All Of Mathematics Is A Tale About

The Mathematical Facts Worthy Of Being Studied Are Those Which, By Their Analogy With Other Facts, Are Capable Of Leading Us To The Knowledge Of A Physical Law. They Reveal The Kinship Between Other Facts, Long Known, But Wrongly Believed To Be Strangers To One Another.

In The Old Days When People Invented A New Function They Had Something Useful In Mind. Now, They Invent Them Deliberately Just To Invalidate Our Ancestors' Reasoning, And That Is All They Are Ever Going To Get Out Of Them.

Mathematicians Do Not Deal In Objects, But In Relations Between Objects; Thus, They Are Free To Replace Some Objects By Others So Long As The Relations Remain Unchanged. Content To Them Is Irrelevant: They Are Interested In Form Only.

Consider Now The Milky Way. Here Also We See An Innumerable Dust, Only The Grains Of This Dust Are No Longer Atoms But Stars; These Grains Also Move With Great Velocities, They Act At A Distance One Upon Another, But This Action Is So Slight At Great Distances That Their Trajectories Are Rectilineal; Nevertheless, From Time To Time, Two Of Them May Come Near Enough Together To Be Deviated From Their Course, Like A Comet That Passed Too Close To Jupiter. In A Word, In The Eyes Of A Giant, To Whom Our Suns Were What Our Atoms Are To Us, The Milky Way Would Only Look Like A Bubble Of Gas.

It Is Often Said That Experiments Should Be Made Without Preconceived Ideas. That Is Impossible. Not Only Would It Make Every Experiment Fruitless, But Even If We Wished To Do So, It Could Not Be Done. Every Man Has His Own Conception Of The World, And This He Cannot So Easily Lay Aside. We Must, Example, Use Language, And Our Language Is Necessarily Steeped In Preconceived Ideas. Only They Are Unconscious Preconceived Ideas, Which Are A Thousand Times The Most Dangerous Of All.