately, measuring analytic skill is not something many HR people are good at.
@Geoff .... so true! Today in fact it seems so correct for me. I have to knock some other door to get fund and at least a six month leave from my normal duty to complete a research work which so earnestly I wish to get done.
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