Why math is painful to read
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Why math is painful to read
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Re: Why math is painful to read
#2Math is such a big field that there tends to be much re-use of symbols, and the limited number of Greek letters doesn't help (no one likes multiple-Greek-letter variable names).
To a mathematician reading the technical literature from his own field, there's no issue with math notation. Problems sometimes come up when a specialist in one field tries to read literature from a different specialty -- but that's true in computer programming too.
This may come as a surprise, but math notation is much more uniform than computer-science notation, including all the efforts to create generic program listings. Mathematicians are reluctant to invent new symbols, but it's sometimes necessary.
When I read computer program source code, I often wish the programmer had summarized the meaning of the code using math notation. I guess that puts me in a position precisely opposite that of the author.
Re: Why math is painful to read
#3Math is such a big field that there tends to be much re-use of symbols, and the limited number of Greek letters doesn't help (no one likes multiple-Greek-letter variable names). To a mathematician reading the technical literature from his own field, there's no issue with math notation. Problems sometimes come up when a specialist in one field tries to read literature from a different specialty -- but that's true in c…
For me, it's because each symbol, letter or number is given a ton of weight in mathematical equations. You can't skip over and fill in the gaps later - like reading a book, for example - you have to understand each aspect before the whole makes sense.
Re: Why math is painful to read
#4Math is such a big field that there tends to be much re-use of symbols, and the limited number of Greek letters doesn't help (no one likes multiple-Greek-letter variable names). To a mathematician reading the technical literature from his own field, there's no issue with math notation. Problems sometimes come up when a specialist in one field tries to read literature from a different specialty -- but that's true in c…
To a mathematician reading the technical literature from his own field, there's no issue with math notation.
Let me guess. You haven't learned much differential geometry?
Constants vary by 2 pi depending on which definition you use. Functions might come before or after their arguments. And, of course, you leave everything that is "unambiguous" out of the notation. (Proving that it truly is unambiguous is sometimes nontrivial for the novice.) Guessing what notation a particular author is using has been known to be daunting, even for other differential geometers.
In grad school I remember sitting down with homework sets and spending longer working out what the question said than I did in solving it once understood.
That said, if you tried to actually write everything out explicitly, it would be impossible to read anything. Compact notation is a necessary evil - without it your brain simply can't think certain things. With it, it is hard for anyone else to figure out what you just said.
Re: Why math is painful to read
#5The article "How to Write Mathematics"
http://www.math.uga.edu/~azoff/courses/halmos.pdf
by the late Paul Halmos, a mathematician famous for his clarity of writing, suggests procedures for making mathematical papers less painful to read.
Re: Why math is painful to read
#6I could not disagree more with that post. Mathematical notation is to Python what English would be to Assembly Language: it is a high-level notation, designed see the forest rather than getting lost in the trees.
Take Maxwell's equations for example. After learning vector calculus and becoming familiar with its notation, you notice how much meaning you can extract at a quick glance from the four Maxwell's equations expressed in differential form. The purpose is not to obfuscate the meaning (like the author implies when he mentions the Obfuscated C contest) but rather to reveal/encapsulate as much information as possible in a way that is easy to understand at a glance. You want to know what would happen if magnetic monopoles existed? Easy to do by just adding a few extra terms that are obvious by symmetry of the equations.
And, if you go beyond vector calculus, you can see an even clearer picture by writing
F = dA (equivalent to the usual two homogeneous Maxwell Equations)
d*F = *J (equivalent to the remaining two)Re: Why math is painful to read
#7I could not disagree more with that post. Mathematical notation is to Python what English would be to Assembly Language: it is a high-level notation, designed see the forest rather than getting lost in the trees. Take Maxwell's equations for example. After learning vector calculus and becoming familiar with its notation, you notice how much meaning you can extract at a quick glance from the four Maxwell's equations e…
> Mathematical notation is to Python what English would be to Assembly Language
Which is often the problem, it looks specific, but it really isn't; it's full of implicit assumptions about what the reader should know. It's not executable, because it's a language designed for being written by hand rather than executed by a computer.
However, the teaching of math would greatly benefit from an explicit executable form that makes no assumptions, i.e. a programming language. Gerry Sussman makes this case, he's pretty convincing.
Re: Why math is painful to read
#8Math is such a big field that there tends to be much re-use of symbols, and the limited number of Greek letters doesn't help (no one likes multiple-Greek-letter variable names). To a mathematician reading the technical literature from his own field, there's no issue with math notation. Problems sometimes come up when a specialist in one field tries to read literature from a different specialty -- but that's true in c…
To a mathematician reading the technical literature from his own field, there's no issue with math notation. Let me guess. You haven't learned much differential geometry? Constants vary by 2 pi depending on which definition you use. Functions might come before or after their arguments. And, of course, you leave everything that is "unambiguous" out of the notation. (Proving that it truly is unambiguous is sometimes no…
There's something to be said for the notion of a glossary or legend... have these been considered at all? Or a notational standards body of some kind, perhaps?
Re: Why math is painful to read
#9I could not disagree more with that post. Mathematical notation is to Python what English would be to Assembly Language: it is a high-level notation, designed see the forest rather than getting lost in the trees. Take Maxwell's equations for example. After learning vector calculus and becoming familiar with its notation, you notice how much meaning you can extract at a quick glance from the four Maxwell's equations e…
> Mathematical notation is to Python what English would be to Assembly Language Which is often the problem, it looks specific, but it really isn't; it's full of implicit assumptions about what the reader should know. It's not executable, because it's a language designed for being written by hand rather than executed by a computer. However, the teaching of math would greatly benefit from an explicit executable form th…
You think that people learning math will find it less confusing if they have no context (which, in mathematics, is largely equivalent to "assumptions") for what they are learning and have to build their mathematical knowledge from first principles? Or am I misunderstanding your post?
Re: Why math is painful to read
#10Disagree strongly. Math is often the easiest and simplest thing that works for the class of problems it exists to solve.
Imagine trying to explain singular value decomposition without the notion of a matrix. No group theory, no well-formed concept of a linear transformation or function. You wouldn't be able to do it. No one even had such ideas before generations of mathematical machinery had been built.
I've come to the conclusion that there are lifer languages (/frameworks/technologies) and start-today languages. C, Lisp, Scala, and Haskell are lifer languages. (Now q, for those who've worked in finance, is even more of a hardcore lifer language.) You'll probably find them painful when you start out, but enjoy them a lot more once you really get them. Python, on the other hand, is much more of a start-today language. It's more modular, you can get up to speed quickly, and it's good enough the vast majority of the time. Also a start-today language is Java, with an IDE. PHP is a start-today language. This also applies to tools. Emacs or vi (both being keyboard-driven) are lifer tools. Unix is a lifer tool. IDEs are start-today tools.
This isn't about "better" or "worse". It's about tradeoffs: with a lifer language, it takes at least a few months before you get it (you start off hating it, because it feels weird and limited) but it becomes immensely powerful once you understand it in a deep way. Start-today languages give you a lot of power to start, but then the learning curve flattens. What's better is unclear. The lifer languages have a lot of benefits and depth, but in fast-moving disciplines like web programming I'd prefer that the frameworks be start-today rather than lifer-oriented.
Math is an extreme lifer language.