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Part 1

ON THE THEORY OF THE INFINITE IN MODERN THOUGHT

TWO INTRODUCTORY STUDIES

E. F. JOURDAIN

DOCTOR OF THE UNIVERSITY OF PARIS VICE-PRINCIPAL, ST. HUGH’S HALL, OXFORD

LONGMANS, GREEN AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY, AND CALCUTTA 1911

Of the two papers here reproduced, the first was given in 1905 to a meeting of the women science students in Oxford; the second, in 1908, to the Philosophical Society of this College. They are printed by request, with the author’s apologies for their incompleteness. The lecture form has been retained. I am indebted to my brother, Mr. P. Jourdain, for help in preparing the first lecture, and for his revision of the text.

E. F. JOURDAIN.

ST. HUGH’S HALL, OXFORD, _January, 1911_.

CONTENTS

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THE PROBLEM OF THE FINITE AND THE INFINITE 1

PRAGMATISM AND A THEORY OF KNOWLEDGE 31

THE PROBLEM OF THE FINITE AND THE INFINITE

The influence of mathematics on philosophy and vice versâ can be inferred from the historical progress of both studies, though it has not been possible till about within the last fifteen years to give a logical explanation for the relations between them. As long as it was believed, according to the Kantian view, that the science of mathematics was based on intuitions of time and space, the alliance between philosophy and mathematics could not be proved to be closer than that between philosophy and experimental science, although the historical fact remained that philosophy and mathematics exercised a mutual stimulus, and developed at the same periods of history.

But mathematics, as now defined, is independent of intuitions of space and time, and also of axioms and hypotheses.[1] Mathematics, as now understood, is based, like formal logic, on the prerequisites of thought, not on the notions of space and time. Here there is no definition of number or space, but the conception of number and space,[2] which is more complicated, can be derived from them. All other complicated mind processes can, in the same way, be reduced to the simple elements of the prerequisites of thought.

Such a science might exist out of conditions of time and space as we know them. It is a science of relations rather than of mere number. Founded, then, on the laws of symbolic logic, it is a valuable aid and illustration to philosophy; philosophy, on the other hand, can imagine lines for the exercise of the constructive power involved in mathematics. It is the object of this paper to show that the close though apparently accidental union of philosophy and mathematics throughout the history of thought can now be explained, and that the problems with which pure mathematics is now concerned are those which lie at the core of philosophic thought and speculation. (Symbolic logic has developed to meet the new demands made upon it. It does not now reduce itself to the syllogism, as Aristotle thought it did; the prerequisites of thought are shown to be manifold instead of single.[3])

The use of the word philosophic in this connection suggests a necessity for further definition. Philosophy is held to include at least two great branches--Metaphysics and Ethics. The influence of mathematics is most evident on the metaphysical side of philosophy; in fact, the grouping of mathematics and metaphysics as allied sciences tends to bring out the essential distinction between metaphysics and ethics, and--though not by any means to imply a break in their real relation--to show where this has been misunderstood. No philosophy has been equally strong on both sides; they represent different forms of activity of the human mind; but it is still true, and from the conditions always must be, that an ethical system grows out of metaphysics as practice follows precept and conduct implies belief. The new definition of mathematics does not touch these consequences; it merely marks the limits within which philosophy on the metaphysical side can submit to, or rest upon, the conclusions of mathematics.

As to the historical relation between metaphysics and mathematics, the subject is so vast that we shall only attempt a very rapid generalisation of its results on the growth of the conception of the Finite and the Infinite. (Of course, there are many other sides of the relation which might be studied.) The general result of the inquiry has been, as far as we can judge, that metaphysics has exercised an inspiring force on mathematics, and mathematics has defined and strengthened the conceptions of metaphysics at every critical stage in the history of philosophy. But where metaphysics has been treated as the proof of science, where it has been laid down as the foundation for exact knowledge, the results have not corresponded with the truth of experience, and the quality of thought has become degenerate. Progress depends on the right perception of the relations between the sciences and parts of philosophy.

Such progress is especially evident in the early Greek and in the modern periods, while the large period from the Christian era to the Renaissance gives examples of the unfortunate reversal of the parts of metaphysics and science and consequent confusion of thought.[4]

The problems of the metaphysician are no doubt in a sense always the same; but this is equally true of the problems of any other science. The methods by which the problems are attacked and the adequacy of the solutions they receive vary, from age to age, in close correspondence with the general development of science. Every great metaphysical conception has exercised its influence on the general history of science, and in return every important movement in science has affected the development of metaphysics. The metaphysician could not if he would, and would not if he could, escape the duty of estimating the bearing of the great scientific theories of his time upon our ultimate conceptions of the nature of the world as a whole. Every fundamental advance in science thus calls for a restatement and reconsideration of the old metaphysical problems in the light of the new discovery.

* * * * *

During the Greek period mathematics was the only branch of science which was at all developed, and its development coincided with the age of the philosophers. Thus when Plato spoke of science he always meant mathematics. And even later, when the physical sciences had begun to develop, Aristotle put mathematical ideas into close connection with metaphysical ones when he stated that they occupied the middle term between the ideal and the sensible. Both Plato and Aristotle referred to and depended upon mathematical proofs and illustrations of philosophical questions. During this Greek period the conception of Infinity took shape. The pre-Platonic notion, reproduced again later in the decline of Socratic theory by the Stoics, was that the Infinite was the aggregate of the Finite; the Platonic and Aristotelian theory, that, namely, of the most vigorous moment of Greek thought, was that the Infinite was _more_ than the aggregate of the Finite; that it had a self-determined existence from which the Finite had been derived. Existence, as known to man, was treated as a compromise between the Finite and the Infinite.

Neo-Platonism altogether separated the Infinite from the Finite. In the Alexandrine metaphysics, which represented a decadent stage of philosophy and its deviation from the sciences, the conception of the Infinite became less clear and logical; it diverged from the view which had been affected by mathematical thought, and tended to assimilate to itself the ideas of perfection and universality, which, philosophically speaking, are conceptions distinct from that of Infinity--universality referring to a common principle of unity, and perfection involving the moral ideal. Real progress was deferred by the too rapid coherence of ideas only partially analysed and understood. Thinkers passed quickly from the exclusive contemplation of subject to that of object and back again,[5] each new period negativing all previous experience, till the result was the exclusion of an imperfectly analysed Relative and Finite from an insufficiently apprehended Absolute and Infinite.

After the Christian era Greek philosophy drifted off into scholasticism and lost touch of reality, the grammar of Aristotelian logic replacing the vital connection of ideas. St. Anselm, it is true, attempted to find a rational proof of the existence of God, and identified Him with the Infinite of Greek thought; but St. Thomas Aquinas led away the argument to a discussion as to how far form and matter, separately considered, shared in the quality of Infinity. (He thought form did, but not matter.) An overpowering sense of mystery, joined to a premature desire for definition without scientific analysis, sapped the vigour of mediæval thought.

Throughout the middle ages, then, we see the conditions of the Greek period reversed: philosophy during the second period is not, as in the first, engaged in giving a stimulus to the efforts of pure reason; rather the intuitions of philosophy are treated as axiomatic, and a false superstructure of knowledge, alien to experience and reality, is erected upon these foundations. Philosophy, in fact, is used as a general basis for science. The parts of philosophy and mathematics, correctly though imperfectly seen by the Greeks, are in the second period exchanged, and the result is confusion of idea. The notion of the Infinite, as in the Alexandrine metaphysics, is held to include perfection and universality, and does not exist as a conception apart from these.

After the Renaissance, the scholastic philosophy falling into disuse, the attempt to find an explanation of the Cosmos, a synthesis of the universe, was abandoned, and replaced by the Cartesian idea--the inference of existence from thought, and the limitation of the sphere of inquiry to that which could be known by the ego. New scientific and mathematical discoveries kept pace with this new analysis and development of thought,[6] and the surer ground in philosophy was definitely allied with the work of the mathematical mind. The philosophical thesis developed from “The Infinite is the negation of the Finite,” to “The Infinite presupposes the Finite and does not exclude it.” The problem of the Finite and the Infinite became the great idea of the age, and there was a reversion to the Greek notion of existence as a compromise between the two, and almost the hint of a coming explanation of them. In the decline of Cartesian philosophy, when it drifted off into Pantheism, there was only a vague conception of the Infinite, and we trace a tendency to identify the notion of Infinity with that of the Cosmos. In mediæval thought the idea of the Infinite had become confused with that of the Perfect and Universal; in the modern period the effort to give a concrete expression to the notions of Infinity, Perfection, and Universality diverted the ideas from their relation to the Creator and applied them to the Creation.

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.[7] 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.[8] 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.[9]

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.