 The logicist thesis appears to have been first advocated in the late seventeenth century by Gottfried Leibniz. Later, the idea was defended in much greater detail by Gottlob Frege. During the critical movement of the 1820s, mathematicians such as Bernard Bolzano, Niels Abel, Louis Cauchy, and Karl Weierstrass succeeded in eliminating much of the vagueness and many of the contradictions present in the mathematics of their day. By the mid- to late-1800s, William Hamilton had gone on to introduce ordered couples of reals as the first step in supplying a logical basis for the complex numbers and Karl Weierstrass, Richard Dedekind, and Georg Cantor had all developed methods for founding the irrationals in terms of the rationals. Using work done by H.G. Grassmann and Richard Dedekind, Guiseppe Peano had then gone on to develop a theory of the rationals based on his now famous axioms for the natural numbers. By Frege’s day, it was thus generally recognized that large parts of mathematics could be derived from a relatively small set of primitive notions.