INAUGURAL ADDRESS BEFORE THE BRITISH ASSOCIATION. 511 



and by reflection. The apparent difference in the 

 mode of first apprehending them and in their ul- 

 timate cogency arises from the difference of the 

 ideas themselves, from the preponderance of 

 quantitative over qualitative considerations in 

 mathematics, from the notions of absolute equal- 

 ity and identity which they imply. 



If we turn to the other question, " How did 

 the world at large acquire and improve its idea 

 of number and of figures ? How can we span the 

 interval between the savage who counted only 

 by the help of outward objects, to whom fifteen 

 was ' half the hands and both the feet,' and New- 

 ton or Laplace ? " The answer is the history 

 of mathematics and its successive developments, 

 arithmetic, geometry, algebra, etc. The first and 

 greatest step in all this was the transition from 

 number in the concrete to number in the abstract. 

 This was the beginning not only of mathematics 

 but of all abstract thought. The reason and 

 mode of it was the same as in the individual. 

 There was the same general influx of evidence, 

 the same unsought-for experimental proof, the 

 same recognition of general laws running through 

 all manner of purposes and relations of life. No 

 wonder, then, if, under such circumstances, math- 

 ematics, like some other subjects, and perhaps 

 with better excuse, came after a time to be 

 clothed with mysticism ; nor that, even in mod- 

 ern times, they should have been placed upon an 

 a priori basis as in the philosophy of Kant. 



Number was so soon found to be a principle 

 common to many branches of knowledge that it 

 was readily assumed to be the key to all. It 

 gave distinctness of expression, if not clearness 

 of thought, to ideas which were floating in the 

 untutored mind, and even suggested to it new 

 conceptions. In " the one " " the all," " the 

 many in one " (terms of purely arithmetic ori- 

 gin), it gave the earliest utterance to men's first 

 crude notions about God and the world. In " the 

 equal," "the solid," "the straight," and "the 

 crooked," which still survive as figures of speech 

 among ourselves, it supplied a vocabulary for 

 the moral notions of mankind, and quickened 

 them by giving them the power of expression. 

 In this lies the great and enduring interest in the 

 fragments which remain to us of the Pythagorean 

 philosophy. 



The consecutive processes of mathematics 

 led to the consecutive processes of logic, but it 

 was not until long after mankind had attained to 

 abstract ideas that they attained to any clear no- 

 tion of their connection with one another. The 

 leading ideas of mathematics became the leading 



ideas of logic. The " one " and the " many " 

 passed into the " whole " and its " parts ; " and 

 thence into the "universal" and the "particu- 

 lar." The fallacies of logic, such as the well- 

 known puzzle of Achilles and the tortoise, par- 

 take of the nature of both sciences. And perhaps 

 the conception of the infinite and the infinitesi- 

 mal, as well as of negation, may have been in 

 early times transferred from logic to mathemat- 

 ics. But the connection of our ideas of number 

 is probably anterior to the connection of any of 

 our other ideas. And, as a matter of fact, ge- 

 ometry and arithmetic had already made con- 

 siderable progress when Aristotle invented the 

 syllogism. 



General ideas there were besides those of 

 mathematics — true flashes of genius which saw 

 that there must be general laws to which the uni- 

 verse conforms, but which saw them only by oc- 

 casional glimpses, and through the distortion of 

 imperfect knowledge ; and, although the only 

 records of them now remaining are the inadequate 

 representations of later writers, yet we must still 

 remember that to the existence of such ideas is 

 due not only the conception but even the pos- 

 sibility of physical science. But these general 

 ideas were too wide in their grasp, and in early 

 days at least were connected to their subjects of 

 application by links too shadowy, to be thorough- 

 ly apprehended by most minds, and so it came to 

 pass that one form of such an idea was taken as 

 its only form, one application of it as the idea 

 itself; and philosophy, unable to maintain itself 

 at the level of ideas, fell back upon the abstrac- 

 tions of sense, and, by preference, upon those 

 which were most ready to hand, namely, those 

 of mathematics. Plato's ideas relapsed into a 

 doctrine of numbers ; mathematics into mysti- 

 cism, into Nco-Platonism, and the like. And so, 

 through many long ages, through good report 

 and evil report, mathematics have always held 

 an unsought-for sway. It has happened to this 

 science, as to many other subjects, that its warm- 

 est adherents have not always been its best 

 friends. Mathematics have often been brought 

 into matters where their presence has been of 

 doubtful utility. If they have given precision to 

 literary style, that precision has sometimes been 

 carried to excess, as in Spinoza and perhaps Des- 

 cartes ; if they have tended to clearness of ex- 

 pression in philosophy, that very clearness has 

 sometimes given an appearance of finality not 

 always true; ' if they have contributed to defin- 

 itiveness in theology, that definitiveness has often 

 1 For example, in Herbart's "Psychologic." 



