Mathematicians From England

"Nor were they so absurd in their conceptions about Gravity, as to think that it was done by the virtue of any point within the Earth, or of a Center, to which all heavy Bodies placed any where tended; but they thought it was done by the power of the whole Matter in the Terrestrial Globe attracting all things to it self: And as the power of the is composed of the powers of the several parts combin'd together, so they believed that the Gravity towards the whole Earth, resulted from the Gravity towards each single part of it. ...[T]hey believ'd there was a Gravity towards the Moon and Sun, acting in the same manner as it does towards the Earth; and that each Planet, like a Stone, whirl'd in a sling, was kept in its Orbit by the same principle, and for the same reason revolving always about us."

"From some things mention'd by Diogenes Laertius concerning Plato, which also are obscurely hinted at in his Timæus I am apt to believe with Galileo that the divine Philosopher suppos'd the Mundane Bodies, when they were first formed, were moved with a Rectilinear motion (by the means of Gravity,) but after that they had arrived to some determined places, they began to revolve by degrees in a Curve, the Rectilinear Motion being chang'd into a Curvilinear one."

"My design in publishing this Book, was, that the Celestial Physics, which the most sagacious Kepler had got the scent of, but the Prince of Geometers Sir Isaac Newton, brought to such a pitch as surprises all the World, might, by my... illustrating, become easier to such as are desirous of being acquainted with Philosophy and Astronomy."

"For Pythagoras as he was passing by a Smith's Shop, took occasion to observe, that the Sounds the Hammers made, were more accute or grave in proportion to the weights of the Hammers; afterwards stretching Sheeps Guts, and fastning various Weights to them, he learn'd that here likewise the Sounds were proportional to the Weights. Having satisfy'd himself of this, he investigated the Numbers, according to which Consonant Sounds were generated. Whether the whole of this Story be true, or but a Fable, 'tis certain Pythagoras found out the true ratio between the sound of Strings and the Weights fasten'd to them."

"Pythagoras... applied the proportion he had thus found by experiments, to the Heavens, and from thence learn'd the Harmony of the Spheres. And, by comparing these Weights with the Weights of the Planets, and the intervals of the Tones, produced by the Weights, with the interval of the Spheres; and lastly, the lengths of Strings with the Distances of the Planets from the Center of the Orbs; he understood, as it were by the Harmony of the Heavens, that the Gravity of the Planets towards the Sun (according to whose measures the Planets move) were reciprocally as the Squares of their Distances from the Sun."

"[T]he Opinion of the Ancients concerning Gravity... they were perswaded that Gravity was not an affection of Terrestrial Bodies only, but of the Celestial also, that all Bodies gravitate towards one another; and that the Planets are retained in their Orbits by the force of Gravity, and lastly, that the Gravity of the Planets towards the Sun are reciprocally as the Squares of their Distances from it. What the industry and skill of the Moderns have added to these inventions of the Ancients, the following Pages do declare at large."

"[W]ith this one problem... three years ago, after the above was published, Gottfried Leibniz... produced a construction... in terms of his system—not without the blemish of paralogism."

"After I had taken holy orders, I returned to the college, and went on with my own studies there, particularly the mathematicks, and the Cartesian philosophy; which was alone in vogue with us at that time. But it was not long before I, with immense Pains, but no assistance, set myself, with the utmost zeal, to the Study of Sir Isaac Newtons wonderful discoveries in his ', one or two of which Lectures I had heard... read in the publick Schools, though I understood them not at all... Being indeed greatly excited thereto by a Paper of Dr. Gregory’s when he was Professor in Scotland; wherein he had given the most prodigious Commendations to that work, as not only right in all things, but in a manner the Effect of a plainly divine genius, and had already caused several of his Scholars to keep Acts, as we call them, upon several branches of the Newtonian Philosophy; while we at Cambridge, poor Wretches, were ignominiously studying the fictitious Hypotheses of the Cartesian, which Sir Isaac Newton had also himself done formerly, as I have heard him say."

"Mr Issac Newton in addition to the geometric figure in any orbit of a projectile sought also to find the measure of the (tending to a given centre) of the body borne in that orbit, from whatever cause that force may arise, be it from a deeper mechanical one or from a law imposed by the supreme creator of all things. He inquires geometrically into the law of centripetal force of a body moved in the circumference of a circle with the force tending to a given point either on the circumference or anywhere outside it or inside it, or even infinitely removed. By the same method he seeks the law of centripetal force tending to the centre of a plane nautical spiral (that is one that the radii cut in a given angle) which will drive a body in that spiral. Also the law of centripetal force that would make a body rotate in an ellipse when the centre of the ellipse coincides with the centre of forces. If the ellipse is changed into a hyperbola and the centripetal force into a centrifugal one the same things apply to the hyperbola. Also the resolution of the same problem when the centre of forces coincides with either focus of the ellipse shows that the law of centripetal force is reciprocally in the duplicate ratio of the distance [as the inverse square of the distance]; others had long before shown that this was the one and only law that would satisfy the other phenomenon observed by Kepler in the motion of the planets. These results also apply to the hyperbola and the parabola when the centre of forces is situated in a focus of the conic section."

"But... Kepler’s problem was to be resolved, to find the position of a body moved in an elliptical orbit at a given time. As concerns an algebraic resolution... adapted to the construction of tables, we... also have produced a work not... to be ashamed of."

"[W]e have come into the age where questions that were once cosmographical are being transformed into geometrical problems. For it has now been shown not only that the areas which bodies driven in a circuit describe by the radii drawn to the centre of forces are in immobile planes and proportional to the times, but also conversely that every body moved in this way is impelled by a centripetal force that tends towards the aforesaid point. By this proposition alone the Ptolemaic system is destroyed, for the primary planets by radii drawn to the Earth describe areas in no way proportional to the times, while with the radii drawn to the sun it is established that they run over areas sufficiently proportional to the times."

"After Kepler’s bold and fruitful efforts to advance natural philosophy by the help of geometry, there should have appeared any philosopher and particularly a geometer, namely Descartes, who should leave this one narrow path and try to investigate the causes of things logically, or rather, sophistically. What is to be said of him who while certainly learned in geometry would build his cosmic system (which he valued so highly and of which he boasted so grandiloquently) from vortices, without previously examining whether bodies carried around by a vortex at different distances from the centre would have periodic times whose squares were as the cubes of the distances from the centre? But he was intoxicated by easier and less composite laws, and, not applying his geometric ability in the slightest, fell into errors from which we were at length liberated by the aid of geometers."

"Since two or more mutually gravitating bodies describe orbits around a common immobile centre of gravity, and since by common consent there is an immense difference between the quantity of matter in the sun and that in the Earth, it is clear that neither the sun nor, much less, the sun in the company of five planets can revolve around an immobile Earth. Thus is shown not only the falsity but the impossibility of the ."

"But, since the law of centripetal force employed by nature is to be discovered from its symptoms, the indisputably elliptical orbit and the sesquialteral ratio of the periodic times and the distances from the centre of forces, the same great Newton solved not only the universal problem of determining the trajectory and the motion in it for any given centripetal force, but also its converse. After this universal problem had been solved the sequel was to find other [quantities] in the geometric figure that are measures of physical qualities; for example, that the periodic times in ellipses are in the sesquiplicate ratio of the transverse axes [the squares of the times are as the cubes of the axes], and as many other things similar to these as possible. Also, for instance, to compare this force, which we experience in the planets, with another given force near to us, namely gravity. But also the new philosophy was to concern itself with movable elliptical orbits, in which the line of apsides either advances or retires. Also, for instance, a more exact [theory] of rectilinear descent and of the motion of pendulous bodies than the Huygenian one, since that supposes the centre to be infinitely removed. Therefore also, other s different from the common one and variously devised according as the pendulum oscillates inside or outside the surface of the Earth. And let that suffice for this problem. But also on account of the mutual actions of bodies moving around a centre the orbits usually turn out to be deformed, and also an investigation of these actions and of the deformity arising from them, whence arise many minor inequalities of the planets, such as the motion of the nodes, the variation of maximum latitude, and other things in the moon."

"[T]he Physics, it is all taken out of the above mention'd Authors; but is here intermix'd with Astronomy, in such places as seem'd proper and convenient; the Geometry to be met with in it, I have either borrowed elsewhere, and quoted... or delivered it Lemmatrically."

"In the past many very base Remus’s leapt over the walls of the astronomical city, but now the geometers have so fortified it with a ditch and a rampart that the portals of the sun receive those whom impartial Appollonius has loved and whom Kepler, Wren, Wallis and Newton have borne to the aetherial regions, and accordingly the profane, that is ungeometrical men, are exiled and depart from the grove and wander away over the whole heaven."

"[W]ithin the memory of ourselves and... our fathers, philosophers began to extend the limits of geometry in order to found the kingdom of astronomy. This they have carried out... with such success that now no one can be received into astronomical citizenship who is not a visiting citizen in the most abstruse geometry and has not arisen from the patrician, that is the geometrical, family of philosophers."

"[W]hat sharpness of mind was employed by John Kepler... when, from there being just five regular solids... he inferred that the number of the planets was six, and by inscription of spheres within these solids and circumscription of spheres around them related the distances and ratios of the orbits. It can scarcely be said with what power of prophecy and by what labours he succeeded in arriving at that great theorem of the elliptical planetary orbits with a common focus at the sun... in such a way that the areas that the radius vector of the planet from the sun traverses are proportional to the times. Nevertheless... so great a man... owned himself unequal to... solving directly the problem of determining for a given time the place of the planet in the elliptical orbit. Here geometry, his goddess-mother, was of no avail... But... he brought forward a conjecture of great use, namely, that the squares of the periodic times are in the same ratio as the cubes of the distances between the planets and the sun. Finally, he discovered a marvellous property of bodies by which in the minimally resisting ether they seek each other and as it were attract. From this he also deduced the tides in a clear but brief discourse in his immortal Commentaries on the star Mars, and was as it were a prophet and a precursor of a great geometer born among the English."

"Although in every age there have been those who cultivated astronomy, either by... observations... or by theories and systems made up according to the state of understanding of any period, or by a talent for exposition, yet the lucubrations of all these astronomers do not reveal the ways of the heaven any more than they reveal the skill and experience of their progenitors in geometrical matters."

"[T]he easier, simpler and less composite these theories are, the more they will be consonant not only with what has already been observed... but with what is yet to be observed, and with the very machine of the world which was constructed by the supreme creator with the greatest simplicity."

"[I]t is by the help of geometry that all the arts necessary for improving life,.. as geography,.. rules of navigation,.. determining of times... [etc.], have been carried to such an incredible pinnacle of distinction."

"[G]lory has been reserved to our era and to the English people, who since the instauration of the sciences have made such advances... And passing over the immense labours undergone by the most fruitful astronomers of our people... [H]ow easy and how exact... how geometrical, astronomy has been left to us by that most acute geometer... or astronomer, the Right Reverend Dr Seth sometime Bishop of Salisbury, who while he was among men adorned this chair. How geometrically and acutely he determined the positions and species of the orbit and other related matters, following Kepler and substituting as mean motion the angle at the other focus (which he accordingly called that of the mean motion) in place of the areas to the sun that the radius vector describes and as it were sweeps out. Content with this artifice he did not detain himself over the solution of Kepler’s problem, in which the division of the area of an ellipse in a given ratio by a straight line through a focus is required. But, being a most perspicacious man, he was conscious of what delays arose hence in the construction of tables, and, in order to show the world that astronomy was to be advanced by the help of geometry whatever hypotheses it depended upon, he accomplished the same astronomical problems geometrically from the circular hypothesis."

"After great and fruitful efforts both in the purer geometry and the more intricate and complex physics, the most skilful geometer Sir Christopher Wren, who among other luminaries of the University of Oxford graced this professorship, solved the following problem: To find the law of gravity or centripetal force by which several bodies moved around a common centre of forces are driven, given that the squares of the periodic times are as the cubes of the radii, as was observed by Kepler for the planets moved around the sun. The most renowned Wren found that the required law of gravity was such that the centripetal forces were reciprocally as the squares of the distances from the centre of forces, and that no other law would agree with what was observed."

"It is... most gratifying to my soul that, after spending a large part of my life in other universities, I can be linked in the University of Oxford with the prince of geometers Dr John Wallis as colleague and the most scholarly Dr as successor."

"[T]hose who are less vers'd in the more abstruse parts of Geometry, or less concerned about the Physical parts, may pass over, and only read the Astronomy separately and distinct..."

"The Celestial Physics, or Physical Astronomy, is not only the first in dignity of all inquiries into Nature... but the first in order, because it is the easiest."

"... He was among the first Europeans to acquire a working knowledge of a North American language—in this case, —and by means of it to understand and record indigenous culture at the time of first contact with Europeans. Outgoing and amiable, he made friends with the people, hunted and feasted with them, learned their methods of agriculture, canoe building, and fishing, and clearly enjoyed much about their way of life. As a general rule, he recorded what he saw with the detachment of a physicist and the engagement of a linguist and ethnologist, describing rather than judging religious practices and cultural ceremonies that were completely alien to him, and observing in context the details of Algonquian life."

"There is an herb which is sowed apart by itself & is called by the inhabitants Uppówoc: In the it has divers names, according to the several places & countries where it grows and is used: The Spaniards generally call it Tobacco. The leaves thereof being dried and brought into powder: they use to take the fume or smoke thereof by sucking it through pipes made of clay into their stomach and head; from whence it purges superfluous & other gross , opens all the pores & passages of the body: by which means the use thereof, not only preserves the body from obstructions; but also if any be, so that they have not been of too long continuance, in short time breaks them: whereby their bodies are notably preserved in health, & know not many grievous diseases wherewithal we in England are oftentimes afflicted."

"By the end of the 16th century with the work of Viète and especially during the first half of the 17th century in the work of Harriot, Fermat, and Descartes, mathematicians began to treat algebra more symbolically, eventually adopting a notation that readily lends itself to making algebraic computations."

"On two different occasions recently, (male) mathematicians asked me in all innocence: But you surely never suffered any discrimination?"

"... Gelfand amazed me by talking of mathematics as if it were poetry. He tried to explain to me what von Neumann had been trying to do and what the ideas were behind his work. That was a revelation for me — that one could talk about mathematics that way. It is not just some abstract and beautiful construction but is driven by the attempt to understand certain basic phenomena that one tries to capture in some idea or theory. If you can’t quite express it one way, you try another. If that doesn’t quite work, you try to get further by some completely different approach. There is a whole undercurrent of ideas and questions."

"Was I ever discriminated against? There are two kinds of discrimination: explicit and implicit. For the most part, explicit discrimination did not affect me much. However, in retrospect, implicit discrimination—for example, the fact that I was so isolated as a postdoc because I could not share in college life—as well as my own internalized misogyny, did have a significant effect, though I hardly noticed this at the time. Another important factor, and one that I was aware of, was pervasive but not overt: it was very rare that women became professional scientists in Britain at the time, largely because science (and particularly “hard” as opposed to “life” science) was considered such a very unfeminine thing to do. ... These days, when most of the obvious barriers to women’s participation in mathematics have been removed, there still remain very strong and insidious internal barriers, shown in such phenomena as stereotype threat or imposter syndrome. The prejudices that lead to people accepting as completely normal that women should not get degrees at Cambridge (they first could get Cambridge degrees in 1948) are very strong and do not disappear immediately when the external barrier is removed. ... In the 1960s there were, of course, very visible manifestations of the idea that academic life is not for women. At the time, most Ivy League universities in the States did not admit women, and in Britain almost all the colleges at the most prestigious universities (Oxford and Cambridge) were single sex."

"The past few years have seen several exciting developments in the field of symplectic geometry, and a beginning has been made towards solving many important and hitherto inaccessible problems. The new techniques which have made this possible have come both from the calculus of variations and from the theory of elliptic partial differential operators. This paper describes some of the results that obtained using elliptic methods, and then shows how applied these elliptic techniques to develop a new approach to , which has important applications in the theory of 3- and 4-manifolds as well as in symplectic geometry."

"Symplectic geometry is the geometry of a closed skew-symmetric form. It turns out to be very different from the with which we are familiar. One important difference is that, although all its concepts are initially expressed in the smooth category (for example, in terms of differential forms), in some intrinsic way they do not involve derivatives. Thus symplectic geometry is essentially topological in nature. Indeed, one often talks about symplectic topology. Another important feature is that it is a 2-dimensional geometry that measures the area of complex curves instead of the length of real curves."

"Over the past 15 years symplectic geometry has developed its own identity, and can now stand alongside traditional Riemannian geometry as a rich and meaningful part of mathematics. The basic definitions are very natural from a mathematical point of view: one studies the geometry of a skew-symmetric bilinear form ω rather than a symmetric one. However, this seemingly innocent change of symmetry has radical effects. For example, one dimensional measurements vanish since ω(v, v) = −ω(v, v) by skew-symmetry. ... The theory has two faces. There are two kinds of geometric subobjects in a symplectic manifolds, hypersurfaces and Lagrangian submanifolds that appear in dynamical constructions, and even-dimensional symplectic submanifolds that are closely related to Riemannian and complex geometry. As we shall see, the analog of a geodesic in a symplectic manifold is a two-dimensional surface called a ."

"… By relating personal stories, historical examples and mathematical analogies, Cheng explains how, when we rely on simplistic concepts like female and male, and the crusty logic that accompanies those concepts, we cannot have good conversations. As Cheng puts it: “If we object to the idea that ‘men are better,’ it’s not that helpful to declare instead that ‘women are better.’ It pits men and women against each other and sets up a prescriptive framework rather than a descriptive one.” She motivates us to strip away consistent triggers for dumb fights that lead nowhere. What would she have us strip away? This is where Cheng becomes a logician. She wants to carefully think through our associations with the word “success” as they relate to gender."

"How to Bake 𝜋 is a success at explaining what mathematics is and how it is done, using simple, appealing language. It should be a rewarding read for mathematicians and nonmathematicians alike. ...[T]eachers will find plenty to borrow for... classrooms... Cheng frequently strips away technical details in order to show the big picture... [T]he book’s frequent digressions... topology, Arrow’s theorem, fair-division problems, s, the Poincaré Conjecture, the Riemann Hypothesis... [etc.]"

"As a category theorist, Cheng researches relationships. She uses this focus on relationships to address the problem of the divisiveness of arguments around gender equality. She abstracts the ideas and reframes the discussion based on relevant character traits that she demonstrates do not have to be linked to gender. She looks for assumptions that have been made, seeks to discard them, and discovers fundamental relationships. In order to better articulate these relationships, she invents new terminology as a way of preventing futile divisive arguments. These new terms are ingressive and congressive. She defines ingressive behavior as “going into things” where the focus is on the self and is more competitive, individualistic, and adversarial. She defines congressive behavior as “bringing things together” where the focus is on community and is more collaborative, interdependent, and cooperative. She gives many examples to illuminate her definitions. ... Cheng is deeply interested in making mathematics accessible to everyone."

"I love to focus on how to make mathematical activities more congressive... so when I'm teaching art students I do a lot of activities where there is no right and wrong... [W]e're not trying to achieve an answer, we're trying to explore... [W]e build something. It's more craft-like. ...[M]aybe we're trying to build s, but it doesn't really matter if you didn't... [A]long the way we discover how triangles fit together and... how versatile an equilateral triangle is... and that you can make all sorts of shapes... and some... are Platonic solids... [E]veryone can explore... and... in the world of , this is... "low floor/high ceiling" activities where there's a very low floor to entry and a very high ceiling, so if someone really does want to go far they can, but... there's no real failure, because everyone has done something. ...[I]f we do more of that, then we will stop putting off congressive people from mathematics."

"A lot of programs for good mathematicians at a young age are... competitions and... problem solving... and Olympiads, and that is very ingressive..."

"The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions. As teachers and professors of math, we should welcome all questions and understand that answering questions is what helps students learn. ... “Is Math Real?” treats mathematical topics in a unique and original way. Discussions on number patterns, Platonic solids, math history, ethnomathematics, and mathematical structures presents the reader with a plethora of ideas on how one can envision mathematics."

"If you hate the idea of being... told you're wrong, then you get put off math at a very early age because it's the one subject where you start being told you're wrong a lot, and... if you don't like that... you'll move off into some subject where... you can create things..."

"I've invented some new words... ingressive to replace masculine and congressive to replace feminine... Ingressive is a character trait... a behaviour... about going forward... not being waylaid about what people say... being competitive and winning. Congressive is about bringing things together and... shedding light and understanding... helping people... and maybe we are presenting mathematics at school in a very ingressive way, because it's often about being right... getting the right answer."

"What if the pieces of string aren't really... string, but they're s? ...[M]aybe ...early diagnosis of Alzheimer's may come from looking at... the tangledness of brain cells that mutate... So an abstract way of telling whether it's tangled is... useful."

"I'm not interested in being right... I'm not interested in winning... but I hate losing, and I don't like being wrong. ...But if it's a situation when nobody is going to lose because we're all trying to understand something together, then there's no risk of losing, and... we can all gain from it."

"Math, unfortunately is presented in this very ingressive way, despite the fact that when you get to the research level, it's very congressive."

"It... [abstract structure] is a beautiful thing, and sometimes all that matters is that it's a beautiful thing."

"[I]f you know where the s are, you can... use things better, you can make things better. You can improve them. You can fix them when they go wrong."

"I'm not interested in playing sport... because I hate the idea of losing, and I'm not interested in winning, so there's no upside and there's only potential downside..."

"[T]hen I started my PhD and discovered that in higher dimensional category theory... the braids show the coherence of the structure inside some higher dimensional categories, and I didn't know this when I first drew this picture, and then I... said "Wow!" I was studying braids before I was even studying braids..."