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:
“For it is not the bare words but the scope of the writer that gives the true light, by which any writing is to be interpreted; and they that insist upon single texts, without considering the main design, can derive no thing from them clearly.”
—Thomas Hobbes (1579–1688)
“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)
“As men’s habits of mind differ, so that some more readily embrace one form of faith, some another, for what moves one to pray may move another to scoff, I conclude ... that everyone should be free to choose for himself the foundations of his creed, and that faith should be judged only by its fruits.”
—Baruch (Benedict)
“The same indignation that is said to have cleared the temple once will clear it again. The question is not about the weapon, but the spirit in which you use it. No man has appeared in America, as yet, who loved his fellow-man so well, and treated him so tenderly. He lived for him. He took up his life and he laid it down for him.”
—Henry David Thoreau (1817–1862)