Mathematics and mysticism

"Now this ratio of the single to the double arises, no doubt, from the ternary number, since one added to two makes three; but the whole which these make reaches to the senary, for one and two and three make six. And this number is on that account called perfect, because it is completed in its own parts: for it has these three, sixth, third, and half; nor is there any other part found in it, which we can call an aliquot part. The sixth part of it, then, is one; the third part, two; the half, three. But one and two and three complete the same six. And Holy Scripture commends to us the perfection of this number, especially in this, that God finished His works in six days, and on the sixth day man was made in the image of God. And the Son of God came and was made the Son of man, that He might re-create us after the image of God, in the sixth age of the human race."

"Of the perfection of the number six, which it the first of the numbers which is composed of its aliquot parts. These works are recorded to have been completed in six days... because six is a ,—not because God required a protracted time... but because the perfection of the works was signified by the number six."

"What English mathematicians were most intrigued by and, at times, embarrassed about was the explanation Ramanujan offered regarding his methodology for arriving at solutions. A devout Hindu, Ramanujan claimed he derived most of his solutions with the help of the goddess Namgiri. He claimed that "an equation for me has no meaning unless it expresses a thought of God," and the quantity 2n-1 stood for "the primordial God and several divinities." ...Ramanujan was unable to provide any explanation of his "methodology" in formal mathematical terms."

"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."

"Throughout history philosophers and mystics have sought a compact key to universal wisdom, a finite formula or text which, when known and understood, would provide the answer to every question. The Bible, the Koran, the mythical secret books of Hermes Trismegistus, and the medieval Jewish Cabala have been so regarded. Sources of universal wisdom are traditionally protected from casual use by being hard to find, hard to understand when found, and dangerous to use, tending to answer more and deeper questions than the user wishes to ask. Like God the esoteric book is simple yet undescribable, omniscient, and transforms all who know It. The use of classical texts to foretell mundane events is considered superstitious nowadays, yet, in another sense, science is in quest of its own Cabala, a concise set of natural laws which would explain all phenomena. In mathematics, where no set of axioms can hope to prove all true statements, the goal might be a concise axiomatization of all “interesting” true statements."

"The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction."

"Number mysticism was not original with the Pythagoreans. The number seven, for example, had been singled out for special awe, presumably on account of the seven wandering stars or planets from which the week (hence our names for the seven days of the week) is derived. The Pythagoreans were not the only people who fancied that the odd numbers had male attributes and the even female... Many early civilizations shared various aspects of numerology, but the Pythagoreans carried number worship to its extreme..."

"When a man counts one, two, three, he is not only doing mathematics, he is on the path to the mysticism of numbers in Pythagoras and Vitruvius and Kepler, to the Trinity and the signs of the Zodiac."

"Figuration can be one of the values or qualities that the Divine Artificer uses to decorate or adorn the invisible first principles of the creation (the word kosmos for 'universe' means 'adornment'). Thus arithmetic and geometric figuration, with the resultant harmonic (musical) figurations, are the most essential tools that Plato posits the Divine Craftsman uses to adorn the 'likeness' of the perfect cosmos, which he is attempting to delineate is the psycho-cosmogony of the Timeaus."

"At Babylon... we see a practical polytheism... combined with the application of the exact sciences, and the gods of heaven subjected to the laws of mathematics. This strange association is to us almost incomprehensible, but it must be remembered that at Babylon a number was a very different thing from a figure. Just as in ancient times and, above all, in Egypt, the name had a magic power, and ceremonial words formed an irresistible incantation, so here the number possesses an active force, the number is a symbol, and its properties are sacred attributes. Astrology is only a branch of mathematics, which the heavens have revealed to mankind by their periodic movements."

"A number of aspects of mathematics are not much talked about in contemporary histories of mathematics. We have in mind business and commerce, war, number mysticism, astrology, and religion. In some instances, writers, hoping to assert for mathematics a noble parentage and a pure scientific experience, have turned away their eyes. Histories have been eager to put the case for science, but the Handmaiden of the Sciences has lived a far more raffish and interesting life than her historians allow."

"To what extent does mathematics itself function as a religion. Insofar as the "laws of mathematics" are properties possessed by certain shared concepts, they resemble doctrines of an established church. An intelligent observer seeing mathematicians at work and listening to them talk, if he himself does not study or learn mathematics, might conclude that they are devotees of exotic sects, pursuers of esoteric keys to the universe. Nonetheless... theologians notoriously differ in their assumptions about God... mathematics seems to be a totally coherent unity with complete agreement on all important questions: especially with the notion of proof, a procedure by which a proposition about an unseen reality can be established... different mathematicians, using different methods, working in different centuries, will find the same answers. Can we conclude that mathematics is a form of religion, and in fact the true religion?"

"All thinges which are, & have beyng, are found under a triple diversitie generall. For, either, they are demed Supernaturall, Naturall, or of a third being. Thinges Supernaturall are immateriall, simple, indivisible, incorruptible, & unchangeable. Things Naturall are materiall, compounded, divisible, corruptible, and changeable. Thinges Supernaturall, are, of the minde onely, comprehended: Things Naturall, of the sense exterior, are able to be perceived. In thinges Naturall, probabilitie and conjecture hath place: But in things Supernaturall, chief demonstration, & most sure Science is to be had. By which properties & comparasons of these two, more easily may be described the state, condition, nature and property of those thinges, which, we before termed of a third being: which, by a peculier name also, are called Thynges Mathematicall. For, these, beyng (in a maner) middle, betwene thinges supernaturall and naturall, are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall: But are thinges immateriall: and neverthelesse, by materiall things hable somewhat to be signified. And though their particular Images, by Art, are aggregable and divisible, yet the generall Formes, notwithstandyng, are constant, unchangeable, untransformable, and incorruptible. Neither of the sense can they, at any tyme, be perceived or judged. Nor yet, for all that, in the royall mynde of man, first conceived. But, surmountyng the imperfection of conjecture, weenyng and opinion, and commyng short of high intellectuall conception, are the Mercurial fruite of Dianœticall discourse, in perfect imagination subsistyng. A mervaylous neutralitie have these thinges Mathematicall, and also a strange participation betwene thinges supernaturall, immortall, intellectual, simple and indivisible: and thynges naturall, mortall, sensible, compounded and divisible. Probabilitie and sensible prose, may well serve in thinges naturall: and is commendable: In Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor yet the testimony of sense, any whit credited: But onely a perfect demonstration, of truthes certaine, necessary, and invincible: universally and necessaryly concluded: is allowed as sufficient for an Argument exactly and purely Mathematical. ... Neither Number, nor Magnitude, have any Materialitie. First, we will consider of Number, and of the Science Mathematicall, to it appropriate, called Arithmetike: and afterward of Magnitude, and his Science, called Geometrie. ...How Immateriall and free from all matter, Number is, who doth not perceave? yea, who doth not wonderfully wonder at it? For, neither pure Element, nor Aristotele's Quinta Essentia, is hable to serve for Number, as his propre matter. Nor yet the puritie and simplenes of Substance Spirituall or Angelicall, will be found propre enough thereto. And therefore the great & godly Philosopher Anitius Boetius, sayd... All thinges (which from the very first originall being of thinges, have bene framed and made) do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator."

"Newton's goal was incomparably more vast than the discovery of the "mathematical principles of natural philosophy." Newton wished to penetrate the divine principles beyond the veil of nature, and beyond the veils of human record and received revelation as well. His goal was the knowledge of God, and for achieving that goal he marshaled the evidence from every source available to him: mathematics, experiment, observation, reason, revelation, historical method, myth, the tattered remnants of ancient wisdom. ...one result of the restricted interests of modernity has been to look askance at Newton's biblical, chronological, and alchemical studies: to consider his pursuit of prisca sapientia [ancient wisdom] as irrelevant. None of those was irrelevant to Newton, for his goal was considerably more ambitious than a knowledge of nature. His goal was Truth, and for that he utilized every possible resource."

"The Pythagoreans... were fascinated by certain specific ratios, and especially by those which relate what we call today the arithmetic [A = \frac{a + b}{2}], geometric [G = \sqrt{ab}], and harmonic [H = \frac{2ab}{a + b}] means. ... The particular ratios between these means in which the Pythagoreans were interested were...A:G = G:HThe other relationship is expressed bya:A = H:bThe Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the ns by Pythagoras himself after having been taken prisoner in Egypt. s lay at the heart of the Pythagorean theory of music. If a string is divided into 12 parts, the ratio 12:6, or 2:1, gives us the octave. If the arithmetic and harmonic means of 12 and 6 are now taken, we haveA = \frac{6 + 12}{2} = 9andH = \frac{2 \times 6 \times 12}{6 + 12} = 8The [perfect proportions] 9:6 and 12:8 both equal to 3:2, correspond to the fifth in the theory of music. Similarly... 8:6 and 12:9, both equal to 4:3, corresponding to the fourth. In this way, certain intervals, crucial in the theory of music, were all obtained by ratios involving the numbers 1, 2, 3, and 4, which came to be of mystical significance for they also represented the perfect triangle, yielding the 10, the sum of 1, 2, 3, and 4. These ratios of the musical fifth and fourth were used by the Pythagoreans to obtain the whole tone of the ,\frac{3}{2} : \frac{4}{3} = 9:8and the semi-tone,\frac{4}{3} : (\frac{9}{8})^2 = 256:243."

"Throughout man's history there has been a close connection between the natural numbers and the activities of the divine Creator. The natural numbers have always been seen by some at least in every civilization to be symbolic of deep esoteric religious beliefs. The Babylonians had a hierarchy of sixty gods each of which was associated with one of the first sixty natural numbers. ...In ancient India, we find religious significance assigned to each of the first 101 numbers, and the Mayan civilization the first thirteen numbers all represented gods. There is something of a parallel in other sciences. Astronomy grew from astrology and chemistry from alchemy. It is therefore not surprising that number theory, first developed by the Pythagoreans, was associated with their religious beliefs and practices."

"In 1882 Mittag-Leffler suggested that Cantor's work should be translated into French, but he was more positive toward Cantor's "mathematics" than his "philosophy." Hermite and Poincaré agreed that French readers of Cantor might object to "research which is at the same time philosophical and mathematical and where arbitrariness has an excessive place." Hermite thought that Cantor's work was more German metaphysics than mathematics. Perhaps for that reason he suggested as translator a Jewish priest at San-Sulpice, observing that "[Cantor's] philosophical turn of mind will not be an obstacle for a translator who knows Kant." During the translation work, Hermite and Poincaré made suggestions for cuts in the philosophical parts. And even some of the mathematics seemed too abstract to them; they observed, as Poincaré remarked, that "higher infinities" seemed to "have a whiff of form without matter, which is repugnant to the French spirit.""

"We left him after the battle of Prague in 1620. He remained in Prague until December, and then took up his winter quarters with a portion of the Duke of Bavaria's troops left in the extreme south of Bohemia. At this time a new and strange influence had come within his life. In that wonderfully productive winter spent with the Bavarian troops, while active operations were impossible, Descartes heard much of what was going on in the literary and scientific world. Amongst other things, he heard of that strange brotherhood of which we have so often read and yet of which we know so little—the Order of the Rosicrucians. They, it was said, taught a new wisdom, the hitherto undiscovered science. This was enough to excite Descartes' interest: Germany was thoroughly aroused; something had been discovered which was to be kept to the few initiated ones. The same Descartes who was in the habit of disdaining the work of others, began to think he had been precipitate in his judgments. Here was he searching for the Truth patiently and with difficulty, and there were men who declared the way had been opened to them. If these were simple imposters, then it was the duty of any honest man to expose their imposition; but if on the issue which to him was all-important, they had found any light, then, as he told his friend, how despicable would it be in him to disdain to be taught anything out of which he might obtain new knowledge. He made it his duty to discover a member of this learned body, in order to discuss the matter with him and subsequently settle his own conclusions. [footnote] The treatise which Descartes specially dedicates to the Order, is that which was written in 1619-20 and never published, the Polybii cosmopolitani Thesaurus mathematicus, which "sets forth the true means of solving this science, and in which it is demonstrated that nothing further can be supplied by the human mind"; "it is specially designed to relieve the pains of those who, entangled in the Gordian knot of the sciences, uselessly waste the oil of their genius." It is dedicated to all learned men, and more especially to the illustrious Brethren of the Rosicrucian Order in Germany. This may have been the treatise that in his journal he promises, if he can obtain sufficient books, and if it seems worthy of publication, on 23rd September 1620, though why the date should be thus fixed, we do not know. Probably when the time came he did not consider it "worthy," and now all is lost excepting the simple title."

"The word witr literally means "odd number," and tradition says: "God is odd, He loves the odd." Musalmāns pay the greatest respect to an odd number. It is considered unlucky to begin any work, or to commence a journey on a day, the date of which is an odd number."

"The work of Copernicus and Kepler is the work of men searching the universe for the harmony their commingled religious and scientific beliefs assured them must exist, and in aesthetically satisfactory mathematical form. To men who were convinced that an omnipotent being, designing a mathematical universe, would certainly prefer the superior features of the new theory, there could only be one conclusion—the heliocentric theory was true. ... There was a mystic element in their thinking that now seems anomalous in great scientists. ... But despite the presence of mystic, poetic, and religious influences, Copernicus and Kepler were thoroughly rational in rejecting any speculations or conjectures that did not agree with observations. What distinguishes their work from medieval vaporizings is not only the mathematical framework of their theories but their insistence of making the mathematics fit reality. In addition, the preference they showed for a simpler mathematical theory is a thoroughly modern scientific attitude. ...their willingness to think in terms of, and indeed their preference for, a heliocentric theory, despite the arguments ...against the reasonableness of such a view in their time, marks them as heroes in the intellectual battles man has fought to attain a secure dwelling place for science."

"When one compares the pre-Greek and Greek understanding of the concepts of mathematics... [one] notes the sharp transition from the concrete to the abstract... One advantage of treating abstraction is the gain in generality. ...Another advantage ...abstracting ..frees the mind from burdensome and irrelevant details ...The emphasis ...was part and parcel of their outlook on the entire universe. ...Pythagoreans and Platonists maintained that truths could be established only about abstractions. ...impressions received by the senses are inexact, transitory, and constantly changing ...truth, by its very meaning, must consist of permanent, unchanging, definite entities and relationships. ...man may rise to the contemplation of ideas. These are eternal realities and the true goal of thought, whereas mere "things are the shadows..." Thus Plato would say that... reality is in the universal type or idea... Beauty, Justice, Intelligence, Goodness, Perfection, and the State, are independent of the superficial appearances... of the flux of life... of... biases and warped desires... they are ...constant and invariable, and knowledge concerning them is firm and indestructible. ...physical or sensible objects suggest the ideas just as diagrams of geometry suggest abstract geometrical concepts... but one must not lose himself in trivial and confusing minutiae. The abstractions of mathematics possessed a special importance for the Greeks. ...to pass from a knowledge of the world of matter to the world of ideas, man must train ...These highest realities blind the person ...The study of mathematics helps make the transition from darkness to light. ...man learns to pass from concrete figures to abstract forms ...this study purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas. ...to lift the mind above mundane considerations and enable it to apprehend the final aim of philosophy, the idea of the Good."

"Plus one, minus one, plus one, minus one, etc. By adding the first two terms, the next two, and so forth, the result is converted into another for which each term is zero. Grandi, the Italian Jesuit, had concluded the possibility of creation from this series; because the result is always equal to ½, he saw the unborn fraction of infinitely many zeros, or nothingness. It was thus that Leibnitz saw in his binary arithmetic the image of creation. He imagined that Unity represented God, and Zero the void; and that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in this system of numeration. This conception was so pleasing to Leibnitz that he communicated it to the Jesuit, [Claudio Filippo] Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, since he was very fond of the sciences, to Christianity. I mention this merely to show how childhood prejudices may lead astray even the greatest men."

"A fifteenth century trend toward Platonic and Pythagorean mysticism, accompanying the growth of humanism over... earlier Scholastic philosophy, encouraged the previously denied use of the infinite and infinitesimal in geometry. As a result... mathematics was viewed as independent of the senses, not bound by empirical investigations, and thus free to use the infinitite and infinitesimal, provided that no inconsistencies resulted. By the early seventeenth century this view opened up bold new approaches."

"No one has attempted a language or characteristic which includes at once both the arts of discovery and of judgement, that is, one whose signs and characters serve that same purpose that arithmetical signs serve for numbers, and algebraic quantities for quantities taken abstractly. ...since God has bestowed these two sciences on mankind, he has sought to notify us that a far greater secret lies hidden in our understanding, of which these are but the shadows. ...When I... took up logic and philosophy... I once raised a doubt concerning the categories. I said that just as we have categories or classes of simple concepts, we ought also to have a new class of categories in which propositions or complex terms themselves may be arranged in their natural order. For I had not even dreamed of demonstrations at that time and did not know that the geometricians do exactly what I was seeking when they arrange propositions in an order such that one is demonstrated from the other. ...I necessarily arrived at this remarkable thought... that a kind of alphabet of human thoughts can be worked out and that everything can be discovered and judged by a comparison of the letters of this alphabet and an analysis of the words made from them. ...I wrote a Dissertation of the Art of Combinations...published... in 1666, and in which I laid my remarkable discovery before the public. This dissertation was... such as might be written by a youth just out of the schools... not yet conversant with the real sciences. For mathematics was not cultivated in those parts; if I had spent my childhood in Paris, as did Pascal, I might have advanced these sciences earlier. ... Why... no mortal has ever essayed so great a thing—this has often been an object of wonder to me. ...these considerations should have occurred from the very first, just as they occurred to me as a boy interested in logic, before I even touched on ethics, mathematics, or physics, solely because I always looked for first principles. The true reason for straying from the portal of knowledge is... that principles usually seem dry and not very attractive... Yet I am most surprised at the failure of three men to undertake so important a thing—Aristotle, Joachim Jung, and René Descartes. For when Aristotle wrote the Organon and the Metaphysics he laid open the inner nature of concepts with great skill. Joachim Jung... is a man... of such rare judgement and breadth of mind that I cannot think of anyone, not even excepting Descartes himself, from whom a great revival of science might better have been expected, if only he had been known and supported. ...As for Descartes ...since he had aimed at his own excessive applause, he seems to have broken off the thread of his investigation and to have been content with metaphysical meditations and geometrical studies by which he could draw attention to himself. ...he did not adequately think through the full reason and force of the thing. For had he seen a method of setting up a reasonable philosophy with the same unanswerable clarity as arithmetic, he would hardly have used any way other than this to establish a sect of followers, a thing which he so earnestly wanted. For by applying this method of philosophizing, a school would from its very beginning, and by the very nature of things, assert its supremacy in the realm of reason in a geometrical manner and could never perish nor be shaken until the sciences themselves die through the rise of a new barbarism among mankind. As for me, I kept this line of thought. ...For this is what I finally discovered ...Nothing more is necessary to establish the characteristic which I an attempting, at least to the point sufficient to build the grammar of this wonderful language and a dictionary for the most frequent cases, or what amounts to the same thing, nothing more is necessary to set up the characteristic numbers for all ideas than to develop a philosophical and mathematical 'course of studies'... based on a certain new method which I can set forth... a few selected men could finish the matter in five years. It would take them only two, however, to work out by an infallible calculus the doctrines most useful for life, that is, those of morality and metaphysics."

"As soon as his fevered imagination had cooled, he determined at once to produce his invention, and notes on the 23rd February (1620) he was thinking of finding a publisher; but presently he changed his intention; and this treatise... is by Baillet suspected to have been possibly the Olympica... Descartes had heard much of the Rosicrucians,—a hidden confraternity who were believed to have attained some mysterious key to natural knowledge apart from theology, and who were supposed to be spread all through society. A considerable literature of attack and of apology as regards this sect then occupied public interest. Baillet tells us all that was then known about them. In a MS. called Cartesii liber de studio bonœ mentis ad Musœum, Descartes confessed that he had done all he could to find out a member of the brotherhood and learn what he might of their magic secrets, but was completely and permanently unsuccessful. Nevertheless, he had talked so much about it at the time, that he found himself set down as a Rosicrucian, and had some difficulty in clearing himself of the imputation. But in the winter of 1619-20 he had not yet given up hopes of finding out this mystery, and the title of a book found among Leibnitz's transcripts gives us the clue to the lost treatise of this date. It was Polybii cosmopolitani Thesaurus mathematicus (I translate the sequel), 'in which are set forth the true means of solving all the difficulties of this science, and there is demonstrated that, as regards it, nothing further can be supplied by the human mind; with the intention of challenging the delay, and exposing the rashness of those who promise to show new marvels in all the sciences, as well as to relieve the torture (Iabores cruciabiles) of many, who, entangled in some of the s of this science, night and day spend uselessly the oil of their genius,—now offered to the learned of all the world, and especially celeberrimis in Germania Fratribus Roseœcrucis.' ... This interesting though confused title shows clearly what Descartes' inventum mirabile was... simply the solution of all geometrical problems by algebraical symbols. What agitated his mind so greatly was that the discovery would not cease there, but that by means of this new and improved calculus he could apply mathematical demonstration to all the realm of nature. ... But at this time he had only simplified his mathematics so as to make it a general method of investigation. It remained for him to likewise so simplify nature as to make it capable of submitting to his analysis. ...He ...soon turned aside to , to make trial of his new method of solving problems on Faulhaber and other mathematicians of distinction. The story of is repeated, mutatis mutandis, in the case of Faulhaber. He first despised, and then sarcastically challenged, the young inquirer, who on this occasion, however, showed considerable self-confidence, and not only solved the problems proposed, but showed general methods of doing so, and even of determining the solubility of various new problems, or the reverse. He also solved the problems proposed by Peter Roten [Roth] in reply to a challenge of Faulhaber in his algebra. These successes must have made Descartes feel assured of his inventum mirabile as far as mathematics went. But he presently suspended further study..."

"Now Faulhaber was a real Rosicrucian and a very ardent one, and one is justified in assuming, in spite of Baillet's denials, that Descartes found in him the man he was seeking, and that through him Descartes entered into direct contact with the intellectual atmosphere of the Rosecrucians. Might not such contact, however fleeting, have a declining influence upon the moral lines and aims of the philosopher's life? May we not even ask ourselves whether at its origin. Descartes' great idea did not permit the supposition that he intended—an intention that became hazier as time went on—fearlessly to transpose to the plane of everyday reason and of the most widespread common sense the design followed on the plane of alchemic mysteries by the naive Rosicrucians, and in doing so render it much less "uplifted," but much more efficacious—mathematics replacing the Cabala leading to universal knowledge, the hermetic sciences and their occult qualities giving place henceforth to geometric physics and the art of mechanics, as the elixir of life to the laws of rational medicine?"

"As nothing on earth is more conservative than religion, we have still a world of symbolism existing amongst us which is far older than our sects and books, our creeds and articles, a relic of a forgotten pre-historic past. Untold ages before writing was invented, it is believed that men attempted to express their ideas in visible forms. Yet how can a savage, who is unable to count his fingers up to five, and has no idea of abstract number, apart from things, whose habits and thoughts are of the earth, earthy, form a conception of the high and holy One who inhabiteth eternity? Even under the highest forms of ancient civilisation, abundant proofs exist that the imagination of men, brooding over the idea of the Unseen and the Infinite, were bounded by the things which were presented in their daily experience, and which most moved their passions, hopes and fears. Through these, then, they attempted to embody such religious ideas as they felt. They could not teach others without visible symbols to assist their conceptions; and emblems were rather crutches for the halting, than wings to help the healthy to soar. Mankind in all ages has clung to the visible and tangible. The people care little for the abstract and unseen."

"Hardy had a] profound conviction that the truths of mathematics described a bright and clear universe, exquisite and beautiful in its structure, in comparison with which the physical world was turbid and confused. It was this that made his friends... think that in his attitude to mathematics there was something which, being essentially spiritual, was near to religion."

"In our modern era, God and mathematics are usually placed in totally separate arenas of human thought. But... this has not always been the case, and even today many mathematicians find the exploration of mathematics akin to a spiritual journey. The line between religion and mathematics becomes indistinct. In the past, the intertwining of religion and mathematics has produced useful results and spurred new areas of scientific thought. ...In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics attempt to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek "ideal," immutable truths and then often try to apply these truths to the real world. Some atheists claim another similarity: mathematics and religion are the most powerful evidence of the inventive genius of the human race. Of course, there are also many differences between mathematics and religion."

"Only a fool would try to compress several millennia's blending of mathematics and religion. We proceed."

"For such purposes as these a very slender knowledge of geometry, and a small portion of arithmetic would suffice;—but as for any considerable amount thereof, and great progress in it, we must inquire how far they tend to this,—namely, to make us apprehend more easily the idea of the good:—and we say that all things contribute thereto, which compel the soul to turn itself to that region in which is the happiest portion of true being, which it must by all means perceive. ...this science has an entirely opposite nature to the words employed in it by those who practise it. ...They speak somehow most absurdly, and necessarily so, since all the terms they use seem to be with a view to operation and practice,—such as squaring, producing, adding, and such like sounds; whereas on the other hand, the whole science should be studied for the sake of real knowledge. ...Is this then further to be agreed on? ...That [it be studied] with a view to the knowledge of eternal being, and not of that which is subject to generation and destruction? ...It would have a tendency, therefore... to draw the soul to truth, and to cause a philosophic intelligence to direct upwards [the thoughts] which we now improperly cast downwards. ...We must give special orders, that the inhabitants of that fine state of yours should by no means omit the study of geometry, since even its by-works are not inconsiderable. ...it is not altogether a trifle, but rather difficult to persuade that by these branches of study some organ of the soul in each individual, is purified and rekindled like fire, after having been destroyed and blinded by other kinds of study,—an organ, indeed, better worth saving than ten thousand eyes, since by that alone can truth be seen."

"As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space."

"It arouses our innate knowledge. ..takes away the forgetfulness and ignorance [of our former existence] that we have from birth,. . . fills everything with divine reason, moves our souls towards Nous,. . . and through the discovery of pure Nous leads us to the blessed life."

"[T]he soul has its essence in mathematical ideas, and it has a prior knowledge of them. .. and brings of them to light when it is set free of the hindrances that arise from sensation. For our sense-perceptions engage the mind with divisible things... and...every divisible thing is an obstacle to our returning upon ourselves. ... Consequently when we remove these hindrances. . we become knowers in actuality..."

"An equation for me has no meaning unless it expresses a thought of God."

"It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosophers, except in France. Philosophical works on the subject, such as Cohen's Princip der Infinitesimalmethode und seine Geschichte, are vitiated, as regards the constructive theory, by an undue mysticism inherited from Kant, and leading to such results as the identification of intensive magnitude with the extensive infinitesimal."

"It seems necessary to say something at the outset in justification of the scientific as against the mystical attitude. Metaphysics, from the first, has been developed by the union or the conflict of these two attitudes. Among the earliest Greek philosophers, the Ionians were more scientific and the Sicilians more mystical. But among the latter, Pythagoras, for example, was in himself a curious mixture of the two tendencies: the scientific attitude led him to his proposition on right-angled triangles, while his mystic insight showed him that it is wicked to eat beans. Naturally enough, his followers divided into two sects, the lovers of right-angled triangles and the abhorrers of beans; but the former sect died out, leaving, however, a haunting flavour of mysticism over much Greek mathematical speculation, and in particular over Plato's views on mathematics. Plato, of course, embodies both the scientific and the mystical attitudes in a higher form than his predecessors, but the mystical attitude is distinctly the stronger of the two, and secures ultimate victory whenever the conflict is sharp. Plato, moreover, adopted from the the device of using logic to defeat common sense, and thus to leave the field clear for mysticism—a device still employed in our own day by the adherents of the classical tradition. The logic used in defence of mysticism seems to me faulty as logic..."

"Mathematics has ceased to seem to me non-human in its subject matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal. I think that the timelessness of mathematics has none of the sublimity that it once seemed to me to have, but consists merely in the fact that the pure mathematician is not talking about time. I can no longer find any mystical satisfaction in the contemplation of mathematical truth. ... One effect of the War was to make it impossible to go on living in a world of abstraction. ... I have no longer the feeling that intellect is superior to sense, and that only Plato's world of ideas gives access to the 'real' world. I used to think of sense, and of thought which is built on sense, as a prison... I now think of sense, and of thoughts built on sense, as windows... I think that we can, however imperfectly, mirror the world, like Leibniz's monads; and I think it is the duty of the philosopher to make himself as undistorting a mirror as he can. ...to recognize such distortions ...Of these, the most fundamental is that we view the world from the here and now, not with that large impersonality which theists attribute to the Deity. To achieve such impartiality is impossible for us, but we can travel a certain distance towards it. To show the road to this end is the supreme duty of the philosopher."

": Prithee, no more prattling; go; I'll hold. This is the third time; I hope good luck lies in odd numbers. Away, go! They say there is divinity in odd numbers, either in nativity, chance, or death. Away!"

"One thing that particularly interested Plato was the mysticism of numbers. In his Republic (Book VIII) he speaks in an obscure fashion of a certain mystic number, but he does not make clear what this number is. He calls it "the lord of better and worse births"... One theory is that 60, or 12,960,000 is the Platonic number. This number played an important part in the mysticism of the Hindus and the Babylonians, and it is possible that Pythagoras found it on the banks of the Euphrates, if he really studied there, and that he took it with him to Crotona, passing it on to his disciples, who, in turn, told it to Plato and his followers. ...More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch..."

"Num. xxxiii. 1. 'Build me here seven alters and prepare me here seven oxen and seven rams.' The ancients were very superstitious about certain numbers, supposing that God delighted in odd numbers.Around his waxen image first I wind Three woollen fillets, of three colours joined; Thrice bind about his thrice devoted head, Which round the sacred alter thrice is led. Unequal numbers please the gods."

"In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the s: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the ... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the ... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology."

"Leibniz discovered his future with the help of a Nuremberg society of alchemists. ...In private ...he exhibited an avid interest in the subject throughout his life—as did many of his contemporaries, such as Isaac Newton. Indeed, he was so certain that he would one day soon discover the means to turn lead into gold that at one point he fretted that the resulting oversupply of the yellow metal might drive down the price and thus deprive him of hard-earned profits."

"Among the first of those who bade adieu to the Scholastic creed was the Cardinal Nicolas Cusanus, a man of rare sagacity and an able mathematician; who arranged and republished the Pythagorean Ideas, to which he was much inclined, in a very original manner, by the aid of his Mathematical knowledge. He considered God as the unconditional Maximum, which at the same time, as Absolute Unity, is also the unconditional Minimum, and begets of Himself and out of Himself, Equality and the combination of Equality with Unity (Son and Holy Ghost). According to him, it is impossible to know directly and immediately this Absolute Unity (the Divinity); because we can make approaches to the knowledge of Him only by the means of Number or Plurality. Consequently he allows us only the possession of very imperfect notions of God, and those by mathematical symbols. It must be admitted that the Cardinal did not pursue this thought very consequently, and that his view of the universe which he connected with it, and which represented it as the Maximum condensed, and thus become finite, was very obscure. Nor was he more successful in his view of the one-ness of the Creator and of Creation, or in his attempt to explain the mysteries of the Trinity and Incarnation, by means of this Pantheistic Theism. Nevertheless, numerous profound though undeveloped observations on the faculty of cognition, are found in his writings, interspersed with his prevailing Mysticism. For instance, he observes, that the principles of knowledge possible to us are contained in our ideas of Number (ratio explicata) and their several relations; that absolute knowledge is unattainable to us (precisio veritatis inattingibilis, which he styled docta ignorantia), and that all which is attainable to us is a probable knowledge (conjectura). With such opinions he expressed a sovereign contempt for the Dogmatism of the Schools."

"Neo-Pythagoreans, like Nicomachus of Gerasa... and Iamblichus... revel in... number mysticism. ...The purpose of the layman's Introduction to Arithmetic... is to explain in a manner intelligible to every one, the wonderful and divine properties of numbers. In a very entertaining way he tells about s... square, rectangular and s... gnomic numbers and spatial numbers, about ratios and the distinction between multiple and epimoric."

"The development of the theory of numbers had its root in the number mysticism of Pythagoras, attained a strictly scientific character in the theory of the even and the odd, and led finally to the systematic establishment of the theory of the divisibility and the proportion of numbers as found in Book VII of the Elements."

"Quite aside from the fact that mathematics is the necessary instrument of natural science, purely mathematical inquiry in itself... by its special character, its certainty and stringency, lifts the human mind into closer proximity with the divine than is attainable through any other medium. Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means. ...The connection between mathematics of the infinite and the perception of God was pursued most fervently by Nicholas of Cusa, the thinker who... intoned the new melody of thought which with Leonardo, Bruno, Kepler, and Descartes gradually swells into a triumphant symphony. He recognizes that the scholastic form of thinking, Aristotelian logic, which is based on... the excluded third, cannot... think the absolute, the infinite... every kind of "rational" theology is rejected, and "mystic" theology takes its place. ...The true love of God is amor de intellectualis [intellectual love]. ...Cusanus does not refer to the mystic form of contemplation, but rather to mathematics and its symbolic method. ...This urge finds its simplest expression in the sequence of numbers, which can be driven beyond any place by repeated addition of one."

"In certain of their aspects, as well as in the attitudes of their practitioners, there is a close analogy between religion and mathematics. To some of its practitioners, religion represents an ideal essentially superpersonal, existing independently of its devotees; to others, it affords rules by which to live, having no intellectual content whatever. And like religion, mathematics traces its beginnings back to prehistoric primitive cultures, as a result of which the numbers with which we count, the so-called "natural numbers"... are frequently regarded in a mystical fashion not ordinarily associated... Also like religion, mathematics furnishes a fascinating study from an anthropological viewpoint, only the meagre beginnings of which are to be found in antropological literature—a gap which I am hoping to help fill..."

"Socrates, in the Phædrus, delivers to us three characters who are elevated from sense, because they fill up and accomplish the primary life of the soul, i.e. the philosopher, the lover, and the musician. But the beginning and path of elevation to the lover, is a progression from apparent beauty, using as excitations the middle forms of beautiful objects. But to the musician, who is allotted the third seat, the way consists in a transition from sensible to invisible harmonies, and to the reasons existing in these. So that to the one, sight is the instrument of reminiscence, and to the other, hearing. But to him who is by nature a philosopher, from whence and by what means is reminiscence the prelude of intellectual knowledge, and an excitation to that which truly is, and to truth itself. For this character also, on account of its imperfection, requires a proper principle: for it is allotted a natural virtue, an imperfect eye, and a degraded manner. It must therefore be excited from itself; and he who is of such a nature, rejoices in that which is. But to the philosopher, says Plotinus, the mathematical disciplines must be exhibited, that they may accustom him to an incorporeal nature, and that afterwards using these as figures, he may be led to dialectic reasons, and to the contemplation of all the things which are. And thus it is manifest, from hence, that the mathematics are of the greatest utility to philosophy."

"Whatever to imperfect natures appears difficult and arduous in obtaining the true knowledge of the gods, the mathematical reasons render, by their images, credible, manifest, and certain. Thus, in numbers, they indicate the significations of super-essential properties, but they evince the powers of intellectual figures, in those figures which fall under cogitation. Hence it is, that Plato, by mathematical forms teaches us many and admirable sentences concerning the gods, and the philosophy of the Pythagoreans; using these as veils, conceals from vulgar inspection the discipline of divine sentences. For such is the whole of the Sacred and Divine Discourse, that of Philolaus in his Bacchics, and the universal method of the Pythagoric narration concerning the Gods. But it especially refers to the contemplation of nature, since it discloses the order of those reasons by which the universe is fabricated, and that proportion which binds, as Timæus says, whatever the world contains, in union and consent; besides it conciliates in amity things mutually opposing each other, and gives convenience and consent to things mutually disagreeing, and exhibits to our view simple and primary elements, from which the universe is composed, on every side comprehended by commensurability and equality, because it receives convenient figures in its proportions, and numbers proper to every production, and finds out their revolutions and renovations, by which we are enabled to reason concerning the best origin, and the contrary dissolution of particulars. In consequence of this, as it appears to me, Timaeus discloses the contemplation concerning the nature of the universe, by mathematical names, adorns the origin of the elements with numbers and figures, referring to these their powers, passions, and energies; and esteeming as well the acuteness as the obtruseness of angles, the levity of sides, or contrary powers, and their multitude and paucity to be the cause of the all-various mutation of the elements."