The Steampunk Satyricon

Thursday, January 22, 2009

Proof: Be Not Afraid

"For years, mainstream thinking about math anxiety assumed that people fear math because they are bad at it. However, a growing body of research shows a much more complicated relationship between math ability and anxiety."
From "A real fear: it's more than stage fright..." by Paul Ruffins, published in Diverse Issues in Higher Education, March 8, 2007
Too many people feel distanced from science because of one thing: Math. Or, more precisely, their math anxiety. It's fine to talk about theories and experiments and genius discoveries, but none of it means dick until some math is attached to it; the nasty, intimidating, complicated kind of math with lots of Greek letters and odd symbols and practically no actual numbers. Science is grounded in math, and math has as its foundation THE PROOF.
Even if you take math classes every semester in high school (shockingly, many people don't), very little time is spent in class specifically discussing the nature of mathematical proofs. Teachers show them to students all the time and expect students to learn them, but while showing proofs is often a big part of math and science classes, teachers often seem to skip over the whole "What is a proof?" thing. Nobody tells you that most of science and pretty much all of math is about proofs. It's an incredibly important subject that educators hope you'll just sort of pick up as you go along.
There are basically three parts to any proof: The hypothesis (the thing we are trying to prove is true), the arguments (the relevant ideas we will stack like bricks to see if they support the truth of the hypothesis) and the conclusion (where we learn whether or not the arguments show the hypothesis to be true or false).
The arguments are things we know to be true because they have already been proven. Bad arguments ruin good proofs. Consider this proof that penguins can fly from Antonella Cupillari's book The Nuts and Bolts of Proofs:
  • Penguins are birds.
  • All birds are able to fly.
  • Therefore penguins are able to fly.
The hypothesis is false, but, based on an invalid argument, we are led to believe that it's true. Consider an actual news event that recently occurred in the UK:
  • A girl died.
  • The girl had recently been given a vaccine.
  • Conclusion: The vaccine is fatal.
A minor panic ensued in England because of people employing this proof who never once considered whether or not the argument was valid (i.e. represented a direct causal link between the observed event and the conclusion drawn). It was not valid. They were idiots.
Like any other proof, a math proof is a logical progression of steps, but to say that no intuition or insight is involved is wrong. There's a reason some people are better at it than others. It's why we are in awe of geniuses like Russian mathematician Grigory Perelman who solved the Poincaré conjecture which, according to the Clay Math Institute, is one of the millennium's seven most difficult math hypotheses.
It's about understanding that facing real science means facing math, facing proofs and doing it fearlessly. Genius is NOT a prerequisite. I'll prove it.