PRESIDENTIAL ADDRESS. 517 



entirely unknown to the discoverers of those theorems. It has been the task of 

 mathematicians under the lead of such men as Cauehy, Riemann, Weierstrass, 

 and G. Cantor, to carry out the work of reconstruction of mathematical analysis, 

 to render explicit all the limitations of the truth of the general theorems, and to 

 lay down the conditions of validity of the ordinary analytical operations. 

 Physicists and others often maintain that this modern extreme precision amounts 

 to an unnecessary and pedantic purism, because in all practical applications of 

 Mathematics only such functions are of importance as exclude the remoter possi- 

 bilities contemplated by theorists. Such objections leave the true mathematician 

 unmoved ; to him it is an intolerable defect that, in an order of ideas in which 

 absolute exactitude is the guiding ideal, statements should be made, and processes 

 employed, both of which are subject to unexpressed qualifications, as conditions 

 of their truth or validity. The pure mathematician has developed a specialised 

 conscience, extremely sensitive as regards sins against logical precision. The 

 physicist, with his conscience hardened in this respect by the rough-and-tumble 

 work of investigating the physical world, is apt to regard the more tender organ 

 of the mathematician with that feeling of impatience, not unmingled with con- 

 tempt, which the man of the world manifests for what he considers to be over- 

 scrupulosity and unpractically. 



It is true that we cannot conceive how such a science as Mathematics could 

 have come into existence apart from physical experience. But it is also true that 

 physical percepts, as given directly in unanalysed experience, are wholly unfitted 

 to form the basis of an exact science. Moreover, physical intuition fails alto- 

 gether to afford any trustworthy guidance in connection with the concept of the 

 infinite, which, as we have seen, is in some form indispensable in the formation 

 of a coherent system of mathematical analysis. The hasty and uncritical exten- 

 sion 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 pitfall for the unwary student in the present. The notions derived from 

 physical intuition must be transformed into a scheme of exact 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 data 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 the case of geometry, in which it has been shown to be 

 possible 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 well-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 supposed to exist in the scheme known as 

 hyperbolic geometry, or in that known as elliptic geometry, would necessarily 

 entail the existence of a corresponding inconsistency in Euclid's set of axioms. 

 The three geometries therefore, from the logical point of view, are completely on 

 a par with one another. An interesting mathematical 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. Poincare, for example, that 

 the question which is the true scheme has no meaning; that it is, in fact, entirely 

 a matter of convention and convenience which of these geometries is actually 

 employed in connection with spatial measurements. To decide between them by 

 a crucial test is impossible, because our space perceptions are not sufficiently exact 

 in the mathematical sense to enable us to decide between the various axioms of 

 parallels. Whatever 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 



