Hilbert's Problem Set
Copyright 2005 by Jeff Suzuki
So what's the secret to being a great filker? I don't know, but I do know this: all the great artists are prolific. This is true in mathematics (Euler with 800+ publications), music (Mozart with 500+ compositions), writing (Shakespeare's 30 plays and hundreds of poems), and art (Durer's woodprints). “Quality over quantity” is a fallacy: it's quality through quantity.
In any case, this is an example of quantity. It so happens that I can sing two songs: one of them is Yankee Doodle and the other one isn't. (OK, it's a weak joke, but don't blame me; that one is attributed to Ulysses S. Grant's) Seriously, I learned the song Barret's Privateers shortly after the birth of our twins, and subsequently wrote a filk to it (Bester's Telepaths). Shortly afterwards I was at a mathematics education workshop, and began writing this one.
Note that the punchline centers around the proof of the Riemann hypothesis. Little did I know that within days of completing the song, the Riemann hypothesis would be proven, though apparently not by “some punk kid,” but instead by someone who had been working on the problem for many years. I suppose I ought to rewrite the last few stanzas, but there is a longstanding tradition in the creative community that you never, never, never try to update your work. Given the reaction to George Lucas's updating of the Star Wars trilogy and Spielberg's disarming of the soldiers in ET, the “leave it as it was first produced” seems to be a good rule.
Oh the year was nineteen double-oh. How I wish I was in Paris now! When a letter from David Hilbert came, With problems unsolved for us to tame, Chorus God damn them all! I was told We'd solve these things for fame, not gold We would do some math, finds some proofs! I'm a broken man as I sit and fret On the last of Hilbert's problem set. David Hilbert looked around How I wish I was in Paris now! For twenty-three problems, all unsolved To test our mettle, and resolve Chorus The CH was the very first. How I wish I was in Paris now! Cohen proved that this was true, And likewise its converse was too! Chorus Hilbert asked of pyramids How I wish I was in Paris now! Could their volume be figured finitely? Dehn showed that this could not be, Chorus Hilbert's sixth conundrum asked How I wish I was in Paris now! Could the physicists write their axiom set Von Neumann said “Yeah, sure, you bet!” Chorus One by one the problems fell. How I wish I was in Paris now! Gelfond, Artin, Godel too By twenty-oh-oh we'd solved a few! Chorus The Riemann was my special task. How I wish I was in Paris now! Do the zeros all have real part one-half, The answer could change all of math. Chorus So here I am in my mumble-mumble year How I wish I was in Paris now! I've spent nearly all my life on it, And I hear it's been solved by some punk kid, Chorus
Notes
In 1900 in Paris, David Hilbert presented to an international conference of mathematicians a list of 23 unsolved problems for mathematicians to focus their efforts on.
The continuum hypothesis (CH) was the first of the Hilbert problems. Cohen proved that both the CH and its converse were consistent with the axioms of set theory.
The equality of the areas of two parallelograms can be shown using a finite dissection (this proposition, in Euclid, Book I, was what first convinced Newton that geometry was worth studying). However to show that two tetrahedra have the same volume it is necessary to use the method of exhaustion (essentially, you have to use calculus). Hilbert's third problem asked whether it was possible to prove the equality of volumes in a finite number of steps; Max Dehn proved that this was impossible.
Gelfond solved the 7th problem (is ab is transcendental if a was algebraic and b irrational---it is); Artin solved the 9th problem (generalization of the reciprocity laws); Godel answered the 2nd problem (the consistency of the axioms of logic---they are, but they won't be complete in that there will always be unprovable but true statements).
Louis de Branges de Bourcia announced a proof in June 2004 (though at last check, this proof has not yet been published; like Fermat's Last Theorem, it may take a few years before all the i's are crossed and t's dotted). Dr. Bourcia has been working on the problem for several decades, so he hardly qualifies as a “punk kid”...still, this last verse reflects the fear common to many in academia that something they've been working on for years and years will be discovered, independently, by someone else (Darwin and Wallace is the best example).