“When we examine the classical set-theoretic foundations of mathematics, we see that the only sets that play a role are sets of restricted type; at the risk of understatement, only sets of rank < ω + ω. Further examination reveals four fundamental principles about sets used: the existence of an infinite set; the existence of the power set of any set; every property determines a subset of any set; and the axiom of choice.”
“These doubts did not halt mathematical creation. Technicians working on the superstructure did not drop their tools and scurry down to the basement because some of their underpinning needed reinforcin...”
Foundations of mathematics
“The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without an...”
Foundations of mathematics
“As soon as I have put it into order I intend to write and if possible to publish a work on parallels. At this moment, it is not yet finished, but the way which I have followed promises me with certain...”
Foundations of mathematics
“I should regard it as a great misfortune if you were to allow yourself to be deterred by the 'clamors of the Bœotians' from explaining your views of geometry. From what Lambert has said and [Ferdinand...”
Foundations of mathematics
“The world is totally connected. Whatever explanation we invent at any moment is a partial connection, and its richness derives from the richness of such connections as we are able to make. ...mathemat...”
Foundations of mathematics
