326 



'.CIENCE. 



[N. S. Vol. IX. No. 218. 



are more than self-cousistent — they are true or 

 false ; and the axioms ia accordance with which 

 the reasoning is conducted correspond to uni- 

 versal properties of space. But suppose that 

 we confine our attention to algebraical analy- 

 sis — to what the treatise before us includes 

 under the terms ordinary algebra and universal 

 algebra. Are the definitions of ordinary algebra 

 merely self-consistent conventions? Are its 

 propositions merely formal without any objec- 

 tive truth ? Are the rules according to which it 

 proceeds arbitrary selections of the mind ? If 

 the definitions and rules are arbitrary, what 

 is the chance of their applying to anything use- 

 ful? The theory of probabilities informs us 

 that the chance must be infinitesimal, and'the 

 author admits that the entities created by the 

 conventions must have properties which bear 

 some affinity to the properties of existing things, 

 if the algebra so founded is to be of any im- 

 portance. The author says ' some aflinity ; ' it 

 may be asked how much ? Unless the aflBnity 

 or correspondence is perfect, how can the one 

 apply to the other ? How can this perfect cor- 

 respondence be secured, except by the conven- 

 tions being real definitions, the equations true 

 propositions, and the rules expressions of uni- 

 versal properties? In the last sentence quoted, 

 Mr. Whitehead makes a large concession to the 

 realist view ; it is only necessary to change the 

 sentence into — "In the case of any algebra 

 worthy of scientific attention the definitions 

 and propositions refer exactly or approximately 

 to the world of existing things." 



M. Laisant, in his recent work, ' La Mathe- 

 matique,' refers to the formal view of mathe- 

 matical science when discussing the theory of 

 fractions, p. 35. He opposes it, as marching in 

 the direction opposite to progress, and as a 

 survival of the spirit of the sophist. 



The realist view of mathematical science has 

 commended itself to me ever since I made an 

 exact analysis of Relationship and devised a 

 calculus which provides a notation for any 

 relationship, can express in the form of an 

 equation the relationship existing between any 

 two persons, and provides rules by means of 

 which a single equation maybe transformed, or 

 a number of equations combined so as to yield 

 any equation involved in their being true 



simultaneously. The notation is made to fit 

 the subject, and the rules for manipulation are 

 derived from universal physiological laws and 

 the more arbitrary laws of marriage. A very 

 real basis, yet the analysis has all the character- 

 istics of a calculus, and throws light by com- 

 parison on several points in ordinary algebra. 



But what is the subject of which ordinary 

 algebra is the analysis ? Quantity ; and in 

 space we have the most complex kind of 

 quantity ; so that if space can be analyzed, the 

 analysis will serve for any less complex kind of 

 quantity. Mr. Whitehead admits that, as a 

 matter of history, mathematics has till recently 

 been the science of number, quantity and the 

 space of common experience. But "the intro- 

 duction of the complex quantity of ordinary 

 algebra, an entity which is evidently based 

 upon conventional definitions, gave rise to the 

 wider mathematical science of to-day. Ordi- 

 nary algebra, in its modern development, is a 

 large body of propositions interrelated by de- 

 ductive reasoning and based upon conven- 

 tional definitions which are generalizations of 

 fundamental conceptions." 



The imaginary quantity, more generally the 

 complex quantity, of ordinary algebra is the 

 foundation upon which the formalist builds his 

 theory ; if it can be shown that it is rot an 

 entity based upon conventional definitions, but 

 corresponds to a reality, then his whole super- 

 structure falls down. The complex quantity 

 first arises in analysis in the solution of the 

 quadratic equation. The general form of the 

 root consists of a quantity independent of the 

 radical sign and a quantity affected by the rad- 

 ical sign. When the quantity under the radical 

 sign is negative the root is said to be imagi- 

 nary, because it appears to he incapable of direct 

 addition to the part independent of the radical 

 sign. In certain papers recently published I 

 have shown at length that the root of a quad- 

 ratic equation may be versor in nature or scalar 

 in nature. If it is versor in nature, then the 

 part affected by the radical involves the axis 

 perpendicular to the plane of reference, and 

 this is so, whether the radical involves the 

 square root of minus one or not. In the former 

 case the versor is circular, in the latter hyper- 

 bolic. When the root is scalar in its nature 



