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Archive for December, 2017

Mathematical Humor

Many years ago I picked up a copy of Martin Gardner’s book A Gardner’s Workout.  In his book, Gardner reviews a collection of writing by the late Ralph Boas, a former professor and department chair at Northwestern University.

In his review of of this collection, Gardner says:

In Princeton’s Fine Hall, Boas recalls, someone once posted a “Scale of Obviousness”:

If Wedderburn says it’s obvious, everybody in the room has seen it ten minutes ago.

If Bohnenblust says it’s obvious, it’s obvious.

If Bochner says it’s obvious, you can figure it out in half an hour.

If von Neumann says it’s obvious, you can prove it in three months if you’re a genius.

If Lefschetz says it’s obvious, it’s wrong.

What I love about this little bit of mathematical humor is that the idea of an “obvious” proposition in mathematics is so inherently subjective.  In fact, in grad school (when I first read this) we had our own joke, which was that if we didn’t know how to justify some part of the proof we were working on, we should just write “clearly.”  Or “the proof is left to the reader.”

On top of the humor is the historical nature of this little joke.  The people mentioned are all well-known mid-century mathematicians from Princeton University faculty.

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Back around 2000, I found a copy of Neal Koblitz’s text A Course in Number Theory and Cryptography at the Borders bookstore in Bangor, Maine.  I only worked my way through the first chapter, but was fascinated with these ideas.  I found Professor Koblitz’s website which, at the time, had a tutorial section on finite fields and elliptic curve cryptography (this may have been on the Certicom website, I can’t remember now).  I moved on to other forms of digital cryptography, like the Diffie-Hellman Key Exchange and RSA Cryptosystem, but always appreciated Prof. Koblitz’s work.  Recently, we dressed up for Halloween as a number and I chose to be the number 4.  As part of my costume, I drew the addition table for the Galois Field of order 4GF(4)=^{GF(2)[x]}/_{x^2+x+1}, and did a lot of thinking that week about the element a, which was defined as the root of the equation 0=x^2+x+1 in GF(2)

This past week, I decided to look at the mathematics behind Bitcoin and blockchain, and lo and behold, it is Finite Fields and Elliptic Curve Cryptography – I don’t know why it took me so long to find this out, but now I’m excited about these topics.  I am a little skeptical about the current “Bitcoin bubble.”  I’m not sure that these valuations are sustainable, but from everything I’ve read, the blockchain algorithm behind Bitcoin is revolutionary and the mathematics is “supercool.”

Here’s a graph of the equation y=x+1 in GF(2).

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