4o6 



NATURE 



[February 25, 1892 



Implicit Axioms. — Are then the axioms explicitly enunciated 

 in treatises the only foundations of geometry? One can be 

 assured to the contrary when one sees that, after having succes- 

 sively abandoned them, there still remain some propositions 

 common to theorems of Euclid, Lowatchewski, and Riemann. 

 These propositions ought to rest on some premisses, as geometers 

 admit, although they do not state them. It is interesting to try 

 to liberate them from classical demonstrations. 



Stuart Mill has made the assertion that every definition 

 contains an axiom, since, in defining it, the existence of the 

 object defined is implicitly affirmed. This is going too far : it 

 is seldom that one gives a definition in mathematics without 

 following it by the demonstration of the existence of the object 

 defined, and when it is omitted, it is generally because the 

 reader can easily supply it. It must not be forgotten that the 

 word existence has not the same sense when it is the question of 

 a mathematical creation as when we have to do with a material 

 object. A mathematical creation exists, provided that its 

 definition involves no contradiction either in itself or with the 

 properties previously admitted. 



But if Stuart Mill's remark cannot be applied to all definitions, 

 it is none the less true for some of them. 



A plane is sometimes defined in the following manner: — 

 The plane is a surface such that the straight line which joins 

 any two points in it lies altogether in the surface. 



This definition manifestly hides a new axiom : we could, it is 

 true, alter it, and that would be better, but then it would be 

 necessary to enunciate the axiom more explicitly. 



Other definitions give place to reflections no less important. 



Such is, for example, that of the equality of two figures : two 

 figures are equal when they can be superposed ; to superpose 

 them it is necessary to displace one until it coincides with the 

 other ; but how must it be displaced ? If we ask, we should be 

 answered that it ought to be done without changing its shape, 

 and in the manner of an invariable solid. The "reasoning in a 

 circle " would then be evident. 



In truth, this definition implies nothing. It would have no 

 meaning for a being who lived in a world where there were only 

 fluids. If it seems clear to us, it is that we are accustomed to 

 the properties of natural solids that do not differ greatly from 

 those of ideal solids whose dimensions are all invariable. 



Meanwhile, however imperfect it may be, this definition 

 implies an axiom. 



The possibility of the movement of an invariable figure is not 

 a truth evident by itself, or at least it is only one in the same 

 way as the posttilattim cP Euclide, and not as an analytical a 

 priori judgment would be. 



Moreover, in studying the definitions and the demonstrations 

 of geometry, we see that one is obliged to admit, without de- 

 monstrating it, not only the possibility of this movement, but 

 even some of its properties. 



This results, first of all, from the definition of the straight line. 

 Many defective definitions have been given, but the true one is 

 that which is understood in a,ll the demonstrations where the 

 straight line is in question : 



"It may happen that the movement of a constant figure is 

 such that all points of a line belonging to this figure remain 

 immovable while all the points situated outside this line are 

 displaced. Such a line will be called a straight line." 



We have in this enunciation purposely separated the definition 

 from the axiom that it implies. 



Several proofs, such as those relating to the equality of 

 triangles which depend on the possibility of letting fall a perpen- 

 dicular from a point on a line, assume propositions that are 

 not enunciated, since we must admit that it is possible to carry 

 a figure from one place to another in a certain manner. 



The Fourth Geometry. — Among these implicit axioms, there 

 is one which seems to me worth mentioning, not only because it 

 has given rise to a recent discussion,^ but because in abandoning 

 it, one can construct a fourth geometry, as coherent as those of 

 Euclid, Lowatchewski, and Riemann. 



To demonstrate that we can always raise from a point. A, a 

 perpendicular to a straight line, AB, a straight line, AC, is con- 

 sidered movable round the point A, and in the first instance 

 coinciding with the fixed line AB ; and it is made to turn 

 round the point A until it lies in the prolongation of AB. 



' See MM. Renouvier, Lechalas, Calinon, Revue Philosophigue, June 

 1889 ; Critique Phitosophiquc, September 30 and November 30, 1889 ; Revue 

 Philosophique, 1890, p. 158. See especially the discussion on the " postulate 

 of perpendicularity." 



NO. II65, VOL. 45] 



We thus assume two propositions : first, that such a rota- 

 tion is possible, and then that it can be continued until the two 

 lines are in one straight line. 



If the first point be admitted, and the second rejected, we 

 are led to a series of theorems still more curious than those of 

 Lowatchewski and Riemann, but equally free from contradiction. 



I will quote only one of them, and that not the most singular : 

 A true straight line can be perpendictilar to itself. 



The Theorem of Lie. — The number of implicit axioms intro- 

 duced in classical demonstrations is greater than it need be, and it 

 would be interesting to reduce them to a minimum. We can 

 ask ourselves, in the first place, if this reduction is possible, if 

 the number of necessary axioms, and imaginable geometries is 

 not infinite. 



M. Sophus Lie's theorem dominates all this discussion : it can 

 be thus stated :— 



Let us suppose that the following premisses are admitted : — 



(1) Space has n dimensions. 



(2) The movement of an invariable figure is possible. 



(3) To determine the position of this figure in space, / con- 

 ditions are necessary. 



The number of geometries compatible with these premisses tvill 

 be limited. 



I can even add that, if n be given, a higher limit to / can be 

 assigned. 



If, then, the possibility of movement be admitted, only a 

 finite number (and that a restricted one) of geometries can be 

 invented. 



The Geometries of Riemann. — However, this result seems 

 to be contradicted by Riemann, because this investigator con- 

 structed an infinite number of different geometries, and the one 

 which generally bears his name is only a particular case. 



Everything depends, he says, on the way in which we define 

 the length of a curve. But there are an infinite number of ways 

 of defining this length, and each of these can become the starting 

 point of a new geometry. 



That is perfectly true ; but most of these definitions are in- 

 compatible with the movement of an invariable figure, which is 

 supposed possible in Lie's theorem. These geometries of 

 Riemann, so interesting on many grounds, can only then remain 

 purely analytical, and do not lend themselves to demonstrations 

 analogous to those of Euclid. 



The A'ature of Axioms. — Most mathematicians regard the 

 geometry of Lowatchewski only as a simple logical curiosity ; 

 some of them, however, have gone further. Since several 

 geometries are possible, is it certain that ours is the true one ? 

 Experience, doubtless, teaches us that the sum of the angles of 

 a triangle is equal to two right angles ; but this is only because 

 we operate on too small triangles ; the difference, according to 

 Lowatchewski, is proportional to the surface of the triangle ; 

 will it not become sensible if we work with larger triangles, or 

 if our means of measurement grow more accurate ? Euclidian 

 geometry would only then be a provisional geometry. 



To discuss this question, we ought in the first instance to 

 inquire into the nature of geometrical axioms. 



Are they synthetical conclusions a priori, as Kant used to 

 say? 



They would appeal to us then with such force, that we could 

 not conceive the contrary proposition, nor construct on it a 

 theoretical edifice. There could not be a non-Euclidian 

 geometry. 



To convince oneself of it, let us take a true synthetical a 

 priori condnsion ; for example, the following : — 



If an infinite series of positive whole numbers be taken, all 

 different from each other, there will always be one number that 

 is smaller than all the others. 



Or this other, which is equivalent to it : — 



If a theoretn be true for the number i, and if it has been shown 

 to be true for n + i, provided that it is true for n, then it will 

 be true for all positive whole numbers. 



Let us next try to free ourselves from this conclusion, and, 

 denying these propositions, to invent a false arithmetic analogous 

 to the non-Euclidian geometry. We find that we cannot; we 

 shall be even tempted in the first instance to regard these conclu- 

 sions as the results of analysis. 



Moreover, let us resume our idea of the indefinitely thin 

 animals : surely we can scarcely admit that these beings, if they 

 have minds like ours, would adopt Euclidian geometry, which 

 would be contrary to all their experience. 



Ought we, then, to conclude that the axioms of geometry are 



