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.