Integers

"For some time past there has been going the rounds of the men about town the slang phrase "Twenty-three." The meaning attached to it is to "move on," "get out," "goody-bye, glad you are gone," "your move" and so on. To the initiated it is used with effect in a jocular manner. It has only a significance to local men and is not in vogue elsewhere. Such expressions often obtain a national use, as instanced by "rats!" "cheese it," etc., which were once in use throughout the length and breadth of the land. Such phrases originated, no one can say when. It is ventured that this expression originated with Charles Dickens in the Tale of Two Cities. Though the significance is distorted from its first use, it may be traced. The phrase "Twenty-three" is in a sentence in the close of that powerful novel. Sidney Carton, the hero of the novel, goes to the guillotine in place of Charles Darnay, the husband of the woman he loves. The time is during the French Revolution, when prisoners were guillotined by the hundred. The prisoners are beheaded according to their number. Twenty-two has gone and Sidney Carton answers to — Twenty-three. His career is ended and he passes from view."

"I first heard of the 23 Enigma from William S. Burroughs, author of Naked Lunch, Nova Express, etc. According to Burroughs, he had known a certain Captain Clark, around 1960 in Tangier, who once bragged that he had been sailing 23 years without an accident. That very day, Clark’s ship had an accident that killed him and everybody else aboard. Furthermore, while Burroughs was thinking about this crude example of the irony of the gods that evening, a bulletin on the radio announced the crash of an airliner in Florida, USA. The pilot was another Captain Clark and the flight was Flight 23."

"Today crowds gather around the Flatiron Building to admire its architecture and place in New York history, but back in the early part of the 20th century, men gathered there for a vastly different reason. As many New Yorkers know, the Flatiron sits at the intersection of Broadway and Fifth Avenue, directly across from Madison Park; the layout of the streets and the park, combined with the building’s placement, can create gusts of wind strong enough to lift women’s skirts. Back in an era when showing any part of one’s legs was risqué, men would gather on 23rd Street hoping to catch a glimpse of a woman’s ankle or maybe even a little more. … While it isn’t used heavily today, some say the phrase “23 skidoo” came from this phenomenon. Popular in the early part of the 20th century, getting the “23 skidoo” refers to either leaving an area quickly or being forced to leave. Apparently, the effect of the wind at this intersection was well known and crowds of men would gather in hopes of seeing some skin."

"WHERE DID "23" — THE MEANING OF WHICH IS GET OUT FIRST ORIGINATE?; There are a Lot of People Who Lay Claim to the Latest Slang Term of the Day. THE INTERPRETATION OF IT IS "SKIDDOO," OF COURSE But its Origin is Shrouded in Mystery, and it May be that "Police Gazette" Readers Can Throw Some Light on the Subject."

"One of the popular New York City myths is that the slang term "twenty-three skidoo" comes from the Flatiron Building at Twenty-Third Street and Broadway/Fifth Avenue. Tourist buses pass by this spot; they have to talk about something. The area has high winds, lifting women's skirts up. Allegedly, an Officer Kane told some naughty boys to "twenty-three skidoo" from the scene. Scram! Beat it! Go away! The problem here is that I've found articles about "twenty-three" in 1899. The Flatiron Building was completed in 1902. One theory is that "23" is the number of the last victim in the then-popular play version of Charles Dickens's novel A Tale of Two Cities, titled The Only Way. "Skidoo" probably comes from "skedaddle," a term made popular during the Civil War. I have several articles that credit the vaudeville actor Billy B. Van with combining the two slang terms into "twenty-three skidoo." No doubt, the slang phrase was popular. No doubt, it was used at 23rd Street."

"The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power.’"

"He sometimes spoke of "zero" as the symbol of the absolute (Nirguna Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words and which is completely beyond the reach of the human mind. According to Ramanujan the appropriate symbol was the number "zero" which is the absolute negation of all attributes."

"My confidence in our shared future is grounded in my respect for India’s treasured past—a civilization that has been shaping the world for thousands of years. Indians unlocked the intricacies of the human body and the vastness of our universe. And it is no exaggeration to say that our information age is rooted in Indian innovations—including the number zero."

"England has forty-two religions and only two sauces."

"The poor fatherless baby of eight months is now the utterly broken-hearted and crushed widow of forty-two."

"Rule forty-two. All persons more than a mile high to leave the court."

"Yet still to choose a brat like you, To haunt a man of forty-two, Was no great compliment!""

"There was one who was famed for the number of things He forgot when he entered the ship: His umbrella, his watch, all his jewels and rings, And the clothes he had bought for the trip.He had forty-two boxes, all carefully packed, With his name painted clearly on each: But, since he omitted to mention the fact, They were all left behind on the beach."

""Forty-two," said Deep Thought, with infinite majesty and calm."

"Legolas: Final count, forty-two. Gimli: Forty-two? Oh, that's not bad for a pointy-eared elvish princeling. Hmph! I myself am sitting pretty on forty-THREE."

"Variations of two earlier meters [is the variation] of a mātrā-vṛtta. For example, for [a meter of] three [morae], variations of two earlier meters, one and two, being mixed three happens. For [a meter] of four [morae], variations of meters of two morae [and] of three morae being mixed, five happens. For [a meter] of five [morae] variations of two earlier [meters] of three morae [and] of four morae. being mixed, eight is obtained. In this way, for [a meter] of six morae, [variations] of four morae [and] of five morae being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, [variations of a meter] of seven morae [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas."

"The sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."

"1, 2, 3, 5, 8, who do we decapitate? 13, 21, rinse, repeat, regret more fun 34, 55, shiny girls, collapse the hive 89, 144, Vermin, honey, we want more"

"Now everybody hop on the one, the sounds of the two It's the third eye vision, five-side dimension The eighth light, is gonna shine bright tonight"

"1, 1, 2, 3, 5, 8, 13, 21 Mathematics is the language of nature"

"It ["Lateralus"] was originally titled 9-8-7. For the time signatures. Then it turned out that 987 was the 17th step of the Fibonacci sequence (in which each integer is equal to the sum of the preceding two). So that was cool."

"Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique pariete circundatus, ut sciret, quot ex eo paria germinarentur in uno anno: cum natura eorum sit per singulum mensem aliud par germinare; et in secundo mense ab eorum natiuitate germinant. Quia suprascriptum par in primo mense germinat, duplicabis ipsum, erunt paria duo in uno mense. Ex quibus unum, scilicet primum, in secundo mense geminat; et sic sunt in secundo mense paria 3 ; ex quibus in uno mense duo pregnantur; et geminantur in tercio mense paria 2 coniculorum ; et sic sunt paria 5 in ipso mense; ex quibus in ipso pregnantur paria 3; et sunt in quarto mense paria 8; ex quibus paria 5 geminant alia paria 5: quibus additis cum parijs 8, faciunt paria 13 in quinto mense; ex quibus paria 5, que geminata fuerunt in ipso mense, non concipiunt in ipso mense, sed alia 8 paria pregnantur; et sic sunt in sexto mense paria 21; cum quibus additis parijs 13, que geminantur in septimo , erunt in ipso paria 34 ; cum quibus additis parijs 21, que geminantur in octauo mense, erunt in ipso paria 55; cum quibus additis parjis 34, que geminantur in nono mense, erunt in ipso paria 89; cum quibus additis rursum parijs 55, que geminantur in decimo mense 144; cum quibus additis rursum parijs 89, que geminantur in undecimo mense, erunt in ipso paria 233. Cum quibus etiam additis parijs 144 , que geminantur in ultimo mense, erunt paria 377; et tot paria peperit suprascriptum par in prefato loco in capite unius anni. Potes enim uidere in hac margine, qualiter hoc operati fuimus, scilicet quod iunximus primum numerum cum secundo, uidelicet 1 cum 2; et secundum cum tercio; et tercium cum quarto; et quartum cum quinto, et sic deinceps, donec iunximus decimum cum undecimo, uidelicet 144 cum 233; et habuimus suprascriptorum cuniculorum summam, uidelicet 377 ; et sic posses facere per ordinem de infinitis numeris mensibus."

"First person: He started with one, and adding one to itself, got two. Second person: Yeah, that sounds pretty simple. First person: Ah, but then he had one next to one next to two, so he said to himself, "I'll take the last two numbers in the list, add one next to the two"— Second person: The one next to the two. First person: —and that'll give him three. Second person: Oh. First person: And then the three next to the two would give him five— Second person: Five, yeah. First person: —the five next to the three would give him eight— Second person: Eight. First person: —and he kept playing with the numbers. Thirteen, twenty-one, thirty-four, fifty-five, eighty-nine— Second person: [laughs] Sounds like a waste of time. First person: Well, that's what I thought, y'know, that he didn't have anything better to do. But it turns out that the curiosity that had him playing with this kind of "adding up of numbers"—what's now remembered as the Fibonacci numbers—turns out to have a curious and very surprising relationship to botany and classical art. Second person: How so? First person: I don't lie, but there seems to be examples of Fibonacci numbers all over the place in nature."

"Make me, one, copy and paste Make me, one, copy and paste Make me, two, copy and paste Make me, Fibonacci Make me, three, copy and paste Make me, five, copy and paste Make me, eight, copy and paste Make me, Fibonacci"

"Black Then White are All I see In my infancy Red and yellow then came to be Reaching out to me Lets me seeThere is So Much More and Beckons me To look through to these Infinite possibilities As below, so above and beyond, I imagine Drawn outside the lines of reason Push the envelope Watch it bend"

"The title track "Lateralus" makes use of the "Fibonacci sequence". The Fibonacci numbers go 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and when plotted as squares, a "golden spiral" can be drawn. Fibonacci numbers are prevalent in nature: in pine cones, arrangement of leaves, the centre of a sunflower, and so on. The number of syllables in each line in "Lateralus" are all Fibonacci numbers. "Black (1) Then (1) White are (2) All I see (3) In my infancy (5) Red and yellow then came to be (8) Reaching out to me (5) Let's me see (3)". "Ride the spiral" most likely refers to the "golden spiral"."

"Things consisting of fewer Principles are more accurate, than those understood by Addition [of more principles], as Arithmetic is more accurate than Geometry. ...That Science is more accurate which consists of fewer Principles, than what is only to be understood by Addition, as Arithmetic is more accurate than Geometry. I say by Addition, as Unity is a Being understood without Position, but a Point is to be understood only by Position."

"Every quantity is recognized as quantity through the one, and that by which quantities are primarily known is the one itself; therefore the one is the source of number as number."

"More knowable than the number is the unit; for it is prior and the source of every number."

"We find also the Famous ', Mathematician to the Prince of Orange, having defined Number to be, That by which is explained the quantity of every Thing, he becomes so highly inflamed against those that will not have the Unit to be a Number, as to exclaim against Rhetoric, as if he were upon some solid Argument. True it is that he intermixes in his Discourses a question of some Importance, that is, whether a Unit be to Number, as a Point is to a Line. But here he should have made a distinction, to avoid the confusing together of two different things. To which end these two questions were to have been treated apart; whether a Unit be Number, and whether a Unit be to Number, as a Point is to a Line; and then to the first he should have said, that it was only a Dispute about a Word, and that an Unit was, or was not a Number, according to the Definition, which a Man would give to Number. That according to Euclid's Definition of Number; Number is a Multitude of Units assembled together: it was visible, that a Unit was no Number. But in regard this Definition of Euclid was arbitrary, and that it was lawful to give another Definition of Number, Number might be defined as Stevin defines it, according to which Definition a Unit is a Number; so that by what has been said, the first question is resolved, and there is nothing farther to be alleged against those that denied the Unit to be a Number, without a manifest begging of the question, as we may see by examining the pretended Demonstrations of Stevin. The first is, The Part is of the same Nature with the whole, The Unit is a Part of a Multitude of Units, Therefore the Unit is of the same Nature with a MuItitude of Units, and consequently of Number. This Argument is of no validity. For though the part were always of the same nature with the whole, it does not follow that it ought to have always the same name with the whole; nay it often... has not the same Name. A Soldier is part of an Army, and yet is no Army... a Half-Circle is no Circle... if we would we could not... give to Unit more than its name of Unit or part of Number. The Second Argument which Stevin produces is of no more force. If then the Unit were not a Number, Subtracting one out of three, the Number given would remain, which is absurd. But... to make it another Number than what was given, there needs no more than to subtract a Number from it, or a part of a Number, which is the Unit. Besides, if this Argument were good, we might prove in the same manner, that by taking a half Circle from a Circle given, the Circle given would remain, because no Circle is taken away. ... But the second Question, Whether an Unit be to Number, as a Point is to a Line, is a dispute concerning the thing? For it is absolutely false, that an Unit is to number as a point is to a Line. Since an Unit added to number makes it bigger, but a Line is not made bigger by the addition of a point. The Unit is a part of Number, but a Point is no part of a Line. An Unit being subtracted from a Number, the Number given does not remain; but a point being taken from a Line, the Line given remains. Thus doth Stevin frequently wrangle about the Definition of words, as when he perplexes himself to prove that Number is not a quantity discreet, that Proportion of Number is always Arithmetical, and not Geometrical, that the Root of what Number soever, is a Number, which shews us that he did not properly understand the definition of words, and that he mistook the definition of words, which were disputable, for the definition of things that were beyond all Controversy."

"What Science can be more accurate than Geometry? What Science can afford Principles more evident, more certain, yea I will add, more simple than Geometrical Axioms, or exercises a more strictly accurate Logic in drawing its Conclusions? But Aristotle and ' affirm that Unity (they had more rightly said Numbers) the Principle of Arithmetic, is more simple than a Point which is the Principle of Geometry, or rather of Magnitude. Because a Point implies Position, but Unity does not. A Point, says Aristotle, and Unity are not to be divided, as Quantity: Unity requires no Position, a Point does. But this Comparison of a Point in Geometry with Unity in Arithmetic is of all the most unsufferable, and derives the worst Consequences upon Mathematical Learning. For Unity answers really to some Part of every Magnitude, but not to a Point: Thus if a Line be divided into six equal Parts, as the whole Line answers to the Number six, so every sixth Part answers to Unity, but not to a Point which is no Part of this Right Line. A Point is rightly termed Indivisible, not Unity. (For how ex. gr. can \frac{2}{6} + \frac{4}{6} equal Unity, if Unity be indivisible, and incomposed, and represent a Point) but rather only Unity is properly divisible, and Numbers arise from the Division of Unity. A Geometrical Point is much better compared to a Cypher or Arithmetical Nothing, which is really the Bound of every Number, coming between it and the Numbers next following, but not as a Part. A Cypher being added to or taken from a Number does neither encrease nor diminish it; from it is taken the Beginning of Computation, while itself is not computed; and it bears a manifest Relation to the principal Properties of a Geometrical Point. Nor is that altogether unexceptionable, which is said of Position; for a Point taken universally is not less indeterminate, and void of Position, than Unity taken the same Way: But Unity taken particularly implies a definite Position, and all other particular Circumstances, as well as a particular Point. Lastly, the Accuracy of Arithmetic and Geometry is so far from being different that it is altogether the same, drawn from the same Principles, and employed about the same Things. I might here annex many Observations and Consequences drawn from hence; but Not to be too tedious and prolix, I judge it will appear plain enough to every one who duly weighs what I have suggested, that, in reality, Number (at least that treated of by Mathematicians) differs nothing from continued Magnitude it self, nor seems to have any other Properties (Composition, Division, Proportion, and the like) than either from, or in respect to it, as it represents, or supplies its Place; nor consequently that it is any Species of Quantity distinct from Magnitude, or the Object of any Science but Geometry (which is conversant about Magnitude in general): In sum, that Number includes in it every Consideration pertaining to Geometry. Therefore the Element Writer (whatsoever Ramus can object, who taunts him with that Name) did not unadvisedly, in inserting Arithmetical Speculations among the Elements of Geometry, nay rather he did great Service to the Mathematics, and merited highly in not permitting these Sciences to be separated from one another, as if they were separate in Nature, but assigning to Arithmetic a suitable Place in Geometry."

"[A]s the great extreme of dimension is sublime, so the last extreme of littleness is in the same measure sublime... when we attend to the infinite divisibility of matter, when we pursue animal life into these excessively small, and yet organized beings... when we push our discoveries yet downward... in tracing which the imagination is lost as well as the sense; we become amazed and confounded at the wonders of minuteness; nor can we distinguish in its effects this extreme of littleness from the vast itself. For division must be infinite as well as addition; because the idea of a perfect unity can no more be arrived at, than that of a complete whole, to which nothing can be added."

"I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to eternity there would still remain finite parts which were undivided. ... Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is... getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows that since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri]. ... There is no difficulty in the matter because unity is at once a square, a cube, a square of a square, and all the other powers [dignitā]; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider the two square numbers, 9 and 4; then 3 is the mean proportional between 9 and 1 [\frac{1}{3} = \frac{3}{9}]; while 2 is a mean proportional between 4 and 1 [\frac{1}{2} = \frac{2}{4}]; between 9 and 4 we have 6 as a mean proportional [\frac{4}{6} = \frac{6}{9}]. A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18 [\frac{8}{12} = \frac{18}{27}]; while between 1 and 8 we have 2 and 4 intervening [\frac{1}{2} = \frac{4}{8}]; and between 1 and 27 there lie 3 and 9 [\frac{1}{3} = \frac{9}{27}]. Therefore we conclude that unity is the only infinite number. These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common."

"When do centuries end?—at the termination of years marked '99 (as common sensibility suggests), or at the termination of years marked '00 (as the narrow logic of a peculiar system dictates)?... the source of all our infernal trouble about the ends of centuries may be laid at the doorstep of a sixth-century monk named , or (literally) Dennis the Short. ...Dennis neglected to begin time with year zero, thus discombobulating all our usual notions of counting. During the year that Jesus was one year old, the time system that supposedly started with his birth was two years old. (Babies are zero years old until their first birthday; modern time was already one year old at its inception.) The absence of a year zero also means that we cannot calculate algebraically (without making a correction) through the B.C.-A.D. transition. ...The problem of centuries starts from Dennis's unfortunate decision to start with year one, rather than year zero... logic and sensibility do not coincide, and since both have legitimate claims upon our decision, the great and recurring debate about century boundaries simply cannot be resolved. ...One might argue that humans, as creatures of reason, should be willing to subjugate sensibility for logic; but we are, just as much, creatures of feeling. And so the debate has progressed at every go-round."

"We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As apeiron in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself."

"Aristotle observes that the One is reasonably regarded as not being itself a number, because a measure is not the things measured, but the measure or the One is the beginning (or principle) of number. This doctrine may be of Pythagorean origin; has it; Euclid implies it when he says that a unit is that by virtue of which each of existing things is called one, while a number is 'the multitude made up of units'; and the statement was generally accepted. According to Iamblichus, (an ancient Pythagorean, probably not later than Plato's time) defined a unit as 'limiting quantity'... or, as we might say, 'limit of fewness', while some Pythagoreans called it 'the confine between number and parts', i.e. that which separates multiples and submultiples. Chrysippus (third century B.C.) called it 'multitude one',... a definition objected to by Iamblichus as a contradiction in terms, but important as an attempt to bring 1 into the conception of number."

"The first definition of number is attributed to Thales, who defined it as a collection of units... following the Egyptian view. The Pythagoreans 'made number out of one' [Aristotle, Metaph. A. 5, 986 a 20]; some of them called it 'a progression of multitude beginning from a unit and a regression ending in it' [, p. 18. 3-5]. Stobaeus credits Modoratus, a Neo-Pythagorean of the time of Nero, with this definition.) Eudoxus defined number as a 'determinate multitude'... has yet another definition, 'a flow of quantity made up of units'... Aristotle gives a number of definitions equivalent to one or other of those just mentioned, 'limited multitude', 'multitude (or combination) of units', 'multitude of indivisibles', 'several ones'... 'multitude measurable by one', 'multitude measured', and 'multitude of measures' (the measure being the unit)."

"Let us take as the basis of our consideration first of all a thought-thing 1 (one)."

"The observable efforts of Greek philosophy were generally directed toward the resolution of multiplicity into unity. Empedocles is reported to have said, "the universe is alternately in motion and at rest—in motion when love is making one out of many, or strife is making many out of one, and at rest in the intermediate periods of time." Even here, where two states are posited, the unifying impulse is obviously felt to be the more desirable. Hence it is very natural that the Pythagoreans should have considered the monad as the first principle from which the other numbers flow. Itself not a number, it is an essence rather than a being and is sometimes, like the duad, designated as a potential number, since the point, though not a plane figure, can originate plane figures. As first originator, the monad is good and God. It is both odd and even, male and female... It is the basis and creator of number... In short, it is always taken to represent all that is good and desirable and essential, indivisible and uncreated."

"Nicomachus gives three definitions of number. ...The third stream of quantity composed of units Philoponus explains as another attempt to distinguish the particular kind of quantum treated in Arithmetic. The Unit was conceived either mystically as an Idea whose "essence" passes in some way into concrete individuals and even into the Ideas to organize them, or spatially and temporally as the boundary of individuals. The former conception gave rise to fantastic speculations on the cosmic meaning of number, examples of which Nicomachus has... given us; the latter gave rise to the s..."

"Those numbers... independent of the particular things which happen to undergo counting—of what are these... ? To pose this question means to raise the problem of "scientific" arithmetic or logistic. ...we are no longer interested in the requirements of daily life ...now our concern is rather with understanding the very possibility of this activity, with understanding... that knowing is involved and that there must... be a corresponding being which possesses that permanence of condition which first makes it capable of being "known." But the soul's turning away from the things of daily life, the changing of the direction... the "conversion" and "turning about"... leads to a further question... What is required is an object which has a purely noetic character and which exhibits at the same time... the countable... This requirement is exactly fulfilled by the "pure" units, which are "nonsensual," accessible only to the understanding, indistinguishable from one another, and resistant to all participation. The "scientific" arithmetician and logistician deals with numbers of pure monads. And... Plato stresses emphatically that there is "no mean difference" between these and the ordinary numbers. ...Only a careful consideration of the fact... forces us into the further supposition that there must indeed be a special "nonsensual" material to which these numbers refer. The immense propaedeutic importance... within Platonic doctrine is immediately clear, for is not a continual effort made in this doctrine to exhibit as the true object of knowing that which is not accessible to the senses? Here we have indeed a "learning matter"... "capable of hauling [us] toward being". It forces the soul to study, by thought alone, the truth as it shows itself by itself. ...ability to count and to calculate presupposes the existence of "nonsensual" units. Thus an unlimited field of "pure" units presents itself to the view of the "scientific" arithemetician and logistician."

"And yet this dianoetic quarry, as it is brought in especially by the mathematicians, must first be handed over to the dialecticians for proper use (Euthydemus 290 C; Republic 531 C-534 E). Only dialectic can open up the realm of true being, can give ground for the powers of the ' and can reveal Being and the One and the Good as they are—beyond all time and all opposition—in themselves and in truth."

"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."

"To the Pythagoreans, numbers were both living entities and universal principles, permeating everything from the heavens to human ethics. In other words, numbers had two distinct, complementary aspects. On the one hand, they had a tangible physical existence; on the other, they were abstract prescriptions on which everything was founded. For instance, the monad (the number 1) was understood both as a generator of all other numbers, an entity as real as water, air, and fire that participated in the structure of the physical world, and as an idea—the metaphysical unity at the source of all creation."

"A theory of the world was gradually developed on the fundamental conception of the unity of all things. Beneath the changes which seem to occur... there must be... some unchanging and unchangeable essence. A hidden thread of unity must run through all phenomena. To find that unchangeable essence, to discover that secret binding unity, became the quest of alchemy, the art which rested on the conception of the one and the all."

"Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd."

"If simple unity could be adequately perceived by the sight or by any other sense, then, there would be nothing to attract the mind towards reality any more than in the case of the finger... But when it is combined with the perception of its opposite, and seems to involve the conception of plurality as much as unity, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks "What is absolute unity?" This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of reality."

"I come now to the capital work of Hilbert which he communicated to the Congress of Mathematicians at Heidelberg... of which...an English translation due to Halsted appeared in The Monist. ...the author's aim is analogous to that of Russell, but on many points he diverges from his predecessor. "But," he says, "on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number. "We fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite." ...what Hilbert says of the principles of logic in the usual exposition applies likewise to the logic of Russell. So for Russell logic is prior to arithmetic; for Hilbert they are 'simultaneous.' We shall find... other differences still greater... I prefer to follow step by step the development of Hilbert's thought... "Let us take as the basis of our consideration first of all a thought-thing 1 (one)." Notice that in so doing we in no wise imply the notion of number, because it is understood that 1 is here only a symbol and that we do not at all seek to know its meaning. "The taking of this thing together with itself respectively two, three or more times ..." Ah! this time it is no longer the same; if we introduce the words 'two,' 'three,' and above all 'more,' 'several,' we introduce the notion of number; and then the definition of finite whole number which we shall presently find, will come too late. Our author was too circumspect not to perceive this begging of the question. So at the end of his work he tries to proceed to a truly patching-up process. Hilbert then introduces two simple objects 1 and =, and and considers all the combinations of these two objects, all the combinations of their combinations, etc. It goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them. Afterwards he separates these combinations into two classes, the class of the existent and the class of the non-existent... entirely arbitrary. Every affirmative statement tells us that a certain combination belongs to the class of the existent; every negative statement tells us that a certain combination belongs to the class of the non-existent. Note now a difference of the highest importance. For Russell any object whatsoever, which he designates by x, is an object absolutely undetermined and about which he supposes nothing; for Hilbert it is one of the combinations formed with the symbols 1 and =; he could not conceive of the introduction of anything other than combinations of objects already defined. ... Russell is faithful to his point of view, which is that of comprehension. He starts from the general idea of being, and enriches it more and more while restricting it, by adding new qualities. Hilbert on the contrary recognizes as possible beings only combinations of objects already known; so that (looking at only one side of his thought) we might say he takes the viewpoint of extension."

"But as the Pythagoreans define a point to be unity having position, let us consider what they mean. That numbers, indeed, are more immaterial and more pure than magnitudes, and that the principle of numbers is more simple than the principle of magnitudes, is manifest to every one: but when they say that a point is unity endued with position, they appear to me to evince that unity and number subsist in opinion: I mean monadic number. On which account, every number, as the pentad and the heptad, is one in every soul, and not many; and they are destitute of figure and adventitious form. But a point openly presents itself in the phantasy, subsists, as it were, in place, and is material according to intelligible matter. Unity, therefore, has no position, so far as it is immaterial, and free from all interval and place: but a point has position, so far as it appears seated in the bosom of the phantasy, and has a material subsistence. But unity is still more simple than a point, on account of the community of principles. Since a point exceeds unity according to position; but appositions in incorporeals produce diminutions of those natures, by which the appositions are received."

"But for the present we desire to contemplate, if possible, the nature of the pure and true one, which is not one from another, but from itself alone. It is therefore here requisite, to transfer ourselves on all sides to one itself, without adding any thing to its nature, and to acquiesce entirely in its contemplation; being careful lest we should wander from him in the least, and fall from one into two. But if we are less cautious we shall contemplate two, nor in the two possess the one itself; for they are both posterior to unity. And one will not suffer itself to be numerated with another, nor indeed to be numbered at all: for it is a measure free from all mensuration. Nor is it equal to any others, so as to agree with them in any particular, or it would inherit something in common with its connumerated natures; and thus this common something, would be superior to one though this is utterly impossible. Hence neither essential number, nor number posterior to this, which properly pertains to quantity can be predicated of one: not essential number whose essence always consists in intellection; nor that which regards quantity, since it embraces unity, together with other things different from one. For the nature pertaining to number which is inherent in quantity, imitating the nature essential to prior numbers, and looking back upon true unity, procures its own essence neither dispersing nor dividing unity, but while it becomes the duad, the one remains prior to the duad, and is different from both the unities comprehended by the duad, and from each apart. For why should the duad be unity itself? Or one unity of the duad rather than another, be one itself? If then neither both together, nor each apart is unity itself, certainly unity which is the origin of all number, is different from all these; and while it truly abides, seems after a manner not to abide. But how are those unities different from the one? And how is the duad in a certain respect one? And again, is it the same one, which is preserved in the comprehension of each unity? Perhaps it must be said that both unities, participate of the first unity, but are different from that which they participate: and that the duad so far as it is a certain one participates of one itself, yet not every where after the same manner: for an army, and a house are not similarly one; since these when compared with continued quantity, are not one, either with respect to essence, or quantity. Are then the unities in the pentad, differently related to one, from those in the decad? But is the one contained in the pentad, the same with the one in the decad? Perhaps also if the whole of a small ship, is compared with the whole of a large one, a city to a city, and an army to an army, there will be in these the same one. But if not in the first instance, neither in these. However, if any farther doubts remain, we must leave them to a subsequent discussion."

"But let us return to unity itself, asserting that it always remains the same, though all things flow from it as their inexhaustible fountain. In numbers, indeed, while unity abides in the simplicitly of its essence, number producing another is generated according to this abiding one. But the one which is above beings, much more abides in ineffable station. But while it abides, another does not produce beings, according to the nature of one: for it is sufficient of itself to the generation of beings. But as in numbers the form of the first monad is preserved in all numbers, in the first and second degree while each of the following numbers do not equally participate of unity; so in the order of things, every nature subordinate to the first, contains something of the first, as it were his vestige or form in its essence. And in numbers, indeed, the participation of unity produces their quantity. But here the vestige of one gives essence to all the series of divine numbers, so that being itself, is as were the footstep of ineffable unity."

"Arithmetic is indeed more accurate than Geometry, for its Principles respect Simplicity. Unity implies no Position, a Point does, and a Point requiring Position is the Principle of Geometry, Unity of Arithmetic."

"Since each number is with respect to its own kind one and without parts but with respect to its own material, as it were, divisible into parts, though not with respect to all of the material either; but rather what is ultimate [in it, i.e., the unit] is without parts even in the material, and in this ultimate thing [counting or calculation and, above all, partitioning] comes to a stop."

"God, creating the universe, neither made it perfectly like Himself, nor perfectly unlike, for He, being One, has made the world as not one, from the diverse multiplicity of its innumerable parts, ordaining, nevertheless, that they should collect into a certain unity by their exact contiguity. The upper world has no connexion with this subject; the lower, and elementary world, owes this contiguity to the weight divinely impressed on its parts, aided by the subtle fluidity of some of its simple bodies. It is by this quality, with which the matter of the four elements is more or less invested, that they are separated from one another, and each transported to its proper place, as the generation of compounds, and the beauty of the universe requires."

"In geometry we begin with the point, which is indimensional. This is the beginning of the first dimensional form, the line, and by movement the point generates the line. Now Nicomachus had a similar idea of the nature of multitude and number; they form a series, as it were a moving stream, which proceeds out of unity, the monad. Just as the point is not part of the line (for it is indimensional, and the line is defined as that which has one dimension), but is potentially a line, so the monad is not a part of multitude nor of number, though it is the beginning of both, and potentially both. The monad is unity, absence of multitude, potentiality; out of it the dyad first separates itself and 'goes forward' and then in succession follow the other numbers."

"All of the sons of Adam are part of one single body, They are of the same essence. When time afflicts us with pain In one part of that body All the other parts feel it too. If you fail to feel the pain of others You do not deserve the name of man."

"We were so foolish, my friend, before you [] said what you did, that we had an opinion about me and you that each of us is one, but that we would not both be one (which is what each of us would be) because we are not one but two. But now, we have been instructed by you that if two is what we both are, two is what each of us must be as well... Then its not entirely necessary, as you said it was a moment ago, that whatever is true of both is also true of each, and that whatever is true of each is also true of both."

"I cannot satisfy myself that, when one is added to one, the one to which the addition is made becomes two, or that the two units added together make two by reason of the addition. I cannot understand how, when separated from the other, each of them was one and not two, and now, when they are brought together the mere juxtaposition or meeting of them should be the cause of their becoming two..."

"Number is of two kinds, the Intellectuall (or immateriall) and the Scientiall. The Intellectual is that eternall substance of number, which Pythagoras and his Discourse concerning the Gods asserted to be the principle most providentiall of all Heaven and Earth, and the nature which is betwixt them. Moreover, it is the root of divine Beings, and of Gods, and of Dœmons. This is that which is termed the principle, fountain, and root of all things, and defined it to be that which before all things exists in the divine mind; from which and out of which all things are digested into order, and remain numbred by an indissoluble series. For all things which are ordered in the world by nature according to an artificiall course in part and in whole appear to be distinguished and adorn'd by Providence and All-creating Mind, according to Number; the exemplar being established by applying (as the reason of the principle before the impression of things) the number præexistent in the Intellect of God, maker of the world. This only in intellectual, & wholly immaterial, really a substance according to which as being the most exact artificiall reason, all things are perfected, Time, Heaven, Motion, the Stars, and their various revolutions."

"Sciential Number is that which Pythagoras defines as the extension and production into act of the seminall reasons which are in the Monad, or a heap of Monads, or a progression of multitude beginning from Monad, and a regression ending in Monad. ... They make a difference between the Monad and One, conceiving the Monad to be that which exists in intellectualls; One, in numbers [or as Moderatus expresseth it, Monad among numbers, One amongst things numbred, one being divisible into infinite; thus Numbers and things numbred differ, as incorporealls and bodies] in like manner Two is amongst numbers. The Duad is indeterminate; Monad is taken according to equality and measure, Duad according to excess and defect: mean and measure cannot admit more and lesse, but excesse and defect (seeing that they proceed to infinite) admit it; therefore they call the Duad indeterminate, holding Number to be infinite, not that number which is separate and incorporeall, but that which is not separate from sensible things."

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

"In the tenth book Euclid deals with certain irrational magnitudes; and since the Greeks possessed no symbolism for surds, he was forced to adopt a geometrical representation. Propositions 1 to 21 deal generally with incommensurable magnitudes. The rest of the book, namely, propostions 22 to 117, is devoted to the discussion of every possible variety of lines which can be represented by \sqrt{(\sqrt{a} \pm \sqrt{b})}, where a and b denote commensurable lines. There are twenty-five species of such lines, and that Euclid could detect and classify them all is in the opinion of so competent an authority as Nesselmann the most striking illustration of his genius. No further advance in the theory of incommensurable magnitudes was made until the subject was taken up by Leonardo and Cardan after the interval of more than a thousand years. In the last proposition of the tenth book [prop. 117] the side and diagonal of a square are proved to be incommensurable. The proof is so short and easy that I may quote it. If possible let the side be to the diagonal in a commensurable ratio, namely, that of two integers, a and b. Suppose this ratio reduced to its lowest terms so that a and b have no common divisor other than unity, that is, they are prime to one another. Then (by Euc. I, 47) b^2 = 2a^2; therefore b^2 is an even number; therefore b is an even number; hence, since a is prime to b, a must be an odd number. Again, since it has been shown that b is an even number, b may be represented by 2n; therefore (2n)^2 = 2a^2; therefore a^2 = (2n)^2; therefore a^2 is an even number; therefore a is an even number. Thus the same number a must be both odd and even, which is absurd; therefore the side and the diagonal are incommensurable. Hankel believes that this proof is due to Pythagoras, and this is not unlikely. This proposition is also proved in another way in Euc. X, 9, and for this and other reasons it is now usually believed to be an interpolation by some commentator on the Elements."

"Fermat's Last Theorem is to the effect that no integral values of x, y, z can be found to satisfy the equation xn+yn=zn if n is an integer greater than 2. ...It is possible that Fermat made some... erroneous supposition, though it is perhaps more probable that he discovered a rigorous demonstration. At any rate he asserts definitely that he had a valid proof—demonstratio mirabilis sane—and the fact that no theorem on the subject which he stated he had proved has been subsequently shown to be false must weigh strongly in his favour; the more so because in making the one incorrect statement in his writings (namely, that about binary powers) he added that he could not obtain a satisfactory demonstration of it. … [I]t took more than a century before some of the simpler results which Fermat had enunciated were proved, and thus it is not surprising that a proof of the theorem which he succeeded in establishing only towards the close of his life should involve great difficulties. ...I venture however to add my private suspicion that continued fractions played a not unimportant part in his researches, and as strengthening this conjecture I may note that some of his more recondite results—such as the theorem that a prime of the form 4n+1 is expressible as the sum of two squares—may be established with comparative ease by properties of such fractions."

"Prime numbers belong to the exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoys simple, elegant description, yet lead to extreme—one might say unthinkable—complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times... vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes..."

"In... 1859 Bernhard Riemann... presented a paper to the [Berlin] Academy... "On the Number of Prime Numbers Less Than a Given Quanitity." ...Riemann tackled the problem with the most sophisticated mathematics of his time... inventing for his purposes a mathematical object of great power and subtlety. ...[H]e made a guess about that object, and then remarked:One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective...That ... guess lay almost unnoticed for decades. Then... gradually seized... imaginations... until it attained the status of an overwhelming obsession. ...The Riemann Hypothesis... remained an obsession all through the twentieth century and remains one today, having resisted every attempt at proof and disproof. [It is] now the great white whale of mathematical research."

"A prime number is one (which is) measured by a unit alone. Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος."

"(a + b) \times (a - b) = a^2 - b^2...The difference of two square numbers is always a product, and divisible both by the sum and by the difference of the roots of those two squares; consequently the difference of two squares can never be a prime number."

"Leonhard Euler stated that 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 would never penetrate."

"And this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long. Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long."

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated."

"Hodge cohomology, algebraic de Rham cohomology, crystalline cohomology, the étale ℓ-adic cohomology theories for each prime number ℓ ... A strategy to encapsulate all the different cohomology theories in algebraic geometry was formulated initially by Alexandre Grothendieck, who is responsible for setting up much of this marvelous cohomological machinery in the first place. Grothendieck sought a single theory that is cohomological in nature that acts as a gateway between algebraic geometry and the assortment of special cohomological theories, such as the ones listed above—that acts as the motive behind all this cohomological apparatus."

"It has the very commendable aim of contributing towards stressing the cultural side of mathematics. ...there appears the widespread interchange of the definitions of excessive and defective numbers. ...it is stated that Euclid contended that every perfect number is of the form 2n-1(2n -1). It is true that Euclid proved that such numbers are perfect whenever 2n - 1 is a prime number but there seems to be no evidence to support the statement that he contended that no other such numbers exist. ...it is stated that the arithmetization of mathematics began with Weierstrass in the sixties of the last century. The fact that this movement is much older was recently emphasized by H. Wieleitner... it is stated that the arithmos of Diophantus and the res of Fibonacci meant whole numbers, and... we find the statement that in the pre-Vieta period they were committed to natural numbers as the exclusive field for all arithmetic operations. On the contrary, operations with common fractions appear on some of the most ancient mathematical records."

"The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics. A striking example is that of the distribution of prime numbers. The solution of this problem lies in finding a general formula which tells us the number of primes that lie in any given numerical interval. ...Edmund Landau ...wrote two large volumes analyzing this problem without solving it, using the most advanced mathematics known at the time. Even in the elementary aspects of mathematics we are thus dealing with complex topics which make great demands on our mathematical skills."

"Although I firmly believe that there is no such thing as a stupid question, there can indeed be stupid answers. 42 is an example. Not only is this a poor ripoff of Doug Adams' Hitchhiker's Guide, but it isn't even a prime number. Everyone surely knows that numerical answers to profound questions are always prime. (The correct answer is 37.)"

"There is another lesser-known "quote" of Erdos that I know first-hand he did not say, since I made it up! ... I gave a talk about Erdos and number theory, and I tried to explain how marvelous the Erdos--Kac theorem is... [Overhead Slide] Einstein: "God does not play dice with the universe." I then said orally: "I would like to think that Erdos and Kac replied..." [Overhead Slide] Erdos and Kac: Maybe so, but something is going on with the primes. ...Somehow, the San Diego newspaper picked this up the next day, and attributed it as a real quote of Erdos."

"I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras... I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as "startling". ...Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request that you go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly..."

"The present work is inspired by Edmund Landau's famous book, Handbuch der Lehre von der Verteilung der Primzahlen, where he posed two extremal questions on cosine polynomials and deduced various estimates on the distribution of primes using known estimates of the extremal quantities. Although since then better theoretical results are available for the error term of the prime number formula, Landau's method is still the best in finding explicit bounds. In particular, Rosser and Schonfeld used the method in their work "Approximate formulas for some functions of prime numbers"."

"I would teach the world how the Greeks proved, more than 2,000 years ago, that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics – when humankind realised that, by pure thought alone, it could prove eternal truths of the universe. Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers."

"We don't naturally have some magic formula to try and find these prime numbers. In fact trying to find where the next prime number is represents one of the biggest mysteries in the whole of mathemaics. ...The challenge if the primes is somehow of the ultimate puzzle in the whole of mathematics."

"The Elements contains thirteen books by Euclid, and two, of which it is supposed that Hypsicles and Damascius are the authors. ...The seventh, eighth, ninth books are on the theory of numbers, or on arithmetic. In the ninth book is found the proof to the theorem that the number of primes is infinite."

", eleven years younger than Archimedes, was a native of Cyrene. He... measured the obliquity of the ecliptic and invented a device for finding prime numbers. ...In his old age he lost his eyesight, and on that account is said to have committed suicide by voluntary starvation."

"Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation."

"Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. ...[I]n the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony... would explain why outwardly the primes look so chaotic. The metamorphosis... where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there."

"Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians."

"Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist."

"The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, You have no idea how much poetry there is in a table of logarithms.'"

"Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built."

"In the spring of 1997, Connes went to Princeton to explain his new ideas to the big guns: Bombieri, Selberg and Sarnak. Princeton was still the undisputed Mecca of the Riemann Hypothesis... Selberg had become godfather to the problem... a man who had spent half a century doing battle with the primes. Sarnak... [had] recently joined forces with ... one of the undisputed masters of the mathematics developed by Weil and Grothendieck. Together they had proved that the strange statistics of random drums that we believe describe the zeros in Riemann's landscape are definitely present in the landscapes considered by Weil and Grothendieck. ...It was Katz who, some years before, had found the mistake in Wiles's first erroneous proof of Fermat's Last Theorem. And finally there was Bombieri... the undisputed master of the Riemann Hypothesis. He had earned his for the most significant result to date about the error between the true number of primes and Gauss's guess - a proof of... the 'Riemann Hypothesis on average'. ...Bombieri, like Katz, has a fine eye for detail."

"The primes are the atoms of arithmetic, the indivisible building blocks from which all other numbers are constructed. In their chaotic distribution lies a profound order, a secret language waiting to be deciphered. To study them is to chase the fundamental rhythm of mathematics itself."

"My money 360, you only 180. Line complete."

"And said unto , Build me here seven altars, and prepare me here seven oxen and seven rams."

"The seven liberal arts do not adequately divide ; but, as says, seven arts are grouped together (leaving out certain other ones), because those who wanted to learn philosophy were first instructed in them. And the reason why they are divided into the and is that "they are as it were paths (viae) introducing the quick mind to the secrets of philosophy.""

"The tree that thou sawest, which grew, and was strong, whose height reached unto the heaven, and the sight thereof to all the earth; Whose leaves were fair, and the fruit thereof much, and in it was meat for all; under which the beasts of the field dwelt, and upon whose branches the fowls of the heaven had their habitation: It is thou, O king, that art grown and become strong: for thy greatness is grown, and reacheth unto heaven, and thy dominion to the end of the earth. And whereas the king saw a watcher and an holy one coming down from heaven, and saying, Hew the tree down, and destroy it; yet leave the stump of the roots thereof in the earth, even with a band of iron and brass, in the tender grass of the field; and let it be wet with the dew of heaven, and let his portion be with the beasts of the field, till seven times pass over him; This is the interpretation, O king, and this is the decree of the most High, which is come upon my lord the king: That they shall drive thee from men, and thy dwelling shall be with the beasts of the field, and they shall make thee to eat grass as oxen, and they shall wet thee with the dew of heaven, and seven times shall pass over thee, till thou know that the most High ruleth in the kingdom of men, and giveth it to whomsoever he will. And whereas they commanded to leave the stump of the tree roots; thy kingdom shall be sure unto thee, after that thou shalt have known that the heavens do rule. Wherefore, O king, let my counsel be acceptable unto thee, and break off thy sins by righteousness, and thine iniquities by shewing mercy to the poor; if it may be a lengthening of thy ."

"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..."

"He (Gilgamesh) crossed the first mountain with him (), but the cedars were not revealed to his heart. The second mountain, the third mountain, the fourth mountain, the fifth mountain, (and) the sixth mountain. When he was crossing the seventh mountain, the cedars were revealed to his heart."

"ARTS, Liberal, or Seven Liberal. The distinction between the liberal arts and the practical arts on the one hand, and philosophy on the other, originates in Greek education and philosophy. In the Republic (Bk. xi.) of Plato, and the Politics (viii. 1) of Aristotle, the 'liberal arts' are those subjects that are suitable for the development of intellectual and moral excellence, as distinguished from those that are merely useful or practical. The distinction was always made, by the Greek theorists, between music, literature in the form of grammar and rhetoric, and the mathematical studies, and that higher aspect of the liberal discipline termed philosophy. Philosophy was sometimes called the liberal art par excellence."

"I have now established... that the human encephalos does not increase after the age of seven, at highest. This has been done, by measuring the heads of the same young persons, from infancy to adolescence and maturity; for the slight increase in the size of the head, after seven (or six) is exhausted by the development to be allowed in the bones, muscles, integuments, and hair."

"Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts."

"Writers differ with respect to the apophthegms of the Seven Sages, attributing the same one to various authors."

"A person who circumambulates this House (the Ka’bah) seven times and performs the two Rak’at Salat (of Tawaaf) in the best form possible will have his sins forgiven."

"Peter came to Jesus and asked, "Lord, how many times shall I forgive my brother when he sins against me? Up to seven times?" Jesus answered, "I tell you, not seven times, but seventy-seven times (or seventy times seven).""

"The folly of Interpreters has been, to foretell times and things by this Prophecy, as if God designed to make them Prophets. By this rashness they have not only exposed themselves, but brought the Prophecy also into contempt. The design of God was much otherwise. He gave this and the Prophecies of the Old Testament, not to gratify men's curiosities by enabling them to foreknow things, but that after they were fulfilled they might be interpreted by the event, and his own Providence, not the Interpreters, be then manifested thereby to the world. For the event of things predicted many ages before, will then be a convincing argument that the world is governed by providence. For, as the few and obscure Prophecies concerning Christ’s first coming were for setting up the Christian religion, which all nations have since corrupted; so the many and clear Prophecies concerning the things to be done at Christ’s second coming, are not only for predicting but also for effecting a recovery and re-establishment of the long-lost truth, and setting up a kingdom wherein dwells righteousness. The event will prove the Apocalypse; and this Prophecy, thus proved and understood, will open the old Prophets, and all together will make known the true religion, and establish it. For he that will understand the old Prophets, must begin with this; but the time is not yet come for understanding them perfectly, because the main revolution predicted in them is not yet come to pass. In the days of the voice of the seventh Angel, when he shall begin to sound, the mystery of God shall be finished, as he hath declared to his servants the Prophets: and then the kingdoms of this world shall become the kingdom of our Lord and his Christ, and he shall reign for ever, Apoc. x. 7. xi. 15. There is already so much of the Prophecy fulfilled, that as many as will take pains in this study, may see sufficient instances of God’s providence: but then the signal revolutions predicted by all the holy Prophets, will at once both turn men’s eyes upon considering the predictions, and plainly interpret them. Till then we must content ourselves with interpreting what hath been already fulfilled. Amongst the Interpreters of the last age there to scarce one of note who hath not made some discovery worth knowing; and thence I seem to gather that God is about opening these mysteries. The success of others put me upon considering it; and if I have done any thing which may be useful to following writers, I have my design."

"There are Seven Seals to be opened, that is to say, Seven mysteries to know, and Seven difficulties to overcome, Seven trumpets to sound, and Seven cups to empty. The Apocalypse is, to those who receive the nineteenth degree, the of that Sublime Faith which aspires to God alone, and despises all the pomps and works of Lucifer. ...[T]raditions are full of Divine Revelations and Inspirations: and Inspiration is not of one Age nor of one Creed. Plato and Philo, also, were inspired."

"Seven scores, seven scores, seven hundreds of saints, And seven thousands and seven ten scores, November a number implored, Though martyrs good they came."

"Strange, indeed, that you should not have suspected that your universe and its contents were only dreams, visions, fiction! Strange, because they are so frankly and hysterically insane—like all dreams: a God who could make good children as easily as bad, yet preferred to make bad ones; who could have made every one of them happy, yet never made a single happy one; who made them prize their bitter life, yet stingily cut it short; who gave his angels eternal happiness unearned, yet required his other children to earn it; who gave his angels painless lives, yet cursed his other children with biting miseries and maladies of mind and body; who mouths justice, and invented hell—mouths mercy, and invented hell—mouths Golden Rules and forgiveness multiplied by seventy times seven, and invented hell; who mouths morals to other people, and has none himself; who frowns upon crimes, yet commits them all; who created man without invitation, then tries to shuffle the responsibility for man's acts upon man, instead of honorably placing it where it belongs, upon himself; and finally, with altogether divine obtuseness, invites his poor abused slave to worship him!"

"The expression artes liberales, chiefly used during the Middle Ages, does not mean arts as we understand the word at this present day, but those branches of knowledge which were taught in the schools of that time. They are called liberal (Latin liber, free), because they serve the purpose of training the free man, in contrast with the artes illiberales, which are pursued for economic purposes; their aim is to prepare the student not for gaining a livelihood, but for the pursuit of science in the strict sense of the term, i.e. the combination of philosophy and theology known as . They are seven in number and may be arranged in two groups, the first embracing grammar, rhetoric, and dialectic, in other words, the sciences of language, of oratory, and of logic, better known as the artes sermocinales, or language studies; the second group comprises arithmetic, geometry, astronomy, and music, i.e. the mathematico-physical disciplines, known as the artes reales, or physicae."

"[W]hile these patterns, the constellations, remained unchanging over time, there were seven objects, or ‘heavenly bodies’, that seemed to move across the skies with a life of their own. They were given the name ‘planet’... ‘wanderer’... These... were the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn... The number seven has long been held to have a certain mystical significance. There are seven days of the week reflecting the seven days of creation in the Bible. The Seven Deadly Sins are balanced by the Seven Heavenly Virtues. In Islam there are seven levels in heaven and the same number in hell. Rome was founded upon Seven Hills. It has been said that Isaac Newton divided the rainbow into seven colours in order to imitate the seven notes in a musical scale. Over time, each of the seven heavenly bodies came to be associated with a particular day of the week and with one of the gods from ancient mythology..."

"Give me a child till he is seven years old, and I will make him what no one will unmake. ...Give me a child until he is 7 and I will show you the man."

"In regard to Philolaus, we are told... that he derived geometrical determinations (the point, the line, the surface, the solid) from the first four numbers, so he derived physical qualities from five, the soul from six; reason, health, and light, from seven; love, friendship, prudence, and inventive faculty from eight. Herein (apart from the number schematism) is contained the thought that things represent a graduated scale of increasing perfection; but we hear nothing of any attempt to prove this in detail, or to seek out the characteristics proper to each particular region."

"I John... was in the isle that is called , for the word of God, and for the testimony of Jesus Christ. I was in the Spirit on the Lord's day, and heard behind me a great voice, as of a trumpet, Saying, I am Alpha and Omega, the first and the last: and, What thou seest, write in a book, and send it unto the seven churches... And I turned to see the voice that spake with me. And being turned, I saw seven golden candlesticks; And in the midst of the seven candlesticks one like unto the Son of man... His head and his hairs were white like wool, as white as snow; and his eyes were as a flame of fire... and his voice as the sound of many waters. And he had in his right hand seven stars: and out of his mouth went a sharp two-edged sword: and his countenance was as the sun shineth in his strength. And when I saw him, I fell at his feet as dead. And he laid his right hand upon me, saying... I am he that liveth, and was dead; and, behold, I am alive for evermore, Amen; and have the keys of hell and of death. Write the things which thou hast seen... The mystery of the seven stars which thou sawest in my right hand, and the seven golden candlesticks. The seven stars are the angels of the seven churches: and the seven candlesticks which thou sawest are the seven churches."

"And I saw another mighty angel come down from heaven, clothed with a cloud: and a rainbow was upon his head, and his face was as it were the sun, and his feet as pillars of fire: And he had in his hand a little book open: and he set his right foot upon the sea, and his left foot on the earth, And cried with a loud voice, as when a lion roareth: and when he had cried, seven thunders uttered their voices. And when the seven thunders had uttered their voices, I was about to write: and I heard a voice from heaven saying unto me, Seal up those things which the seven thunders uttered, and write them not. And the angel which I saw stand upon the sea and upon the earth lifted up his hand to heaven, And sware by him that liveth for ever and ever, who created heaven, and... the earth... and the sea, and the things which are therein, that there should be time no longer: But in the days of the voice of the seventh angel, when he shall begin to sound, the mystery of God should be finished, as he hath declared to his servants the prophets."

"The number 108 is actually the average distance that the sun is in terms of its own diameter from the earth; likewise, it is also the average distance that the moon is in terms of its own diameter from the earth. It is owing to this marvelous coincidence that the angular size of the sun and the moon, viewed from the earth, is more or less identical. It is easy to compute this number. The angular measurement of the sun can be obtained quite easily during an eclipse. The angular measurement of the moon can be made on any clear full moon night. An easy check on this measurement would be to make a person hold a pole at a distance that is exactly 108 times its length and confirm that the angular measurement is the same. Nevertheless, the computation of this number would require careful observations. Note that 108 is an average and due to the ellipticity of the orbits of the earth and the moon the distances vary about 2 to 3 percent with the seasons. It is likely, there- fore, that observations did not lead to the precise number 108, but it was chosen as the true value of the distance since it is equal to 27 x 4, because of the mapping of the sky into 27 naksatras. The diameter of the sun is roughly 108 times the diameter of the earth, but it is unlikely that the Indians knew this fact."

"108 angulas make a dhanus, a measure [used] for roads and city-walls . . ."