Mathematics and mysticism

Quotes

A-B

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.
- Augustine of Hippo, On the Trinity (ca. 417) Bk. IV, as quoted in The Works of Aurelius Augustine, Bishop of Hippo A New Translation (1873) Vol. 7, p. 116, ed. Marcus Dods, Tr. Arthur West Hadden.

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 perfect number,—not because God required a protracted time... but because the perfection of the works was signified by the number six.
- Augustine of Hippo, City of God (ca. 430) Bk 11, Ch. 30; Tr. Marcus Dods (1888) Vol. 1, p. 474.

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.
- Zaheer Baber, The Science of Empire: Scientific Knowledge, Civilization, and Colonial Rule in India (1996) p. 234.

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.
- John D. Barrow, The Artful Universe (1995)

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.
- Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1949).

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...
- Carl B. Boyer, A History of Mathematics (1968)

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.
- Jacob Bronowski, "The Reach of Imagination" (1967)

C-D

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.
- Keith Critchlow Forward, The Theology of Arithmetic by Iamblichus (1988) Tr. Robin Waterfield, p. 12.

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.
- Franz Cumont, Astrology and Religion among the Greeks and Romans (1911) p. 30.

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.
- Philip J. Davis, Reuben Hersh, The Mathematical Experience (1980) "Underneath the Fig Leaf," p.89.

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?
- Philip J. Davis, Reuben Hersh, The Mathematical Experience (1980) "Underneath the Fig Leaf," pp.111-112.

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.
- John Dee, The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara (1570) Tr. Henry Billingsley

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.
- Betty Jo Teeter Dobbs, The Janus Faces of Genius: The Role of Alchemy in Newton's Thought (2002)

E-G

The Pythagoreans... were fascinated by certain specific ratios, and especially by those which relate what we call today the arithmetic [ $A={\frac {a+b}{2}}$ $A={\frac {a+b}{2}}$ ], geometric [ $G={\sqrt {ab}}$ $G={\sqrt {ab}}$ ], and harmonic [ $H={\frac {2ab}{a+b}}$ $H={\frac {2ab}{a+b}}$ ] means. ...
The particular ratios between these means in which the Pythagoreans were interested were... $A:G=G:H$ The other relationship is expressed by $a:A=H:b$ The Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the Babylonians by Pythagoras himself after having been taken prisoner in Egypt.
Ratios 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 have $A={\frac {6+12}{2}}=9$ and $H={\frac {2\times 6\times 12}{6+12}}=8$ The [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 triangular number 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 diatonic scale, ${\frac {3}{2}}:{\frac {4}{3}}=9:8$ and the semi-tone, ${\frac {4}{3}}:({\frac {9}{8}})^{2}=256:243$ .
- Graham Flegg, Numbers: Their History and Meaning (1983)

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.
- Graham Flegg, Numbers: Their History and Meaning (1983)

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."
- Loren Graham, Jean-Michel Kantor, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity (2009)

H-K

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.
- Elizabeth Sanderson Haldane, Descartes: His Life and Times (1905)

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.
- Thomas Patrick Hughes, "Witr," A Dictionary of Islam (1885)

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.
- Morris Kline, Mathematics and the Physical World (1959)

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.
- Morris Kline, Mathematics for the Nonmathematician (1967)

L

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.
- Pierre-Simon Laplace, Essai Philosophique sur les Probabilitésas (1814) p. 82, partially quoted in Tobias Dantzig, Number: The Language of Science (1930) p. 15 and in Richard Courant, Herbert Robbins What is Mathematics? (1941) revised by Ian Stewart (1996)

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.
- Reinhard Labenbacher, David Pengelley, Mathematical Expeditions: Chronicles by the Explorers (1999)

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.
- Gottfried Wilhelm Leibniz, "On the General Characteristic," (ca. 1679) as quoted in Philosophical Papers and Letters (1956) ed., Leroy E. Loemaker

M-N

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 Gordian knots 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 Ulm, to make trial of his new method of solving problems on Faulhaber and other mathematicians of distinction. The story of Isaac Beeckman 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...
- John Pentland Mahaffy, Descartes (1880)

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?
- Jacques Maritain, Le Songe de Descartes, Suivi de quelques Essais (1932) as translated in The Dream of Descartes: together with some other Essays (1944) Tr. Jacques and Mabelle L. Andison, p. 18.

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.
- John Newton, "The Assyrian Grove and other Emblems," Appendix in Thomas Inman, Ancient Pagan and Modern Christian Symbolism (1884)

O-P

[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.
- Oxford Magazine (Jan 22, 1948) Obituary for G. H. Hardy

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.
- Clifford A. Pickover, The Loom of God: Tapestries of Mathematics and Mysticism (2009)

Only a fool would try to compress several millennia's blending of mathematics and religion. We proceed.
- Clifford A. Pickover, The Loom of God: Tapestries of Mathematics and Mysticism (2009)

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.
- Plato, Republic (c. 380 BC) Bk VII., Ch. IX-X. Tr. (1849) Benjamin Jowett pp. 216-217

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.
- Robert Potts, Euclid's Elements of Geometry (1845) Introduction pp. iii-iv

Q-R

An equation for me has no meaning unless it expresses a thought of God.
- Srinivasa Ramanujan as quoted by Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan (1991)

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.
- Bertrand Russell, The Principles of Mathematics (1903) Vol. 1.

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 Eleatics 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...
- Bertrand Russell, Our Knowledge of the External World (1914)

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.
- Bertrand Russell, "The Retreat from Pythagoras," My Philosophical Development (1959)

S

Falstaff: 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!
- William Shakespeare, The Merry Wives of Windsor (1597, 1602)

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...
- David Eugene Smith, History of Mathematics (1925) Vol. 1.

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.
- Thomas Stackhouse, Stackhouse's History of the Holy Bible (1743) as quoted from the 1842 edition, p. 316.

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 Platonic solids: 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 golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.
- Alexey Stakhov, Samuil Aranson, The “Golden” Non-Euclidean Geometry: Hilbert's Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant (2016)

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.
- Matthew Stewart, The Courtier and the Heritic (2006)

T-Z

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.
- Wilhelm Gottlieb Tennemann, Grundriss der Geschichte der Philosophie für den akademischen Unterricht (1812) translated as A Manual of the History of Philosophy (1852) Tr. Rev. A. Johnson, J. R. Morel, citing Nicolai Cusani Opera, Paris. 1514. 3 vols. fol.; Basil. 1665, 3 vols. fol. De Doctâ Ignorantiâ, tom. III. Apologia Doctæ Ignorantiæ, lib. I. De Conjecturis libb. II De Sapientiæ libb III.

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 triangular numbers... square, rectangular and polygonal numbers... gnomic numbers and spatial numbers, about ratios and the distinction between multiple and epimoric.
- Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) Tr. Science Awakening (1954) by Arnold Dresden.

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.
- Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) Tr. Science Awakening (1954) by Arnold Dresden.

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.
- Hermann Weyl, "The Open World: Three Lectures on the Metaphysical Implications of Science," (1932) as quoted in Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics (2009) ed. Peter Pesic.

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...
- Raymond Louis Wilder, "The Nature of Modern Mathematics," The Evolution of the Science of Numbers, Quarterly Review of the Michigan Alumnus, Vol. 65, 1958-1959, p. 302.

The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements (1792)

Proclus (c. 450 AD) as translated by Thomas Taylor, Vol. 1

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.

Mysticism in Modern Mathematics (1910)

Hastings Berkeley, source

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.
- Preface

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...
- Preface

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.
- Introductory, with footnote

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.
- Introductory

"The Philosophy of M. Henri Bergsen" (1913)

George Santayana

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.

"Mysticism and Logic" (July, 1914)

Bertrand Russell, Hibbert Journal (July, 1914) as quoted in Mysticism and Logic: And Other Essays (1919) p. 12

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.

Milton and Jakob Boehme: A Study of German Mysticism in Seventeenth-century England (1914)

Margaret Lewis Bailey, source

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, Matthias Bernegger, Joachim Jungius, Theodor Haak, Samuel Hartlib, John Dury, 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.

Problems of Mysticism and Its Symbolism (1917)

Herbert Silberer, source

In reference to the symbol and image language, which was comprehensible only to the initiated, we think naturally of the ancient mysteries. The religious societies of the oldest Christians, in the centuries when Christianity belonged in the Roman Empire to the forbidden cults, found a possibility of existence before the law in the form of licensed societies, i.e., as guilds, burial unions, and corporations of all sorts. The primitive Christians were not the only forbidden sects that sought and found this recourse. Under the disguise of schools, trade unions, literary societies, and academies, there existed in the jurisdiction of the Roman Empire, and later inside of the world church, organizations that before the law were secular societies, but in the minds of the initiated were associations of a religious character. Within these associations there appeared very early a well developed system of symbols, which were adopted for the purpose of actually maintaining, through the concealment necessitated by circumstances, their unions and their implements and customs—symbols that they chose as cloaks and that in the circle of the initiated were explained and interpreted according to the teachings of their cult.
Valuable monuments of this symbolism are preserved in the vast rock temples that are found in Egypt, Syria, Asia Minor, Sicily, and the Apennine peninsula, in Greece, France, and on the Rhine, and these vaults, which in part also served the early Christians as places of worship, show in their images and records and in their architectural form so close a resemblance that they must be acknowledged as the characteristic of a great religious cult extending over many lands, which has had consistent traditions for the use of such symbols and for the production of these structures.
Many of these symbols, it should be noted in passing, are borrowed from those tokens and implements of the building corporations, which were necessary to the completion of their buildings. An important part was played even in the early Christian symbolism by the sacred numbers and the figures corresponding to them, a group of educational symbols which we find likewise in the pythagorean and platonic schools. It is known that the symbolical language of the subterranean rock temples, some of which were used by the earliest Christians for their religious worship, are closely connected with the pythagorean and platonic doctrines. From the year 325 A.D. on, every departure from the beliefs of the state church was considered a state offense. So those Christians who retained connection with the ancient philosophic schools were persecuted. In the religious symbol language of the church, the sacred numbers naturally began to disappear from that time. In the writings of Augustine begins the war on the symbolic language, whose use he declared a characteristic of the gnostics. In spite of the suppression the doctrines of the sacred numbers continued through all the centuries in religious use, in quiet but strong currents which flowed beside the state church. The sect names, which were invented by polemic theology for the purpose of characterizing methods that were regarded as imitations of the gnostics, are of the most varied kinds; it may be enough to remember that in all those spiritual currents, that like the old German mysticism, the earlier humanism, the so-called natural philosophy, etc., show a strong influence of platonic thinking, the doctrines of the sacred numbers recur, in a more or less disguised form, but yet clearly recognizable.
As the old number symbolism constitutes a part of the hieroglyphics of alchemy, I shall pause a moment to consider them. The use of mathematical and geometrical symbols proceeds from the use of the simplest forms, points and lines, but in all cases where the object is not a representation in the flat but in space, both the points and lines are replaced by plastic forms, i.e., forms of cylinders, spheres, bars, rings, cubes, etc. From this point it was but a short step to the use of trees, leaves, flowers, implements, and other things that showed similarities in form. ...
For symbolism, too, which served as the characterization of the forms of organization and the building up of the fraternity into degrees... in place of lines and points are found plastic forms which were at their disposal in carpenters' squares, crossed bars, etc.
As the circle symbolized the all and the eternal or the celestial unity of the all, and the divinity, so the number one, the single line, the staff or the scepter, represented the terrestrial copy of the power, the ruling, guiding, sustaining and protecting force of the personality that had attained freedom on earth. ...
Duality, the Dyas, represents in contrast to the celestial being the divided terrestrial being that is dominated by the antagonism of things and is only a transitory, imperfect existence; the opposites, fluid and solid, sulphur and mercury, dry and wet, etc.
Tetragrammaton
from Petrus Alphonsi,
Dialogi contra Iudaeos
(ca. 1109 A.D.)
Tr: "Dialogue against the Jews"

In the symbol of the trinity, which frequently occurs in the form of a triangle... is shown how the divided and sensuous nature is led by the higher power of the number 3 to a harmony of powers and to a new unity. The symbol of reason attaining victory over matter becomes visible. A representation of trinity is possible by means of the conventional cross. We can see in it two elements of lines which by their unification or penetration give the third as the point of intersection. More generally the cross is conceived as quinity (fiveness)—i.e., 4+1ness (in alchemy four elements which are collected about the quinta esentia). A cross in which unity splits into duality so that trinity results, is Y, which is called the forked cross. From unity grows duality, that is, nature divides into spirit and matter, into active and passive, necessity and freedom. The divided returns through trinity to unity.

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 loggia 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."
- citing Dr. Ludwig Keller (de.Wikisource), De Anfänge der Renaissance, Die, und die Kultgesellschaften des Humanismus im XIII. und XIV. (1903) [Tr. The Beginnings of the Renaissance and the Cult Societies of Humanism in the 13th and 14th centuries] p. 38.

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."

Mathematics
Mathematicians (by country)	Abel • Anaxagoras • Archimedes • Aristarchus of Samos • Averroes • Arnold • Banach • Cantor • Cartan • Chern • Cohen • Descartes • Diophantus • Erdős • Euclid • Euler • Fourier • Gauss • Gödel • Grassmann • Grothendieck • Hamilton • Hilbert • Hypatia • Lagrange • Laplace • Leibniz • Milnor • Newton • von Neumann • Noether • Penrose • Perelman • Poincaré • Pólya • Pythagoras • Riemann • Russell • Schwartz • Serre • Tao • Tarski • Thales • Turing • Weil • Weyl • Wiles • Witten
Numbers	1 • 23 • 360 • e • π • Fibonacci numbers • Irrational number • Negative number • Number • Prime number • Quaternion • Octonion
Concepts	Abstraction • Algorithms • Axiomatic system • Completeness • Deductive reasoning • Differential equation • Dimension • Ellipse • Elliptic curve • Exponential growth • Infinity • Integration • Geodesic • Induction • Proof • Partial differential equation • Principle of least action • Prisoner's dilemma • Probability • Randomness • Theorem • Topological space • Wave equation
Results	Euler's identity • Fermat's Last Theorem
Pure math	Abstract algebra • Algebra • Analysis • Algebraic geometry (Sheaf theory) • Algebraic topology • Arithmetic • Calculus • Category theory • Combinatorics • Commutative algebra • Complex analysis • Differential calculus • Differential geometry • Differential topology • Ergodic theory • Foundations of mathematics • Functional analysis • Game theory • Geometry • Global analysis • Graph theory • Group theory • Harmonic analysis • Homological algebra • Invariant theory • Logic • Non-Euclidean geometry • Nonstandard analysis • Number theory • Numerical analysis • Operations research • Representation theory • Ring theory • Set theory • Sheaf theory • Statistics • Symplectic geometry • Topology
Applied math	Computational fluid dynamics • Econometrics • Fluid mechanics • Mathematical physics • Science
History of math	Ancient Greek mathematics • Euclid's Elements • History of algebra • History of calculus • History of logarithms • Indian mathematics • Principia Mathematica
Other	Mathematics and mysticism • Mathematics education • Mathematics, from the points of view of the Mathematician and of the Physicist • Philosophy of mathematics • Unification in science and mathematics

Quotes

A-B

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.
- Augustine of Hippo, On the Trinity (ca. 417) Bk. IV, as quoted in The Works of Aurelius Augustine, Bishop of Hippo A New Translation (1873) Vol. 7, p. 116, ed. Marcus Dods, Tr. Arthur West Hadden.

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 perfect number,—not because God required a protracted time... but because the perfection of the works was signified by the number six.
- Augustine of Hippo, City of God (ca. 430) Bk 11, Ch. 30; Tr. Marcus Dods (1888) Vol. 1, p. 474.

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.
- Zaheer Baber, The Science of Empire: Scientific Knowledge, Civilization, and Colonial Rule in India (1996) p. 234.

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.
- John D. Barrow, The Artful Universe (1995)

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.
- Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1949).

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...
- Carl B. Boyer, A History of Mathematics (1968)

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.
- Jacob Bronowski, "The Reach of Imagination" (1967)

C-D

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.
- Keith Critchlow Forward, The Theology of Arithmetic by Iamblichus (1988) Tr. Robin Waterfield, p. 12.

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.
- Franz Cumont, Astrology and Religion among the Greeks and Romans (1911) p. 30.

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.
- Philip J. Davis, Reuben Hersh, The Mathematical Experience (1980) "Underneath the Fig Leaf," p.89.

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?
- Philip J. Davis, Reuben Hersh, The Mathematical Experience (1980) "Underneath the Fig Leaf," pp.111-112.

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.
- John Dee, The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara (1570) Tr. Henry Billingsley

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.
- Betty Jo Teeter Dobbs, The Janus Faces of Genius: The Role of Alchemy in Newton's Thought (2002)

E-G

The Pythagoreans... were fascinated by certain specific ratios, and especially by those which relate what we call today the arithmetic [ $A={\frac {a+b}{2}}$ $A={\frac {a+b}{2}}$ ], geometric [ $G={\sqrt {ab}}$ $G={\sqrt {ab}}$ ], and harmonic [ $H={\frac {2ab}{a+b}}$ $H={\frac {2ab}{a+b}}$ ] means. ...
The particular ratios between these means in which the Pythagoreans were interested were... $A:G=G:H$ The other relationship is expressed by $a:A=H:b$ The Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the Babylonians by Pythagoras himself after having been taken prisoner in Egypt.
Ratios 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 have $A={\frac {6+12}{2}}=9$ and $H={\frac {2\times 6\times 12}{6+12}}=8$ The [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 triangular number 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 diatonic scale, ${\frac {3}{2}}:{\frac {4}{3}}=9:8$ and the semi-tone, ${\frac {4}{3}}:({\frac {9}{8}})^{2}=256:243$ .
- Graham Flegg, Numbers: Their History and Meaning (1983)

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.
- Graham Flegg, Numbers: Their History and Meaning (1983)

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."
- Loren Graham, Jean-Michel Kantor, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity (2009)

H-K

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.
- Elizabeth Sanderson Haldane, Descartes: His Life and Times (1905)

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.
- Thomas Patrick Hughes, "Witr," A Dictionary of Islam (1885)

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.
- Morris Kline, Mathematics and the Physical World (1959)

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.
- Morris Kline, Mathematics for the Nonmathematician (1967)

L

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.
- Pierre-Simon Laplace, Essai Philosophique sur les Probabilitésas (1814) p. 82, partially quoted in Tobias Dantzig, Number: The Language of Science (1930) p. 15 and in Richard Courant, Herbert Robbins What is Mathematics? (1941) revised by Ian Stewart (1996)

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.
- Reinhard Labenbacher, David Pengelley, Mathematical Expeditions: Chronicles by the Explorers (1999)

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.
- Gottfried Wilhelm Leibniz, "On the General Characteristic," (ca. 1679) as quoted in Philosophical Papers and Letters (1956) ed., Leroy E. Loemaker

M-N

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 Gordian knots 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 Ulm, to make trial of his new method of solving problems on Faulhaber and other mathematicians of distinction. The story of Isaac Beeckman 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...
- John Pentland Mahaffy, Descartes (1880)

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?
- Jacques Maritain, Le Songe de Descartes, Suivi de quelques Essais (1932) as translated in The Dream of Descartes: together with some other Essays (1944) Tr. Jacques and Mabelle L. Andison, p. 18.

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.
- John Newton, "The Assyrian Grove and other Emblems," Appendix in Thomas Inman, Ancient Pagan and Modern Christian Symbolism (1884)

O-P

[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.
- Oxford Magazine (Jan 22, 1948) Obituary for G. H. Hardy

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.
- Clifford A. Pickover, The Loom of God: Tapestries of Mathematics and Mysticism (2009)

Only a fool would try to compress several millennia's blending of mathematics and religion. We proceed.
- Clifford A. Pickover, The Loom of God: Tapestries of Mathematics and Mysticism (2009)

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.
- Plato, Republic (c. 380 BC) Bk VII., Ch. IX-X. Tr. (1849) Benjamin Jowett pp. 216-217

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.
- Robert Potts, Euclid's Elements of Geometry (1845) Introduction pp. iii-iv

Q-R

An equation for me has no meaning unless it expresses a thought of God.
- Srinivasa Ramanujan as quoted by Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan (1991)

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.
- Bertrand Russell, The Principles of Mathematics (1903) Vol. 1.

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 Eleatics 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...
- Bertrand Russell, Our Knowledge of the External World (1914)

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.
- Bertrand Russell, "The Retreat from Pythagoras," My Philosophical Development (1959)

S

Falstaff: 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!
- William Shakespeare, The Merry Wives of Windsor (1597, 1602)

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...
- David Eugene Smith, History of Mathematics (1925) Vol. 1.

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.
- Thomas Stackhouse, Stackhouse's History of the Holy Bible (1743) as quoted from the 1842 edition, p. 316.

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 Platonic solids: 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 golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.
- Alexey Stakhov, Samuil Aranson, The “Golden” Non-Euclidean Geometry: Hilbert's Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant (2016)

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.
- Matthew Stewart, The Courtier and the Heritic (2006)

T-Z

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.
- Wilhelm Gottlieb Tennemann, Grundriss der Geschichte der Philosophie für den akademischen Unterricht (1812) translated as A Manual of the History of Philosophy (1852) Tr. Rev. A. Johnson, J. R. Morel, citing Nicolai Cusani Opera, Paris. 1514. 3 vols. fol.; Basil. 1665, 3 vols. fol. De Doctâ Ignorantiâ, tom. III. Apologia Doctæ Ignorantiæ, lib. I. De Conjecturis libb. II De Sapientiæ libb III.

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 triangular numbers... square, rectangular and polygonal numbers... gnomic numbers and spatial numbers, about ratios and the distinction between multiple and epimoric.
- Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) Tr. Science Awakening (1954) by Arnold Dresden.

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.
- Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) Tr. Science Awakening (1954) by Arnold Dresden.

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.
- Hermann Weyl, "The Open World: Three Lectures on the Metaphysical Implications of Science," (1932) as quoted in Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics (2009) ed. Peter Pesic.

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...
- Raymond Louis Wilder, "The Nature of Modern Mathematics," The Evolution of the Science of Numbers, Quarterly Review of the Michigan Alumnus, Vol. 65, 1958-1959, p. 302.

The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements (1792)

Proclus (c. 450 AD) as translated by Thomas Taylor, Vol. 1

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.

Mysticism in Modern Mathematics (1910)

Hastings Berkeley, source

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.
- Preface

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...
- Preface

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.
- Introductory, with footnote

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.
- Introductory

"The Philosophy of M. Henri Bergsen" (1913)

George Santayana

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.

"Mysticism and Logic" (July, 1914)

Bertrand Russell, Hibbert Journal (July, 1914) as quoted in Mysticism and Logic: And Other Essays (1919) p. 12

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.

Milton and Jakob Boehme: A Study of German Mysticism in Seventeenth-century England (1914)

Margaret Lewis Bailey, source

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, Matthias Bernegger, Joachim Jungius, Theodor Haak, Samuel Hartlib, John Dury, 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.

Problems of Mysticism and Its Symbolism (1917)

Herbert Silberer, source

In reference to the symbol and image language, which was comprehensible only to the initiated, we think naturally of the ancient mysteries. The religious societies of the oldest Christians, in the centuries when Christianity belonged in the Roman Empire to the forbidden cults, found a possibility of existence before the law in the form of licensed societies, i.e., as guilds, burial unions, and corporations of all sorts. The primitive Christians were not the only forbidden sects that sought and found this recourse. Under the disguise of schools, trade unions, literary societies, and academies, there existed in the jurisdiction of the Roman Empire, and later inside of the world church, organizations that before the law were secular societies, but in the minds of the initiated were associations of a religious character. Within these associations there appeared very early a well developed system of symbols, which were adopted for the purpose of actually maintaining, through the concealment necessitated by circumstances, their unions and their implements and customs—symbols that they chose as cloaks and that in the circle of the initiated were explained and interpreted according to the teachings of their cult.
Valuable monuments of this symbolism are preserved in the vast rock temples that are found in Egypt, Syria, Asia Minor, Sicily, and the Apennine peninsula, in Greece, France, and on the Rhine, and these vaults, which in part also served the early Christians as places of worship, show in their images and records and in their architectural form so close a resemblance that they must be acknowledged as the characteristic of a great religious cult extending over many lands, which has had consistent traditions for the use of such symbols and for the production of these structures.
Many of these symbols, it should be noted in passing, are borrowed from those tokens and implements of the building corporations, which were necessary to the completion of their buildings. An important part was played even in the early Christian symbolism by the sacred numbers and the figures corresponding to them, a group of educational symbols which we find likewise in the pythagorean and platonic schools. It is known that the symbolical language of the subterranean rock temples, some of which were used by the earliest Christians for their religious worship, are closely connected with the pythagorean and platonic doctrines. From the year 325 A.D. on, every departure from the beliefs of the state church was considered a state offense. So those Christians who retained connection with the ancient philosophic schools were persecuted. In the religious symbol language of the church, the sacred numbers naturally began to disappear from that time. In the writings of Augustine begins the war on the symbolic language, whose use he declared a characteristic of the gnostics. In spite of the suppression the doctrines of the sacred numbers continued through all the centuries in religious use, in quiet but strong currents which flowed beside the state church. The sect names, which were invented by polemic theology for the purpose of characterizing methods that were regarded as imitations of the gnostics, are of the most varied kinds; it may be enough to remember that in all those spiritual currents, that like the old German mysticism, the earlier humanism, the so-called natural philosophy, etc., show a strong influence of platonic thinking, the doctrines of the sacred numbers recur, in a more or less disguised form, but yet clearly recognizable.
As the old number symbolism constitutes a part of the hieroglyphics of alchemy, I shall pause a moment to consider them. The use of mathematical and geometrical symbols proceeds from the use of the simplest forms, points and lines, but in all cases where the object is not a representation in the flat but in space, both the points and lines are replaced by plastic forms, i.e., forms of cylinders, spheres, bars, rings, cubes, etc. From this point it was but a short step to the use of trees, leaves, flowers, implements, and other things that showed similarities in form. ...
For symbolism, too, which served as the characterization of the forms of organization and the building up of the fraternity into degrees... in place of lines and points are found plastic forms which were at their disposal in carpenters' squares, crossed bars, etc.
As the circle symbolized the all and the eternal or the celestial unity of the all, and the divinity, so the number one, the single line, the staff or the scepter, represented the terrestrial copy of the power, the ruling, guiding, sustaining and protecting force of the personality that had attained freedom on earth. ...
Duality, the Dyas, represents in contrast to the celestial being the divided terrestrial being that is dominated by the antagonism of things and is only a transitory, imperfect existence; the opposites, fluid and solid, sulphur and mercury, dry and wet, etc.
Tetragrammaton
from Petrus Alphonsi,
Dialogi contra Iudaeos
(ca. 1109 A.D.)
Tr: "Dialogue against the Jews"

In the symbol of the trinity, which frequently occurs in the form of a triangle... is shown how the divided and sensuous nature is led by the higher power of the number 3 to a harmony of powers and to a new unity. The symbol of reason attaining victory over matter becomes visible. A representation of trinity is possible by means of the conventional cross. We can see in it two elements of lines which by their unification or penetration give the third as the point of intersection. More generally the cross is conceived as quinity (fiveness)—i.e., 4+1ness (in alchemy four elements which are collected about the quinta esentia). A cross in which unity splits into duality so that trinity results, is Y, which is called the forked cross. From unity grows duality, that is, nature divides into spirit and matter, into active and passive, necessity and freedom. The divided returns through trinity to unity.

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 loggia 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."
- citing Dr. Ludwig Keller (de.Wikisource), De Anfänge der Renaissance, Die, und die Kultgesellschaften des Humanismus im XIII. und XIV. (1903) [Tr. The Beginnings of the Renaissance and the Cult Societies of Humanism in the 13th and 14th centuries] p. 38.

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."

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The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements (1792)

Mysticism in Modern Mathematics (1910)

"The Philosophy of M. Henri Bergsen" (1913)

"Mysticism and Logic" (July, 1914)

Milton and Jakob Boehme: A Study of German Mysticism in Seventeenth-century England (1914)

Problems of Mysticism and Its Symbolism (1917)

See also

External links

Quotes

A-B

C-D

E-G

H-K

L

M-N

O-P

Q-R

S

T-Z

The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements (1792)

Mysticism in Modern Mathematics (1910)

"The Philosophy of M. Henri Bergsen" (1913)

"Mysticism and Logic" (July, 1914)

Milton and Jakob Boehme: A Study of German Mysticism in Seventeenth-century England (1914)

Problems of Mysticism and Its Symbolism (1917)

See also

External links