In my introductory post I talked about fake difficulties. It will take some time and several more posts before I can say what I really mean by that notion, but this post will get me a bit closer. So far I have mentioned that if you can’t solve a problem because you haven’t been bothered to look up what the words mean, then your difficulties are not genuine. A more interesting category of fake difficulties is a failure to grasp a few basic logical principles. I call this a fake difficulty even though for some people it is genuinely difficult; the reason I do so is that when mathematicians consider a problem to be hard, it is not for basic logical reasons. To put that another way, with a little bit of practice one can make basic logical deductions completely mechanically, and it is absolutely essential to learn how to do so. It is a simple skill (which is not to say that no work is needed), and it underlies all mathematical reasoning. Trying to understand university-level mathematics without a secure grasp of basic logic is like trying to learn long multiplication without knowing your tables — only a lot harder.
What am I talking about when I use the phrase “basic logic”? I am talking about having a good understanding of the following.
Logical connectives. The main ones are “and”, “or”, “not” and “implies”.
Quantifiers. The phrases “for all” and “there exists” come up a lot in mathematics and you have to be capable of dealing with sentences like this: for every there exists such that for every
Relationships between statements. Given a statement, you should have no trouble forming its negation, its converse and its contrapositive. Of course, for that you need to know what those three things are.