I don’t trust uncountable sets

I have a mathematical confession: I don’t trust uncountable sets. Some time ago on MathOverflow somebody asked what a reasonable definition of “infinite permutation” would be. The first answer that comes to mind is a bijection . The set of all such bijections does form a group, but not only is it uncountably generated, it [...]

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Halmos on writing and education

John Ewing wrote up a nice collection of quotes from Paul Halmos for the Notices of the AMS; let’s meditate on his words. For example: The best notation is no notation; whenever possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is [...]

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IMO 2009 and proof systems

The problems from IMO 2009 are now available. I haven’t had much time to work on them, though. There are two classical geometry problems, which I already know I won’t attempt. While I am well aware that classical geometry often requires a great deal of ingenuity, I am also aware of the existence of the [...]

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Exceptional structures

Recently Isabel Lugo asked about problems that are hard for intermediate values of some parameter, and in discussing the question I got to thinking about exceptional structures in mathematics such as the sporadic groups. In 2006 David Corfield asked about how “natural” the sporadic simple groups are at the n-Category cafe. In that discussion and [...]

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Mathematical historical fiction

Bill Gasarch is right – writing technical posts is tiring! (I’ve been trying to finish the next GILA post for days.) So I’ll share some more thoughts instead. Today’s thought was triggered by David Corfield: In the first of the above posts I mention Leo Corry’s idea that professional historians of mathematics now write a [...]

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I hate axioms

(A more appropriate title for this post would probably be “I hate Bourbaki,” but I like it as is.) I spend a lot of my free time reading research papers, usually in combinatorics; those tend to require the least background. Today I decided to read everything I could find written by one of the great [...]

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