For example, it's quite important for me to know when I might need to compute an integral, and what I'd do with it. But when it comes to banging out the symbolic manipulations, remembering tables of what integrates to what, recalling integration strategies for common kinds of integrals, etc., Mathematica is more skilled than I am, so I defer to its expertise, and haven't tried to keep any of that stuff in my head since high school.

I also generally have trouble getting Mathematica to do all of the work I need to do to come up with error bounds on numerical schemes.

Appealing to symmetry or applying Green's theorem (e.g.) will save time over "mechanical processes". Also avert mistakes and be more comprehensible.

{if you don't know what I mean: what's ∫sin x · cos^55 x · abs|x|^131 · x^444 · exp(-x^2) ?}

Moreover only conceptual understanding allows the development or use of certain things, like SVD, wavelets, eigendecomposition, game theory, theory of distributions, topology, single-crossing curves...

I'm hesitant to endorse any paradigm that aims to make math more intuitive, since at the end of the day that's a lie.

Is it a lie? It sounds like maybe you just didn't experience the intuition personally.

math ... will contradict your intuition

To me the point is to re-ground intuitions in true mathematical facts. For example I intuitively think of the reals as transcendentals ∪ algebraics or as the completion of the rationals, since I learned counter-intuitive things about the reals that disfavour the "possibly nonterminating decimal" view.

paradigm ... pace

Great distinction. Grand changes are often proposed when small changes might do.

But actually inventing that time saving machinery is different from applying it.

On the other hand, maybe it's okay that people don't really understand what math is (as mathematicians understand it). I don't see any evidence that society is falling apart due to lack of appreciation for pure math.

Einstein's special theory of relativity was mainly an intuitive interpretation of Lorenz's discovery of Maxwell's equations' invariance under Lorenz transformation. The insight "c is constant in every frame of reference", would not have been possible without Lorenz mechanically working out what sort of transformation would leave Maxwell's equations invariant.

Dirac predicted the existence of the positron solely based on a mechanical process of finding out what sort of equation satisfied the symmetries observed in nature.

My point being that many great intellectual advances have been made by people who trusted the process more than themselves, and that's one of the cornerstones of mathematical thinking: trust the process more than yourself.

  > I think "mathematics is a mechanical process for solving
  > problems without an intuitive grasp of their nature"
  > is a very good definition, with which many professional
  > mathematicians would agree. 

No professional mathematician of my acquaintance (and there are many, including three winners of the Fields medal) would agree with that. Every professional mathematician I know would say that mathematics is a creative subject requiring insight, intuition, rigorous logic and occasional luck.

Blindly turning handles just doesn't get results - the search space is way too big to chance across stuff regularly unless guided by some feel for what's going on. Listen to Wiles talk about his proof of FLT, or Gowers talk about the process of doing math.

I'm amazed that you make the claim you do, and am intrigued to know what there is in your background that has led you to that conclusion.

For reference, I'm a PhD in Pure Math, have an Erdos number of 2 (of the second type of 3), and regularly meet with groups of professional mathematicians. I don't tell you this to create a "Proof by Authority" argument, but to give you some background as to my personal experience.

When I'm investigating a physical system, doing the mathematics often tells me something qualitatively different from what I was expecting, and I find that my expectations were wrong more often than I screwed up calculations. I don't mean to trivialize what goes into doing the calculations - what I mean is that I'm constantly solving math problems that force me to revise a flawed understanding of a system.

That's what I mean by "mechanical" - I have to remain disciplined and resist the temptation to reason by analogy to something I may not even understand completely.

If you care, I came to applied mathematics via nuclear engineering.

It's a lot more complicated than that, but setting up the equations is the doing of math - solving the equations is just manipulation, and that's using, not doing. The difference is important - conflating the two leads to many misunderstandings.

I don't want to say that doing math is not creative. Far from it. But mathematicians strive to make themselves unemployed. They prove theorems once and for all, so you don't have to for each right triangle why it has this curious properties about the sum of squares.

Eliminating the need for creativity takes a lot of creativity.

What I want to say is, that the process of doing mathematics is solving and understanding problems. And since we are rarely interested in concrete solutions to concrete problems, we build up theories and algorithms to solve whole classes of problems. Building up those theories is a highly creative process, but ultimately, a problem can only be seen as `solved' if we find a mechanical process for eliminating the need for creativity.

Let me give you an example: There are lots of interesting questions you can ask about linear recurrence sequences (e.g. the Fibonacci sequence), like what happens if we add to sequences? Or when we interleave them? Or when we only pick every n-th element. Or when we want to find out the i-th, without having to calculate every element that comes before.

Solving those kinds of problems requires lots of thinking.

But if you apply even more thinking, you can come up with generating functions. They are a tool that will enable you to solve all those problems really easily. (And enable you to spare your creativity for much harder problems. That's progress!)