Mathematical reading

A big part of the practice of mathematics is the struggle to put our own internal mathematical universes into order, trying to fit new understanding in with old.

To do this by reading what other people have written is difficult, because their way of structuring things can be very different from the way you think.

Here are some of the things I keep in mind when trying to read things other people have written.

1. Always know why you are reading what you are reading and what you want to get out of something before you read it.

   It's hard to read maths, so you shouldn't have to motivate yourself at the same time. I would suggest that in your explorations of the mathematical world, you follow your own sequence of questions and interests. You should, therefore, have a definite idea of why the next book/paper you read is worth reading - because it promises to answer some question you have.

   Q. Why are you reading that book about Hodge structures?

   A. "I always wanted to know about Hodge structures."

   How will you figure out when you "know about Hodge structures"? (1) When you've seen the definition? (2) When you've seen a few examples? (3) When you have finished the book and have a complete command of the theory? With this attitude, you're most likely to achieve (1) and then forget it before you hear the words used again.

   A. "My supervisor said I needed to know about Hodge structures."

   Did s/he say why? See previous comment.

   A. "I heard that Hodge structures could be used to study moduli spaces of varieties, and have a vague idea that the words 'period map' are involved. I want to know how this works."

   Then you might get somewhere. But only if the book you're reading goes that far. Which brings me to...

2. Read backwards.

   If you know what you want to get out of reading a book/paper, it helps to know that the book/paper will explain it (or at least something closely related which you can modify for your own purposes). So look through the book and see where it is explained.

   It's possible (though unlikely) that the explanation will be self-contained and near the end of the book, in which case you have saved yourself potentially months of wading through preliminary reading.

   More likely, the explanation is dependent on a strict subset of what the book covers. You can figure out which subset by working backwards. Read the explanation/proof and do not be put off by the fact that you don't understand most of the words. Make a note of what seem like the most important words/facts that you don't understand and look back through the book to see where they're explained (again, probably using words you don't understand). Iterate this process until you do understand.

   Of course, most books are not written to be read this way and develop notation from the outset, so reading backwards is hard. It's important to distinguish between notation you don't understand (which is usually easier to fix) and concepts you don't understand.

   If you read this way, by the time you get down to something you can understand, you will have at least an inkling of how it's going to be used later and you can keep this in mind as you proceed forwards. Which brings me to...

3. Always have an example in mind that you want to understand while reading.

   While you are reading, continually refer back to this example. See if you can simplify what you are reading in the special case you're interested in (usually notation is developed to allow the greatest possible generality and may be unnecessarily complicated for any given situation). If you don't see the relevance of a particular section to your particular example, skip it. You can always come back to it if you realise it was relevant (and then you'll know why).

   BEWARE: If you read this way you will often end up with certain assumptions about how things work which are only artefacts of your particular example. This usually comes out in the wash, when you apply your newfound understanding to a new example and start making deductions which you realise are false. So it's good to (a) be aware of this (and always question your assumptions), (b) read the same thing a couple of times with (very) different examples in mind.