384 



NA TURE 



[May 27, 1909 



(Lord Kelvin), and Maxwell, and those of many living 

 show this ; and in Germany one of the greatest influences 

 in ptire mathematics was Riemann, who is usually con- 

 trasted with Weierstrass as a type of the creative, as 

 opposed to the critical, genius.' But in this article we 

 are only concerned with questions in the theory of know- 

 ledge, with the principles of mathematics, and the basis 

 of their application to physics, and, through these ques- 

 tions, with the relevance of mathematics to our whole 

 civilisation and, what is still more important, to our whole 

 lives.' 



The critical inquiries into the nature and purposes of 

 our conceptions in physics, which have been mentioned 

 above, have put in a clear light the fact, which seems to 

 have been overlooked by Laplace in that flush of 

 enthusiasm which a mathematician can so readily under- 

 stand, and which, without the excuse of the sudden 

 illumination brought about in the eighteenth century by 

 the development of mathematics, is still overlooked by the 

 cruder physicists, that the " world " with which we have 

 to deal in theoretical (mathematical) mechanics, for 

 example, is but a mathematical scheme the function of 

 which it is to imitate by logical consequences of the proper- 

 ties assigned to it by definition certain processes of nature 

 as closely as possible. Thus our " dynamical world " may 

 be called a model of reality, and must not be confused 

 with the reality itself. 



That this model of reality is constructed solely out of 

 logical conceptions results from our conclusion that mathe- 

 matics is based on logic, and on logic alone ; that such a 

 model is possible is indeed surprising, and the surprise only 

 goes when we follow up in history the growth of the 

 application of mathematics to physics. The need for com- 

 pleting facts of nature in thought was, no doubt, first felt 

 as a practical need — the need that arises because we feel 

 it convenient to be able to predict certain kinds of future 

 events. Thus, with a purely mathematical model of the 

 solar system, we can tell, with an approximation which 

 depends upon the completeness of the model, the relative 

 positions of the sun, stars, and planets several years ahead 

 of time; this enables us to publish the "Nautical 

 Almanac," which is so useful to sailors, and makes up to 

 us, in some degree, for our inability '* to grasp this sorry 

 scheme of things entire . . . .Tnd re-mould it nearer to the 

 heart's desire." 



The need of the completion of facts in thought is not 

 merely practical ; it is also intellectual. The striving after 

 logical completeness, whether in generality of results or 

 consistency of its own premisses or those of its models of 

 reality,' is accompanied by a feeling of aesthetic pleasure 

 or of intellectual honesty, or of both. We may say that 

 mathematics has an aesthetic and a moral value. 



Mathematics is relevant to those who go down to the 

 sea in ships, to those who stay on dry land and build 

 bridges or locomotives, and to those who observe the sun's 

 corona during a total eclipse to find out what the sun 

 is made of. Mathematics is relevant to the philosopher, 

 for not only has it investigated and does it investigate its 

 own foundations, but also it explains what is meant by 

 the philosophers' own phrases, such as " the postulate of 

 the comprehensibility of nature " (which seems to be the 

 postulate that a purely logical model is possible), and the 

 " laws of uniformity, continuity, and causality." And 

 lastly, mathematics, besides being relevant to resthetics and 

 morals in the above sense, is of moral significance in 



^ On a closfr consideration, this distinction bre.iks down almost entirely. 

 Aprjrt from the numerous instances which can be quoted of particularly 

 critical work by Riemann and particularly creative work by Weierstrass, 

 surely it is always true both that there should be no creation without 

 criticism (otherwise we run the risk of building castles in the air) and 

 that th're cannot be any relevant criticism which does not add to our 

 knowledge, and is in so far creative. 



2 C/: A. Voss, "tjber das Wesen der Mathematik." Pp. 3-4. (Leipzig 

 and Berlin : B. G. Teubner, 1508.) 



3 I have tried to show by some ex.Tmples that we can and ousht to 

 examine the details of our models with the aid of the most refined 

 conceptions of modern mathematics, in order to be certain that the models 

 are loeically cnsislent (" On some Points in the Foundation of Mathema- 

 tical Physics," T'n- Monist, April, 1908, vol. xviii., pp. 217-26: if. 

 Voss, ftfi. cit., pp. 7T-2). An example of the results of critical inves- 

 tigation into applied mathematics is the discovery— which has also 

 obvious practical results in the avoidance of labour doomed to iinfruit- 

 fulness-by Poincar^ of limits of validity for certain of Laplace's 



fo; 



NO. 2065, VOL. So] 



another respect. Since the basis of mathematics is logic, 

 and logic alone,' all those personal, national, and historical 

 questions which are from time to time mixed up with 

 mathematics — however essential some of them may be to 

 the understanding of certain points and to education — show 

 themselves, when looked at from a higher plane of truth, 

 to be irrelevant. Philip E. B. Jourdain. 



THE IRON AND STEEL INSTITUTE. 

 'T'HE fortieth annual general meeting of the Iron and 

 Steel Institute was held at the Institution of Civil 

 Engineers on May 13 and 14, under the chairmanship 

 of Sir Hugh Bell, who retains the oflice of president for 

 another year, and will be succeeded next May by his 

 Grace the Duke of Devonshire. The report of the council 

 for the past year shows that the affairs of the institute 

 are in a prosperous condition. Five Carnegie research 

 scholarships had been awarded, and Mr. Carnegie had 

 presented 11,000 dollars, the income of which would assist 

 in meeting clerical expenses and those incurred in issuing 

 special memoirs. 



The proceedings on May 13 opened with three papers, 

 taken together lor discussion, dealing with corrosion and 

 protection of iron and steel. The paper by Mr. W. H. 

 Walker, of Boston, U.S.A., contains the fundamental 

 conceptions involved in the modern electrolytic theory of 

 the corrosion of iron, develops this theory from the facts 

 now known, and shows that the older carbonic-acid theory 

 can be, and is, included therein, and points out some of 

 the practical applications of this theory to the problem of 

 corrosion. Mr. Allerton S. Cushman, of the United States 

 Department of .'\griculture, contributed a paper on the 

 preservation of iron and steel. The author favours the 

 view of corrosion as an electrochemical phenomenon, and 

 deals with the questions of the production of a metal 

 highly resistant to corrosion, of protective coatings, and 

 of the passive condition which iron is capable of assuming. 

 It seems to be a fact that carefully made open-hearth 

 metal, in which the ordinary impurities are cut down to 

 mere traces, and in which the heat treatment has been 

 carefully controlled, is much more resistant to corrosion 

 than the ordinary types of metal with a comparatively 

 high percentage of impurities. The preservation of iron 

 and steel by application of other metals to the surface, 

 and of paint and other coatings, is fully discussed, and 

 certain experiments having the object of determining their 

 relative values under ordinary weathering conditions, which 

 are now being carried out in America, are described and 

 illustrated with pliotographs. Mr. J. Cruickshank Smith, 

 of London, contributed a paper on physical tests for pro- 

 tective coatings for iron and steel. Tests are described for 

 examining the following points: — that the proper propor-" 

 tion of pigment and vehicle has been obtained with the 

 minimum of free oil space in the dry film ; the smallness 

 and uniformity of size of the pigmentary particles ; the 

 possession of the property of minimum tendency of the 

 pigment and vehicle to separate ; the determination of the 

 thickness and uniformity of the film and its strength and 

 elasticity ; the permeability and hardness of the film. 



An important paper on the solubility of steel in sulphuric 

 acid was contributed by Messrs. E. Heyn and O. Bauer, 

 of Gross-Lichtcrfelde. This paper contains 120 pages of 

 matter, together with plates, and can only be briefly 

 noticed here. The authors' researches show that the 

 transition from the martensite of hardened steel to pearlite 

 of annealed steel is not continuous through the inter- 

 mediate stage of tempering as has been hitherto supposed. 

 There is an intermediate metastable form to which the 

 authors have given the name of "osmondite," in honour 

 of Osmond. The fact is shown by the curve of solubility 

 in dilute sulphuric acid attaining a sharply defined maxi- 

 mum at 400° C. The researches dealt with the influences 

 of the quenching and tempering of steel on its solubility, 

 of quenching and re-heating soft mild steel, and of the 

 quenching temperature: the influence of cold working and 

 annealing on the solubility of mild steel, and of the 



1 Mathematics is a wonderfully refined sy)iifwlic (for the importance 

 of this character, see Voss, c/. cit, pp. 25-26) losic, the product of 

 thousands of minds, and so adapted as to spare all waste of thought 

 on unessenlials. 



