Timothy Gowers, University of Cambridge mathematician and Fields Medalist, is teaching an analysis class this term, and fortunately for me, he’s blogging about it. Analysis IA is part of the first-year math major sequence at the University of Cambridge, and it is a rigorous approach to calculus at the undergraduate level. I am teaching a similar analysis class this semester, and although Gowers says that his posts are for his students, they’ve been useful for me as well. I have taught this class before, but it’s always good to see how someone with much more experience than I have approaches the subject.
In his first post about the class, Gowers gives a big picture overview of the course:
“One of the messages I want to get across is that in a sense the entire course is built on one axiom, namely the least upper bound axiom for the real numbers. I don’t really mean that, but it would be correct to say that it is built on one new axiom, together with other properties of the real numbers that you are so familiar with that you hardly give them a second’s thought.
If I want to say that more precisely, then I will say that the course is built on the following assumption: there is, up to isomorphism, exactly one complete ordered field. If the phrase ‘complete ordered field’ is unfamiliar to you, it doesn’t matter, though I will try to explain what it means in a moment. Roughly speaking, this assumption is saying that there is exactly one mathematical structure that has all the arithmetical and order properties that you would expect of the real numbers, and also satisfies the least upper bound axiom. And that structure is the one we call the real numbers.”
I also like his section about the difference between abstract and concrete in mathematics. The emphasis is mine.
“Up to now, you will have been used to thinking of the real numbers as infinite decimals. In other words, the real number system is just out there, an object that you look at and prove things about. But at university level one takes the abstract approach. We start with a set of properties (the properties of ordered fields, together with the least upper bound axiom) and use those to deduce everything else. It’s important to understand that this is what is going on, or else you will be confused when your lecturers spend time proving things that appear to be completely obvious, such as that the sequence 1/n converges to 0. Isn’t that obvious? Well, yes it is if you think of a real number as one of those things with a decimal expansion. But it takes quite a lot of work to prove, using just the properties of a complete ordered field, that every real number has a decimal expansion, and rather than rely on all that work it is much easier to prove directly that 1/n converges to 0.”
I sometimes struggle to articulate the big picture to my students effectively, and Gowers is great at making that broad vision clear.
The rest of the posts also have some similar gems. From How to work out proofs in Analysis I:
“For some reason, Analysis I contains a number of proofs that experienced mathematicians find easy but many beginners find very hard. I want to try in this post to explain why the experienced mathematicians are right: in a rather precise sense many of these proofs really are easy, in the sense that if you just repeatedly do the obvious thing you will solve them. Others are mostly like that, with perhaps one smallish idea needed when the obvious steps run out. And even the hardest ones have easy parts to them.”
This post also talks about some of the common “moves” we use when we do basic analysis proofs in the context of teaching a computer to do them.
Gowers mentions that his colleague Vicky Neale wrote blog posts after each analysis class when she taught it last year. I have not perused them yet, but I look forward to getting some ideas from them. In the past, I have also found Terry Tao’s blog helpful for understanding and teaching analysis, particularly the measure theory notes.
I love teaching analysis, and I’m very glad that I get to benefit from Gowers’ (and Neale’s and Tao’s) experience, especially when I’m trying to explain how individual theorems fit into the subject as a whole.
Evelyn, I’d like to ask you some questions about this statement:
>there is exactly one mathematical structure that has all the arithmetical and order properties that you would expect of the real numbers, and also satisfies the least upper bound axiom. And that structure is the one we call the real numbers.
My understanding is that calculus can be built up in two ways – the conventional way with limits and so on, and then non-standard analysis, which includes infinitesimals. I would have thought (I don’t have any recent experience studying analysis) that the number system which includes the infinitesimals is a different “mathematical structure that has all the arithmetical and order properties that you would expect of the real numbers, and also satisfies the least upper bound axiom”, but it is different from the reals. What am I missing here? (I studied surreal numbers a bit, long ago. Do they not satisfy the least upper bound axiom?)
I don’t know much about nonstandard analysis myself, but it’s my understanding that something goes wrong with the least upper bound property. The infinitesimals don’t have the least upper bound property, but I don’t know if there’s a way around that in nonstandard analysis. It is my understanding that nonstandard analysis is in some sense the “same” as standard analysis, i.e. the same theorems are true in each one. So perhaps the “up to isomorphism” part of Gowers’ quote comes in here.
The transfer principle says that first order statements about the real numbers are also valid for the hyperreals. However, the least upper bound axiom is not a first order property, since it involves quantifying over arbitrary bounded subsets of the reals.
I haven’t done Analysis in a while, but the professor who taught me Analysis mentioned that he was going to teach the Least Upper Bound Axiom as a result, rather than an assumption. I forget exactly what he said was interchangeable. It had something to do with making a different assumption, which lead to Least Upper Bound. However, you still needed to make an axiom to replace it, so it’s not like he was working with less Axioms than a standard Analysis course.