Still on track

Just a quick progress report. I am still just about keeping up with the work but I can see that next week I will probably get behind because of the Christmas holidays. Never mind.

Book 5 of mathematical logic (formal proof) was fine, just a bit boring, but there was nothing in it that caused too much hair pulling. Previously in book 4 we set about constructing a language of proof that had a specific structure that could be checked mechanically. That book was about taking a restricted set of basic symbols (the alphabet, if you like) and deciding what were terms and formulas (the equivalent of words and sentences). A term is an expression in mathematics, something like (x+y) and formulas are based around atomic formulas such as x=y that are then linked with the connectives such as or, and, implies etc. This is all very well, but a long string of symbols has to mean something (much as a sentence in English has to make sense). Meaning is obtained from interpreting the symbols (i.e. + means add!) and determining truth or falsity of formulas depending on their interpretations and the domains on which they operate. Some formulas are always true because of the logical construction of their connectives (what they call tautology) , others will depend on the given interpretation. An important idea is logical consequence - that a formula is true because it is true in every interpretation that certain other formulas are true.

Book 5 moves on to how we can construct proofs from formulas. Now we have something that looks more like sentences making up paragraphs and paragraphs making up chapters etc. With the aim of making everything systematic, formulas are arranged like lines of a basic computer program. In order to 'derive' one line from another we need certain rules of proof and these rules are introduced by analogy with everyday mathematics. For example, we need to start by making certain assumptions so there is a rule for introducing a formula that is an assumption. The difficulty for the new initiate is that it all looks like gobbledygook at the moment. You can follow the rules and do the exercises but it doesn't really make much sense yet. That will happen in the next two books when we try to get all this working for proofs in number theory.

In the mean time I have begun the geometry book GE3 on two-dimensional lattices. It's a bit of light relief to think about parallelograms and vectors.

I am looking forward to the next number theory/logic tutorial at the beginning of January but I must try and get that TMA written up so that I can hand it in then!