Rogers — The Logical Basis of Mathematics. 189 



knowledge. A different view is taken by Mill, who claims, as I think rightly, 

 that the apodictic certainty of mathematics is in the inferences, not in the 

 premisses — is, in a word, logical. The strongest objection to the whole 

 Absolutist position is this, that you cannot claim to have a perfect knowledge 

 of any part of the external world without knowing the whole scheme of 

 things — in a word, without omniscience. Kant endeavoured to get out of the 

 difficulty by separating the form of intuition from the matter of knowledge. 

 Space and Time are the objects of Pure Mathematics ; and here absolute 

 certainty is possible. But this does not remove the difficulty, as Kant 

 himself recognizes occasionally. One cannot ' think away ' every property 

 from Body except its extension; Matter possesses a priori intensive qualities 

 as well. The Kantian theory of Space as the a pi-iori intuition is really a 

 survival of the Cartesian doctrine that Extension is the essence of matter. 

 The real explanation of the superior certainty of Pure Mathematics, if it is 

 to be identified, as it was by Kant, with the mathematics of Space and 

 Time, is that the spatial properties of matter are simpler and more easily 

 measurable than its other properties, and, owing to their homogeneity, can 

 be expressed precisely by a set of logical axioms, as is shown in modern 

 Logical Geometry. They are more amenable to Deductive treatment. This 

 is verified historically, for geometry is the earliest form of Deductive Science. 



That the premisses of Applied Mathematics are hypothetical no one can 

 deny. This is true not only of the propositions, but of the terms used in these 

 propositions. In Dynamics and Statics we assume the existence of absolutely 

 permanent particles of matter. The concept ' particle of matter ' is quite 

 ideal ; ' atoms,' ' corpuscles,' and so forth, are all ideal ; they can be used 

 logically, however, as conceptions. In Hydrostatics and Hydrodynamics we 

 postulate the existence of perfect fluids, which no physicist believes in. In fact, 

 these sciences are simply types of logical geometry, possessing three or more 

 dimensions. Even were our postulates true, physical measurement could 

 never approach the logical precision of our assumptions. Then in elementary 

 optics we assume that light proceeds from indivisible points in straight lines. 

 Experience proves that it does neither one nor the other. Light proceeds 

 from space-filling bodies ; and we are now told that its path is not rectilinear. 



The old distinction between Pure and Applied Mathematics is thus some- 

 what illusory. All mathematics is Pure in the sense that it is ideal or 

 hypothetical ; in other words, it proceeds by Logic, as Mr. Bertrand Eussell 

 has pointed out. Again, most„if not all, mathematics is Applied in the sense 

 that the axioms and premisses are suggested by experience, and in some cases 

 can be verified by a return to experience. In this sense the Arithmetic of 

 finite numbers and Euclidean geometry are Applied Mathematics. 



R.I. A. PROC, VOL. XXVII., SECT. A. ' [27] 



