CONCEPTIONS AND METHODS OF MATHEMATICS 463 



What are its disadvantages? I can see only two. First that, as has 

 been already remarked, the idea of drawing necessary conclusions 

 is a slightly vague and shifting one. Secondly, that it lays exclusive 

 stress on the rigorous logical element in mathematics and ignores 

 the intuitional and other non-rigorous tendencies which form an 

 important element in the great bulk of mathematical work concern- 

 ing which I shall speak in greater detail later. 



IV. Geometry an Experimental Science 



Some of you will also regard it as an objection that there are 

 subjects which have almost universally been regarded as branches 

 of mathematics but are excluded by this definition. A striking 

 example of this is geometry, I mean the science of the actual space 

 we live in; for though geometry is. according to Peirce's definition, 

 preeminently a mathematical science, it is not exclusively so. Until 

 a system of axioms is established mathematics cannot begin its work. 

 Moreover, the actual perception of spatial relations, not merely 

 in simple cases but in the appreciation of complicated theorems, is 

 an essential element in geometry which has no relation to mathe- 

 matics as Peirce understands the term. The same is true, to a con- 

 siderable extent, of such subjects as mechanical drawing and model- 

 making, which involve, besides small amounts of physics and math- 

 ematics, mainly non-mathematical geometry. Moreover, although the 

 mathematical method is the traditional one for arriving at the truth 

 concerning geometric facts, it is not the only one. Direct appeal to 

 the intuition is often a short and fairly safe cut to geometric results; 

 and on the other hand experiments may be used in geometry, just 

 as they are used every day in physics, to test the truth of a proposi- 

 tion or to determine the value of some geometric magnitude. 1 



We must, then, admit, if we hold to Peirce's view, that there is 

 an independent science of geometry just as there is an independent 

 science of physics, and that either of these may be treated by math- 

 ematical methods. Thus geometry becomes the simplest of the 

 natural sciences, and its axioms are of the nature of physical laws, 

 to be tested by experience and to be regarded as true only within 

 the limits of error of observation. This view, while it has not yet 

 gained universal recognition, should, I believe, prevail, and geo- 

 metry be recognized as a science independent of mathematics, just 

 as psychology is gradually being recognized as an independent 

 science and not as a branch of philosophy. 



The view here set forth, according to which geometry is an ex- 

 perimental science like physics or chemistry, has been held ever 



1 I am thinking of measurements and observations made on accurately con- 

 structed drawings and models. A famous example is Galileo's determination of 

 the area of a cycloid by cutting out a cycloid from a metallic sheet and weighing it. 



