MATHEMATICS IN THE NINETEENTH CENTURY 483 



without a derivative, and Hankel and Cantor, by means of their 

 principle of condensation of singularities, could construct analytic 

 expressions for functions having in any interval however small an 

 infinity of points of oscillation, an infinity of points in which the 

 differential coefficient is altogether indeterminate, or an infinity of 

 points of discontinuity. Another rude surprise was Cantor's dis- 

 covery of the one to one correspondence between the points of a 

 unit segment and a unit square, followed up by Peano's example 

 of a space-filling curve. 



These examples and many others made it very clear that the 

 ideas of a curve, a surface region, motion, etc., instead of being clear 

 and simple, were extremely vague and complex. Until these notions 

 had been cleared up, their admission in the demonstration of an 

 analytical theorem was therefore not to be tolerated. On a purely 

 arithmetical basis, with no appeal to our intuition, Weierstrass 

 develops his stately theory of functions which culminates in the 

 theory of Abelian and multiply periodic functions. 



But the notion of rigor is relative and depends on what we are 

 willing to admit either tacitly or explicitly. As we observed, Gauss, 

 whose rigor w r as the admiration of his contemporaries, freely ad- 

 mitted geometrical notions. This Weierstrass would criticise. On 

 the other hand, Weierstrass has made a grave oversight: he no- 

 where shows that his definitions relative to the number he introduces 

 do not involve mutual contradictions. If he replied that such con- 

 tradictions would involve contradictions in the theory of positive 

 integers, one might ask what assurance have we that such contradic- 

 tions may not actually exist. A flourishing young school of mathe- 

 matical logic has recently grown up under the influence of Peano. 

 They have investigated with marked success the foundations of 

 analysis and geometry, and in particular have attempted to show 

 the non-contradictoriness of the axioms of our number-system by 

 making them depend on the axioms of logic, which axioms we must 

 admit, to reason at all. 



The critical spirit, which in the first half of the century was to 

 be found in the writings of only a few of the foremost mathematicians, 

 has in the last quarter of the century become almost universal, at 

 least in analysis. A searching examination of the foundation of 

 arithmetic and the calculus has brought to light the insufficiency of 

 much of the reasoning formerly considered as conclusive. It became 

 necessary to build up these subjects anew. The theory of irrational 

 numbers invented by Weierstrass has been supplanted by the more 

 flexible theories of Dedekind and Cantor. Stolz has given us a sys- 

 tematic and rigorous treatment of arithmetic. The calculus has 

 been completely overhauled and arithmetized by Thomae, Harnack, 

 Peano, Stolz, Jordan, and Valle'e-Poussin. 



