our present-day notions of atomic structure present already in Newton's undiscovered ocean of truth? Could these conceptions have been deduced by him by a sufficient intending of mind from the mathematical, physical and dynamical relationships known to him? I take it that they could not and that only by the creative thought of Newton's nineteenth century successors did these parts of the ocean of truth actually come into existence.

Otherwise it would appear that there was mental determinism and that Clerk Maxwell, Hertz, Planck, Einstein, J. J. Thomson and others thought as they did think because Newton's mind worked in the way it did. And nothing in the results of modern biology seems to suggest that. I have referred already to the way in which mutations of form arise with all the appearance of spontaneity or lack of causation. It must have occurred to many biologists to attempt to predict the evolutionary career of some organic stock or other but beyond suggesting that certain specific forms are in process of extinction, or that some bodily parts of an organic species are becoming vestigial and tend to disappear nothing of the sort has, I believe, been attempted. No biologist has ventured to predict the appearance of a mutation--the essential step to a process of transformism. Now one admits the incredible complexity of the physico-chemical systems in which such mutations arise and we may well despair of laying bare the physical antecedents of a mutation-supposing that there are such. But the overwhelming impression that most biologists have, in thinking about these matters at all, is that of the spontaneity of appearance of the mutation. I admit that there is evidence that environmental changes may induce mutations; expose an organism to some environmental stimuli and mutations may arise but what we have to deal with here are active, functional adaptations of the organism, ways in which it responds to the external change. But that the particular nature of the response is a function, in the mathematical sense, of the particular

change in the environment does not seem to be established, nor do I think that it is likely.

Probably we must generalize "responses responses" in the widest possible way. I have argued elsewhere that the most various kinds of behaviour are of essentially the same organic nature. An "adaptation," I take it, is not merely a change in colour or form that renders an animal less conspicuous to its enemies or prey, or confers upon it some useful means of finding food or shelter, or of avoiding its foes. These changes are of much significance in hypotheses of the means of transformism and so they are the things that we usually think about when we speak of adaptations. But temporary variations of functioning (such as the process of sweating when one becomes warm) are also adaptations. So is the behaviour of a man who takes an umbrella with him on an unpromising morning, or that of the skipper of a vessel when he shortens sail in anticipation of bad weather.

Again the invention and use of a new tool is an adaptation and so also is the discovery of a new mathematical device (say tables of logarithms). It is quite true that some of these organic modes of behaviour are transmitted by heredity so that they become integral in the life-processes of the race while others would disappear on the death of the individual in which they are evolved. That, however, is because the one kind of adaptation (mutations) is characteristic of the racial life passage (it is germinal) while the other appears in the individual life-passage (it is somatic). It would disappear if it were not preserved by tradition.

So I can make no essential distinction between morphological or organic" adaptations of functioning and those changes of ways of mental operation that we call scientific discoveries. The strengthening of the muscles of the fingers and wrists of a pianist; the formation of a skin callus on some part of the hand in consequence of the persistent holding of a tool; manual dexterity in some repetitional mechanical operation;

facility in arithmetical work; the finding of some new mathematical relationships (say Maxwell's four thermodynamic potentials) and so on-all these seem to me to be processes that have the same significance. In each of them there is something creative or new, some means whereby the organism becomes the better able to oppose the tendency to inert-materiality. This is, of course, pure Bergsonism: Maxwell's thermodynamic functions or the Christoffel tensor analysis" are means of acting on nature. Sooner or later, someone endeavours to give even the most abstruse of mathematical results a "physical meaning" and sooner or later also, these results receive "applications" in industry. All "intended" thinking, I take it. aims at establishing inter-connexions between events in nature, All general discoveries are mental adaptations-something really new in organic behaviour.

And if that is so we must, I think, regard Newton's ocean of truth as amorphous in structure. The relations that are to be discovered in it are only in it in the sense that they come into existence with the thought that makes the relations. Our knowledge of nature, as Eddington says, is a knowledge of form and not of content, but even the form is carved out from a nature that may have any forms-or as many as are implicit within the limitations of the human mind. At any moment in human history, then, our description of nature is complete, that is to say, what more is in it than that which we know has still to be made by us. I feel that, as I have stated this there is something paradoxical in it, but my meaning will, I trust, be plain.

Finally, I return to Mr. Whitehead's saying-perhaps that which is urging nature forwards is in the future as well as in the present and past; and in the future that may be as well as in future that will be. I take this to be literally true. The impetus is certainly in the remotest past as well as in the present, since we inherit modes of acting on nature which have passed" only in the sense that they came into existence one

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THE LIMITATIONS OF KNOWLEDGE.

after the other in "time," but which nevertheless endure in that they constitute our present life-mechanisms. The impetus is in the future that will be, surely, because many of the things that we do are done in order that some change, or condition clearly thought about but which has not yet happened, or does not yet exist, will come about as the result of our acting. So from the wolf we have bred the sheep dog and we are rearing rustless wheat and potatoes that are immune to disease. These organisms were not discovered in nature nor did they exist there in the literal sense they were made, if my interpretation of the meaning of mutations is a right one. The impetus is in the future that may be, because in seeking for something we make something else. There is an ideal communistic state that may be and that is ardently desired by some. In seeking to make it. our present-day idealists are, without doubt, making some other society, the form of which is not discoverable.

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

on December 19th, 1921, at 8 P.M.

IV. PHYSICAL SPACE AND HYPERSPACES.

By F. TAVANI.

I.

I SHALL consider space under two aspects: 1. As object of perception, and by this I mean the physical space in which are the objects with qualities which we perceive. Such a space is to objects like a frame in which they either move or are at rest. 2. Besides such space, perceived with and through the bodies which it contains, we have also the idea of a space possessing the means for fixing and determining the position of a body with respect to another body. Space as the locus of entities, by means of which the position of something in it is determined, is a concept, a concept to which a percept may but does not necessarily correspond. The study of space as a concept belongs chiefly to analytical geometry of one, two, three, four, ..., n dimensions.

The properties of this space, taken in their most general expressions, are not objects of perception, they become such only for the particular case in which the dimensions are three. Thence the distinction between dimensions and hyperdimensions while they are all concepts, to those concepts within the space of three dimensions perceptions correspond, but for 4, 5, ..., n dimensional spaces nothing in the world of perception corresponds. Hyperdimensions thus remain abstract entities suggested, not by perception, but by the possibility of generalizing the properties of the three-dimensional space to which a perception corresponds.

The object of this paper is to show that hyperdimensional space is physically real, meaning by hyperdimensional space a space defined by the characteristic that through any of its

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