We have found it necessary to give very full proofs, because otherwise — Principia Mathematica

"The above proposition [1 + 1 = 2] is occasionally useful."

"But why all these pages to prove that 1 + 1 = 2?""Hm … how shall I put it? Its the price you pay for being truly certain."

"He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European one."

"It takes a huge chunk of this volume just to prove that 1 + 1 = 2. And a large part of that proof revolves around the problems of the finite and the infinite, and the paradoxes that Cantors work had thrown up."

"As a work in mathematics, Principia Mathematica soon became obsolete. Symbolic logic is also a field of philosophy, and so the study of major works from the past has a special role unlike that of the study of the history of mathematics."

"The present work was originally intended by us to be comprised in a second volume of The Principles of Mathematics. With that object in view, the writing of it was begun in 1900. But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions. It therefore became necessary to make our book independent of The Principles of Mathematics."