Despite being moderately good at mathematics, even managing to scrape together a PhD, there are certain topics that are always brick walls to me so that I find it hard to get started even at the most elementary level. In algebraic topology I always had problems with spectral sequences but they're not so elementary and are notoriously tricky. But in logic I can barely get off the ground. Here's an example of a sentence from an introduction to linear logic that baffles the hell out of me "One of the most important properties of the proof-rules for Classical Logic is that the cut-rule is redundant". This is one of the most ridiculous things I have read in mathematics writing. If it's redundant then don't study it. Excise it from the list of derivation rules and don't bother with it every again.
I'm sure than when set theorists first tried to write down the axioms that became ZF they found lots of redundant axioms. Over the years they were whittled them down to the list we have today so that I bet you can't even name the axioms that were jettisoned for redundancy. Not so in "Gentzen style" logic. Every document ever written on the subject seems to introduce this rule and then with a flourish they show how it can be eliminated. They develop the whole subject and then proceed to demolish the earlier work by showing how they can rewrite everything they did earlier without using this rule. The only explanation I can come up with is that authors of books on logic are paid by the word and that this allows them a few extra chapters for very little work.
Of course the problem here must be me. I'm sure there's a perfectly good reason for harping on about the cut rule, I just don't see it. And I think this points to a difficulty with trying to read mathematics texts outside of academia. When you're a student you're sharing extratextual material all the time. People tell you that such and such a result is interesting because it has an application somewhere and then you go and read the formal details knowing what the motivation was. Or someone will give an informal seminar where they write equations on the board, but speak much more informally between the equations. But most mathematics texts hide this material from you and present the bare facts. This is fine when you're in an academic environment, but I have to confess to finding it difficult when working on my own.
One thing I'd love to see online is the equivalent of the reading seminars we did during my PhD work. Each week we'd read a chapter and then have a seminar to discuss what we'd just read. Does anyone do this? Blogs seem like a great way to do this but I've seen no evidence of groups working like this.
Anyway, I shall try to persist until I see the light...
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