IIPRAGMATISM AND A THEORYOF KNOWLEDGE
Thequestion before us is the relation of Pragmatism to a body of knowledge.(a) One question at issue between the Idealist[15]and the Pragmatist has to do with the way in which each defines knowledge and gauges its ultimate aim. Both say that knowledge is relative, but one school asserts that the human mind slowly and laboriously uncovers or discovers what Goethe calls the “Living garment of Deity,”i.e.the world of nature, and comes into a heritage of scientific truth which increasingly corresponds to the subject of his faith; the other claims that we live in a self-evolving universe in which in the course of long ages a new heaven and a new earth may be created which are not foreseen or implied in present conditions. In other words, the Idealist finds the Divine in human life; he finds in his own small corner of the universe the microcosm and symbol of Infinity: the Pragmatist considers that nothingiswhich is not a result of human action, and lowers the Divine element to the result of individual human activity. A compromise between the two ideas on new and interesting lines has recently been made by Bergson. The Christian doctrine of Immanence and Transcendence also combines them.Now the increase of a body of knowledge would seem to depend on the comparison of the successful working out of hypotheses with the discrepancies from theory that from time to time appear. Taken together, proofs and discrepancies point to the evidence of a larger law. This is Hegel’s logic, and the principle, so far as it is here implied, is not denied in modern times, for no one wishes to found a logic on a study of discrepancies as such. Even W. James says, “Whenever you once place yourself at the point of view of any higher synthesis you see exactly how it does, in a fashion, take up opposites into itself.”[16]In fact, without the notion of unity, that of discrepancy could not exist: there must be a background on which the differences appear. The ultimate unity is symbolised in the Idealist doctrine of an Absolute.The Absolute of Idealistic thought is not, however, now conceived of (as the Pragmatist would have us believe) as an abstract unity, but as one involving a social bond, and hence relations which can be described as personal, if we remember that the Personality of the Absolute transcends our notion of human personality. Such a conception of the term Absolute, a new reading of the theory of the One and the Many, has been led up to by Bradley and Royce by methods of logic, and without any reference to dogma. It has been conveniently expressed by Taylor. The argument is briefly that ultimate Reality must be One, Many, and Personal.“For our conclusion that mere truth cannot be the same thing as ultimate reality was itself based upon the principle that only harmonious individuality is finally real, and this is the very principle employed by the intellect itself whenever it judges one thought-construction relatively higher or truer than another.”[17]And again:—“If we speak of existence as a society, then we must be careful to remember that the individual unity of a society is just as real a fact of experience as the individual unity of the members which compose it, and that when we call the Absolute a society rather than a self, we do not do so with any intention of casting doubt upon its complete spiritual unity as an individual experience.”[18]The Absolute has been stated in modern thought to be One, Many, Real, and Personal or Social, and these terms of its qualification have been successively arrived at.W. James’s words ring hollow when he attempts to dissociate such a conception from the reality of which it is the crown and inclusive symbol, and type and essence. “I personally,” he says, “give up the Absolute. I find it entangles me in metaphysical paradoxes that are inacceptable.” He allows that there may be a God, though limited in power and goodness, “one helper amongst others,primus inter paresin the midst of all the shapers of the great world’s fate.” In such a system, as H. Jones has pointed out, “there is neither in the universe nor in God any principle to inspire or guide, or in any way to bring about the amelioration desired. The process is guided by no end. The universe begins by being an aggregate of accidents, pluralistic, discontinuous, irrational, and, of itself, cannot become otherwise. There is nothing actual within to change its character.… God is himself finite, helpless to bring about this great change, a part, and no more, of a universe broken in fragments.”Another form, and a very scholarly one, of the argument against the existence of an Absolute has been stated by Bax in the “Roots of Reality.” He appears to have reached the conclusion that thetelos, the goal of human thought, is not an Absolute involving any notion of fixity, but that it may be conceived of as a “moving synthesis.” He argues that everything of which we are conscious in the universe is seen against a background which itself moves, and is only realisable or distinguishable if it shifts upon something relatively motionless behind it. He concludes, therefore, that by analogy there is no Absolute, since what we perceive always implies something against which we perceive it; thus that there is no goal by which and at which the spirit of man can find rest. On his theory we could never claim to reach the conception of an Absolute, though he admits the progressive character of human thought, and the increasing reach, lucidity, and depth of the human mind. The true answer to this argument is that it proves exactly what it sets out to disprove. As it is acknowledged that only the permanent or the relatively permanent can produce the phenomena of change, sothe appearance of the goal of thought as a moving synthesis would presuppose an Absolute as a ground reality.[19]If in truth we were able to apprehend entirely the source of all life and the background of all experience, we might say that it did not exist for usas an Absolute, but the fact that whatever we perceive postulates an unending series behind it, carries with it the proof of an Absolute Infinite. (This conclusion is led up to by the mathematician’s idea of the series of all finite and transfinite ordinal numbers.) Some part of this argument has been already suggested in Ormond’s “Foundations of Knowledge,” and so far was used by Mr. Illingworth in the “Doctrine of the Trinity.”[20]“From a deeper metaphysical point of view it is the concept of evolution itself that must submit to the determination of knowledge, for it will be found that in so far as it becomes epistemologically necessary to ground relative processes in an Absolute experience, just so far will it become necessary also to connect the evolutionary aspect of the world itself with a ground reality that is stable, and involves the flux of change only as transcending and including it.”[21]The further answer that any judgment, even the Pragmatist’s “judgment of value,” implies an Absolute has been stated in his Oxford Lectures[22]by Professor H. Jones.(b) The next point we should like to work out is the relation of fact to law. The Pragmatist denies scientific law and also logic, and makes his appeal to facts. No conclusion can be drawn from that denial except by the use of logic itself. If he consistently denied logic, his position would be unassailable by logic, but he uses the method he denies, and is thus open to attack. On the subject of the Laws of Science the Pragmatist points out truly that there is no actual continuity between a fact and a law. But laws are concepts, the result of mental activities; they are themselves subject to the laws of logic. “They were means, and you make them ends,” complains the Pragmatist. That is just what nature herself does. She perfects means, such as the means of supporting life, and then these become ends. Language, again, is at first a means, and then becomes an end. So does any science change its character to the onlooker. A law, too, though it generalises facts, is a limit on absolute generalisation. It thus stands midway between the abstraction and the fact. The Pragmatist, however, opposes to law what he calls a new fact—what should rather be called a hypothesis. He asserts that in every event, action, experiment, there is a margin unseen and unrecognised by us; that at every moment, therefore, the unknown, the unexpected, may take shape and voice and denounce all our careful and reasoned conclusions. “Why should the sun rise to-morrow because he has risen to-day and yesterday?” asks the Pragmatist. “We are making an enormous assumption,” he says, “in claiming the uniformity of Nature and the principle of causality.” The Idealist answers that the Pragmatist makes a larger assumption in doubting the truth of the principles, which though relative and not absolute, still do work out in practice, than the Idealist does in his act of faith. In fact, the act of faith is rational as well as natural; it is the act of doubting that is in this case due to a mere scholastic quibble. It is the Idealist and not the Pragmatist who makes his appeal to the truth of facts. Each day that the sun goes on rising finds the Idealist in a better philosophical position and the Pragmatist in a worse, except on the assumption that the link between man and the external world is a false imagination. Let us emphasise:—It is the Pragmatist who quibbles with logic, and the Idealist who appeals to facts.(c) Now there are certain facts and certain deductions from facts, well known to mathematicians, which we should like to quote here as having a bearing on the theory of the Absolute, because they deal with aspects of Infinity, and mark a connection between the world as we know it and the concepts of the philosopher. All have the support of science, and furnish the Idealist philosopher with examples which support his theories, and strengthen his position in the face of the Pragmatist attack. They have to do with the theory of Infinity as shown in:—I.The Indefinite Regress.II.Infinite series.III.Dimensions in space and time.Before entering upon them we must repeat that the question of number and series in mathematics is independent of the assumptions of space and time. As a science, mathematics could exist outside them: order is not necessarily spatial or temporal. Our conclusions, therefore, cannot be attacked on the ground that they are based on Euclidean conceptions of space: they are based on the laws of logic.I.The Indefinite RegressHume and, later, Kant argued that by the principle of association when we think of one quality of a thing the others are naturally brought before our minds, and thus that we get into the habit of attributing to the notion of the thing a certain group of qualities. And it is true that we do attend to a thing all at once, including in the notion of it all the qualities which we know belong to it.Now experience, according to Leibniz, gives us an example of a unity which embraces a multiplicity of detail. Thus a thing is one substance as embodying an individual experience, and its qualities belong to it in the same sense as the constituents of experience belong to the single experience. These qualities are in relation. (The Pragmatist denies the existence of relations as part of a higher unity.[23]) But they are not only relation, since relation always implies something more than itself. Let us take the example of number. Numbers could never have been counted if there had not been things to count. Now suppose each quality could be analysed into a new relation, we should still not get rid of the quality. At each stage there remains a quality in relation, and this goes on to Infinity. Such a constant subdivision perhaps results from our finite experience seizing facts in a disjointed way. When we analyse a law in its working, we always do seem to come to this Indefinite Regress. Now it has been the reproach against metaphysics, as uttered by the Pragmatist, that there is no correspondence in scientific fact to this road into Infinity.W. James asserts: “But in point of fact, nature doesn’t make eggs by making first half an egg, then a quarter, then an eighth, &c., and adding them together. She either makes a whole egg at once or none at all, and so of all her other units. It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come, that Zeno’s paradox gives trouble. And it gives trouble then only if the successive steps of change be infinitely divisible.”[24]The sphere of change, however, one would answer, includes all nature, and science in its discoveries acts on the hypothesis that these steps of change may be infinitely divisible. Royce held to it firmly that any consistent attempt to make an orderly arrangement of the terms of an infinite whole must lead to the Indefinite Regress. And he further shows the connection with the fact that an infinite series can be adequately represented by a part of itself.In the Boyle Lecture, delivered in Oxford in 1908, on the properties of radium, two facts emerged which show that the Indefinite Regress is now recognised in science.First, that in the region of experiment we become aware of groups of elements allied to radium, which seem, in the number of individuals in their groups, to follow a simple arithmetical progression.Secondly, that radio-active elements lose in activity at a certain rate, which always represents an exact proportion of the mass which remains. The tremendous disintegrating force slackens in exact relation to the time which passes, so that the smaller the morsel the less the relative loss of mass. Here, then, is the Indefinite Regress. In the world of fact as well as of ideas we are dealing with aspects of Infinity.[25]II.Infinite SeriesThere are other aspects of Infinity which we can get at by studying series, and which in the conception of series of series give strength and point to the philosophic conception of an Absolute.Prof. C. Keyser develops this thought, and shows (in two recent articles, January and April 1909, in theHibbert Journal) that certain theological dogmas, such as the doctrine of the Trinity, and certain attributes of the Divine Being, such as Omniscience and Omnipresence, are entirely conceivable by the human mind if regarded without the paralysing limitations of the Finite. He shows that in our mathematical formulæ which have to do with infinite series we have the exact replica of what to the lay, non-mathematical mind seem to be the paradoxes of the Athanasian Creed. He first shows that in a mathematical analogy points of view about an Infinite Being, even if partially discordant, may all be true if regard is had to His Infinity.[26]Further, he shows that certain assumptions, such asthe whole is greater than its part, are inapplicable to Infinite Being. The conception of a Trinity in Unity in which “none is afore or after other, none is greater or less than another, but the whole three persons are co-eternal together and co-equal” is rationally conceivable by the mathematician who is familiar with the theory of manifolds.[27]We have, he shows, three infinite manifolds:—E of the even integers.O of the odd ones.F of the fractions having integers for their terms.No two of these have a single element in common, yet the three together constitute one manifold M, that is exactly equal in wealth of elements toeachof its infinite components.Again, there is the apparent opposition between the Omniscience of God and the freedom of man. The antithesis disappears if we realise that from the point of view of Infinites the dignity and power of Omniscience remain the same, even if some part of experience is not yet drawn into the sphere of Omniscience.[28]Here we have the present conceived of as a moving plane separating the unknown from the known. The “past” can be said to be known, though its content changes every instant. This is the real answer to W. James’s cry that he could accept an Absolute if it had even the fragment of an “other.” There can be this “other,” and yet the Absolute still remains an Absolute.The doctrine ofOmnipresencefollows from the argument of the Continuum (which is the aggregate of all real numbers). Thus the number of points in space of infinite dimension is no greater than the number of points in any part of space as known to us. The whole is incarnate in every part, because to each part, in however small an atom, corresponds a point in the universal whole, and the number of points in a space of infinite dimensions is equal to the number of points in a straight line however small.And this is true not merely of points but also of forces. “The Universe is dynamic, charged throughout with innumerable modes of motion. Each point, however, of any moving thing—an ion of gas, a vibrating fibre of brain—is represented by a corresponding point in S (a small typical atom), and so within the tiny sphere—indeed, in every room, however small—the whole dynamics of the universe is depicted completely and co-enacted by motion of points and transformation of point configurations. There in miniature proceed at once the countless play and interplay of every kind of motion, small and large, simple and complex, the quivering dance of the molecule, the wave and swing of universal æther.”[29]III.Dimensions in SpaceThere is another argument, one relating to the theories of time and space, which greatly affects the conception of omnipresence. This is the argument of the many dimensions, called by Keyser the “radiant concept of hyper-space, which only a generation ago was regarded, even by mathematicians—most adventurous of men—as being purposeless and vain, but which meanwhile has advanced so rapidly to commanding position that even the following statement by Poincaré, in his recent address before the International Mathematical Congress at Rome on ‘L’Avenir des Mathématiques,’ is well within the limits of conservatism: ‘Nous sommes aujourd’hui tellement familiarisés avec cette notion que nous pouvons en parler, même dans un cours d’université, sans provoquer trop d’étonnement.’ The fact is that the doctrine already exists in a vast and rapidly growing literature, flourishes in all the scientific languages of the world, and in its essential principles has become for mathematics as orthodox as the multiplication table.”The present position of the theory is briefly this: If there did not exist a fourth dimension, we could not be aware of a third as such, and so on. Are we then looking out upon a third dimensional world, and realising it as such because we are mentally capable of conceiving dimensions beyond it? Our world sensibly contains one dimensional and two dimensional facts—the first such as a time series, for which one number is sufficient to fix a point, and the second such as a plane where position can be fixed by two numbers. Does our world contain facts of other dimensions?“All particles of air are four-dimensional in magnitude when, in addition to their position in space, we also consider the variable densities which they assume, as they are expressed by the different heights of the barometer in the different parts of the atmosphere. Similarly all conceivable spheres in space are four-dimensional magnitudes, for their centres form a three-dimensional point-aggregate, and around each centre a one-dimensional totality of spheres, the radii of which can be expressed by every numerical magnitude from zero to infinity. Further, if we imagine a measuring-stick of invariable length to assume every conceivable position in space, the positions so obtained will constitute a five-dimensional aggregate. For in the first place one of the extremities of the measuring-stick may be conceived to assume a position at every point of space, and this determines for one extremity alone of the stick a three-dimensional totality of position, and, secondly, as we have seen above, there proceeds from every such position of this extremity a two-dimensional totality of directions, and by conceiving the measuring-stick to be placed lengthwise in every one of these directions, we shall obtain all the conceivable positions which the second extremity can assume, and consequently the dimensions must be 3 + 2 or 5 …” &c., &c.[30]Mathematicians have for long done problems in the seventh and eighth dimensions. They have told us that you cannot tie a knot in the second dimension, because there is no up or down, and the threads would not cross—nor in the fourth, because the knot would pull out in a new direction and would not hold. But it has only lately been realised that fourth and other dimensions may be actual fact in the world round us. Of course, from the point of view of a point there are only three dimensions to be known, but to a line in the same space there are five, to the surface probably six. Our intelligence at present does not go beyond the point; but if we could think of space from the point of view of a solid, worlds upon worlds would rise before our view.Of the fourth dimension we can discover some facts by analogy. We can count the edges of its typical figure, and apply thought to determining some of its conditions. But a more interesting subject of research is the inquiry into the light thrown by the theory of four dimensions on the determination of certain atoms in chemistry, that are known to be distinct elements, but could only be determined actually in another dimension.[31]“In chemistry, the molecules of a compound body are said to consist of the atoms of the elements which are contained in the body, and these are supposed to be situated at certain distances from one another and to be held in their relative positions by certain forces. At first the centres of the atoms were conceived to lie in one and the same plane. But Wislicenus was led by researches in paralactic acid to explain the differences of isomeric molecules of the same structural formulæ by the different positions of the atoms inspace. In fact, four points can always be so arranged in space that every two of them may have any distance from each other; and the change of one of the six distances does not necessarily involve the alteration of any other.“But suppose our molecule consists of five atoms? Four of these may be so placed that the distance between any two of them can be made what we please. But it is no longer possible to give the fifth atom a position such that each of the four distances by which it is separated from the other atoms may be what we please. On the contrary, the fourth distance is dependent on the three remaining distances, for the space of experience has only three dimensions. If, therefore, I have a molecule which consists of five atoms, I cannot alter the distance between two of them without at least altering some second distance. But if we imagine the centres of the atoms placed in a four-dimensioned space, this can be done; all the ten distances which may be conceived to exist between the five points will then be independent of one another. To reach the same result in the case of six atoms we must assume a five-dimensional space, and so on.”[32]Here we see that if chemistry as a science is bound to take account of all its facts, the scientist is confronted with a problem of dimensions that is really a problem of Infinity applied not, as in the other cases quoted, to number, but to space.And there is a reason which explains why the same problem tends to appear in these different ways. Both time and space can be most correctly thought of asseries: the former known to us as possessing one direction, though possibly involving more, and the latter three, though possibly involving more. Time is not a thing nor a condition, but it is the way in which we are enabled to apprehend the relations of actions to one another. The assumption of the Pragmatist, that a different date in history is a new condition which might affect a chemical experiment, is meaningless, unless by that he intends to say that at the different date new conditions prevailed.The general conclusion of recent thought is then to establish the Idealist position more strongly by an appeal to mathematical argument. This argument is strengthened by finding at the present time some support in scientific fact and experiment. The Idealist therefore appeals to fact, and his position rests ultimately on a truth which has its aspects of conformity with scientific experiment and with logical or mathematical proof.
Thequestion before us is the relation of Pragmatism to a body of knowledge.
(a) One question at issue between the Idealist[15]and the Pragmatist has to do with the way in which each defines knowledge and gauges its ultimate aim. Both say that knowledge is relative, but one school asserts that the human mind slowly and laboriously uncovers or discovers what Goethe calls the “Living garment of Deity,”i.e.the world of nature, and comes into a heritage of scientific truth which increasingly corresponds to the subject of his faith; the other claims that we live in a self-evolving universe in which in the course of long ages a new heaven and a new earth may be created which are not foreseen or implied in present conditions. In other words, the Idealist finds the Divine in human life; he finds in his own small corner of the universe the microcosm and symbol of Infinity: the Pragmatist considers that nothingiswhich is not a result of human action, and lowers the Divine element to the result of individual human activity. A compromise between the two ideas on new and interesting lines has recently been made by Bergson. The Christian doctrine of Immanence and Transcendence also combines them.
Now the increase of a body of knowledge would seem to depend on the comparison of the successful working out of hypotheses with the discrepancies from theory that from time to time appear. Taken together, proofs and discrepancies point to the evidence of a larger law. This is Hegel’s logic, and the principle, so far as it is here implied, is not denied in modern times, for no one wishes to found a logic on a study of discrepancies as such. Even W. James says, “Whenever you once place yourself at the point of view of any higher synthesis you see exactly how it does, in a fashion, take up opposites into itself.”[16]In fact, without the notion of unity, that of discrepancy could not exist: there must be a background on which the differences appear. The ultimate unity is symbolised in the Idealist doctrine of an Absolute.
The Absolute of Idealistic thought is not, however, now conceived of (as the Pragmatist would have us believe) as an abstract unity, but as one involving a social bond, and hence relations which can be described as personal, if we remember that the Personality of the Absolute transcends our notion of human personality. Such a conception of the term Absolute, a new reading of the theory of the One and the Many, has been led up to by Bradley and Royce by methods of logic, and without any reference to dogma. It has been conveniently expressed by Taylor. The argument is briefly that ultimate Reality must be One, Many, and Personal.
“For our conclusion that mere truth cannot be the same thing as ultimate reality was itself based upon the principle that only harmonious individuality is finally real, and this is the very principle employed by the intellect itself whenever it judges one thought-construction relatively higher or truer than another.”[17]
And again:—
“If we speak of existence as a society, then we must be careful to remember that the individual unity of a society is just as real a fact of experience as the individual unity of the members which compose it, and that when we call the Absolute a society rather than a self, we do not do so with any intention of casting doubt upon its complete spiritual unity as an individual experience.”[18]
The Absolute has been stated in modern thought to be One, Many, Real, and Personal or Social, and these terms of its qualification have been successively arrived at.
W. James’s words ring hollow when he attempts to dissociate such a conception from the reality of which it is the crown and inclusive symbol, and type and essence. “I personally,” he says, “give up the Absolute. I find it entangles me in metaphysical paradoxes that are inacceptable.” He allows that there may be a God, though limited in power and goodness, “one helper amongst others,primus inter paresin the midst of all the shapers of the great world’s fate.” In such a system, as H. Jones has pointed out, “there is neither in the universe nor in God any principle to inspire or guide, or in any way to bring about the amelioration desired. The process is guided by no end. The universe begins by being an aggregate of accidents, pluralistic, discontinuous, irrational, and, of itself, cannot become otherwise. There is nothing actual within to change its character.… God is himself finite, helpless to bring about this great change, a part, and no more, of a universe broken in fragments.”
Another form, and a very scholarly one, of the argument against the existence of an Absolute has been stated by Bax in the “Roots of Reality.” He appears to have reached the conclusion that thetelos, the goal of human thought, is not an Absolute involving any notion of fixity, but that it may be conceived of as a “moving synthesis.” He argues that everything of which we are conscious in the universe is seen against a background which itself moves, and is only realisable or distinguishable if it shifts upon something relatively motionless behind it. He concludes, therefore, that by analogy there is no Absolute, since what we perceive always implies something against which we perceive it; thus that there is no goal by which and at which the spirit of man can find rest. On his theory we could never claim to reach the conception of an Absolute, though he admits the progressive character of human thought, and the increasing reach, lucidity, and depth of the human mind. The true answer to this argument is that it proves exactly what it sets out to disprove. As it is acknowledged that only the permanent or the relatively permanent can produce the phenomena of change, sothe appearance of the goal of thought as a moving synthesis would presuppose an Absolute as a ground reality.[19]
If in truth we were able to apprehend entirely the source of all life and the background of all experience, we might say that it did not exist for usas an Absolute, but the fact that whatever we perceive postulates an unending series behind it, carries with it the proof of an Absolute Infinite. (This conclusion is led up to by the mathematician’s idea of the series of all finite and transfinite ordinal numbers.) Some part of this argument has been already suggested in Ormond’s “Foundations of Knowledge,” and so far was used by Mr. Illingworth in the “Doctrine of the Trinity.”[20]
“From a deeper metaphysical point of view it is the concept of evolution itself that must submit to the determination of knowledge, for it will be found that in so far as it becomes epistemologically necessary to ground relative processes in an Absolute experience, just so far will it become necessary also to connect the evolutionary aspect of the world itself with a ground reality that is stable, and involves the flux of change only as transcending and including it.”[21]
The further answer that any judgment, even the Pragmatist’s “judgment of value,” implies an Absolute has been stated in his Oxford Lectures[22]by Professor H. Jones.
(b) The next point we should like to work out is the relation of fact to law. The Pragmatist denies scientific law and also logic, and makes his appeal to facts. No conclusion can be drawn from that denial except by the use of logic itself. If he consistently denied logic, his position would be unassailable by logic, but he uses the method he denies, and is thus open to attack. On the subject of the Laws of Science the Pragmatist points out truly that there is no actual continuity between a fact and a law. But laws are concepts, the result of mental activities; they are themselves subject to the laws of logic. “They were means, and you make them ends,” complains the Pragmatist. That is just what nature herself does. She perfects means, such as the means of supporting life, and then these become ends. Language, again, is at first a means, and then becomes an end. So does any science change its character to the onlooker. A law, too, though it generalises facts, is a limit on absolute generalisation. It thus stands midway between the abstraction and the fact. The Pragmatist, however, opposes to law what he calls a new fact—what should rather be called a hypothesis. He asserts that in every event, action, experiment, there is a margin unseen and unrecognised by us; that at every moment, therefore, the unknown, the unexpected, may take shape and voice and denounce all our careful and reasoned conclusions. “Why should the sun rise to-morrow because he has risen to-day and yesterday?” asks the Pragmatist. “We are making an enormous assumption,” he says, “in claiming the uniformity of Nature and the principle of causality.” The Idealist answers that the Pragmatist makes a larger assumption in doubting the truth of the principles, which though relative and not absolute, still do work out in practice, than the Idealist does in his act of faith. In fact, the act of faith is rational as well as natural; it is the act of doubting that is in this case due to a mere scholastic quibble. It is the Idealist and not the Pragmatist who makes his appeal to the truth of facts. Each day that the sun goes on rising finds the Idealist in a better philosophical position and the Pragmatist in a worse, except on the assumption that the link between man and the external world is a false imagination. Let us emphasise:—It is the Pragmatist who quibbles with logic, and the Idealist who appeals to facts.
(c) Now there are certain facts and certain deductions from facts, well known to mathematicians, which we should like to quote here as having a bearing on the theory of the Absolute, because they deal with aspects of Infinity, and mark a connection between the world as we know it and the concepts of the philosopher. All have the support of science, and furnish the Idealist philosopher with examples which support his theories, and strengthen his position in the face of the Pragmatist attack. They have to do with the theory of Infinity as shown in:—
I.The Indefinite Regress.
II.Infinite series.
III.Dimensions in space and time.
Before entering upon them we must repeat that the question of number and series in mathematics is independent of the assumptions of space and time. As a science, mathematics could exist outside them: order is not necessarily spatial or temporal. Our conclusions, therefore, cannot be attacked on the ground that they are based on Euclidean conceptions of space: they are based on the laws of logic.
I.The Indefinite Regress
Hume and, later, Kant argued that by the principle of association when we think of one quality of a thing the others are naturally brought before our minds, and thus that we get into the habit of attributing to the notion of the thing a certain group of qualities. And it is true that we do attend to a thing all at once, including in the notion of it all the qualities which we know belong to it.
Now experience, according to Leibniz, gives us an example of a unity which embraces a multiplicity of detail. Thus a thing is one substance as embodying an individual experience, and its qualities belong to it in the same sense as the constituents of experience belong to the single experience. These qualities are in relation. (The Pragmatist denies the existence of relations as part of a higher unity.[23]) But they are not only relation, since relation always implies something more than itself. Let us take the example of number. Numbers could never have been counted if there had not been things to count. Now suppose each quality could be analysed into a new relation, we should still not get rid of the quality. At each stage there remains a quality in relation, and this goes on to Infinity. Such a constant subdivision perhaps results from our finite experience seizing facts in a disjointed way. When we analyse a law in its working, we always do seem to come to this Indefinite Regress. Now it has been the reproach against metaphysics, as uttered by the Pragmatist, that there is no correspondence in scientific fact to this road into Infinity.
W. James asserts: “But in point of fact, nature doesn’t make eggs by making first half an egg, then a quarter, then an eighth, &c., and adding them together. She either makes a whole egg at once or none at all, and so of all her other units. It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come, that Zeno’s paradox gives trouble. And it gives trouble then only if the successive steps of change be infinitely divisible.”[24]
The sphere of change, however, one would answer, includes all nature, and science in its discoveries acts on the hypothesis that these steps of change may be infinitely divisible. Royce held to it firmly that any consistent attempt to make an orderly arrangement of the terms of an infinite whole must lead to the Indefinite Regress. And he further shows the connection with the fact that an infinite series can be adequately represented by a part of itself.
In the Boyle Lecture, delivered in Oxford in 1908, on the properties of radium, two facts emerged which show that the Indefinite Regress is now recognised in science.
First, that in the region of experiment we become aware of groups of elements allied to radium, which seem, in the number of individuals in their groups, to follow a simple arithmetical progression.
Secondly, that radio-active elements lose in activity at a certain rate, which always represents an exact proportion of the mass which remains. The tremendous disintegrating force slackens in exact relation to the time which passes, so that the smaller the morsel the less the relative loss of mass. Here, then, is the Indefinite Regress. In the world of fact as well as of ideas we are dealing with aspects of Infinity.[25]
II.Infinite Series
There are other aspects of Infinity which we can get at by studying series, and which in the conception of series of series give strength and point to the philosophic conception of an Absolute.
Prof. C. Keyser develops this thought, and shows (in two recent articles, January and April 1909, in theHibbert Journal) that certain theological dogmas, such as the doctrine of the Trinity, and certain attributes of the Divine Being, such as Omniscience and Omnipresence, are entirely conceivable by the human mind if regarded without the paralysing limitations of the Finite. He shows that in our mathematical formulæ which have to do with infinite series we have the exact replica of what to the lay, non-mathematical mind seem to be the paradoxes of the Athanasian Creed. He first shows that in a mathematical analogy points of view about an Infinite Being, even if partially discordant, may all be true if regard is had to His Infinity.[26]
Further, he shows that certain assumptions, such asthe whole is greater than its part, are inapplicable to Infinite Being. The conception of a Trinity in Unity in which “none is afore or after other, none is greater or less than another, but the whole three persons are co-eternal together and co-equal” is rationally conceivable by the mathematician who is familiar with the theory of manifolds.[27]
We have, he shows, three infinite manifolds:—
E of the even integers.
O of the odd ones.
F of the fractions having integers for their terms.
No two of these have a single element in common, yet the three together constitute one manifold M, that is exactly equal in wealth of elements toeachof its infinite components.
Again, there is the apparent opposition between the Omniscience of God and the freedom of man. The antithesis disappears if we realise that from the point of view of Infinites the dignity and power of Omniscience remain the same, even if some part of experience is not yet drawn into the sphere of Omniscience.[28]
Here we have the present conceived of as a moving plane separating the unknown from the known. The “past” can be said to be known, though its content changes every instant. This is the real answer to W. James’s cry that he could accept an Absolute if it had even the fragment of an “other.” There can be this “other,” and yet the Absolute still remains an Absolute.
The doctrine ofOmnipresencefollows from the argument of the Continuum (which is the aggregate of all real numbers). Thus the number of points in space of infinite dimension is no greater than the number of points in any part of space as known to us. The whole is incarnate in every part, because to each part, in however small an atom, corresponds a point in the universal whole, and the number of points in a space of infinite dimensions is equal to the number of points in a straight line however small.
And this is true not merely of points but also of forces. “The Universe is dynamic, charged throughout with innumerable modes of motion. Each point, however, of any moving thing—an ion of gas, a vibrating fibre of brain—is represented by a corresponding point in S (a small typical atom), and so within the tiny sphere—indeed, in every room, however small—the whole dynamics of the universe is depicted completely and co-enacted by motion of points and transformation of point configurations. There in miniature proceed at once the countless play and interplay of every kind of motion, small and large, simple and complex, the quivering dance of the molecule, the wave and swing of universal æther.”[29]
III.Dimensions in Space
There is another argument, one relating to the theories of time and space, which greatly affects the conception of omnipresence. This is the argument of the many dimensions, called by Keyser the “radiant concept of hyper-space, which only a generation ago was regarded, even by mathematicians—most adventurous of men—as being purposeless and vain, but which meanwhile has advanced so rapidly to commanding position that even the following statement by Poincaré, in his recent address before the International Mathematical Congress at Rome on ‘L’Avenir des Mathématiques,’ is well within the limits of conservatism: ‘Nous sommes aujourd’hui tellement familiarisés avec cette notion que nous pouvons en parler, même dans un cours d’université, sans provoquer trop d’étonnement.’ The fact is that the doctrine already exists in a vast and rapidly growing literature, flourishes in all the scientific languages of the world, and in its essential principles has become for mathematics as orthodox as the multiplication table.”
The present position of the theory is briefly this: If there did not exist a fourth dimension, we could not be aware of a third as such, and so on. Are we then looking out upon a third dimensional world, and realising it as such because we are mentally capable of conceiving dimensions beyond it? Our world sensibly contains one dimensional and two dimensional facts—the first such as a time series, for which one number is sufficient to fix a point, and the second such as a plane where position can be fixed by two numbers. Does our world contain facts of other dimensions?
“All particles of air are four-dimensional in magnitude when, in addition to their position in space, we also consider the variable densities which they assume, as they are expressed by the different heights of the barometer in the different parts of the atmosphere. Similarly all conceivable spheres in space are four-dimensional magnitudes, for their centres form a three-dimensional point-aggregate, and around each centre a one-dimensional totality of spheres, the radii of which can be expressed by every numerical magnitude from zero to infinity. Further, if we imagine a measuring-stick of invariable length to assume every conceivable position in space, the positions so obtained will constitute a five-dimensional aggregate. For in the first place one of the extremities of the measuring-stick may be conceived to assume a position at every point of space, and this determines for one extremity alone of the stick a three-dimensional totality of position, and, secondly, as we have seen above, there proceeds from every such position of this extremity a two-dimensional totality of directions, and by conceiving the measuring-stick to be placed lengthwise in every one of these directions, we shall obtain all the conceivable positions which the second extremity can assume, and consequently the dimensions must be 3 + 2 or 5 …” &c., &c.[30]
Mathematicians have for long done problems in the seventh and eighth dimensions. They have told us that you cannot tie a knot in the second dimension, because there is no up or down, and the threads would not cross—nor in the fourth, because the knot would pull out in a new direction and would not hold. But it has only lately been realised that fourth and other dimensions may be actual fact in the world round us. Of course, from the point of view of a point there are only three dimensions to be known, but to a line in the same space there are five, to the surface probably six. Our intelligence at present does not go beyond the point; but if we could think of space from the point of view of a solid, worlds upon worlds would rise before our view.
Of the fourth dimension we can discover some facts by analogy. We can count the edges of its typical figure, and apply thought to determining some of its conditions. But a more interesting subject of research is the inquiry into the light thrown by the theory of four dimensions on the determination of certain atoms in chemistry, that are known to be distinct elements, but could only be determined actually in another dimension.[31]
“In chemistry, the molecules of a compound body are said to consist of the atoms of the elements which are contained in the body, and these are supposed to be situated at certain distances from one another and to be held in their relative positions by certain forces. At first the centres of the atoms were conceived to lie in one and the same plane. But Wislicenus was led by researches in paralactic acid to explain the differences of isomeric molecules of the same structural formulæ by the different positions of the atoms inspace. In fact, four points can always be so arranged in space that every two of them may have any distance from each other; and the change of one of the six distances does not necessarily involve the alteration of any other.
“But suppose our molecule consists of five atoms? Four of these may be so placed that the distance between any two of them can be made what we please. But it is no longer possible to give the fifth atom a position such that each of the four distances by which it is separated from the other atoms may be what we please. On the contrary, the fourth distance is dependent on the three remaining distances, for the space of experience has only three dimensions. If, therefore, I have a molecule which consists of five atoms, I cannot alter the distance between two of them without at least altering some second distance. But if we imagine the centres of the atoms placed in a four-dimensioned space, this can be done; all the ten distances which may be conceived to exist between the five points will then be independent of one another. To reach the same result in the case of six atoms we must assume a five-dimensional space, and so on.”[32]
Here we see that if chemistry as a science is bound to take account of all its facts, the scientist is confronted with a problem of dimensions that is really a problem of Infinity applied not, as in the other cases quoted, to number, but to space.
And there is a reason which explains why the same problem tends to appear in these different ways. Both time and space can be most correctly thought of asseries: the former known to us as possessing one direction, though possibly involving more, and the latter three, though possibly involving more. Time is not a thing nor a condition, but it is the way in which we are enabled to apprehend the relations of actions to one another. The assumption of the Pragmatist, that a different date in history is a new condition which might affect a chemical experiment, is meaningless, unless by that he intends to say that at the different date new conditions prevailed.
The general conclusion of recent thought is then to establish the Idealist position more strongly by an appeal to mathematical argument. This argument is strengthened by finding at the present time some support in scientific fact and experiment. The Idealist therefore appeals to fact, and his position rests ultimately on a truth which has its aspects of conformity with scientific experiment and with logical or mathematical proof.
FOOTNOTES:[15]This word is used here in the most general and inclusive sense as applying to all thinkers who accept the reality of relations as part of a higher Unity.[16]“A Pluralistic Universe,”p.99.[17]Taylor, “Elements of Metaphysics,”p.312.[18]Ibid.,p.350.[19]A succession of what is disconnected is not change. Change is a succession within an identity: if not within the identity, there is no change, only analysis and re-grouping. The closer our knowledge is of ourselves or anything else, the more we see thatchange is the expression in time of an identity.[20]Illingworth, “The Doctrine of the Trinity,”p.6.[21]Ormond, “Foundations of Knowledge,”p.19.[22]1908-10.[23]James, “A Pluralistic Universe,”p.80.[24]“A Pluralistic Universe,”p.230.[25]See A. T. Cameron, “Radio-Chemistry,”p.17: “The curves illustrate two further points. They approach constant value towards the end of a month, but it is seen that they reach a final value only at infinite time. This property is common to all such curves; it illustrates the fact that thelifeof a radio-active element is infinite.” It is explained in the same book (p.31) that “infinity is only a relative term; in this connection it only means a longer time than we can measure.”[26]His theology is not so good as his mathematics; he seems to think that in the Creed we assert our belief in the Incomprehensible, in the sense of that which is “not capable of being seized by the mind,” instead of in that which is “untrammelled by limitations.” The word isimmensus, best translated infinite.[27]Hibbert Journal, 1909,pp.626-28.[28]Hibbert Journal, 1909,p.629.[29]Hibbert Journal, 1909,p.632.[30]Schubert, “Mathematical Essays and Recreations,”pp.70, 71.[31]See Van t’ Hoff,La Chimie dans l’espace, and Schubert, “Mathematical Essays and Recreations,”pp.88-89.[32]Schubert, “Mathematical Essays and Recreations,”pp.88-89. See also Mach, “Conservation of Energy” (trans.Open Court PublishingCo.).
[15]This word is used here in the most general and inclusive sense as applying to all thinkers who accept the reality of relations as part of a higher Unity.[16]“A Pluralistic Universe,”p.99.[17]Taylor, “Elements of Metaphysics,”p.312.[18]Ibid.,p.350.[19]A succession of what is disconnected is not change. Change is a succession within an identity: if not within the identity, there is no change, only analysis and re-grouping. The closer our knowledge is of ourselves or anything else, the more we see thatchange is the expression in time of an identity.[20]Illingworth, “The Doctrine of the Trinity,”p.6.[21]Ormond, “Foundations of Knowledge,”p.19.[22]1908-10.[23]James, “A Pluralistic Universe,”p.80.[24]“A Pluralistic Universe,”p.230.[25]See A. T. Cameron, “Radio-Chemistry,”p.17: “The curves illustrate two further points. They approach constant value towards the end of a month, but it is seen that they reach a final value only at infinite time. This property is common to all such curves; it illustrates the fact that thelifeof a radio-active element is infinite.” It is explained in the same book (p.31) that “infinity is only a relative term; in this connection it only means a longer time than we can measure.”[26]His theology is not so good as his mathematics; he seems to think that in the Creed we assert our belief in the Incomprehensible, in the sense of that which is “not capable of being seized by the mind,” instead of in that which is “untrammelled by limitations.” The word isimmensus, best translated infinite.[27]Hibbert Journal, 1909,pp.626-28.[28]Hibbert Journal, 1909,p.629.[29]Hibbert Journal, 1909,p.632.[30]Schubert, “Mathematical Essays and Recreations,”pp.70, 71.[31]See Van t’ Hoff,La Chimie dans l’espace, and Schubert, “Mathematical Essays and Recreations,”pp.88-89.[32]Schubert, “Mathematical Essays and Recreations,”pp.88-89. See also Mach, “Conservation of Energy” (trans.Open Court PublishingCo.).
[15]This word is used here in the most general and inclusive sense as applying to all thinkers who accept the reality of relations as part of a higher Unity.
[16]“A Pluralistic Universe,”p.99.
[17]Taylor, “Elements of Metaphysics,”p.312.
[18]Ibid.,p.350.
[19]A succession of what is disconnected is not change. Change is a succession within an identity: if not within the identity, there is no change, only analysis and re-grouping. The closer our knowledge is of ourselves or anything else, the more we see thatchange is the expression in time of an identity.
[20]Illingworth, “The Doctrine of the Trinity,”p.6.
[21]Ormond, “Foundations of Knowledge,”p.19.
[22]1908-10.
[23]James, “A Pluralistic Universe,”p.80.
[24]“A Pluralistic Universe,”p.230.
[25]See A. T. Cameron, “Radio-Chemistry,”p.17: “The curves illustrate two further points. They approach constant value towards the end of a month, but it is seen that they reach a final value only at infinite time. This property is common to all such curves; it illustrates the fact that thelifeof a radio-active element is infinite.” It is explained in the same book (p.31) that “infinity is only a relative term; in this connection it only means a longer time than we can measure.”
[26]His theology is not so good as his mathematics; he seems to think that in the Creed we assert our belief in the Incomprehensible, in the sense of that which is “not capable of being seized by the mind,” instead of in that which is “untrammelled by limitations.” The word isimmensus, best translated infinite.
[27]Hibbert Journal, 1909,pp.626-28.
[28]Hibbert Journal, 1909,p.629.
[29]Hibbert Journal, 1909,p.632.
[30]Schubert, “Mathematical Essays and Recreations,”pp.70, 71.
[31]See Van t’ Hoff,La Chimie dans l’espace, and Schubert, “Mathematical Essays and Recreations,”pp.88-89.
[32]Schubert, “Mathematical Essays and Recreations,”pp.88-89. See also Mach, “Conservation of Energy” (trans.Open Court PublishingCo.).