Ask HN: How can I learn to read mathematical notation?

During my time as a PhD student in philosophy, I had the luck of having a mathematician and physicist turned philosopher as my supervisor. He helped me a lot in understanding notation and improving my own mathematical writing style (~ getting silly notation ideas playing around with LaTeX out of my head).

Notation heavily relied on conventions and these differ from field to field. Mathematicians are usually very clear and define everything. They have to be easy to understand, because what they are talking about gets very complicated very fast. Unfortunately, in other areas like logic-oriented theoretical CS there is an unhealthy focus on 'notational precision' (plus shortcuts, making things worse) rather than being understood that can make papers rather incomprehensible at first.

2. Look for textbooks with good mathematical prose.

Get some recommendations. Some good introductory books are relatively short, self-contained, define everything, and avoid symbols in favor of English in many places. Try to get one of those in the area you're interested in.

3. Check your reading style. (kind of obvious advice that is often given, but maybe still helpful)

You may be reading the texts in the wrong way. I did that very often and occasionally still do. Reading a mathematical text takes way more time than any other text and requires active participation. Do the exercises and stop every so often to check your understanding. Constantly use pen and paper and check everything. Draw diagrams.

This is something the MOOC crowd often overlook. It's based on a misunderstanding of what a university gives you: university entry doesn't grant you access to some members-only club where they hide away the knowledge to keep it from the plebs (ignoring the open access movement, at least). Instead it grants you access to an environment in which you can learn effectively.

The idea that MOOCs will threaten the existence of universities is absurd. It's already possible to get access to the sum of human knowledge without being enrolled at a university. It's always been possible. They're called books.

But this stuff is hard, and attending a high-quality institution gives you a huge advantage in learning the material, compared to being self-taught. MOOCs may grant everyone access to lectures, but just as important is learning with high-quality peers. A MOOC is not the equivalent of being in a class with other smart and well-motivated students.

Again, it's always been possible to learn this stuff independently, but your odds of succeeding are far lower that way. It's great that we have lots of freely available learning material online, but I really don't see that MOOCs are going to turn the world on its head the way some people seem to think.

I'm ignoring accreditation here, of course. If you want to be a surgeon, you obviously do not have the option of teaching yourself, but in terms of learning maths/technology (and assuming knowledge - and not a certificate - is your goal), books have always been there. MOOCs are books++, not universities--. (With apologies to the pedants who might prefer the pre-positional operators.)

After enough slow progress (“one page per day” can easily be speed reading), parts of what you are reading become what mathematicians call ‘trivial’, and your reading speed of similar texts increases.

I think there’s an analogy with ‘reading’ a chess position. If you watch the ongoing Carlson-Caruana match on https://youtube.com/watch?v=DgvqBjrusIA, you’ll notice that the commenters can easily go through three or four variants in a minute, and call one position an obvious draw, another clearly winning, etc. The reason they can do that is that they have looked at thousands of similar positions, and remember the essential parts of them.

* Read formulae aloud, using natural languages for symbol names (it is not "sigma of k from i to j", it is a "summation of terms with k from i to j").

* Look at the proofs/construction of these formula and understand them.

* Look at translations of the formulae (applied physics textbooks, more often program code); sometimes it helps to work your way back to the formula from a starting point you are more comfortable with.

* Understand that making up symbol meanings ad hoc is one of the perks of mathematicians - you can find fights within a single math department on how notation should be.