The Axiom of Choice

April 22, 2010

Yes…I’m still alive and blogging.  Just been very busy, as it is coming up to exam-time.  Thankfully I think I can safely say that my exam-sitting days are behind me, but there are a lot of anxious students happy to pay good money to someone who will help them pass theirs!  Anyway, I recently read an interesting blog post about the axiom of choice, and I thought it would be a good topic to give my usual treatment (that is an attempt to explain it to an interested and patient non-mathematician).

Now, unlike other sciences – in which theories and models are often proposed, used for while, and then modified or just plain discredited by subsequent discoveries – mathematics is constantly building on what has gone before.  If sciences were buildings, then mathematics would be a vast complex structure that is constantly being added to: sometimes the foundations would be strengthened, sometimes an extension or a whole new wing would be added, but nothing would ever be demolished.  Physics, on the other hand, would be a lot of separate buildings, some of which have long ago been abandoned, and most of the rest of which are in a constant state of being torn down and rebuilt in the modern style!  This is part of what drew me to the subject in the first place; the fact that if you prove something mathematically, it is definitely true, and will never be disproved.  We still use theorems proved by people that lived thousands of years ago, and it is reassuring to know that anything we prove now will still be being used by people in thousands of years.

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Quadratic Equations! (or: what do mathematicians actually do?)

April 4, 2010

When I tell people that I study mathematics they tend to have one of two reactions:

1. They make some impressed-sounding noise, or mention that they were terrible at maths at school, and then quickly make it clear that they wish to change the subject.

2. They are genuinely interested, and want me to tell them exactly what it is that I study.

The second reaction is the one that I fear most!  And at this point it is usually me who tries to change the subject.  I have an ongoing competition with myself to increase the length of time which I can spend explaining my research to someone before their eyes glaze over and their body language starts to say “I want to be somewhere else now”.  I am currently up to about 15 seconds.  And the subject I am currently working on (somewhere between graph theory and galois theory) is fairly accessible compared to some of the more exotic branches of mathematics!

The main problem in explaining pure mathematics to a non-mathematician is the level of abstraction involved in the subject.  Most people’s view of mathematics is that it deals with numbers, and it is hard for people to imagine what exactly it is that mathematicians do…add and subtract really big numbers?  Many seem to find it difficult to imagine how it could be that all the mathematics that could be done hasn’t been done already.* People rarely encounter abstract mathematics before university; and for good reason, as the transition from dealing with concrete quantities in a familiar setting, to treating those quantities and that setting as merely one very special case in a vast world of abstraction, can be rather bewildering.

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