Kurt Godel's Achievement In Modern Logic Is

The Total Subject Of Mathematics Is Clearly Too Broad For Any Of Us. I Do Not Think That Any Mathematician Since Gauss Has Covered It Uniformly And Fully; Even Hilbert Did Not And All Of Us Are Of Considerably Lesser Width Quite Apart From The Question Of Depth Than Hilbert.

When We Talk Mathematics, We May Be Discussing A Secondary Language Built On The Primary Language Of The Nervous System.

There Is No Point In Being Precise If You Do Not Even Know What You Are Talking About.

It Is Exceptional That One Should Be Able To Acquire The Understanding Of A Process Without Having Previously Acquired A Deep Familiarity With Running It, With Using It, Before One Has Assimilated It In An Instinctive And Empirical Way... Thus Any Discussion Of The Nature Of Intellectual Effort In Any Field Is Difficult, Unless It Presupposes An Easy, Routine Familiarity With That Field. In Mathematics This Limitation Becomes Very Severe.

By And Large It Is Uniformly True In Mathematics That There Is A Time Lapse Between A Mathematical Discovery And The Moment When It Is Useful; And That This Lapse Of Time Can Be Anything From 30 To 100 Years, In Some Cases Even More; And That The Whole System Seems To Function Without Any Direction, Without Any Reference To Usefulness, And Without Any Desire To Do Things Which Are Useful.