Part 2

Kant, who gave a new impulse to some parts of the Cartesian idea, neglected both mathematical proofs and the search for a metaphysical Absolute. In avoiding the subject he helped to perpetuate the vague descriptions of the Finite and the Infinite, uncorrected by mathematical thought, which had been the currency of the philosophy of his age, and which corrupted the philosophy of the succeeding century. The nineteenth century produced nothing more than guesses at truth, which were, perhaps, not very far wrong, and which the present century is engaged in correcting and substantiating. The same vagueness afflicted both mathematics (theory of functions) and philosophy. Fichte, Schelling, and Hegel, particularly Hegel, identify the metaphysical Absolute with reality, infinity, and the universal. The ideas of continuity and infinity are not separated by them from those of perfection and universality, nor from one another, and their nature is not understood.

Leaving aside the French neo-critical School (Renouvier) and the English School (Spencer)--the first of whom deny the Infinite, thus acting in opposition to mathematical reasoning, while the second perpetuate Kant’s error of considering the Infinite, though thinkable, as unknowable (Dr. Caird has pointed out that this position is illogical)--we arrive at a moment in history which is more fruitful in result on the mathematical side, and will, no doubt, have an effect on metaphysics. For, owing to recent discoveries in Germany and England, mathematics is now in a position to give greater support than before to the intuitions of philosophy. Hitherto, philosophers have been reluctant to allow full value to the mathematical conceptions of Infinity, and with some justice, as the notion had not been sufficiently analysed. Philosophers, who never attempted the analysis, have been inclined to accept certain contradictions in their conception as inherent in the nature of Infinity. Within the last twenty-five years Cantor and Dedekind have cleared up the notion of continuity, and Russell has given greater precision to the idea, and has applied this reasoning to philosophy.

Present-day metaphysicians seem to be divided into two groups; on the one side, those who consider in philosophy the value of a theory of being, and, on the other, those who chiefly consider the value of a theory of knowledge, i.e. the Epistemologists. The first group, devoting themselves to psychology, evolution, and history, have no necessary belief in the Infinite. The Epistemologists, whose work is founded on Kant, discuss the theory of knowledge and enumerate the conditions of knowledge. Their argument may not touch, but does not exclude, the notion of the Infinite. The position of the Epistemologist has been made infinitely more secure by recent mathematical work. That of the psychologist remains almost untouched. It is necessary now to examine more closely the mathematical results to which reference has been made.

In general terms it may be said that mathematics has, as a study, led immediately from the nature of the subject to the perception of the Infinite, and to a knowledge of the connection between the Infinite and the Finite. The simplest form in which the idea can be put is stated by St. Augustine, who said that numbers considered individually were finite, but considered as an aggregate were infinite. Before St. Augustine, and after him down the long stream of philosophic thought, the theologian and the philosopher have turned to mathematics for illustrations of the infinitely great and infinitely little, as developed from the concrete processes of arithmetic and geometry. The recurring decimal in arithmetic, the properties of the circle and ellipse in geometry, of the cone in conic sections, and of the surd in algebra, all touch the problem of number and space on the side of Infinity.

In higher mathematics it is possible to start from the idea of the Finite and reach the conception of the Infinite; or to reverse the process, and from the Infinite to deduce the Finite. Thus in the familiar puzzle of the subdivision of the parts of a straight line by halving the remainder, there will be a crowding and a coalescing of the points of division towards one end of the line, the points of division getting infinitely nearer, but the steps will never meet. Here in the centre of a straight line--a limited straight line--we are confronted with the problem of Infinity.

Again, from a series of finite numbers we can gain the notion of an infinite series. Take two series which have a correspondence with one another. If for every element of the one we can choose an element of the other, and of the other there is an element for the one, when at any point we cut off its progress to infinity, this happens:--

One series, if summed up, will give a larger numerical result than the other, and therefore can be said to be greater than the second. Let us call the first series A, and the second B. Let us now imagine the two series, though starting at a definite point, are never cut at the further end. Then to all infinity series B is without certain numbers which series A possesses, and as an infinite series is smaller than series A. But, on the other hand, when neither series is cut, series B retains its correspondence with series A. Thus we attain a definition of an infinite series. It is such that the part, while being less than the whole, has yet a complete correspondence with the whole. The whole is greater than the part, but take away the part from the whole and that which remains corresponds to it in infinity, because the test of summing the series (which would give a contrary result) involves limitation, and thus cannot be applied. Subtraction can take place in Infinity without loss.

By reversing this process, and by starting from the theory of the Infinite, we may gain some idea of the discovery of the Finite. So Dedekind and Russell define finite numbers not only in the usual way as those which can be reached by mathematical induction, starting from 0 and increasing by 1 at each step, but also as those of classes which are not similar to the parts of themselves obtained by taking away single terms. That is the reversal of the process applied just now. Dedekind also has deduced the Finite from the Infinite by a novel process. He predicates a world of thought which we each and all possess, filled with thoughts and things, to each thing corresponding a thought. There are thus two “trans-finite” series in the minds of each and all of us; we cannot say when the series of thoughts and things will end; but they have number, though it is infinite number. (Number exists wherever there is a correspondence, one to one, between two aggregates.) But in this Gedankenwelt, says Dedekind, there is one thing to which there is no corresponding thought: that is the ego. Each man is part of his own world of thought, but there is no thought of himself in his mind corresponding exactly to himself, as a thought in his mind corresponds to another object. Two important results follow from Dedekind’s theory: first, the existence of a finite number one, the number of the ego, as deduced from the Gedankenwelt of two infinite systems; second, by putting together all the Gedankenwelts there are or may be, we get the notion of series of series, which seems to transcend Infinity, and it gives us the conditions which are possibly gathered up in the Absolute. Now the argument from the Finite to the Infinite and the converse process may both be employed in mathematics (or both may be neglected, as in the elementary methods of calculation used in arithmetic). A discussion has taken place in the Hibbert Journal on the relative value of the two methods. Keyser, in an article called the Axiom of Infinity, argued that one method, that of Dedekind, should be exclusively developed. Russell answered him, stating that it was not necessary to hold exclusively to either. If the Finite and the Infinite can in turn be deduced from one another, neither conception can be truly called an axiom. The real axiom is existence, which includes both, and which is defined by mathematicians as that which is not self-contradictory.

Now the problem of Infinity includes also that of continuity; in other words, the problem of number includes that of cardinal and ordinal number. It is time to get to the mathematical definition of number, which we have found as a conception can be attached both to the Finite and to the Infinite. What is number in mathematics?

Take any collection of things--we call that an aggregate. If an aggregate corresponds one to one with another aggregate, they are both said to have a number, and the same number. Subtract from the idea of an aggregate the idea of quality or kind, and order or arrangement, what is left is its cardinal number. If you subtract quality and not order, the result is an ordinal number. This reasoning applies both to finite and infinite aggregates; in fact, the Infinite may be said to possess most of the properties which we attach to the Finite. Two infinite aggregates, for example, can have an ordinal correspondence, and infinite aggregates submit, like finite ones, to arithmetical processes.

The mathematician analyses still more closely the relation between the Finite and the Infinite, as follows:--

He starts from the aggregate, which he analyses into the Finite and the Infinite, and the latter he analyses into the Transfinite and the Absolute. Of these two elements, one only has till just lately been the subject of mathematical treatment--the one called the Transfinite. It is the transfinite subdivision of the Infinite to which the idea of number is applicable, and which is, therefore, in a sense inseparable from the Finite. Infinite numbers or series ought then to be more correctly described as Transfinite. But the processes of mathematics do not end here; they reach up to the idea of the Absolute Infinite, the conception of which has been attained in recent years by mathematical work. The results of this work may now be briefly summarised.

I. The Absolute appears to have the same relation to the Transfinite as the Transfinite to the Finite. If the Finite deals with numbers, and the Transfinite with series of numbers, the Absolute deals with series of series. Thus there are at least two examples of the Infinite within our grasp which lead up to the idea of the Absolute. One is the class of all classes of propositions; the other is the series of all worlds of thought, in Dedekind’s sense.

II. The Finite, Transfinite, and Absolute can be further defined in this way. There is no greatest finite number, but there is a least transfinite number, which has been called Aleph 0, and which can be proved to be greater than any possible finite number, however large, because if there were a last number it must be smaller than the sum of the whole series. There are unending series of Alephs or infinite numbers, which are as distinct from one another in idea as 1 is from 0, and which can no more be derived from one another by a mathematical process than 1 can be derived from 0, but can be reached in the same way by induction. Beyond the Transfinite we cannot discover in the Absolute the idea of least or of greatest.

III. The relation of cardinal and ordinal number also throws some light on the Finite, Transfinite, and Absolute. In the Finite, cardinals and ordinals are parallel to one another; in the Transfinite they strikingly diverge; in the Absolute we cannot trace any connection between cardinals and ordinals, i.e., it is possible to have an ordinal series to which there can be no corresponding cardinal number or type.

IV. If arithmetical processes are applied to the Finite, Transfinite, or Absolute, we get interesting results. We know the effect of addition, multiplication, and raising to a power, on the Finite. The first two processes have been applied to the Alephs; the last has been formulated, but the mathematical results have not yet been brought to a satisfactory conclusion. Broadly speaking, we may say that the raising of an Aleph to a power may make it transcend the Finite and the Transfinite and melt into the Absolute. Thus all mathematical processes which find their goal in the Absolute would find their annihilation there. No finite mathematical conception would be applicable to it.

Now the conception of this Absolute Infinite, of which the aggregate of all ordinal numbers is perhaps a symbol, has been subjected to criticism. Some mathematicians 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. 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.

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. 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:

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.

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.