various objects, the wealth and blending of colours, which call out immediately the response of human appreciation. These features would undoubtedly not be the same if I stood on the hill itself but the modified features are "mere appearance" in Professor Alexander's sense of the term, and are due to external reality, not to mind. The blueness of the mountains and their rugged outline, the blending of the purple and gold of the heather and gorse, which are the elements I consider beautiful are undoubtedly due to the distance from which I am viewing them, but I cannot see that in judging them as beautiful mind has added anything more to them than in judging them as distant. When the mountain is judged as distant the proposition claims to be true; and if the claim is coherent with all other true propositions about the mountain then reality is of this character,—and the mountain is distant. When the mountain is judged to be beautiful, then its beauty according to Alexander lies in the fact that the proposition judged presupposes another proposition in which the mind has in some way united itself with the mountain. In each case the "judging" must be a bare act of assertion. These assertions are distinguishable the one from the other only by their objects which are in the first example an assertion about external reality as such, and in the second an assertion about external reality combined with mind.

One cannot help feeling that this treatment of values gives no account of the experiencing of value. How does the enjoyment of a judgment of truth differ from the enjoyment of a judgment of beauty? Yet as experiences they are certainly different. Would Professor Alexander agree with Professor Perry in describing the "central fact" of value experience as "modes of attitude and impulses," or "as an enacted, sensed, or possibly felt mode of the organism," or is there no such "central fact"?

Professor Alexander's account of truth I find still more 'difficult. It is asserted that the criterion of truth is coherence among propositions, i.e., assertions about reality; but if truth

thus follows reality and is determined by it what is gained by denying coherence to reality itself? (see p. 258). Unless reality itself be coherent how can "truth about reality" be so? It would seem that by thus refusing to admit that it is the nature of reality to be coherent, Alexander is driven, in spite of himself, to treat truth as a subjective character of minds, for he is compelled to admit that a fact which is true to-day may be false to-morrow (p. 263) or even that contradictory facts may be true for two minds at the same time if they live under different conditions. "The once true proposition may turn out even to be erroneous for the newer knowledge, while it remains true and real as such within the narrower range of ancient revealed fact." Professor Alexander's criticism of the pragmatic view of truth is that it gives no account of the nature of reality but states simply "Truth is that which works." He himself would go a step further and say, "Truth works because it is determined by the nature of reality"; but for a satisfactory theory of truth one must I think understand " determined by reality" in a more drastic sense than Alexander himself does.

From the point of view of his theory there can be no distinction between a "judgment of fact" and a "judgment of value," for every judgment must be of the latter type. One feels, however, that there is a real difference in status between truth and the other tertiary qualities. While goodness and

beauty are predominantly subjective in reference, even when universal in form, truth is essentially objective. Goodness and beauty are determined primarily by the efforts and interests of individual subjects; truth is determined primarily by the nature of reality itself. Acts of believing or judging are of course subjective and transient in character; but truth possesses a form of reality timeless and unchangeable in and through which the world of changing existence is revealed

to us.

Meeting of the Aristotelian Society at 21, Gower Street, W.C. 1,

on June 12th, 1922, at 8 p.m.

XI.-GEOMETRY AND REALITY.

By THOMAS GREENWOOD.

FOR many centuries, geometry and mechanics had the most brilliant fortune as rational sciences. But the discovery of nonEuclidian geometry, and the momentous revolution brought about in the field of natural philosophy during these last years, have thrown strong doubts on the self-evidence of the fundamental concepts of geometry, and the basic notions of Newtonian mechanics. It is argued now, that only æsthetical considerations and psychological reasons of formal economy and utility could justify the privileged position of the axioms of these sciences. The practice, for example, of seeing in a distance two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation. If, in pursuance of our habit of thought, we now supplement the propositions of Euclidian geometry by the single proposition that two points on a practically rigid body always correspond to the same distance or line-interval, independently of any changes in position to which we may subject the body, the propositions of Euclidian geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.*

One is led then to suppose that the truths of geometry and mechanics reach a unique level. As Professor Painlevé says, there is no essential difference between geometry and mechanics;†

* Einstein, Relativity, Chap. I.

+ Painlevé, La Mécanique, in Méthode des Sciences, vol. i.

Q

both are experimental sciences, although their developments have been different; the axioms of pure geometry are nothing else but the refined form of the properties of natural bodies. And Böcher writes: "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 the errors of observation."

Professor Einstein has a different opinion, when he writes in his book on Relativity that the concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relations of the ideas involved in it to objects of experience, but only with the logical connexion of the ideas among themselves.† Poincaré had already said: "Geometry is not an experimental science; experience forms merely the occasion for our reflecting upon the geometrical ideas which pre-exist within us."‡

And Professor Eddington answering the question whether it is true to say that "any two sides of a triangle are together greater than the third side," says he is quite unable to say whether this proposition is true or not. "I can deduce it," he continues," from certain other propositions still more elementary, the axioms; if these are true, the proposition is true; if the axioms are not true, the proposition is not true universally; whether the axioms are true or not, I cannot say, and it is outside my province to consider. But for reasons which I do not profess to understand, my friend the physicist is more interested in Euclidian geometry and is continually setting us problems in it."§

All these conflicting opinions are unilateral. For the

*Bulletin Amer. Math. Soc. [2], 2 (1904), p. 124.

+ Einstein, Relativity, Chap. I.

"Foundations of Geometry," in Monist, 9 (1899), p. 41. § Eddington, Space, Time and Gravitation, Chap. I.