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:
“Happy is that mother whose ability to help her children continues on from babyhood and manhood into maturity. Blessed is the son who need not leave his mother at the threshold of the world’s activities, but may always and everywhere have her blessing and her help. Thrice blessed are the son and the mother between whom there exists an association not only physical and affectional, but spiritual and intellectual, and broad and wise as is the scope of each being.”
—Lydia Hoyt Farmer (1842–1903)
“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)
“The aggregate of all knowledge has not yet become culture in us. Rather it would seem as if, with the progressive scientific penetration and dissection of reality, the foundations of our thinking grow ever more precarious and unstable.”
—Johan Huizinga (1872–1945)
“‘Tis not the balm, the sceptre, and the ball,
The sword, the mace, the crown imperial,
The intertissued robe of gold and pearl,
...
Not all these, laid in bed majestical,
Can sleep so soundly as the wretched slave
Who with a body filled and vacant mind
Gets him to rest, crammed with distressful bread.”
—William Shakespeare (1564–1616)