Mathematics and mysticism

"When we consider... peculiarities of mental constitution, reflect upon the nature of the intellectual matrix which brought forth mathematics as a science, and call to mind that universally, and at all times, curious superstitions and extravagant notions appear to have been associated with ideas of number, the hypothesis not unnaturally suggests itself that even now the philosophy of mathematics may not wholly have freed itself from mystical implication. ...it may be that all ratiocinative processes, no matter what the subject, in which the current and continual substitution of symbols (of any kind) for concepts is a prime condition of the effective conduct of the process, are provocative of that attitude of mind. ...and I incline to think that not infrequently the more gifted is the individual for the prosecution of purely symbolic trains of thought the greater is the provocation to this mental attitude."

"Discussion of the principles which underlie... development of mathematical expression has been confined almost exclusively within the inner circle of mathematicians interested in the philosophy of their science. But well-nigh up to the close of last century the explanations of this development—especially with regard to Imaginary Quantity—were almost purely mystical: they really explained nothing, threw no light on the motives and processes of thought which obscurely prompted it, and thus left it as much of an enigma as ever. In recent years, however, a real endeavour has been made to unravel this long-standing difficulty in the development of algebraic symbolism; it is even averred that we have now at last a completely adequate explanation of it. The claim is... that the doctrine has been purged of its mystical implications. But... the claim outruns the performance: the purgation is not thorough..."

"The necessity of ratiocination by symbols does carry in its train a tendency to mysticism from which even the keenest intellects cannot always free themselves. ...[footnote:] Some thinkers have spoken of, or alluded to, a certain 'reaction of language upon thought' which results in error or confusion... It is to this kind of illusion or, to... the mental attitude which makes it possible, that I refer when I speak of mysticism, the mystical tendency or bias."

"Even in pure mathematics, conspicuously the domain of definite and stable conceptions, careful investigation might be found to reveal evidence of an infiltration of the mystical. ...this fact, if so it be, in some measure accounts for the almost insuperable difficulty which many strenuous minds find in mastering even the first principles of the subject, as well as for the purely intellectual repugnance with which they regard it."

"Things must be reduced again to what they seem; it is vain and terrible to take them for what we find they are. M. Bergsen is at bottom an apologist for very old human prejudices, an apologist for animal illusion. His whole labour is a plea for some vague but comfortable faith which he dreads to have stolen from him by the progress of art and knowledge. ...Mr. Bergsen is afraid of space, of mathematics, of necessity, and of eternity; he is afraid of the intellect and the possible discoveries of science; he is afraid of nothingness and death. These fears may prevent him from being a philosopher in the old and noble sense of the word; but they sharpen his sense for many a psychological problem, and make him the spokesman of many an inarticulate soul. Animal timidity and animal illusion are deep in the heart of all of us. Practice may compel us to bow to the conventions of the intellect, as to those of polite society; but secretly, in our moments of immersion in ourselves, we may find them a great nuisance, even a vain nightmare. Could we only listen undisturbed to the beat of protoplasm in our hearts, would not that oracle solve all the riddles of the universe, or at least avoid them. ...it is necessary for the mystic to sally forth and attack his enemy on his own ground. If he refuted physics and mathematics simply out of his own faith, he might be accused of ignorance of the subject. He will therefore study it conscientiously, yet with a certain irritation and haste to be done with it, somewhat as a Jesuit might study Protestant theology. ...in retracing a free inquiry in his servile spirit, he remains deeply ignorant, not indeed of its form, but of its nature and value. ...physical science never solicited of anybody that he should be wholly absorbed in the contemplation of atoms, and worship them; that we must worship and lose ourselves in reality, whatever reality may be, is a mystic aberration, which physical science does nothing to foster. Nor does any critical physicist suppose that what he describes is the whole of the object; he merely notes the occasions on which its sensible qualities appear, and calculates events. Because the calculable side of nature is his province, he does not deny that events have other aspects. ...If he chances to call the calculable elements of nature her substance, as it is proper to do, that name is given without passion; he may perfectly proclaim with Goethe that it is accidents, in the farbiger Abglanz [colorful reflection in Faust], that we have our life. ... The horror of mechanical physics arises, then, from attributing to that science pretensions and extensions which it does not have; it arises from the habits of theology and metaphysics being imported inopportunely into science. Similarly, when M. Bergsen mentions mathematics, he seems to be thinking of the supposed authority it exercises—one of Kant's confusions—over the empirical world, and trying to limit and subordinate that authority, lest movement should somehow be removed from nature, and vagueness from human thought. But nature and human thought are what they are; they have enough affinity to mathematics, as it happens, to suggest that study to our minds, and to give those who go deep into it a great, though partial, mastery over things."

"Even the cautious and patient investigation of truth by science, which seems the very antithesis of the mystic's swift certainty, may be fostered and nourished by that very spirit of reverence in which mysticism lives and moves. ...Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realm, it is insight that first arrives at what is new. ...both intuition and intellect have been developed because they are useful and ...they are useful when they give truth and become harmful when they give falsehood. ...intuition ...seems on the whole to diminish as civilisation increases. It is greater, as a rule, in children than in adults, in the uneducated than in the educated. In advocating the scientific restraint and balance, as against the self-assertion of a confident reliance upon intuition, we are only urging, in the sphere of knowledge, that largeness of contemplation, that impersonal disinterestedness, and that freedom from practical preoccupations which have been inculcated by all the great religions of the world. Thus our conclusion, however it may conflict with the explicit beliefs of many mystics, is, in essence, not contrary to the spirit which inspires those beliefs, but rather the outcome of this very spirit as applied in the realm of thought."

"Logic has been pursued by those of the great philosophers who were mystics. But since they usually took for granted the supposed insight of the mystic emotion, their logical doctrines were presented with a certain dryness, and were believed by their disciples to be quite independent of the sudden illumination from which they sprang. Nevertheless their origin clung to them, and they remained—to borrow a useful word from Mr. Santayana—"malicious" in regard to the world of science and common sense. It is only so, that we can account for the complacency with which philosophers have accepted the inconsistency of their doctrines with all the common and scientific facts which seem best established and most worthy of belief."

"Everyone knows that to read an author simply in order to refute him is not the way to understand him; and to read the book of Nature with a conviction that it is all illusion is just as unlikely to lead to understanding. If our logic is to find the common world intelligible, it must not be hostile, but must be inspired by a genuine acceptance such as is not usually to be found among metaphysicians."

"Among these friends of Andreae's were many whom we find later in other humanistic societies: Wilhelm von der Wense and Tobias Adami, pupils of Campanella, Johann Kepler, discoverer of the laws of planetary motion, , , Theodor Haak, Samuel Hartlib, , and Comenius. That these various societies had deeper motives than those generally ascribed to them is certain. The Italian academies, after the pattern of which the "Order of the Palm" was founded, must have been, to a certain extent at least, secret societies, since neither their organization, their symbolism, their forms, nor the list of membership was communicated to outsiders, and their real aims were concealed while publicity was given to purposes of a genuinely innocent and popular nature. That the German organizations were not the mere language societies they were generally considered is apparent when we look at the activities of their members. They emphasized the study of the mother-tongue, it is true, but there was hardly a writer among them who was not also interested in the study of natural philosophy, in religion, in mathematics or astronomy, so much so, in fact, that to most of them clung the suspicion of heresy—that they were Rosicrucians and as such members of a religious sect highly dangerous to the church and liable of course to persecution. Members of the seventeenth century academies were natural philosophers, reformers, theologians, educators, statesmen, poets, noblemen; such members there were, as Bacon, Giordano Bruno, Comenius, Robert Boyle, J. B. van Helmont, Campanella, Hugo Grotius, Leibniz, Oxenstierna, Valentin Andreae, Spanheim, Pufendorf, Opitz. Throughout the whole list of membership there runs a line of spiritual relationship in the fact of their tolerance for the beliefs of others, a tolerance remarkable for the seventeenth century. With this they united strict opposition to the scholastic method. They were seriously religious, even to the extent of being mystics, but they understood the essence of Christianity differently from the ruling dogma. They treat not only of the relation of man to God, but of man to nature and of men to each other. For them a knowledge incapable of helping mankind had no value; a science shut off from the people in its language is useless; hence their emphasis of the vernacular. To make all knowledge fruitful for the education of the human race and thus lead the race on to a higher stage of development was one of their great ideals. Their turn for the practical led them on in their striving for a general reformation of the whole world. With their keen sense of the significance of fraternal organization, they formed unions which were intended to benefit the whole man and his whole mode of thinking, to influence his whole life. Their activities were in no way directed, as has been claimed, toward "childish play with symbols and signs but toward inclusive spiritual, religious, philosophical, and scientific aims," the carrying out of which, in those times, could be accomplished only under secret organization. The difficulties under which they labored compelled them to proceed with extreme caution, concealing their real interests and exhibiting to the world only what they considered secondary. When the time and place is more propitious for a franker carrying out of their plans and purposes of reform, we shall find them in a country of larger opportunity; we shall find them in England."

"Besides the early Christian ideal, which recognized and encouraged the connection between the teachings of Christ and the ancient wisdom of platonism, there was in early times another which emphasized and endeavored to develop the antithesis more than the connection. From the time when the new Christian state church came to life, and sacrificial religion and the belief in devils and the priesthood were restored, a struggle of life and death developed between the church and the so called philosophic schools. "The fraternity saw that it had to draw down the mask still further over its face than formerly, and the 'House of the Eternal,' the 'Basilika,' the 'Academies,' and the 'Museums' became workshops of stone cutters, latomies, and or innocent guilds, unions, and companies of every variety. But all later greater religious movements and tendencies which maintained the old beliefs, whether they appeared under the names of mysticism, alchemy, natural philosophy, humanism, or special names and disguises, as workshops or societies, have preserved more or less truly the doctrine of the "sacred numbers" and the number symbolism, and found the keys of wisdom and knowledge in the rightly understood doctrine of the eternal harmony of the spheres.""

"Several interesting peculiarities should not be omitted, as for instance, that Leibniz, about 1667, was secretary of an alchemist's society (of so called rosicrucians) in Nuremberg. Leibniz describes alchemy as an "introduction to mystic theology" and identifies the concepts of "Arcana Naturae" and "Chymica.""