From math to policy makers

Many academia mathematicians are happiest when they can find NO real world application to their research.  They want to have nothing to do with reality and live in a world of math logic and symbols.  They bemoan the fact that a significant portion of mathematics is currently applied to other problems.  For example, Pierre-Simon Laplace created Laplace transforms to do research on Probability theory.  Now we use Laplace transforms for many things, including hearing aids and speaker phones.  But some mathematician professors work very hard to make their research applicable to something.  They work to solve a specific problem.  And sometimes they get very meaningful results.  Helpful even!

Okay, but what happens next?  How does the brilliant professor get his information to the hands of other who need it or could use it?  Well, he usually starts by publishing a paper in order to get credit for his work.  But the professor is not a publist.  (In fact he might be quite socially awkward!)  A speaker I heard this past week said something amazing about this process.  He says we (academic professors) lob our papers over the Academic Great Wall and “hope to God someone is there to pick them up and use them!”   I drew a sketch about it my feelings on the topic:

Leaving up the Scaffolding

One of the biggest critiques I had about mathematical proofs is that they are so darn hard to understand.  I want the author to walk me through how they arrived at the correct answer.  Perhaps the proof would teach me something meaningful about theorem or I would arrive at some better understanding.  But there are a great deal of very unenlightening proofs out there.  The proofs I’m talking about usually beautiful structure and clever mathematics which ends in a flourish or a bang when the theorem is proved.  But I have no idea how they came to that conclusion.

You too may find this to be true about math. (or perhaps this is why you hated geometry in high school?) I used to think bitter thoughts about some author’s proofs. Then my real analysis professor said something fascinating.  I think he may have been quoting someone else (any one out there know who he was quoting?).  A masterpiece of architecture would never be opened to the public with it’s scaffolding still up.  You are to see the final product and wonder at it’s beauty as apposed to analyzing how it was built.  For the same reason mathematicians remove the scaffolding from their proofs once they are complete.  They do the final finish work and publish incomprehensible and beautiful proofs.

So the next time you read a mathematical argument and find it incompressible- I challenge you to look for beauty in the proof.  If you can see a glimmer of charm in that proof, then the author has achieved his goal of a beautiful, elegant proof.

The Brilliant Mathematician Reads a Proof.

Please place yourself in the role of the brilliant mathematician. Now I shall teach you how to act when a rogue and unknown proof arrives at your door. (Or as my Modern Algebra professor says, “when someone hands you a problem in a dark alley…”)

Today I will hand you a proof that √2 is irrational. Which you may use as an example as you, the brilliant mathematician follow the steps below.

Step One:

Be wary of the proof. There are multiple ways to get a proof done, but thousands of ways to screw it up. Are you wary yet?

Step Two:

Begin reading the proof. First question the assumptions or underlying assumptions. Are those reasonable assumptions for this conjecture? (note to the mathematical audience: you can safely assume the basic assumptions of Modern Math hold; Euclidean Geometry, Algebra and the like are valid. Because you don’t have all year, you must finish reading this proof today)

Step Three:

You are ready to begin reading the proof. Consider the symbols they will use. For example, in my last proof I let: 0< q < n < m < π/2 It is a good use of your time to wonder why I chose q to be a companion of n and m. Maybe it’s important! (In my case it wasn’t- I didn’t like L yesterday…)

Step Four:

If the proof convinces you of something based on some assumptions and if the proof didn’t quite work out, you, the brilliant mathematician, may require yourself to re-read the pieces that have already been solidly checked. This is not a waste of time. I swear. Mathematicians do it all the time.

Step Five:

If you are convinced, then you may claim, “I find no fault in this proof” And I point out that “There are no errors” is completely different that what you have just claimed. Alternatively, you make exclaim a “Wow” or “Golly!” after reading the proof. But you should never immediately claim there are no errors.

Thus you have completed your first proof and you are beginning to understand how a mathematician is wary of the truth.