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:
“Every person is responsible for all the good within the scope of his abilities, and for no more, and none can tell whose sphere is the largest.”
—Gail Hamilton (1833–1896)
“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)
“If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them.”
—Henry David Thoreau (1817–1862)
“The best laid schemes o’ mice and men Gang aft agley;
An’ lea’e us nought but grief an’ pain, For promis’d joy!”
—Robert Burns (1759–1796)