Mathematical structures, 1

The course is now up and running, and the first set of notes is posted.

I’ll start this post by reiterating something I said in the comments to the preceding post. My view of mathematics is like the mathematics department at the University of Queensland where I was an undergraduate: you come in at the ground floor; you may go down to the basement (set theory, logic, foundations of mathematics) or up to higher floors (algebra, analysis, and thence to group theory, functional analysis, and so on). At the end of the day you return to the ground floor. One of my colleagues reminded me of this, also from Four Quartets:

… the end of all our exploring
Will be to arrive where we started
And know the place for the first time.

What we bring in to the building when we enter is the mathematics that we have had since earliest childhood: the number line, basic properties of counting and measuring, and so on. At the end of the “day”, we have a broader and deeper understanding of these topics. This metaphor tells us that numbers are not unfamiliar to us, and it is quite legitimate in a course like this to start with some assumptions about them, even if these will be refined later on.

The notes

The introductory notes start off by explaining what a maths degree is good for, why it is a good thing to have (I pointed the students to JobsRated, where mathematician came out as best of the 200 jobs ranked, or the Oxford College, Emory University wiki, which discusses careers available to mathematics graduates. Why is this? As well as learning various facts about mathematical objects, our students should learn to reason, to make a convincing argument and present it in public. That is what I hope they will get from the course.

But what is mathematics? The dictionary definition, according to Chambers’, is

mathematics n sing or n pl the science of magnitude and number, the relations of figures and forms, and of quantities expressed as symbols.

This tells what mathematics is, but not how to do it. For the latter, I prefer the statement of Paul Erdős:

The purpose of life is to prove and to conjecture.

While I was writing the first section of notes, I happened to buy a Big Issue from a seller outside Baker Street station. I was wearing a CanaDAM T-shirt; he asked me if I was Canadian, and we got into conversation. I told him that I was a mathematician. He thought that mathematics was the ultimate dry and logical subject, and was quite amazed when I told him, and gave examples, that mathematics has the two features that Erdős highlighted. As well as the strict logical business of getting a proof that works, there is the intuitive, creative business with coming up with a good guess in the first place, and of coming up with an idea about how it might be proved.

Anyway, I take Euclid’s theorem that there are infinitely many primes as an example of an accessible proof. I give the proof, and then see just how watertight it is. (The answer is, not very watertight; the most obvious hole is the statement that every number greater than 1 has a prime factor.)

The last part of the notes is a description of some of the words mathematicians use (definition, theorem, etc.), and some common notation, including the Greek alphabet. I have always noticed that a student trying to read out a piece of mathematics and hitting a squiggle whose name they don’t know loses the thread of the argument; I want to avoid this if possible.

The lectures

I was a bit nervous before the first lecture. Rather, a bit more nervous than usual; a bit of nervousness is necessary to fire me up for a good performance. Apart from anything else, it was a larger audience than I have lectured to before, and I planned to start on the computer, showing the students the module web page, and talking them through lectures (how to behave, the fact that questions are encouraged), office hours, coursework, notes on the web, resources (books and web pages), and many other things.

In the event, it all worked fine. The first lecture was one of the most successful I have ever given, in terms of getting student interaction going. One student in the front row had seen the BBC Horizon programme about Infinity, and asked whether it made any sense to talk about infinitely many primes if, as Doron Zeilberger suggested, there are only finitely many natural numbers.

The students’ questions helped me to see where the notes needed improvement.

One of the students asked whether the material would be on the Queen Mary IDE. I said no, it would be on a public website, and added that all the world would be watching us. Saying this, I felt a bit like the Greek warriors at the siege of Troy in the Iliad, doing their best while the gods sat around watching (and interfering).

Finally

One final thing worth saying here: this module owes a lot to many colleagues and friends, but especially to Thomas Prellberg. It was his idea at first, and he persuaded me, somewhat against my will, to put it on. I felt (and still feel) that the things I am teaching are so important that they should not be “ghettoised” but should form part of the DNA of a mathematics education, present in every course the students take. In fact, I think that we have (at least partly) achieved that; many colleagues were already doing this, and some may be encouraged by what is happening here.

Anyway, Thomas has kept up the pressure, and done a lot to help get the module launched, introducing the small-group tutorials and masterminding the reorganisation of material in other first-year modules. Now it is up to me and my students.

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About Peter Cameron

I count all the things that need to be counted.
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One Response to Mathematical structures, 1

  1. Jon Awbrey says:

    It was always my nature to begin with the axioms and work backward.

    I guess that’s why I spent so much of my life in subterranean realms.

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