Kronecker Quotes
Objections... inspired Kronecker and others to attack Weierstrass' "sequential" definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth-whatever that may be, if anything-precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.
Eric Temple Bell
Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Riemann, the more geometric approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century "American" school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods.
Robin Hartshorne
