SPACE 653 



let us take a true synthetic a priori intuition the following, for 

 instance, which played an important part in the first chapter : If a 

 theorem is true for the number 1, and if it has been proved that it is 

 true of Ti+1, provided it is true of n, it will be true for all positive 

 integers. Let us next try to get rid of this, and while rejecting this 

 proposition let us construct a false arithmetic analogous to non- 

 Euclidean geometry. We shall not be able to do it. We shall be 

 even tempted at the outset to look upon these intuitions as analytical. 

 Besides, to take up again our fiction of animals without thickness, we 

 can scarcely admit that these beings, if their minds are like ours, 

 would adopt the Euclidean geometry, which would be contradicted by 

 all their experience. Ought we, then, to conclude that the axioms of 

 geometry are experimental truths? But we do not make experiments 

 on ideal lines or ideal circles; we can only make them on material 

 objects. On what, therefore, would experiments serving as a founda- 

 tion for geometry be based? The answer is easy. We have seen above 

 that we constantly reason as if the geometrical figures behaved like 

 solids. What geometry would borrow from experiment would be 

 therefore the properties of these bodies. The properties of light and 

 its propagation in a straight line have also given rise to some of the 

 propositions of geometry, and in particular to those of projective 

 geometry, so that from that point of view one would be tempted to 

 say that metrical geometry is the study of solids, and projective 

 geometry that of light. But a difficulty remains, and is unsurmount- 

 able. If geometry were an experimental science, it would not be 

 an exact science. It would be subjected to continual revision. Nay, 

 it would from that day forth be proved to be erroneous, for we 

 know that no rigorously invariable solid exists. Tine geometrical 

 axioms are therefore neither synthetic a priori intuitions nor experi- 

 mental facts. They are conventions. Our choice among all possible 

 conventions is guided by experimental facts; but it remains free, and 

 is only limited by the necessity of avoiding every contradiction, and 

 thus it is that postulates may remain rigorously true even when the 

 experimental laws which have determined their adoption are only ap- 

 proximate. In other words, the axioms of geometry (I do not speak of 

 those of arithmetic) are only definitions in disguise. What, then, are 

 we to think of the question: Is Euclidean geometry true? It has 

 no meaning. We might as well ask if the metric system is true, 

 and if the old weights and measures are false ; if Cartesian co-ordinates 

 are true and polar co-ordinates false. One geometry cannot be more 

 true than another; it can only be more convenient. Now Euclidean 

 geometry is, and will remain, the most convenient: 1st, because it is 

 the simplest, and it is not so only because of our mental habits or 

 because of the kind of direct intuition that we have of Euclidean 

 space; it is the simplest in itself, just as a polynomial of the first de- 



