Spivak and Sussman have made excellent cases as to the problem of notation in mathematics. I have long had much the same opinions as the grand parent so was delighted to see such luminaries in agreement. Now, mathematics is compact but that is not the problem. The problem is ambiguity. You can with time learn to pack the and unpack the extremely dense notation and the upfront costs are worth it but the ambiguity (not to mention differing conventions across branches) is an inexcusable mess. Calculus for example is replete with the abuse of variable binding which is just cruel to the beginner. Quoting: http://mitpress.mit.edu/sicm/book-Z-H-5.html#footnote_Temp_4
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1 In his book on mathematical pedagogy [17], Hans Freudenthal argues that the reliance on ambiguous, unstated notational conventions in such expressions as f(x) and df(x)/dx makes mathematics, and especially introductory calculus, extremely confusing for beginning students; and he enjoins mathematics educators to use more formal modern notation.
2 In his beautiful book Calculus on Manifolds [40], Michael Spivak uses functional notation. On p. 44 he discusses some of the problems with classical notation. We excerpt a particularly juicy passage:
The mere statement of [the chain rule] in classical notation requires the introduction of irrelevant letters. The usual evaluation for D1(fo(g,h)) runs as follows:... This equation is often written simply...
Note that f means something different on the two sides of the equation!
3 This is presented here without explanation, to give the flavor of the notation. The text gives a full explanation.
4 ``It is necessary to use the apparatus of partial derivatives, in which even the notation is ambiguous.'' V.I. Arnold, Mathematical Methods of Classical Mechanics [5], Section 47, p. 258. See also the footnote on that page.
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In my experience I think notation would not make the life of a student much easier if you're learning things at the level Terry Tao writes in his blog, he's not writing college level math for engineers (ie. Calculus, LinAlg, Basic Fourier Analysis and some other topics) but Graduate level math for mathematicians and people interested in pure and/or applied mathematics. Notation is not the problem when you're having a hard time studying Functional Analysis, Algebraic Topology, Advanced Probability (using Lebesgue integral) or trying to understand the proof of the Prime Number theorem.
The problem actually is that notation preference is a matter of personal preference, the physicists love the bra and ket notation I think it is very confusing, maybe because I'm not a physicist. I think any time someone comes with new notation for settled things they just turns the matter worse.
Partial Differential Equations is a perfect example how these things works, there's generally different notations being used by mathematicians, physicists and engineers, at least in my experience, and because every couple of years someone comes with a new notation to "simplify" everything to his field of study, but a introductory note of a page or two in any book is enough to explain the differences between the notations.
The grandparent argument is valid for some math below graduate level and some confusing bits in advanced mathematics but it's generally not a problem for anything above. I actually think notation is a problem for students that are not that much interested in mathematics, high schoolers and some engineering students that I saw in my life generally are confused why some things are written the way they are without any justification. This is a problem with learning methods and if you just give a new arbitrary notation to these students I believe the problem will persist.
I also think that trying to reboot the entire mathematical notation to fit areas such as aerospace control theory, algorithm complexity theory, abstract algebra, biostatistics and thermodynamics, among other fields would probably result in failure, like creating a universal language like esperanto that would never be used.
Right, that's why the focus is on introductory classes. As I noted, yes a sufficiently motivated individual will get used to the notation in time but that doesn't mean things are okay.
These things that seem minor to the expert actually make a big difference before chunking is achieved and can hinder all but the most motivated. If you are taxing short term memory by using unhygenicly bounded variables then no, it is not just a matter of who is interested. If you are not pointing out the difference between higher order functions and regular functions nor separating the notion of function from application then you are causing unnecessary representational couplings that create a lot of friction. These things have real cognitive and physiological costs. The design should streamline thought for expert and novice alike, it should not be arbitrary. And in the absence of anything better we cannot say that the issue of notation is not a problem at high levels. Sure learning is no longer the problem but what of adroit mental manipulations? I tend to agree with Alfred Whitehead who said
"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.".
We can't lament the lack of scientists and engineers on one hand and not try to do reduce uptake friction on the other. There's a real problem with math education if the experts are not trying to relate to the ones who are struggling.
btw braket is a wonderful notation in my book and I'm not a physicist, it is an elegant way of writing sparse vectors.
" 1 In his book on mathematical pedagogy [17], Hans Freudenthal argues that the reliance on ambiguous, unstated notational conventions in such expressions as f(x) and df(x)/dx makes mathematics, and especially introductory calculus, extremely confusing for beginning students; and he enjoins mathematics educators to use more formal modern notation.
2 In his beautiful book Calculus on Manifolds [40], Michael Spivak uses functional notation. On p. 44 he discusses some of the problems with classical notation. We excerpt a particularly juicy passage:
The mere statement of [the chain rule] in classical notation requires the introduction of irrelevant letters. The usual evaluation for D1(fo(g,h)) runs as follows:... This equation is often written simply...
Note that f means something different on the two sides of the equation!
3 This is presented here without explanation, to give the flavor of the notation. The text gives a full explanation.
4 ``It is necessary to use the apparatus of partial derivatives, in which even the notation is ambiguous.'' V.I. Arnold, Mathematical Methods of Classical Mechanics [5], Section 47, p. 258. See also the footnote on that page. "