Part 2

Now the conception of this Absolute Infinite, of which the aggregate of all ordinal numbers is perhaps a symbol,[10] has been subjected to criticism. Some mathematicians[11] think that it exists, but has no number. It is discovered by a logical process, but defies analysis and the application to it of the notion of number. All mathematical conceptions find in it their aim and conclusion. The importance of this theory, its practical importance, lies in the very much simpler mathematical formulæ that can be produced now that the logical process is shown to extend from the Finite to the Absolute Infinite (in the same way that the labour of summing a series arithmetically by statement and addition is shortened by the application of algebraical principles which depend on larger knowledge). Its philosophical importance is great: the Absolute is here, as elsewhere, the goal of human thought, and is the mathematician’s name for the highest power discoverable by human reason.

It would be very interesting to discuss the probable attitude of a Pascal or a Hegel to these mathematical conceptions, if they had been aware of them. Take Pascal’s puzzle of the Finite and the Infinite. He thought that if the Finite could be subtracted from the Infinite, the Infinite would thereby lose some of its quality of infinity. How differently would it have appeared to him had he realised that an aggregate infinite cardinal can have subtracted from it either finite or transfinite terms: if transfinite terms, many different answers result, giving different degrees of transfinity: if only finite terms are taken away, the Infinite remains in its entirety.

How, again, would Hegel have rejoiced in a definition of thought and existence which would bridge over the logical gulf in his system! Hegel asserted that thought and existence were one. He is objected to by many philosophers, who ask where is the _tertium quid_ which makes it possible to reach from one to the other, or predicate their essential unity? But the mathematician defines existence as something which is not self-contradictory. Thought, then, to him is a form of existence, for thought is not self-contradictory; but the two, thought and existence, are not necessarily conterminous.[12] Hence, to say that non-contradiction is a fundamental condition of true thinking is as much as to say that it is a fundamental characteristic of real existence, and he identifies thought with reality.

Dr. Caird remarks that the secular conscience conceives of the Infinite as opposed to the Finite; the religious conscience treats the Infinite as real, presupposed by the illusory Finite. Where does the truth lie? Mathematics does not admit the necessity of adopting either view at the expense of the other.

Metaphysics standing alone produces results that may be disproved, but cannot be proved. Mathematics standing alone produces results that are susceptible of proof. Both are based on logic, and rest on the prerequisites of thought. Together they are a field for the best powers of human reason: metaphysics supplies insight, intuition, imagination; mathematics offers the indubitable proof and translates the ideal into the actual.

But the element in philosophical thought which, employing the psychological method, tends to the discussion of a theory of being rather than that of knowledge, and thus to the realisation of an ethical system rather than to metaphysical discovery, is averse from accepting these conclusions. It remains, therefore, for us to examine the criticism offered by the psychological school on what they call the mathematising of philosophy; and it will be found that the attack deals both with the ground of the alliance and its results.

A typical exponent of this school is Moisant, who, in the _Revue Philosophique_ for January 1905, attacked what he considered to be the characteristic of modern philosophy and also its vice. It will be observed that at the outset he reverses the _rôles_ of philosophy and mathematics as we have apprehended them. Philosophy, he says, should expect to be inspired by mathematics, but should avoid its method. Next, he connects the modern movement with the theories of Leibniz, who aimed at substituting general formulæ for elementary forms of reason and calculation. These short cuts, which seem to the mathematician to liberate the mind from a burden which prevents it from employing its full activity, seem to the psychologist to tend to a mechanical method, in which the thinker is only aware of premises and results, and in which the mathematical concept tends to replace the real idea. Then he attacks the new definition of mathematics as the science of relations, asserting that it still contains notions of space.[13]

Finally, he comes to the real question at issue, and enters into the comparison of a metaphysical and a mathematical problem. He takes as his subject the argument from the known to the unknown. Descartes had said that argument should lead from the known to the unknown, simple to complex, and had defined the first as that which could be known without the help of the second. This logical order of reasoning has been attributed to mathematics, but has been considered to be inapplicable to philosophy. Mathematics, in its recent development, by the argument from the Finite to the Infinite and back again, starts from two propositions, neither of which can be said to be axiomatic, because each in turn can be proved from the other, but in the course of argument from either mathematics makes use of the logical process. The real axiom, as has been shown, is that of _existence_ or _being_. A metaphysical argument has the same root--that of existence--but a metaphysical problem deals with paradoxes, with questions which are sometimes defined as having two answers, each equally correct, and sometimes as yielding no answer at all. The method of thesis, antithesis, and synthesis is in the Hegelian logic applied to their solution.

A mathematical and a metaphysical problem are not, then, problems of the same kind to be solved by the same method; nor is the conception of the mathematical Absolute reached in the same way as that of the metaphysical Absolute. We are even unable to say how far they correspond except in respect of their absoluteness.[14] But the contention of the mathematician to-day and of the epistemologist school of philosophy is not the identity of methods and results in the two sciences. It is the axiom of existence on which they both depend: the law of thought by which all methods are developed, and, above all, the _correlative value of each science to the other_, which allows us, in developing our knowledge from the standpoint of the two sciences, to recognise something of the greatness of the Absolute principle to which they both reach up, and in which their being consists.

FOOTNOTES:

[1] Of course, if we comprehend in our view only elementary geometrical and algebraical science, it is easy to show that they _do_ demand both axioms and intuitions. Take, _e.g._ Euclid I. I., where in the construction it is necessary to employ intuition for the assertion that the arcs really cut one another. There is no logical certainty that they do; in fact, in some other conditions, _e.g._ in those of other space dimensions, they might not.

[2] This is, of course, not the space of experience. Logic and mathematics deal with implications of thought. See B. Russell (_Hibbert Journal_, 1904, pp. 809-12), who has shown that in all pure mathematics it is only the implications that are asserted, not the premiss or the consequence, as mathematicians used formerly to assume.

[3] De Morgan, Peirce, Schröder, and B. Russell have worked out the logic of relations as well as the syllogism.

[4] See Taylor, “Elements of Metaphysics,” p. 13.

[5] See Dr. Caird, “Evolution of Theology in the Greek Philosophers.”

[6] So Galileo, Newton, Huygens were philosophers in science. Descartes, Pascal, Leibniz were mathematicians as well as philosophers.

[7] See S. Augustine, _De Civitate Dei_, Book XII. ch. xix.: “Ita vero suis quisque numerus proprietatibus terminatur, ut nullus eorum par esse cuicumque alteri possit. Ergo et dispares inter se atque diversi sunt, et singuli quique finiti sunt, et omnes infiniti sunt.”

[8] See R. Dedekind, _Was sind und was sollen die Zahlen?_ 1893.

[9] Two transfinite aggregates can have an ordinal correspondence with one another.

[10] See G. Cantor, _Zur Lehre vom Transfiniten_. 1890.

[11] _e.g._ Mr. P. Jourdain, _Philosophical Magazine_. 1904.

[12] The same result is hinted at by Mr. Taylor. Taylor, “Elements of Metaphysics,” p. 22.

[13] Linear order, 1, 2, 3, &c. Circular. A CD B, A CD B.… The latter, it is true, involves the idea of separation. But this idea can be developed from those of inclusion and exclusion, which belong to the fundamental laws of thought.

[14] The Absolute, according to a recent metaphysical thinker, is “a conscious life which embraces the totality of existence, all at once, and in a perfect systematic unity, as the content of its experience.”--Taylor, “Elements of Metaphysics,” p. 60.

II

PRAGMATISM AND A THEORY OF KNOWLEDGE

The question 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 nothing _is_ which 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 pares_ in 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 the _telos_, 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, so _the 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 us _as 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.