DEVELOPMENT OF MATHEMATICAL THOUGHT. 737 



matical thought and enter the region of metaphysics. 

 Like other lines of reasoning which have occupied us 

 in former chapters, the exact and rigid definitions and 

 deductions of arithmetic and geometry lead us up to 

 that other large department of our subject philosophic 

 thought. Many eminent mathematicians of recent years 

 have noticed this tendency, and have urged the mutual 

 help which arithmetic and geometry on this side, logic 

 and psychology on that, may derive from each other. 

 The names of Helmholtz, Georg Cantor, and Dede- 

 kind in Germany ; of M. Tannery and M. Poincare" in 

 France ; of Peano and Veronese in Italy, stand prom- 

 inently forward abroad ; while England can boast of hav- 

 ing cultivated, much earlier, by the hands of De Morgan 

 and Boole, a portion at least of this borderland, and of 

 having in recent years taken up the subject again in 

 an original and independent manner. 1 Cayley, in his 

 address to the British Association in 1883, has said : 

 " Mathematics connect themselves on the one side with 

 common life and the physical sciences ; on the other 



1 I refer to the important but 

 unfinished works of Mr Whitehead 

 on ' Universal Algebra ' (vol. i., 

 1898), and of Mr Bertrand Russell 

 on ' The Principles of Mathematics ' 

 (vol. i., 1903). I must defer a 

 more detailed appreciation of these 

 and other writings of this class, 

 such as those of the late Prof. 

 Ernst Schroder ( ' Algebra der 

 Logik,' 3 vols., 1890-95) and of 

 Prof. Gottlob Frege (see an 

 account of his writings in the 

 appendix to Mr Russell's ' Prin- 

 ciples '). They belong largely to 

 a department of philosophical 

 thought which may be termed 



VOL. II. 



" the Philosophy of the Exact 

 Sciences." This deals with two 

 great questions the logical found- 

 ations of scientific reasoning, and 

 the general outcome and import- 

 ance of scientific thought, not for 

 technical purposes, but in the 

 great edifice of human thought 

 which we may term Philosophy. 

 It deals with what has been 

 called " the Creed of Science " 

 and its value. Stanley Jevons 

 and Prof. Karl Pearson in this 

 country, Prof. Mach in Germany, 

 and M. Poincare" in France, have 

 treated the philosophy of science 

 in one or both of these aspects. 



3 A 



