FOM: Mathematical Intuition

[Joe Shipman]

In Frank Morgan's upcoming "Math Chat" column (to be published on
December 7th at http://www.maa.org ), Timur Dogan poses the question

"Which branches of mathematics or types of problems are the most
counterintuitive?"

My answer was as follows, but I'd be happy to see alternative answers:


>>
Whose intuition?

The most counterintuitive to new learners of mathematics is in my
opinion applications of the Axiom of Choice such as the Banach-Tarski
paradox.

The branch which takes the most surprising and
initially-hard-to-motivate turns as you learn it is number theory.

The branch that is the most difficult for professionals to develop an
intuition for is hard to say, since there are several types of
mathematical intuition (geometrical, algebraic, logical, combinatorial,
arithmetical for example).  But "difficult to get an intuition for" is
not the same thing as "counterintuitive".

If we ask which branch of mathematics most frequently surprises the
professionals within that branch, I would say topology.  My criterion:
in that area, you often find "counterexamples", while in other areas
open questions seem to be very likely to be settled in the direction
originally conjectured.
<<

-- Joe Shipman