September i, 1910J 



NATURE 



289 



indispensable in the formation of a coherent system of 

 mathematical analysis. The hasty and uncritical extension 

 to the region of the infinite, of results which are true and 

 often obvious in the region of the finite, has been a 

 fruitful source of error in the past, and remains as a pit- 

 fall for the unwary student in the present. The notions 

 derived from physical intuition must be transformed into 

 a scheme of e.xact definitions and axioms before they are 

 available for the mathematician, the necessary precision 

 being contributed by the mind itself. A very remarkable 

 fact in connection with this process of refinement of the 

 rough d.nta of experience is that it contains an element of 

 arbitrariness, so that the result of the process is not 

 necessarily unique. The most striking example of this 

 want of uniqueness in the conceptual scheme so obtained 

 is :'ie case of geometry, in which it has been shown to be 

 poss ble to set up various sets of axioms, each set self- 

 consistent, but inconsistent with any other of the sets, 

 and yet such that each set of axioms, at least under 

 suitable limitations, leads to results consistent with our 

 perception of actual space-relations. Allusion is here 

 made, in particular, to the wcU-known geometries of 

 Lobatchewsky and of Riemann, which differ from the 

 geometry of Euclid in respect of the axiom of parallels, 

 in place of which axioms inconsistent with that of Euclid 

 and with one another are substituted. It is a matter of 

 demonstration that any inconsistency which might be sup- 

 posed to exist in the scheme known as hyperbolic geo- 

 metry, or in that known as elliptic geometry, would 

 necessarily entail the existence of a corresponding incon- 

 sistency in Euclid's set of axioms. The three geometries 

 therefore, from the logical point of view, are com- 

 pletely on a par with one another. An interesting mathe- 

 matical result is that all efforts to prove Euclid's axiom 

 of parallels, i.e. to deduce it from his other axioms, are 

 doomed to necessary failure ; this is of importance in 

 view of the many efforts that have been made to obtain 

 the proof referred to. When the question is raised which 

 of these geometries is the true one, the kind of answer 

 that will be given depends a good deal on the view taken 

 of the relation of conceptual schemes in general to actual 

 experience. It is maintained by M. Poincar^, for ex- 

 ample, that the question which is the true scheme has no 

 meaning ; thaj^ it is, in fact, entirely a matter of con- 

 vention and convenience which of these geometries is 

 actually employed in connection with spatial measure- 

 ments. To decide between them by a crucial test is 

 impossible, because our space perceptions are not 

 sufficiently exact in th^ mathematical sense to enable us 

 to decide between the various axioms of parallels. What- 

 ever views are taken as to the difficult questions that arise 

 in this connection, the contemplation and study of schemes 

 of geometry wider than that of Euclid, and some of them 

 including Euclid's geometry as a special case, is of great 

 interest, not only from the purely mathematical point of 

 view, but also in relation to the general theory of know- 

 ledge, in that, owing to the results of this study, some 

 change is necessitated in the views which have been held 

 by philosophers as to what is known as Kant's space- 

 problem. 



The school of thought which has most emphasised the 

 purely logical aspect of Mathematics is that which is re- 

 presented in this country by Mr. Bertrand Russell and 

 Dr. Whitehead, and which has distinguished adherents 

 both in Europe and in .America. The ideal of this school 

 is a presentation of the whole of Mathematics as a 

 deductive scheme in which are employed a certain limited 

 number of indefinables and unprovable axioms, by means 

 of a procedure in which all possibilitv of the illicit 

 intrusion of extraneous elements into the deduction is 

 excluded by the employment of a svmbolism in which each 

 s^'mbol expresses a certain logical relation. This school 

 receives its inspiration from n neculiar form of philosophic 

 realism which, in its revolt from idealism, produces in 

 the adherents of the school a strong tendency to ignore 

 altoC'=t!iei the psychological imnlications in the movements 

 of mathematical thought. This is carried so far that in 

 their writings no explicit 'recognition is made of anv 

 psychological factors in the selection of the indefinables 

 nnd in the formulation of tlie axioms utxin which the 

 • ' structure of Mathem.Ttics is to be based. The 



NO. 2 13 1, VOL. 84] 



actually worked-out part of their scheme has as yet reached 

 only the mere fringe of modern Mathematics as a great 

 detailed body of doctrine ; but to any objection to the 

 method on the ground of the prolixity of the treatment 

 which would be necessary to carry it out far enough to 

 enable it to embrace the various branches of Mathematics 

 in all the wealth of their present development, it would 

 probably be replied that the main point of interest is to 

 establish in principle the possibility only of subsuming 

 Pure Mathematics under a scheme of logistic. It is quite 

 impossible for me here to attempt to discuss, even in out- 

 line, the tenets of this school, or even to deal with the 

 interesting question of the possibility of setting up a final 

 system of definables and axioms which shall suffice for 

 all present and future developments of Mathematics. 



I am very far from wishing to minimise the high philo- 

 sophic interest of the attempt made by the Peano-Russell 

 school to exhibit Mathematics as a scheme of deductive 

 logic. I have myself emphasised above the necessity and 

 importance of fitting the resuhs of mathematical research 

 in their final form into a framework of deduction for the 

 purpose of ensuring the complete precision and the verifi- 

 cation of the various mathematical theories. At the same 

 time, it must be recognised that the purely deductive 

 method is wholly inadequate as an instrument of research. 

 Whatever view may be held as regards the place of psycho- 

 logical implications in a completed body of mathematical 

 doctrine, in research the psychological factor is of para- 

 mount importance. The slightest acquaintance with the 

 history of Mathematics establishes the fact that discoveries 

 have seldom, or never, been made by purely deductive pro- 

 cesses. The results are thrown into a purely deductive 

 form after, and often long after, their discovery. In many 

 cases the purely deductive form, in the full sense, is quite 

 modern. The possession of a body of indefinables, axioms, 

 or postulates, and symbols denoting logical relation, 

 would, taken by itself, be wholly insufficient for the 

 development of a mathematical theory. With these alone 

 the mathematician would be unable to move a step. In 

 face of an unlimited number of possible combinations,^ a 

 principle of selection of such as are of interest, a purposive 

 element, and a perceptive faculty are essential for the 

 development of anything new. In the process of dis- 

 covery, the chains in a sequence of logical deduction do 

 not at first arise in their final order in the mind of the 

 mathematical discoverer. He divines the results before 

 they are established: he has an intuitive grasp of the 

 general line of a demonstration long before he has filled 

 m the details. A developed theory, or even a demonstra- 

 tion of a single theorem, is no more identical with a 

 mere complex of syllogisms than a melody is identical 

 with the mere sum' of "the musical notes employed in its 

 composition. In each case the whole is something more 

 than merely the sum of its parts; it has a unity of its 

 own, and that unity must be, in some measure at least, 

 discerned by its creator before the parts fall completely 

 mto their places. Logic is, so to speak, the grammar of 

 Mathematics ; but a knowledge of the rules of grammar 

 and the letters of the alphabet would not be sufficient 

 equipment to enable a man to write a book. There is 

 much room for individuality in the modes of mathematical 

 discovery. Some great mathematicians have employed 

 largely 'images derived from spatial intuition as a guide 

 to their results ; others appear wholly to have discarded 

 such aids, and were led by a fine feeling for algebraic and 

 other species of mathematical form. A certain tentative 

 process is common, in which, bv the aid of results known 

 or obtained in special cases, generalisations are perceived 

 and afterwards established, which take up into themselves 

 all the special cases so employed. Most mathematicians 

 leave some traces, in the final presentation of their work, 

 of the scaffolding they have employed in building their 

 edifices, some much more than others. _ 



The difference between a mathematical theory in the 

 making and as a finished product is. perhaps, most strik- 

 ingly illustrated by the case of geometry, as presented in its 

 most approved modern shape. It is not too much to say 

 that ceometrv. reduced to a purely deductive form— as 

 presented, for example, bv Hilbert, or by some of the 

 modern Italian school— has no necessary connection with 

 space. The word; "point." "line," "plane" are em- 



