Monday, February 20, 2006

Dilbert, mathematics, proofs, patterns, and biases

Dilbert: I need help making unrealistic assumptions to support a business case for a bad idea.

Dogbert: Easy. There's a hole in the back of our wardrobe closet that leads to a magical world of preposterous business assumptions.

Dilbert: We don't have a wardrobe closet.

Dogbert: Assume we do.

(Dilbert, 2/16/06 , copyright Scott Adams, Inc.)

Human beings are programmed to search for and identify patterns. We're pattern matching animals. Much of mathematics consists of formally and rigorously proving the existence of the identified patterns as they are defined by theorems.

How does Dogbert's statement, "assume we do" relate to mathematical proofs? A new math student learning the techniques of proof writing will see many short proofs that start with either "assume" or "suppose." Given the statement, "a implies b" a proof might be constructed that supposes or assumes "a" and then proceeds to logically show that "b" is true and follows from "a." So, assume we have a wardrobe closet. What will logically follow from this assumption?

As pattern seeking machines, people work very hard at fitting the available information into the patterns they have identified. This is a part of our "hard wiring." In the general day-to-day pattern matching, we do not follow the application of rigorously defined systems of logic that are requirements for mathematical proof. Instead, we attempt to make our observations fit the patterns we have already identified. Cognitively this leads to errors that arise from biases in our decision making processes.

The most common of these biases are:

The belief-bias where we use our established belief system almost exclusively. We believe something to be true so discount (or deny) those things which do not fit.

The confirmation bias where in an effort to support our hypothesis, we see only those things which confirm or support the hypothesis.

There are also illusory correlations that we create out of our belief that different and unrelated observations are actually related.

(Matlin, M.W. (2005) Cognition, (6th ed.) New York, NY:
John Wiley & Sons, Inc., pp 409 - 430)

Now that you have assumed the existence of a wardrobe closet. Where will this lead you? What patterns will you invoke? What landscape will you see? Will the mathematician's wardrobe imply yours? Or will yours imply the mathematicians?


Harvey said...

On the flip side, you have situations where you DO have a wardrobe closet and people tell you to assume that it *doesn't* exist.

I've had people tell me that my wardrobe closet was illusory, even though they're standing close enough to it to get splinters in their noses :-)

vw bug said...

Why do you do this to me? I don't want to think.

MathCogIdiocy said...

vw - you don't have to think. You just have to keep up with those two darling boys. *grin*

Harvey - just offer them a pair of tweezers to pull the splinters out.

Rave said...

"Sometimes a cigar is just a cigar."


MathCogIdiocy said...

hehehe. And sometimes a wardrobe has a magic door. hehehe

Harvey said...

"Assume these tweezers don't exist" :-)

MathCogIdiocy said...

LOL. You'll always be able to one-up me won't you, Harvey?