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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, by 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 strikingly illustrated by the case of geometry, as pre- 

 sented in its most approved modern shape. It is not too much to say that 

 geometry, reduced to a purely deductive form — as presented, for example, by 

 Hilbert, or by some of the modern Italian school — has no necessary connection 

 with space. The words 'point,' 'line,' 'plane' are employed to denote any 

 entities whatever which satisfy certain prescribed conditions of relationship. 

 Various premisses are postulated that would appear to be of a perfectly arbitrary 

 nature, if we did not know how they had been suggested. In that division of 

 the subject known as metric geometry, for example, axioms of congruency are 

 assumed which, by their purely abstract character, avoid the very real difficulties 

 that arise in this regard in reducing perceptual space-relations of measurements to 

 a purely conceptual form. Such schemes, triumphs of constructive thought at its 

 highest and most abstract level as they are, could never have been constructed 

 apart from the space-perceptions that suggested them, although the concepts of 

 spatial origin are transformed almost out of recognition. But what I want to 

 call attention to here is that, apart from the basis of this geometry, mathe- 

 maticians would never have been able to find their way through the details of the 

 deductions without having continual recourse to the guidance given them by 

 spatial intuition. If one attempts to follow one of the demonstrations of a par- 

 ticular theorem in the work of writers of this school, one would find it quite 

 impossible to retain the steps of the process long enough to master the whole, 

 without the aid of the very spatial suggestions which have been abstracted. This 

 is perhaps sufficiently warranted by the fact that writers of this school find it 

 necessary to provide their readers with figures, in order to avoid complete 

 bewilderment in following the demonstrations, although the processes, being 

 purely logical deductions from premisses of the nature I have described, deal only 

 with entities which have no necessary similarity to anything indicated by the 

 figures. 



A most interesting account has been written by one of the greatest mathe- 

 maticians of our time, M. Henri Poincare, of the way in which he was led to some 

 of his most important mathematical discoveries. 1 He describes the process of 

 discovery as consisting of three stages : the first of these consists of a long effort 

 of concentrated attention upon the problem in hand in all its bearings ; during the 

 second stage he is not consciously occupied with the subject at all, but at some 

 quite unexpected moment the central idea which enables him to surmount the 

 difficulties, the nature of which he had made clear to himself during. the first 

 stage, flashes suddenly into his consciousness. The third stage consists of the 

 work of carrying out in detail and reducing to a connected form the results to 

 which he is led by the light of his central idea ; this stage, like the first, is one 

 requiring conscious effort. This is, I think, clearly not a description of a purely 

 deductive process; it is assuredly more interesting to the psychologist than to the 

 logician. We have here the account of a complex of mental processes in which it 

 is certain that the reduction to a scheme of precise logical deduction is the latest 

 stage. After all, a mathematician is a human being, not a logic-engine. Who 

 that has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, 

 Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great 

 artist ? The faculties possessed by such men, varying greatly in kind and degree 

 with the individual, are analogous to those requisite for constructive art. Not 

 every great mathematician possesses in a specially high degree that critical faculty 

 which finds its employment in the perfection of form, in conformity with the 



1 See the Remit du Mois for 1908. 



