Logicians

"Madam, I have come from a country where people are hanged if they talk."

"Now I will have less distraction."

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."

"All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction."

"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be."

"La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. … La chose ne me paroît pas impossible."

"It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful."

"Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of primes which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order."

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities."

"Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta."

"He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum."

"The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery."

"Of no little importance are Euler's labors in analytical mechanics. ...He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus on reading Virgil's lines. "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by P. Maupertius. He wrote also on tides and on sound."

"Somebody said "Talent is doing what others find difficult. Genius is doing easily what others find impossible." ...by that definition, Euler was a genius. He could do the seemingly impossible, and he did it throughout his long and illustrious life. ...Way to Go, Uncle Leonhard!"

"Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!"

"Euler lacked only one thing to make him a perfect genius: He failed to be incomprehensible."

"The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it."

"It is customary to consider Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by Brouncker and Wallis who, in March of 1655, made discoveries which influenced the development of analysis for the next hundred years. Namely, Wallis found an infinite product of rational numbers converging to 4/π and Brouncker gave a remarkable continued fraction for this quantity. ...The only mathematician who understood the importance of these discoveries was Euler. ...he felt that the recovery of the original Brouncker's proof could open up new perspectives for analysis. As usual, Euler was right."

"Following a suggestion by Daniel Bernoulli, Euler gave the first treatment of elastic lines by means of the in the Additamentum I to his Methodus inveniendi (1744...) which carries the title De curvis elasticus. Euler characterized the equilibrium position of an elastic line by the following variational principle: Among all curves of equal length, joining two points where they have prescribed tangents, to determine that which minimizes the value of the expression \int ds/\rho^2 [where \rho is the radius of curvature]. In other words, Euler interpreted an elastic line as an inextensible curve \boldsymbol\zeta with a "" of \int \kappa^2 ds, \kappa [i.e., 1/\rho] being the function of \boldsymbol\zeta, whose positions of (stable) equilibrium are characterized by the minima of the potential energy, i.e., by 's principle of . Thus the problem of the elastic line leads to the isoperimetric problem\int_{\boldsymbol \zeta} \kappa^2 ds \to min \qquad with \int_{\boldsymbol\zeta} ds = L..."

"Euler published so much and in so many different fields that an edited volume is probably the only way (at least at this time) to do him something like justice, since no one person will know enough to span all of his work."

"Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by , was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the s. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of s, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions."

"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, , and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination."

"To the reader of today much in the conception and mode of expression of that time appears strange and unusual. Between us and the mathematicians of the late seventeenth century stands Leonhard Euler... He is the real founder of our modern conception. However non-rigorous he may be in details: he ends and conquers the previous epoch of direct geometric infinitesimal considerations and introduces the period of mathematical analysis according to form and content. Whatever was written after him on the logarithmic series is necessarily based no longer on the already obscured predecessors in the receding mathematical Renaissance, but on Euler's Introductio in analysin infinitorum... in which the entire seventh chapter [De Quantitabus exponentialibus ac Logarithmis] treats of logarithms."

"Read Euler: he is our master in everything."

"He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children."

"If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics, a man whose immense output... is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. ...Moreover, we owe to Euler many of the mathematical symbols in use today, among them i, π, e, and f(x). And as if that were not enough, he was a great popularizer of science..."

"Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)"

"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."

"It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics."

"In 1736, during his first stay in St. Petersburg, Euler tackled the now famous problem of the seven bridges of Königsberg. His contribution to this problem is often cited as the birth of graph theory and topology."

"I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes."

"Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language."

"As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler."

"It is by participation of species that we call every sensible object beautiful. Thus, since everything void of form is by nature fitted for its reception, as far as it is destitute of reason and form it is base and separate from the divine reason, the great fountain of forms; and whatever is entirely remote from this immortal source is perfectly base and deformed. And such is matter, which by its nature is ever averse from the supervening irradiations of form. Whenever, therefore, form accedes, it conciliates in amicable unity the parts which are about to compose a whole; for being itself one it is not wonderful that the subject of its power should tend to unity, as far as the nature of a compound will admit. Hence beauty is established in multitude when the many is reduced into one, and in this case it communicates itself both to the parts and to the whole. But when a particular one, composed from similar parts, is received it gives itself to the whole, without departing from the sameness and integrity of its nature. Thus at one and the same time it communicates itself to the whole building and its several parts; and at another time confines itself to a single stone, and then the first participation arises from the operations of art, but the second from the formation of nature. And hence body becomes beautiful through the communion supernally proceeding from divinity."

"It is now time, leaving every object of sense far behind, to contemplate, by a certain ascent, a beauty of a much higher order; a beauty not visible to the corporeal eye, but alone manifest to the brighter eye of the soul, independent of all corporeal aid. However, since, without some previous perception of beauty it is impossible to express by words the beauties of sense, but we must remain in the state of the blind, so neither can we ever speak of the beauty of offices and sciences, and whatever is allied to these, if deprived of their intimate possession. Thus we shall never be able to tell of virtue's brightness, unless by looking inward we perceive the fair countenance of justice and temperance, and are convinced that neither the evening nor morning star are half so beautiful and bright. But it is requisite to perceive objects of this kind by that eye by which the soul beholds such real beauties. Besides it is necessary that whoever perceives this species of beauty, should be seized with much greater delight, and more vehement admiration, than any corporeal beauty can excite; as now embracing beauty real and substantial. Such affections, I say, ought to be excited about true beauty, as admiration and sweet astonishment; desire also and love and a pleasant trepidation. For all souls, as I may say, are affected in this manner about invisible objects, but those the most who have the strongest propensity to their love; as it likewise happens about corporeal beauty; for all equally perceive beautiful corporeal forms, yet all are not equally excited, but lovers in the greatest degree."

"Perhaps, the good and the beautiful are the same, and must be investigated by one and the same process; and in like manner the base and the evil. And in the first rank we must place the beautiful, and consider it as the same with the good; from which immediately emanates intellect as beautiful. Next to this, we must consider the soul receiving its beauty from intellect, and every inferior beauty deriving its origin from the forming power of the soul, whether conversant in fair actions and offices, or sciences and arts. Lastly, bodies themselves participate of beauty from the soul, which, as something divine, and a portion of the beautiful itself, renders whatever it supervenes and subdues, beautiful as far as its natural capacity will admit. Let us, therefore, re-ascend to the good itself, which every soul desires; and in which it can alone find perfect repose. For if anyone shall become acquainted with this source of beauty he will then know what I say, and after what manner he is beautiful. Indeed, whatever is desirable is a kind of good, since to this desire tends. But they alone pursue true good, who rise to intelligible beauty, and so far only tend to good itself; as far as they lay aside the deformed vestments of matter, with which they become connected in their descent. Just as those who penetrate into the holy retreats of sacred mysteries, are first purified and then divest themselves of their garments, until someone by such a process, having dismissed everything foreign from the God, by himself alone, beholds the solitary principle of the universe, sincere, simple and pure, from which all things depend, and to whose transcendent perfections the eyes of all intelligent natures are directed, as the proper cause of being, life and intelligence. With what ardent love, with what strong desire will he who enjoys this transporting vision be inflamed while vehemently affecting to become one with this supreme beauty! For this it is ordained, that he who does not yet perceive him, yet desires him as good, but he who enjoys the vision is enraptured with his beauty, and is equally filled with admiration and delight. Hence, such a one is agitated with a salutary astonishment; is affected with the highest and truest love; derides vehement affections and inferior loves, and despises the beauty which he once approved. Such, too, is the condition of those who, on perceiving the forms of gods or daemons, no longer esteem the fairest of corporeal forms. What, then, must be the condition of that being, who beholds the beautiful itself?"

"What measures, then, shall we adopt? What machine employ, or what reason consult by means of which we may contemplate this ineffable beauty; a beauty abiding in the most divine sanctuary without ever proceeding from its sacred retreats lest it should be beheld by the profane and vulgar eye? We must enter deep into ourselves, and, leaving behind the objects of corporeal sight, no longer look back after any of the accustomed spectacles of sense. For, it is necessary that whoever beholds this beauty, should withdraw his view from the fairest corporeal forms; and, convinced that these are nothing more than images, vestiges and shadows of beauty, should eagerly soar to the fair original from which they are derived. For he who rushes to these lower beauties, as if grasping realities, when they are only like beautiful images appearing in water, will, doubtless, like him in the fable, by stretching after the shadow, sink into the lake and disappear. For, by thus embracing and adhering to corporeal forms, he is precipitated, not so much in his body as in his soul, into profound and horrid darkness; and thus blind, like those in the infernal regions, converses only with phantoms, deprived of the perception of what is real and true."

"The sensitive eye can never be able to survey, the orb of the sun, unless strongly endued with solar fire, and participating largely of the vivid ray. Everyone therefore must become divine, and of godlike beauty, before he can gaze upon a god and the beautiful itself. Thus proceeding in the right way of beauty he will first ascend into the region of intellect, contemplating every fair species, the beauty of which he will perceive to be no other than ideas themselves; for all things are beautiful by the supervening irradiations of these, because they are the offspring and essence of intellect. But that which is superior to these is no other than the fountain of good, everywhere widely diffusing around the streams of beauty, and hence in discourse called the beautiful itself because beauty is its immediate offspring. But if you accurately distinguish the intelligible objects you will call the beautiful the receptacle of ideas; but the good itself, which is superior, the fountain and principle of the beautiful; or, you may place the first beautiful and the good in the same principle, independent of the beauty which there subsists."

"Pleasure and distress, fear and courage, desire and aversion, where have these affections and experiences their seat? Clearly, either in the Soul alone, or in the Soul as employing the body, or in some third entity deriving from both. And for this third entity, again, there are two possible modes: it might be either a blend or a distinct form due to the blending."

"We may treat of the Soul as in the body — whether it be set above it or actually within it — since the association of the two constitutes the one thing called the living organism, the Animate. Now from this relation, from the Soul using the body as an instrument, it does not follow that the Soul must share the body's experiences: a man does not himself feel all the experiences of the tools with which he is working."

"All teems with symbol; the wise man is the man who in any one thing can read another."

"Withdraw into yourself and look. And if you do not find yourself beautiful yet, act as does the creator of a statue that is to be made beautiful: he cuts away here, he smoothes there, he makes this line lighter, this other purer. ... Cut away all that is excessive, straighten all that is crooked, bring light to all that is overcast, labor to make all one glow or beauty and never cease chiseling your statue, until there shall shine out on you from it the godlike splendor of virtue."

"Hence, as Narcissus, by catching at the shadow, plunged himself in the stream and disappeared, so he who is captivated by beautiful bodies, and does not depart from their embrace, is precipitated, not with his body, but with his soul, into a darkness profound and repugnant to intellect (the higher soul), through which, remaining blind both here and in Hades, he associates with shadows."

"When the soul has descended into generation (from its first divine condition) she partakes of evil, and is carried a great way into a state the opposite of her first purity and integrity, to be entirely merged in which, is nothing more than to fall into a dark mire. ...The soul dies as much as it is possible for the soul to die: and the death to her is, while baptized or immersed in the present body, to descend into matter, and be wholly subjected by it; and after departing thence to lie there til it shall arise and turn its face away from the abhorrent filth. This is what is meant by falling asleep in Hades, of those who have come there."

"[W]hen they write incantations, and utter them as to the stars, not only to [the bodies and] souls of these, but also to things superior to soul, what do they effect? They answer, charms, allurements, and persuasions, so that the stars hear the words addressed to them, and are drawn down; if any one of us knows how in a more artificial manner to utter these incantations, sounds, aspirations of the voice, and hissings, and such other particulars as in their writings are said to possess a magical power. ...They likewise pretend that they can expel disease. And if, indeed, they say that they effect this by temperance and an orderly mode of life, they speak rightly, and conformably to philosophers. But now when they assert that diseases are daemons, and that they are able to expel these by words, and proclaim that they possess this ability, they may appear to the multitude to be more venerable, who admire the powers of magicians; but they will not persuade intelligent men that diseases have not their causes either from labours, or satiety, or indigence, or putrefaction, and in short from mutations which either have an external or internal origin. This, however, is manifest from the cure of diseases. For disease is deduced downward, so as to pass away externally, either through a flux of the belly, or the operation of medicine. Disease, also, is cured by letting of blood and fasting. ...The disease ...[is] something different from the daemon. ...The manner, however, in which these things are asserted by the Gnostics, and on what account is evident; since for the sake of this, no less than of other things, we have mentioned these daemons. ...And this must every where be considered, that he who pursues our form of philosophy, will, besides all other goods, genuinely exhibit simple and venerable manners, in conjunction with the possession of wisdom, and will not endeavour to become insolent and proud; but will possess confidence accompanied with reason, much security and caution, and great circumspection."

"Πλάτων—Πλωτῖνος—Πλήθων"

"Plotinus devoted himself to the methodical task of transforming the direction of studies so that it corresponded to the greatness of the empire and its current problems. To this end, he started again from Plato and the interpretation of Plato; instead, he proposed Aristotle and his method as a subject of study and discussion. When Gallienus, who had recognised him as Caesar, rose to supreme power, he also had the authority and character of official teacher of the empire's philosophy. ('La vita e l'opera di Plotino, “'The Life and Work of Plotinus”', V, p. 37)"

"There were at least two avenues for originality open to Plotinus, even if it was not his intention to say fundamentally new things. The first was in trying to say what Plato meant on the basis of what he wrote or said or what others reported him to have said. This was the task of exploring the philosophical position that we happen to call "Platonism". The second was in defending Plato against those who, Plotinus thought, had misunderstood him and therefore unfairly criticized him. Plotinus found himself, especially as a teacher, taking up these two avenues. His originality must be sought for by following his path."

"The three basic principles of Plotinus' metaphysics are called by him "the One" (or, equivalently, "the Good"), Intellect, and Soul (see V 1; V 9.). These principles are both ultimate ontological realities and explanatory principles. Plotinus believed that they were recognized by Plato as such, as well as by the entire subsequent Platonic tradition. The One is the absolutely simple first principle of all. It is both "self-caused" and the cause of being for everything else in the universe. There are, according to Plotinus, various ways of showing the necessity of positing such a principle."

"The "concept" of the One is not, properly speaking, a concept at all, since it is never explicitly defined by Plotinus, yet it is nevertheless the foundation and grandest expression of his philosophy. Plotinus does make it clear that no words can do justice to the power of the One; even the name, "the One," is inadequate, for naming already implies discursive knowledge, and since discursive knowledge divides or separates its objects in order to make them intelligible, the One cannot be known through the process of discursive reasoning (Ennead VI.9.4). Knowledge of the One is achieved through the experience of its "power" (dunamis) and its nature, which is to provide a "foundation" (arkhe) and location (topos) for all existents (VI.9.6). The "power" of the One is not a power in the sense of physical or even mental action; the power of the One, as Plotinus speaks of it, is to be understood as the only adequate description of the "manifestation" of a supreme principle that, by its very nature, transcends all predication and discursive understanding."

"Less elusive than Plato's was the supra-rationality of his distant disciple, the Egyptian Plotinus (died 270), creator of Neo-Platonism. With him the supra-rational represented an élan, a reaching beyond the clearly seen or clearly known, to the Spirit itself. He had a disciple Porphyry, like himself a sage—and yet a different sage [whose] supra-rationalities hungered for many things from which his rational nature turned askance. But he has a disciple, Iamblicus by name, whose rational nature... prostitutes itself in the service of unreason."

"The synthetic genius of Plotinus enabled him to weave into his system valuable elements from Aristotle and the Stoics. But he was above all a Platonist. He presents the spiritual triad: the One, the Mind, the Soul. From the One comes the Mind, that is, the Nous, which embraces the totality of the knowable or intelligible, to wit, the Cosmos of Ideas. From that, come the Soul of the World and the souls of men. Matter, which is no-thing, gains form and partial reality when informed with soul. Plotinus's attitude toward knowledge of the concrete natural or historic fact, displays a transcendental indifference exceeding that of Plato. Perceptible facts with him are but half-real manifestations of the informing spirit. They were quite plastic, malleable, reducible. Moreover, thoughts of the evil of the multiple world of sense held for Plotinus and his followers a bitterness of ethical unreality which Plato was too great an Athenian to feel."

"Dualistic ethics which find in matter the principle of unreality or evil, diminish the human interest in physical fact. The ethics of Plotinus consisted in purification and detachment from things of sense. This is asceticism. And Plotinus was an ascetic, not through endeavor, but from contempt. He did not struggle to renounce the world, but despised it with the spontaneity of a sublimated temperament. He seemed like a man ashamed of being in the body, Porphyry says of him. Nor did he wish to cure any contemptible bodily ailments, or wash his wretched body."

"Plotinus's Absolute, the First or One, might not be grasped by reason. Yet to approach and contemplate it was the best for man. Life's crown was the ecstasy of the supra-rational and supra-intelligible vision of it. This Plotinean irrationality was lofty; but it was too transcendent, too difficult, and too unrelated to the human heart, to satisfy other men. ...his followers would bring it down to the level of their irrational tendencies. ...There was a tendency to contrast the spiritual and real with the manifold of material nonentity, and a cognate tendency to emphasize the opposition between the spiritual and good, and the material and evil, or between opposing spiritual principles. With less metaphysical people such opposition would take more entrancing shapes in the battles of gods and demons. Probably it would cause ascetic repression of the physical passions. ...within the schools of Neo-Platonism, in the generations after Plotinus... these tendencies flourished, beneath the shelter of his elastic principles. Here three kindred currents made a resistless stream: a transcendental, fact-repelling dialectic; unveiled recognition of the supreme virtue of supra-rational convictions and experiences; and an asceticism which condemned matter and abhorred the things of sense. What more was needed to close the faculties of observation, befool the reason, and destroy knowledge in the end?"

"Thus when Plotinus speaks of "the flight of the alone to the Alone," and the positivist or the empiricist asserts that these words are meaningless, he is right. Yet this does not import that the words are nonsense locutions, mere senseless noises which a makes like a cough or a sneeze though it is possible that this is what the positivist intends. If this were so, it would be impossible to explain why generations of men have quoted those famous words. The explanation is that the words evoke in us a measure of the same experience which the author of them had. Our experience may be but a dim reflection of what was in him bright and clear. Our spirits vibrate faintly in unison with the soul of the great mystic, as a tuning fork vibrates faintly in response to the sound of the clear bell. But it is our own spontaneous experience which is evoked; it is not his experience which is communicated to us. His words are as grappling irons let down into the depths of our subconsciousness, which draw our own inner experiences nearer to the conscious threshold."

"Better a century of tyranny than one day of chaos."

"God does not create pure evil. Rather, in everything that He creates is a wise purpose by virtue of what is good. However, there may be some evil in it for some people, and this is partial, relative evil. As for total evil or absolute evil, the Lord is exonerated of that."

"If God—exalted is He—is Creator of everything, He creates good and evil on account of the wise purpose that He has in that by virtue of which His action is good and perfect."

"Guidance is not attained except with knowledge and correct direction is not attained except with patience."

"This whole religion (of Islam) revolves around knowing the truth and acting by it, and action must be accompanied by patience."

"The more the servant loves his Master, the less will he love other objects and they will decrease in number The less the servant loves his Master, the more will he love other objects and they will increase in number."

"The jihad against the soul is the foundation for the Jihad against the disbelievers and hypocrites."

"The objective of asceticism is to leave all that harms the servants Hereafter and the objective of worship is to do all that will benefit his Hereafter."

"Sins are like chains and locks preventing their perpetrator from roaming the vast garden of tawhid and reaping the fruits of righteous actions."

"What can my enemies do to me? I have in my breast both my heaven and my garden. If I travel they are with me, never leaving me. Imprisonment for me is a chance to be alone with my Lord. To be killed is martyrdom and to be exiled from my land is a spiritual journey."

"A man married a maid-slave who bore him a child. Would that child be free or would he be an owned slave?" "Her child whom she bore from him would be the property of her master according to all the Imams (heads of the four Islamic schools of law) because the child follows the (status) of his mother in freedom or slavery. If the child is not of the race of Arabs, then he is definitely an owned slave according to the scholars, but the scholars disputed (his status) among themselves if he was from the Arabs - whether he must be enslaved or not because when A'isha (Muhammad's wife) had a maid-slave who was an Arab, Muhammad said to A'isha, `Set this maid free because she is from the children of Ishmael.'"

"The power of the Al-Sauds still rested on their alliance with clerics upholding the legacy of the holier-than-thou preacher Ibn Abdelwahhab. Born in 1703, Ibn Abdelwahhab had been inspired by the dogmatic teachings of a literalist, medieval theologian, Ahmad ibn Taymiyya, who belonged to the Hanbali school of jurisprudence, the strictest of the four Islamic schools. A complex character with a rich legacy who had lived at the time of the Crusades and sanctified war against the Christian invaders, Ibn Taymiyya would be quoted mostly for his edicts allowing war against a Muslim ruler in certain cases. He would inspire generations of activist and jihadist Salafists who ignored the nuances of his teachings."

"Ibn Abdelwahhab had taken Ibn Taymiyya’s pronouncements stripping Islam down to absolute monotheism and began to enforce them in Najd. He went further still by declaring war against anyone who didn’t follow his teachings—non-Muslims but also Muslims. The Najdi preacher had taken theology and turned it into a political and military mission. Ibn Abdelwahhab was so extreme that his own father and brother denounced him. He sent missives around the Arabian Peninsula and beyond to scholars and notables of the Muslim world, appealing to them to follow him and what he claimed was the true version of Islam. He was rejected and mocked in scathing responses coming from as far away as Tunisia, where the scholars of Al-Zaytuna, one of the oldest, most important centers of Islamic learning, undid his arguments one by one. The locals in his desert settlement accused him of heresy and tried to kill him."

"Ibn Taymiyya digested the “poison of philosophy” – yet, his brilliant mind turned the poison into honey. This very honey, extracted from the hive of his writings, can accordingly nourish a new era of modern Islamic philosophy. That Ibn Taymiyya himself, no doubt, would have taken umbrage at this sort of labeling of his work demonstrates how rich in irony the history of ideas can actually be!"

"He is a righteous man, famous among Maghrebians for religion and knowledge, and upon him the smelting of ascets and their tranquility"

"They adopted al-Matiti and al-Zawawi ***** as well as Ibn Sahl for every Zawi"

"The foundation of the Muslim League and Minto’s concessions had the effect of dividing the Hindus and Muslims into almost two hostile political camps. A remarkable example of this is afforded by a letter written about 1908 by Mr. Ziauddin Ahmad, later Vice- Chancellor of the Muslim University, Aligarh, to Mr. Abdulla Shuhrawardy, both of whom were then prosecuting their studies in Europe. Abdulla Shuhrawardy shared the national feelings which then characterized Indian students in Europe, and for this he was rebuked by Ziauddin in a letter from which we quote the following extract; “You know that we have a definite political policy at Aligarh, i.e. the policy of Sir Syed. I understand that Mr. Kirshna Varma has founded a society called ‘Indian Home Rule Society’ and: you are also one of its vice-presidents. Do you really believe that the Mohammedans will be profited if Home Rule be granted to India de lene. There is no doubt that this Home Rule is decidedly against the Aligarh policy...What I call the Aligarh policy is really the policy of all the Mohammedans generally—of the Mohammedans of Upper India particularly.” Mr. Asaf Ali wrote to Pandit Shyamji in September, 1909: “I am staying with some Muslim friends who do not like me to associate with nationalists; and, to save many unpleasant consequences, I do not want to irritate them unnecessarily.” Thus the Muslim antagonism to the Freedom Movement of India dates back to its beginning itself. (151ff)"

"Although a big part of the research on paraconsistent logics has so far been motivated by the paradoxes in naive set theory, and to developing alternative paraconsistent mathematics, I do not think that paraconsistent mathematics has real interest - at least not as long as we deal with truth in pure mathematics."

"For me the value of paraconsistent logics is as a potential instrument, to be used when there is a need to draw practical conclusions from an inconsistent body of "knowledge"."