Eleanor F. Jourdain

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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? ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ The doctrine of Omnipresence follows 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”