Abe Dillon wrote:
I don't disagree that infix notation is more readable because humans
have trouble balancing brackets visually.
I don't think it's just about brackets, it's more about
keeping related things together. An expression such as
b**2 - 4*a*c
can be written unambiguously without brackets in a variety
of less-readable ways, e.g.
- ** b 2 * 4 * a c
That uses the same number of tokens, but I think most people
would agree that it's a lot harder to read.
The
main concern of math notation seems to be limiting ink or chalk use at
the expense of nearly all else (especially readability).
Citation needed, that's not obvious to me at all.
Used judiciously, compactness aids readability because it
allows you to see more at once. I say "judiciously" because
it's possible to take it too far -- regexes are a glaring
example of that. Somewhere there's a sweet spot where you
present enough information in a well-organised way in a
small enough space to be able to see it all at once, but
not so much that it becomes overwhelming.
Why is
exponentiation or log not infixed? Why so many different ways to
represent division or differentiation?
Probably the answer is mostly "historical reasons", but I
think there are good reasons for some of those things persisting.
It's useful to have addition, multiplication and exponentiation
all look different from each other. During my Lisp phase, what
bothered me more than the parentheses (I didn't really mind those)
was that everything looked so bland and uniform -- there were no
visual landmarks for the eye to latch onto.
Log not infix -- it's pretty rare to use more than one base for
logs in a given expression, so it makes sense for the base to
be implied or relegated to a less intrusive position.
I can only think of a couple of ways mathematicians represent
division (รท is only used in primary school in my experience)
and one of them (negative powers) is kind of unavoidable once
you generalise exponentiation beyond positive integers.
I'll grant that there are probably more ways to represent
differentiation than we really need. But I think we would lose
something if we were restricted to just one. Newton's notation
is great for when you're thinking of functions as first-class
objects. But Leibnitz makes the chain rule blindingly obvious,
and when you're solving differential equations by separating
variables it's really useful to be able to move differentials
around independently. Other notations have their own advantages
in their own niches.
Something persisting because it works does not imply any sort of
optimality.
True, but equally, something having been around for a long time
doesn't automatically mean it's out of date and needs to be
replaced. Things need to be judged on their merits.
A good way to test this is to find a paper with heavy use of
esoteric math notation and translate that notation to code. I think
you'll find the code more accessible. I think you'll find that even
though it takes up significantly more characters, it reads much quicker
than a dense array of symbols.
In my experience, mathematics texts are easiest to read when the
equations are interspersed with a good amount of explanatory prose,
using words rather than abbreviations.
But I wouldn't want the equations themselves to be written more
verbosely. Nor would I want the prose to be written in a programming
language. Perhaps ironically, the informality of the prose makes
it easier to take in.
I spent a good few weeks trying to make sense of the rather short book
"Universal Artificial Intelligence" by Marcus Hutter because he relies
so heavily on symbolic notation. Now that I grasp it, I could explain it
much more clearly in much less time to someone with much less background
than I had going in to the book.
But your explanation would be in English, not a formal language,
right?
--
Greg
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