The Cardinality of the Continuum

February 24, 2010

A nice grand title to pique your interest!  After some thought and a couple of conversations, I have decided to keep this blog very much aimed at the layman; the thinking being that I don’t particularly want to write hard maths in my spare time, mathematicians don’t particularly want to read hard maths in their spare time, and non-mathematicians definitely don’t want to read hard maths ever.

My PhD supervisor recently appeared on a BBC programme about infinity, which, while good viewing, was rather over-ambitious, and so had to skip over some interesting stuff.  I thought I’d fill in some of the gaps in this post.  So what is this continuum?  Technically a continuum can be anything that is continuous – that is it goes through smooth, infinitesimally gradual transitions, and has no discontinuities or “jumps”.  But in reality the word is rarely used outside of Star Trek and mathematics.  The continuum I will write about is not the space-time continuum, but the real numbers,\mathbb{R}.  In my post onp-adic numbers, I mentioned that completeness is an important mathematical attribute for a space of numbers to possess.  The notion of continuity is even more fundamental, and people often refer to mathematics as having 2 distinct branches: continuous mathematics, and discrete – or discontinuous – mathematics.

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A bunch of people, in a room

February 16, 2010

Partly in a bid to keep the interest of the small band of readers I appear to have gained since my last post, and partly out of sheer laziness, I am again going to dispense with serious mathematics this week, and instead discuss what interesting things can be said about: some people, in a room.

In his Guardian column last Saturday, after having unleashed the full extent of his fury at some poor unsuspecting tabloid for getting a statistic slightly wrong (don’t get me wrong, the media needs more people like him) Ben Goldacre mentions in passing that if there are at least 23 people in a room, the probability that 2 of them will have the same birthday is over 50%.  This is known as “the birthday paradox”, and while not technically being an actual paradox, it is highly counterintuitive, as probabilistic results often tend to be.  The counterintuitivity comes from the fact that people tend to assume the question is: “if I am in a room with some people, what is the probability of someone having the same birthday as me?” If there are 22 other people, then this gives only 22 possibilities.  But if we don’t specify the actual birth-date, the number of pairs of people, and hence the number of possible birthday-matches, becomes23\choose 2(the number of ways of choosing 2 things from 23 things), which is 253.  It is quite easy to believe that there is some likelihood of one of these pairs of people having the same birthday.

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Mathematics in Music

February 12, 2010

Leaving aside the topic ofp-adic numbers (I feel as though I should learn more about it myself before I make any mistakes), I’m going to get back to a subject I hinted at a couple of posts back: the role of mathematics in music.  When I tell people that I studied music before switching to mathematics, they often say something along the lines of that the subjects are very similar/interconnected/both use “the left side of the brain”.   This isn’t really quite as true as seems to be commonly thought: you can certainly find a lot of mathematical patterns and structure in music; but so can you in any art, and indeed – arguably – in anything if you look hard enough!  And while I am personally averse to “side of the brain” arguments, if we are stooping to that level then I would argue that there is a creative right-side element integral to the creation and appreciation of music which is largely absent from mathematics.

However, it is true that many mathematicians are also involved in music – especially classical music – to some degree .  And it is true that the mathematical and logical structure of music is much more apparent and easier to appreciate than with other art-forms.   One of the best ways to see this is to look at tuning systems.

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