People From New York City

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties... of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent... and Newton, introduced transformations other than projection and section."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"One of the first algebraists to accept negative numbers was ... who occasionally placed a negative number by itself on one side of an equation. But he did not accept negative roots. ... gave clear definitions for negative numbers. Stevin used positive and negative coefficients in equations and also accepted negative roots. In his L'Invention nouvelle en l'algèbre (1629), ... placed negative numbers on a par with positive numbers and gave both roots of a quadratic equation, even when both were negative. Both Girard and Harriot used the minus sign for the operation of subtraction and for negative numbers."

"The Hindus introduced negative numbers... The first known use is about 628; he also states the rules for the four operations with negative numbers. Bhāskara points out that the square root of a positive number is twofold, positive and negative. He brings up the matter of the square root of a negative number but says that there is no square root because a negative number is not a square. No definitions, axioms, or theorems are given. The Hindus did not unreservedly accept negative numbers. Even Bhāskara, while giving 50 and -5 as two solutions of a problem, says, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions." However, negative numbers gained acceptance slowly."

"As for negative numbers... most mathematicians of the sixteenth and seventeenth centuries did not accept them... In the fifteenth century and, in the sixteenth, Stifel both spoke of negative numbers as absurd numbers. ...Descartes accepted them, in part. ...he had shown that, given an equation, one can obtain another whose roots are larger than the original one by any given quantity. Thus an equation with negative roots could be transformed into one with positive roots. Since we can turn false roots into real roots, Descartes was willing to accept negative numbers. Pascal regarded the subtraction of 4 from zero as utter nonsense."

"In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers."

"Laplace created a number of new mathematical methods that were subsequently expanded into branches of mathematics, but he never cared for mathematics except as it helped him to study nature."

"Because they [the ancient Greeks] feared infinite processes they missed the limit process. In approximating a circle by a polygon they were content to make the difference smaller than any given quantity, but something positive was always left over. Thus the process remained clear to the intuition; the limit process, on the other hand, would have involved the infinitely small."

"The concept of the infinitely small is involved in the relation of points to a line or the relation of the discrete to the continuous, and Zeno's paradoxes may have caused the Greeks to shy away from this subject."

"The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine. ...Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate."

"The Pythagoreans associated good and evil with the limited and unlimited, respectively."

"Aristotle says the infinite is imperfect, unfinished, and therefore unthinkable; it is formless and confused. Only as objects are delimited and distinct do they have a nature."

"Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length—for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension."

"Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making."

"The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces.""

"When an equation...clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. ...Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions."

"The goal of deriving all the phenomena of nature from a few basic physical laws and the axioms of mathematics had been set by Galileo... In studying curvilinear motions on the earth Galileo had found the parabola to be the basic curve. In the heavens... Kepler... had found the ellipse to be the basic curve. Why this difference? ...since parabola and ellipse are both conic sections there was the provocative suggestion that perhaps some physical law unified these related paths of motion. ... It has often happened in the history of mathematics and science that major problems remained outstanding... great minds... succeeded only in revealing the true difficulties... and in generating an atmosphere of dispair... Then a genius appeared... with ideas that seemed remarkably simple once propounded, clarified the entire situation, dispelled the confusion, restored order, and produced a new synthesis that embraced far more even than the phenomena under consideration. The genius who... picked up the torch of science dropped by Galileo, was Isaac Newton."

"Aristotle had considered the question of whether space is infinite and gave six nonmathematical arguments to prove that it is finite; he foresaw that this question would be troublesome."

"Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians."

"Galileo had provided the methodology for the analysis of motions on and near the earth and had applied it successfully. Copernicus and Kepler had previously obtained the laws of motion of the planets and their satellites. ...But Galileo had succeeded in deriving numerous laws from a few physical principles and... the axioms and theorems of mathematics. ...The Keplerian laws ...were not logically related to each other. Each was an independent inference from observations. ...They seemed to be suspended in the same vacuum in which the planets moved. Galileo's laws had the additional advantage of supplying physical insight. The first law of motion and the law that the force of graviation gives... a downward acceleration of 32 ft/sec2... explain the vertrical rise and fall of bodies, motion on slopes, and projectile motion. Kepler's laws... had no physical basis. ...Kepler tried to introduce the idea of a magnetic force which the sun exerted... But he failed to related the behavior of the planets to the precise laws of planetary motion. ... The new astronomical theory was completely isolated from the theory of motion on earth. ...it bothered mathematicians and scientists who believed that all the phenomena of the universe were governed by one master plan instituted by the master planner—God."

"To avoid any assertion about the infinitude of the straight line, Euclid says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in Euclid's statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point."

"While the mathematicians were still looking askance at the Greek gift of the irrational number, the Hindus of India were preparing another brain-teaser, the negative number, which they introduced about A.D. 700. The Hindus saw that when the usual, positive numbers were used to represent assets, it was helpful to have other number represent debts."

"The relationship of point to line bothered the Greeks and led Aristotle to separate the two. Though he admits points are on lines, he says that a line is not made up of points and that the continuous cannot be made up of the discrete. This distinction contributed also to the presumed need for separating number from geometry, since to the Greeks numbers were discrete and geometry dealt with continuous magnitudes."

"The Hindus saw clearly that if the arithmetic operations... were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers. ...To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens. What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations."

"Henri Poincaré thought the theory of infinite sets a grave malady and pathologic. "Later generations," he said in 1908, "will regard set theory as a disease from which one has recovered."

"As our survey indicates, the Hindus were interested in and contributed to the arithmetical and computational activities of mathematics rather than to the deductive patterns. Their name for mathematics was ganita, which means “the science of calculation”. There is much good procedure and technical facility, but no evidence that they considered proof at all. They had rules, but apparently no logical scruples. Moreover, no general methods or new viewpoints were arrived at in any area of mathematics."

"The famous sixteenth-century algebraist Jerome Cardan called negative roots fictitious, and the founder of modern symbolic algebra, François Viète, discarded negative roots entirely. Descartes, called them false on the ground that they represented numbers less than nothing and so were meaningless."

"Historically, it was Euclidean geometry that, developed to a large extent as a votive offering to the God of Reason, opened men's eyes to the possibility of design and to the possibility of uncovering it by the pursuit of mathematics."

"The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. ...on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries..."

"The use of canon raised numerous questions concerning the paths of projectiles. ...One might determine... what type of curve a projectile follows and.... prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra."

"The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by... Al Khowarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation... When the Moors reached Spain... algebrista... came to mean a bonesetter... and signs reading Algebrista y Sangrador (bonesetter and bloodletter) were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts."

"The chief innovator of symbolism in algebra was François Viète... an amateur in the sense that his professional life was devoted to the law... John Wallis... says that Viète, in denoting a class of numbers by a letter, followed the custom of lawyers who discussed legal cases by using arbitrary names [for the litigants]... and later the abbreviations... and still more briefly A, B, and C. Actually, letters had been used occasionally by the Greek Diophantus and by the Hindus. However, in these cases letters were confined to designating a fixed unknown number, powers of that number, and some operations. Viète recognized that a more extensive use of letters, and, in particular, the use of letters to denote classes of numbers, would permit the development of a new kind of mathematics; this he called logistica speciosa in distinction from logistica numerosa. ...the growth of symbolism was slow. Even simple ideas take hold slowly. Only in the last few centuries has the use of symbolism become widespread and effective."

"The theory of perspective was taught in painting schools from the sixteenth century onward according to principles laid down by the masters... However, their treatises on perspective had on the whole been precept, rule, and ad hoc procedure; they lacked a solid mathematical basis. In the period from 1500 to 1600 artists and subsequently mathematicians put the subject on a satisfactory deductive basis, and it passed from quasi-empirical art to a true science. Definitive works on perspective were written much later by eighteenth-century mathematicians Brook Taylor and J. H. Lambert."

"To the scientists of 1850, Hamilton's principle was the realization of a dream. ...from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. ...they made striking progress ...But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics."

"Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his Arithmetica Infinitorum (1655), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity."

"The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived... A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations."

"[Asked who inspired him to be a vegetarian] I used to sell books, when I was about 18 …. We were selling books, and right next to us was an old man. I mean, he had to be about 80 years old, and he was a nutritionist, but he was a holistic nutritionist, and he was talking to us about how you shouldn’t really eat animals and things of that nature … and he pulled up his shirt and the man had the abs of a 15-year-old. He says to us, “Touch ’em. Punch me,”—whatever—and he’s like, “How old do you think I am?” And we thought he was about 50 years old, and he’s like, “I’m 86 years old.” … We looked at each other, me and my friends, and we were, like, when we’re 86, we want to be like that …. So that’s what started it."

"I’ll do anything that’s going to save animals. Anytime I can encourage people to think twice before [doing] anything to animals … that’s a good thing."

"My decision to leave applied mathematics for economics was in part tied to the widely-held popular belief in the 1960s that macroeconomics had made fundamental inroads into controlling business cycles and stopping dysfunctional unemployment and inflation."

"What truth is not, according to Kuhn, is an accurate representation of the world as it is in itself. Scientific theories represent a world, but one partially constituted by the cognitive activities of the scientists themselves. This is not a commonsensical view, but it has a distinguished philosophical pedigree, associated most strongly with Kant. The Kantian view is that the truths we can know are truths about a ‘phenomenal’ world that is the joint product of the ‘things in themselves’ and the organising, conceptual activity of the human mind. Kuhn, however, is Kant on wheels. Where Kant held that the human contribution to the phenomenal world is invariant, Kuhn’s view is that it changes fundamentally across a scientific revolution. This is what he means by his notorious statement that, after a scientific revolution, ‘the world changes’. This is neither the trivial claim that scientists’ beliefs about the world change, nor the crazy claim that scientists can change the things in themselves simply by changing their beliefs. It is the claim that the phenomenal world changes because the human contribution to it changes."

"For Kuhn, scientific development is neither cumulative nor teleological: science is not moving towards the goal of a theory of everything. His favoured analogy is with biological evolution by natural selection. Organisms develop under selective pressures from the environment, but development, though tending to an increase in complexity and differentiation, has no fixed goal. Moreover, the selective environments that determine which organisms survive and which die off themselves change and are partially constituted by the organisms’ own activities. Biological evolution is not moving towards an ideal organism, and scientific evolution is not moving towards the Truth. Both processes are pushed from behind, not pulled from the front."

"Students in the Department would often attempt to help one another by ‘Liptonating’ (or ‘Liptonising’): expressing the content of one’s interlocutor’s garbled thoughts in far superior terms of clarity, precision, and concision."

"His ability charitably to “summarise” the inchoate and/or confused thoughts of his interlocutors was legendary: Lipton invariably said what you wished you had meant."

"Lipton once gave me some advice that I like quite a lot: if a footnote says something important, put it in the main text where people will read it. If it says nothing important, delete it."

"Be a physical chemist, an analytical chemist, an organic chemist, if you will; but above all, be a chemist."

"If you had the opportunity to save a million people from preventable death, would you do it? … This is not merely a rhetorical question, but one that members of the Congress will have to answer in the present… Right now, legislation has already passed the House of Representatives that would do just that. And it was included in the newly released COVID relief bill that is being negotiated between House Speaker Nancy Pelosi (D-Calif.) and Treasury Secretary Steven Mnuchin. It would require the Treasury Department, which represents our government at the International Monetary Fund (IMF), to support a multi-trillion dollar relief package from the Fund... And they do not cost the U.S. government anything at all — not now, and not at any time in the future. The IMF leadership, and almost all of the 189 member countries — including U.S. allies such as Germany and Canada — are ready to allocate the aid that Congress is considering. The reason it hasn’t already been approved at the IMF is that the U.S. Treasury has said no, and the U.S. — alone — has a veto at the IMF on this matter. .. [I]t’s not at all clear why the Treasury is blocking this desperately needed aid. … Nor is there any reason that it should be a partisan issue... It would take almost no effort to include the House or Senate bill that would unblock Treasury’s hold on the IMF funding…"

"First of all, I think the United States is the greatest country in the world to invest in. And we see that. And we see that money is pouring into the United States for those reasons. So I think we're really going to be focused on economic growth and creating jobs. And that's really going to be the priority."

"Aw!! Did you think this was a personal trip?! Adorable! Do you think the US govt paid for our honeymoon or personal travel?! Lololol. Have you given more to the economy than me and my husband? Either as an individual earner in taxes OR in self sacrifice to your country? I’m pretty sure we paid more taxes toward our day “trip” than you did. Pretty sure the amount we sacrifice per year is a lot more than you’d be willing to sacrifice if the choice was yours. You’re adorably out of touch. Thanks for the passive aggressive nasty comment. Your kids look very cute. Your life looks cute. I know you’re mad but deep down you’re really nice and so am I. Sending me passive aggressive Instagram comments isn’t going to make life feel better. Maybe a nice message, one filled with wisdom and hunanity would get more traction. Have a pleasant evening. Go chill out and watch the new game of thrones. It’s fab!"

"I’m what C.S. Lewis would call a ʽMere Christianʼ, and all of my two books are very, very much meant to be from the point of view of what C.S. Lewis called ‘Mere Christianity’, meaning that I don’t touch upon anything at all where Protestants, Catholics, and Orthodox Christians differ, they express just the basics of the faith, from a basic, ecumenical Christian viewpoint. They only talk about the Christian faith that they have agreement on."