A.— JIATHEMATICS AND PHYSICS. IT 



There is, however, one preliminary remark which I cannot resist 

 the temptation of making. The present is a particularly happy moment 

 for a pure mathematician, since it has been marked by one of the 

 greatest recorded triumphs of pure mathematics. This triumph is the 

 work, as it happens, of a man who would probably not describe himself 

 as a mathematician, but who has done more than any mathematician 

 to vindicate the dignity of mathematics, and to put that obscure and 

 p(M])lexing construction, commonly described as ' physical reality,' in 

 its proper place. 



There is probably less difference between the methods of a physicist 

 and a mathematician tlian is generally supposed. The most striking 

 among them seems to me to be this, that the mathematician is in mucli 

 more direct contact with reality. This may perhaps seem to you a 

 paradox, since it is the physicist who deals with the subject-matter to 

 which the epithet ' real ' is commonly applied. But a very little 

 reflexion will show that the ' reality ' of the physicist, whatever it may 

 be (and it is extraordinarily difficult to say), has few or none of the 

 attributes which common-sense instinctively marks as real. A chair 

 may be a collection of whirling atoms, or an idea in the mind of God. 

 It is no't my business to suggest that one account of it is obviously 

 more plausible than the other. Whatever the merits of either of them 

 may be, neither draws its inspiration from the suggestions of common- 

 sense. 



Neither tlie philosophers nor the pliysicis^ts Iheuiselves liavo ever 

 put forward any very convincing account of what physical reality is, or 

 of how the physicist passes, from the confused mass of fact or sensation 

 from which he starts, to the construction of the objects which he 

 classifies as real. We cannot be said, therefore, to know what the 

 subject-matter of physics is ; but this need not prevent us from under- 

 standing the task which a physicist is trying to perform. That, clearly, 

 is to correlate the incoherent body of facts confronting him with some 

 definite and orderly scheme of abstract relations, the kind of scheme, in 

 short, which he can only borrow from mathematics. 



A mathematician, on the other hand, fortunately for him, is not 

 concerned with this physical reality at all. It is impossible to prove, 

 by mathematical reasoning, any proposition whatsoever concerning the 

 l)hysical world, and only a mathematical crank would be likely now to 

 imagine it his function to do so. There is plainly one way only of 

 ascertaining the facts of experience, and that is by observation. It is 

 not the business of a mathematician to suggest one view of the universe 

 or another, but merely to supply the physicists with a collection of 

 abstract schemes, which it is for them to select from, and to adopt or 

 discard at their pleasure. 



The most obvious example is to be found in the science of geometry. 

 Mathematicians have constructed a very large number of different 

 systems of geometry, Euclidean or non-Euclidean, of one, two, three, or 

 any number of dimensions. All these systems are of complete and 

 equal validity. They embody the results of mathematicians' observa- 

 tions of their reality, a reality far more intense and far more rigid than 

 the dubious and elusive reality of physics. The old-fashioned geometry 

 of Euclid, the entertaining seven-point geometry of Veblen, the space- 



