Scope of Foundations Laid
The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.
A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.
Read more about this topic: Principia Mathematica
Famous quotes containing the words scope of, scope, foundations and/or laid:
“The scope of modern government in what it can and ought to accomplish for its people has been widened far beyond the principles laid down by the old “laissez faire” school of political rights, and the widening has met popular approval.”
—William Howard Taft (1857–1930)
“As the creative adult needs to toy with ideas, the child, to form his ideas, needs toys—and plenty of leisure and scope to play with them as he likes, and not just the way adults think proper. This is why he must be given this freedom for his play to be successful and truly serve him well.”
—Bruno Bettelheim (20th century)
“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
—Gottlob Frege (1848–1925)
“Where were you when I laid the foundation of the earth? Tell me, if you have understanding. Who determined its measurements—surely you know! Or who stretched the line upon it? On what were its bases sunk, or who laid its cornerstone when the morning stars sang together and all the heavenly beings shouted for joy?”
—Bible: Hebrew, Job 38:4 -7.
God, to Job.