[Retric]

I agree it is not as natural as on paper it's only unnatural because you did not spend 12+ years doing it that way. I spent enough time with my old ti86 that it became vary natural to express math on it despite the poor interface.

  I have no problem using this machine to say:
  abs(4/3 - 1) = abs(3/3 + 1/3 - 1) = abs(1/3)
  abs(3/4 - 1) = abs(4/4 - 1/4 - 1) = abs(-1/4)
  abs(1/3) > abs (-1/4), so 3/4 is closer to 1 than 4/3.

[mechanical_fish]

Keep in mind that this is a toy example. The original essay used it because it was a toy, and therefore didn't break up the flow of the writing or scare away the mathematically unsophisticated (this is a common problem faced by essay-writers who are trying to explain complicated things).

Try something harder - at least, say, the quadratic formula. Or, if you really want to appreciate what we're getting at, try a chewy example, like Maxwell's equations:

http://en.wikipedia.org/wiki/Maxwells_equations

These are expressible on the calculator command line, and first-year grad students with Mathematica licenses type them out fairly often, but you're going to discover that Knuth spent a decade of his life inventing TeX for a very good reason. Without proper notation it is hard to reason about math.

You can, of course, use the better class of notation editors to get your computer to display math in proper notation, just as the Wikipedia authors did, but it's a bunch of fiddly work, rather more work than writing out the math by hand. And then you find that you can't make very faint tickmarks or cross-outs or circles on your math. You can't easily draw arrows connecting one line to another.

A useful side-effect of pencil-written mathematics is that the intermediate steps are there in front of you. Watch a professor talk you through a derivation on a chalkboard. Observe that, despite the fact that chalkboards can be erased, using technology that has been available since prehistory, the professor rarely transforms equations by erasing the old ones and replacing them with new ones in-place. That's because recopying the equation after every one or two transformational steps leaves behind a changelog that represents your train of thought. When the problem is almost done but you're trying to track down the sign error you'll be grateful for that.

And, yes, computers have undos, so you can rewind and fast-forward your math. But that's inferior technology for thinking about problems. Ask Tufte: The secret to reasoning about data or logic is to spread it out in front of you in as flat a manner as possible, so that you can move from step to step using nothing but your eye muscles, or defocus your eyes a bit and view the whole problem space in the abstract.

Giant desk-sized iPads may one day render pencils and paper obsolete for serious math, but I'm not convinced that will happen in my lifetime. The hardware is expensive, the software is far more expensive, and paper is cheap, and scanners for digitizing paper are cheap.

[Retric]

People complained just as hard when transitioning from slide rules to calculators and they have some valid points, but the net gains where also clear. It's easy to make a bad interface but there are plenty of great Math interfaces that keep a listing of all previous steps above what your working on both how you enter it and show you how things would look like on a blackboard. There are a also plenty upsides like the ability to cut and paste lines so you can avoid a lot of stupid mistakes like dropping signs. But, far more important from and educational perspective is a students improved comfort using a computer to do advanced math vs. the near phobia that you and many others apparently have.

[mechanical_fish]

Oh, it's not a phobia. Just skepticism. ;)

I don't doubt that one can, and that we will, build a computer system for manipulating higher mathematics that's so much better than a stack of paper, a pencil, and a decent eraser that you won't even own the paper. What I doubt is that it's done yet. But I haven't exactly been looking for it, so maybe I'm wrong. Certainly, once it's done there won't be any problem selling it to me (except for the sad fact that I no longer manipulate equations on an everyday basis).