Math Is Sometimes Called The Science Of

I Was Reminded Of The Sydney Harris Cartoon That Said 'adding Two Numbers That Have Not Been Added Before Does Not Constitute A Mathematical Breakthrough'.

The Trouble With Integers Is That We Have Examined Only The Very Small Ones. Maybe All The Exciting Stuff Happens At Really Big Numbers, Ones We Can't Even Begin To Think About In Any Very Definite Way. Our Brains Have Evolved To Get Us Out Of The Rain, Find Where The Berries Are, And Keep Us From Getting Killed. Our Brains Did Not Evolve To Help Us Grasp Really Large Numbers Or To Look At Things In A Hundred Thousand Dimensions.

Someone Has Said That All The Great Jugglers Are Dead.

Incidentally, When We're Faced With A "prove Or Disprove," We're Usually Better Off Trying First To Disprove With A Counterexample, For Two Reasons: A Disproof Is Potentially Easier (we Need Just One Counterexample); And Nitpicking Arouses Our Creative Juices. Even If The Given Assertion Is True, Our Search For A Counterexample Often Leads To A Proof, As Soon As We See Why A Counterexample Is Impossible. Besides, It's Healthy To Be Skeptical.

Ab=1/4((a+b)^2-(a-b)^2) Is An Amazing Identity, And Unfortunately, I Have To Remind My Current Students How To Prove It.